BGBiogeosciencesBGBiogeosciences1726-4189Copernicus PublicationsGöttingen, Germany10.5194/bg-14-1647-2017Reviews and syntheses: parameter identification in marine planktonic ecosystem modellingSchartauMarkusmschartau@geomar.dehttps://orcid.org/0000-0003-1114-0415WallheadPhiliphttps://orcid.org/0000-0001-5009-5165HemmingsJohnLöptienUlrikehttps://orcid.org/0000-0002-8765-4183KriestIrisKrishnaShubhamWardBen A.SlawigThomasOschliesAndreasGEOMAR Helmholtz Centre for Ocean Research Kiel, Kiel, GermanyNIVA, Norwegian Institute for Water Research, Bergen, NorwayWessex Environmental Associates, Salisbury, UKUniversity of Bristol, School of Geographical Sciences, Bristol, UKChristian-Albrechts-Universität zu Kiel, Department of Computer Science, Kiel, Germanynow at: Met Office, Exeter, UKMarkus Schartau (mschartau@geomar.de) and Phil Wallhead (philip.wallhead@niva.no)29March2017146164717013June201620June201617February201721February2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://bg.copernicus.org/articles/14/1647/2017/bg-14-1647-2017.htmlThe full text article is available as a PDF file from https://bg.copernicus.org/articles/14/1647/2017/bg-14-1647-2017.pdf
To describe the underlying processes involved in oceanic plankton dynamics is
crucial for the determination of energy and mass flux through an ecosystem
and for the estimation of biogeochemical element cycling. Many planktonic
ecosystem models were developed to resolve major processes so that flux
estimates can be derived from numerical simulations. These results depend on
the type and number of parameterizations incorporated as model equations.
Furthermore, the values assigned to respective parameters specify a model's
solution. Representative model results are those that can explain data;
therefore, data assimilation methods are utilized to yield optimal estimates
of parameter values while fitting model results to match data. Central
difficulties are (1) planktonic ecosystem models are imperfect and (2) data
are often too sparse to constrain all model parameters. In this review we
explore how problems in parameter identification are approached in marine
planktonic ecosystem modelling.
We provide background information about model uncertainties and estimation
methods, and how these are considered for assessing misfits between
observations and model results. We explain differences in evaluating
uncertainties in parameter estimation, thereby also discussing issues of
parameter identifiability. Aspects of model complexity are addressed and we
describe how results from cross-validation studies provide much insight in
this respect. Moreover, approaches are discussed that consider time- and
space-dependent parameter values. We further discuss the use of
dynamical/statistical emulator approaches, and we elucidate issues of
parameter identification in global biogeochemical models. Our review
discloses many facets of parameter identification, as we found many
commonalities between the objectives of different approaches, but scientific
insight differed between studies. To learn more from results of planktonic
ecosystem models we recommend finding a good balance in the level of
sophistication between mechanistic modelling and statistical data
assimilation treatment for parameter estimation.
Introduction
The growth, decay, and interaction of planktonic organisms drive the
transformation and cycling of chemical elements in the ocean. Understanding
the interconnected and complex nature of these processes is critical to
understanding the ecological and biogeochemical function of the system as a
whole. The development of biogeochemical models requires accurate
mathematical descriptions of key physiological and ecological processes, and
their sensitivity to changes in the chemical and physical environment. Such
mathematical descriptions form the basis of integrated dynamical models,
typically composed of a set of differential equations that allow credible
computations of the flux and transformation of energy (light) and mass
(nutrients) within the ecosystem (US Joint Global Ocean Flux Study
Planning Report Number 14, Modeling and Data Assimilation, 1992).
Generalized mechanistic
descriptions of how energy is absorbed and how mass becomes distributed in an
ecosystem already exist, such as dynamic energy budget models
or the metabolic theory of ecology
. But these theories still have limitations, and
include incompatible assumptions . So far no
fundamental ecophysiological principle has been further
consolidated beyond the conservation of mass. A consistent theme running through
most ecosystem models is the determination of mass flux of certain
biologically important elements, such as nitrogen, phosphorus, iron, and
carbon (N, P, Fe, and C).
Nonetheless, the precise details of how mass is transformed and allocated
within an ecosystem is far from being established. For this reason, we find a
large variety of plankton ecosystem models that differ in their number of
state variables as well as in their parameterization of individual
physiological and ecological processes.
Mass flux induced by plankton dynamics
Dynamical marine, as well as limnic, ecosystem models usually start from a
description of the build-up of biomass by photoautotrophic organisms
(phytoplankton) as these take up dissolved nutrients from the water column
and exploit light energy by photosynthesis. Phytoplankton biomass, as a
product of primary production, is subsequently removed by natural mortality
(cell lysis due to starvation, senescence, and viral attack), predation by
zooplankton, and vertical export away from surface ocean layers via sinking
of single or aggregated cells and of fecal pellets. Parameterizations of
these three loss processes can be interlinked, e.g. grazing of phytoplankton
aggregates by large copepods. Depending on the trophic levels considered in a
model, the predation among different zooplankton types (e.g. between
herbivores, carnivores, or omnivores) can be explicitly parameterized.
Mortality and aggregation of phytoplankton cells and the excretion of organic
matter (fecal pellets) by zooplankton act as primary sources of dead
particulate organic matter (detritus) that can be exported to depth via
sinking. Exudation by phytoplankton and bacteria can be a major source of
labile dissolved organic matter that represents diverse substrates for
remineralization. The transformation of particulate and dissolved organic
matter back to inorganic nutrients is parameterized as hydrolysis and
remineralization processes. Often hydrolysis and remineralization are assumed
to be proportional to the biomass of heterotrophic bacteria, which is
considered in many models. Heterotrophic bacteria remain unresolved in some
models where microbial remineralization is parameterized only as a function
of concentration and quality of organic substrates.
At some level most models include a parameterization to account for the net
effect of higher trophic levels that are not explicitly resolved. This is
usually formulated as a closure flux back to nutrient pools and whose rates
simply depend on the biomass of the highest trophic level resolved. These
closure assumptions ensure mass conservation while neglecting the actual mass
loss to higher trophic levels like fish, which would be subject to fish
movements and changes in biomass on multi-annual scales rather than seasonal
timescales. Every marine planktonic ecosystem model can thus be described as
a simplification of the dynamics inherent to a system of nutrients,
phytoplankton, zooplankton, detritus, dissolved organic matter, and possibly
bacteria.
In many cases marine ecosystem models are embedded in an existing physical
ocean model set-up that simulates environmental conditions, advection, and
mixing of the biological and chemical state variables. Feedbacks from the
ecosystem model states on physical variables can be relevant
e.g., but are rarely considered in current marine
biogeochemical studies.
Parameters of plankton ecosystem models
One of the most influential model approaches to studying the nitrogen flux
through such a marine plankton ecosystem at a local site was proposed by
. Their model involves 27 parameters and they stressed
the invidious situation of finding a reliable ecosystem model solution by
choosing parameter values that are uncertain or unknown. Laboratory
measurements, as well as ship-based experiments with field samples, can
provide information about the range of typical values for some parameters,
for example the maximum growth rate of photoautotrophs or the maximum
ingestion rate of herbivorous plankton. Other model parameters are extremely
difficult to measure, like exudation rates of dissolved organic carbon by
phytoplankton or by bacteria. Another difficulty is that parameter values
from laboratory experiments are often specific with respect to plankton
species, temperature, and light conditions. Their values may not be directly
applicable for ocean simulations where parameter values need to be
representative of a mixture of different plankton species in a continuously
varying physical environment. For example, for a natural composition of
diverse phytoplankton cells that all differ in their genotypic and phenotypic
characteristics, we may expect values of some model parameters to follow a
distribution rather than having a single fixed value.
In practice, there are always some fixed model parameters that need to be
assigned values, whether they describe the behaviour of fixed plankton
functional types or the distributions of traits in a stochastic community. In
the end, it is the choice of these parameter values that determines a
specific model solution of any ecological or biogeochemical model set-up.
The vital role of observational data
Model solutions of interest are typically those that can simulate and explain
complex data. Model calibration, which can be considered a form of data
assimilation (DA), is the process by which model parameter values are
inferred from the observational data. Optimal parameter values are regarded
as those that generate model results that match observations (minimize the data–model
misfit) but that are also in accordance with the range of values known e.g. from
experiments or from preceding DA studies. To determine optimal parameter
estimates we have to account for uncertainties in data and in model dynamics
as well, which is specified by an error model. Parameter estimates are thus
conditioned by (a) the dynamical model equations, (b) the data, (c) our prior
knowledge about the range of possible parameter values, and (d) the
underlying error model .
Situations can occur where model results that are compared with data are
insensitive to variations of some parameters. Values of those parameters
remain unconstrained by the available data, which is a problem of parameter
identifiability. The availability (type and number) of data thus places
limitations on the number of model parameters whose values become
identifiable, and values of some parameters may never be fully constrained.
This in turn sets restrictions on the complexity of plankton interactions
that can be unambiguously confined during ecosystem model calibration
. Choosing appropriate model complexity is ambiguous and is
still subject to discussion e.g., a situation which sustains large differences
in the level of complexity of current plankton models.
Inferences from data assimilation
Much of the literature on DA in oceanography is focussed on state estimation
e.g..
In these studies, the primary objective is to improve hindcasts, nowcasts, or
forecasts of time-dependent variables such as
chlorophyll a (Chl a). However, many of the DA methods
originally developed for state estimation have more recently been adapted to
estimate static parameters, especially for stochastic models where random
noise is injected into the model dynamics. Stochastic noise offers a
plausible way to represent model error, but it should be noted that it can
lead to violations of mass conservation unless it is injected in certain ways
(e.g. by perturbing growth rate parameters). Deterministic plankton ecosystem
models guarantee mass conservation and have a longer tradition in parameter
estimation for marine ecosystem models, although they imply a less explicit
treatment of model error. To identify and gradually eliminate model
deficiencies it can be helpful to analyse model state and flux estimates
while mass conservation is imposed as a strong constraint. The optimization
of parameter values only ensures that simulation results remain dynamically
and ecologically consistent, which is comparable with those DA approaches in
physical oceanography that produce dynamically and kinematically consistent
solutions of ocean circulation e.g..
Thorough reviews of common DA methods applied in marine biogeochemical
modelling are given by and by
. provide a helpful and
up-to-date overview of mainly sequential DA approaches where state estimation
is combined with parameter estimation. and
discuss how the success of DA results of marine
ecosystem models has been evaluated in the past and how model performance can
be generally assessed. Fundamentals on DA that include aspects relevant to
marine ecosystem and biogeochemical modelling are explained in
and in .
In our review we primarily focus on topics related to parameter
identification, thereby including basic aspects of DA. Parameter
identification in marine planktonic ecosystem modelling is a wide field and
we do not attempt to discuss differences between various DA tools or
techniques. We rather put emphasis on models, including parameterizations of
ecosystem processes, statistical (error) models, model uncertainties, and
structural complexity. We adopt and explain mathematical notation that is
often used for DA studies in operational meteorology and oceanography. On the
one hand we provide background information that should facilitate
intelligibility when studying DA literature. On the other hand we like to
elucidate typical objectives and common problems when simulating a marine
planktonic system. In this manner we hope to support a mutual understanding
between ecologically/biogeochemically and mathematically/statistically
motivated studies.
The paper starts with some theoretical background information
(Sect. ), introducing mathematical notation and depicting
prevalent assumptions that are typically made for parameter identification
analyses and model calibration (Sect. ). We then
branch off from DA theory and discuss the parameters typically dealt with in
plankton ecosystem models. In Sect. we disentangle major
differences between approaches to parameterizing photoautotrophic growth and
briefly discuss simple but common parameterizations of plankton loss rates.
In this context we also address the utilization of data from laboratory and
mesocosm experiments. Error models are described in order to elucidate error
assumptions made in previous ecosystem modelling studies
(Sect. ). This is followed by a description of
different approaches to specifying uncertainties in parameter values
(Sect. ). An example of parameter estimation with
simulations of a mesocosm experiment connects aspects of
Sect. with the theoretical considerations of
Sect. . Thereafter, model complexity is jointly
addressed together with cross-validation in Sect. ,
followed by a review of space–time variations in marine ecosystem model
parameters (Sect. ). Emulator, or surrogate-based,
approaches are briefly explained and exemplified
(Sect. ) before we discuss parameter estimation of
large-scale and global biogeochemical ocean circulation models
(Sect. ). Finally, we summarize the insights that we
gained into parameter identification in Sect. , and we will
briefly address prospects of some marine ecosystem model approaches that
could improve parameter identification.
Theoretical background
The term parameter identification is used broadly to describe parameter
estimation problems, including the specification of uncertainties in
parameter estimates and model parameterizations. It involves the following
procedures.
Parameter sensitivity analyses: the evaluation of how model results
change with variations of parameter values.
Parameter estimation: the calibration of model results by adjusting
parameter values in light of the data.
Parameter identifiability analyses: the specification of parameter
uncertainties in order to reveal structural model deficiencies and shortages
in data availability/information.
All three aspects are interrelated and should not be viewed as mutually
exclusive procedures. For example, before starting with parameter estimation
it is helpful to include information from a preceding sensitivity analysis,
e.g. selecting only parameters to which model results are sensitive.
Likewise, an identifiability analysis complements the sensitivity analysis by
providing information about error margins and possible ambiguities of optimal
parameter estimates.
Statistical model formulationModel states, parameters, and dynamical model errors
The prognostic dynamical equations of a marine ecosystem model can be
expressed as a set of difference equations:
xi+1=Mxi,θe,fi,ηiθη,
with index i representing a particular time step (i.e. ti). The model
state vector xi has dimension
Nx=Ng×Ns where
Ng is the number of spatial grid points and Ns is the
number of model state variables (e.g. phytoplankton biomass). The dynamical
model operator M is typically at least a nonlinear function of the earlier
state xi, a set of ecosystem parameters θe
describing rate constants and coefficients in the dynamical model, and a set
of time- and space-dependent forcings and boundary conditions fi.
If the ecosystem model is coupled “online” with a physical ocean model,
fi includes both physical model forcings (e.g. wind stress) and
ecosystem model forcings (e.g. surface short-wave irradiance). If the physics
is coupled “offline”, fi includes ecosystem model forcings and
physical model outputs (e.g. seawater temperature).
For stochastic dynamical models, M also depends on random noise variables
or dynamical model errors ηi, while for deterministic
models we have ηi= 0. These errors are described by
distributional parameters θη, e.g. location and scale parameters
of a probability density function. Dynamical model errors usually enter the
dynamics additively, multiplicatively, or as time- or space-dependent
corrections to f or θe. They may represent the
individual or combined effects of errors in forcings, boundary conditions,
random variability in model parameters, and structural errors in both the
physical transport model (e.g. due to limited spatial resolution) and the
biological source-minus-sink terms (e.g. due to aggregation of species into
model groups). In the geophysical DA community, error models that explicitly
account for dynamical model errors (noise) are often termed weak constraint models, while those that assume a deterministic model are termed
strong constraintp. 25.
True states and kinematic model errors
To relate the dynamical model output of Eq. () to
observations, it is helpful to first consider how it may relate to a
conceptual and hypothetical true state xt, which is then
imperfectly observed. In this respect we must also consider the averaging
scales. In marine ecosystem modelling there is almost always a large
discrepancy between the spatio-temporal averaging scales of the model, which
define the meaning of the “concentrations” in x, and the averaging
scales of the observations from in situ sampling or remote sensing. For
example, the spatial averaging scale of a model may be defined by a model
grid cell of size 10 km in the horizontal and 10 m in the vertical, while
the averaging scale of the observations might be the 10 cm scale, e.g. of a
Niskin bottle sample. Even with a perfect model, data from fine-scale
observations may diverge from model output due to unresolved sub-grid-scale
variability induced by fluid structures such as eddies and fronts, forming
patches of high next to low concentrations e.g. of nutrients or organic
matter.
A general relationship between the true state and model state can be
expressed as
xt=Tx,ζθζ,
where T is a truth operator, and ζ is a set of random
variables described by distributional parameters θζ. We will
refer to the ζ as kinematic model errors because they
are associated with the model state, while the dynamical model
errors η in Eq. () act to perturb the model
dynamics. The true values of the kinematic model errors therefore define the
potential discrepancy between the target true state and a hypothetical ideal
model output (i.e. with the “true” values of the parameters and, if
applicable, also with the “true” values of the dynamical model errors).
How we interpret and specify Eq. () depends on the
spatio-temporal averaging scales chosen to define the true
state xt, which in turn depends on the objectives of the
modelling study. One approach is to define these averaging scales as equal to
or larger than the shortest space scales and timescales that are fully
resolved by the model. Kinematic model errors ζ may then
represent the integrated effects of the various dynamical sources of model
error, if these are not already accounted for by dynamical model
errors η in Eq. (). Alternatively, the true
state can be defined over scales smaller than those resolved by the model,
possibly at the scales of the observations. This may lead to a simpler model
for observational error (see below), but now the ζ must
account for the unresolved scales, in addition to any error effects in the
model dynamics otherwise not accounted for. With stochastic dynamical models
(η≠ 0), the true state is usually defined on the
scales of the model and assumed to coincide with the model output for some
(θe, η), such that no kinematic error
model is needed.
Data and observational errors
The observation vector y can be related to the true state via
y=Oxt,ϵθϵ,
where O is the generalized observation operator and
ϵ is a set of random observational errors
described by distributional parameters θϵ and accounting for
uncertainties associated with the usage and interpretation of the data. These
include at least the random measurement error due to, for example, instrument
noise. In addition they may include a contribution from
representativeness error due to fine-scale variability, if
xt is defined as an average over larger scales than those
of the observations (see above). Alternatively, if the observations are
preprocessed into estimates on the larger scales of xt,
there may be an undersampling error component due to inexhaustive
coverage of the raw samples. The observation operator O may also contribute
to ϵ, for example if the model output needs to be
interpolated from the model grid to the data coordinates, or if O includes
conversion factors such as chlorophyll a-to-nitrogen
(Chl a : N) ratios.
The simplest possible example of an observational error model assumes
additive Gaussian errors. Equation () then becomes
y=Hxt+ϵ⟶ϵ=y-Hxt,
where H accounts for interpolation and units conversion and
ϵ∼G(0, R) is Gaussian
distributed with mean zero and covariance matrix R.
This may be a reasonable error model for most physical variables and chemical
concentrations with ranges well above zero (e.g. dissolved inorganic carbon
or total alkalinity in the open ocean). However, many nutrients and plankton
biomass variables may vary close to their lower bounds of zero, and display
positive skew in their observational errors. For such variables, a lognormal
observational error model may be more appropriate:
y=Hxt∘expϵ̃-σ̃22⟶ϵ̃=log(y)-logHxt+σ̃22,
where ∘ denotes element-wise multiplication and
σ̃2 denotes the variance in logarithmic space.
The bias-correction term (σ̃2/2) ensures
unbiased errors, but is frequently neglected in practice. The various options
and challenges of defining an appropriate error model are discussed in detail
in Sect. ().
Estimation methodsBasic probabilistic approaches
We now consider how to estimate uncertain parameters Θ given the
data y, where Θ includes all biological
parameters θe and possibly distributional parameters
(θη, θζ, θϵ). There are basically
two probabilistic approaches for doing this: Bayesian estimation and maximum
likelihood estimation. In the Bayesian approach, we treat the parameters as
random variables, and choose parameter values on the basis of their
“posterior probability”, i.e. the conditional probability density of the
parameter values given the data p(Θ|y). The posterior
probability is computed using Bayes' theorem:
p(Θ|y)=p(y|Θ)⋅p(Θ)p(y)∝p(y|Θ)⋅p(Θ),
where p(y|Θ) is the likelihood and p(Θ) is the
unconditional or “prior” distribution of the parameter values. The
proportionality follows in Eq. () because the probability of
the data p(y), otherwise known as the “evidence” for the model, is
independent of the parameter values.
In general the likelihood can be expressed as an integral over probabilities
conditioned on particular values of the model state and true state:
p(y|Θ)=∫∫p(y|xt,Θ)⋅p(xt|x,Θ)⋅p(x|Θ)dxtdx,
where the conditional probabilities p(y|xt, Θ),
p(xt|x, Θ), and p(x|Θ) are specified
by the chosen models for observational error (Eq. ),
kinematic model error (Eq. ), and dynamical model error
(Eq. ) respectively. In practice we are unlikely to require
such a complex expression for numerical evaluation; aggregation of error
terms and redundancy between kinematic and dynamical model error usually
allows simplifications.
The Bayesian approach encourages us to explicitly quantify our prior
knowledge about the parameter values through the prior p(Θ). In marine
ecosystem modelling, we are unlikely to ever consider cases of complete
parameter ignorance, where a parameter value could possibly switch sign or
get incredibly large. Every parameter is expected to have a value that falls
into a credible range; otherwise, the associated parameterization would be
difficult to defend. In some cases, when broad uniform or “uninformative”
priors are assumed, it may not be necessary to specify exact limits of these
distributions as the analyses may become insensitive to these limits once the
range becomes sufficiently broad. There are inherent difficulties with the
concept of “ignorance” priors: for example, a flat prior distribution
over ϕ will correspond to an informative prior for some
function g(ϕ) (see , for further discussion).
In any case, trying to minimize the impact of prior distributions rather
defeats the object of Bayesian estimation, which explicitly aims to
synthesize information from new data with prior information from previous
analyses.
Once the likelihood is formulated and a prior distribution is prescribed,
classical Bayes estimates (BEs) may be computed from posterior mean or
posterior median values of Θ. Assuming the statistical assumptions are
correct, these estimators will minimize the mean square error or mean
absolute error respectively of the parameter estimate Θ^e.g.. To obtain BEs can be computationally
expensive, requiring sophisticated techniques to sample efficiently from the
posterior distribution (e.g. by Markov chain Monte Carlo, MCMC, methods). An
alternative Bayesian estimator, very widely used in geosciences, is the joint
posterior mode or maximum a posteriori (MAP) estimator
e.g., given by maximizing the
posterior probability p(Θ|y) as a function of Θ. Such
estimates are more computationally feasible in large problems where the
search for the maximum of the posterior (or the minimization of its negative
logarithm) can be greatly accelerated by techniques such as the variational
adjoint Chap. 4.
In maximum likelihood (ML) estimation we seek the parameter values
Θ^ML that maximize the probability of the data
given the parameter set, i.e. p(y|Θ). When considered as a
function of Θ, this probability is called the likelihood of
the parameter values L(Θ|y) because it is strictly a probability
of the data, not of the parameter values. Indeed, in ML estimation we do not
need to consider the parameter values as random variables at all; rather they
are considered as fixed, unknown constants. For this reason the “|”s are
sometimes replaced by “;”s to emphasize that, in a non-Bayesian context,
the likelihood is not a conditional probability in the sense of one set of
random variables dependent on another e.g.. In the
ML approach, no prior information on the parameter values is used except
possibly to define upper or lower plausible limits or allowed ranges for the
parameter search .
Historically, Bayesian methods predate ML
methods of by some margin. Fisher introduced ML methods
partly to avoid problems in defining prior ignorance (see above) but also to
avoid the noninvariance property of Bayesian estimators .
This property means that given the BE of one parameter
ϕ^B, the corresponding BE of a nonlinear
function of that parameter g(ϕ) is not simply given by plugging in the
estimate (g^B≠g(ϕ^B)),
while for ML estimates the invariance property does hold
(g^ML=g(ϕ^ML)). We
will see an example of this in Sect. .
Sequential methods
In some problems, assimilating all the data at once from all available
sampling times can be computationally impractical. This is particularly
likely for models with stochastic dynamics (η≠ 0 in
Eq. ), if the data are clustered in time, or if model states
need to be repeatedly updated as new data come in. In such cases a sequential
approach can be expedient. The basic idea is to break the large integration
problem defined by Eq. () into a number of smaller
problems by sequentially assimilating observations in subsets defined by
sampling time. The method comprises a consecutive sequence of two major steps: a forecast step and an
analysis step. If the sequential
algorithm is accurate, it should approximate the posterior parameter
distribution defined by Eqs. ()
and () at times where all available data have been
assimilated.
To see how this works, suppose we know the probability density
p(xjt|y1:j, Θ) of the true state at sampling time tj
(possibly an initial condition) for a given value of the uncertain
parameters Θ and given all the previously assimilated observations y1:j
(possibly null). The probability density at sampling time tj+1 is given
by the forecast density:
pxj+1t|y1:j,Θ=∫pxj+1t|xjt,Θ⋅pxjt|y1:j,Θdxjt.
In general this integral can be approximated by an ensemble of Monte Carlo
simulations, sampling an initial condition from
p(xjt|y1:j, Θ) and
then running the model to the next sampling time tj+1 (possibly
including stochastic dynamical noise, and possibly
accounting for kinematic model error). Next, in the analysis step, the new
observations are assimilated by applying Bayes' theorem:
pxj+1t|y1:(j+1),Θ∝pyj+1|xj+1t,Θ⋅pxj+1t|y1:j,Θ,
which again can be approximated e.g. by Monte Carlo sampling. The forecast
and analysis steps can then be repeated until all the data are assimilated.
Note that Eq. () assumes conditional independence of
the observations, allowing us to write p(yj+1|xj+1t,
Θ) instead of p(yj+1|xj+1t, y1:j,
Θ). This amounts to assuming that the observational errors are
independent between sampling times , which may not be
strictly true if sampling is frequent and if there is a noticeable
contribution from representativeness/undersampling, or from errors in
conversion factors (see Sect. ).
Once the predictive filtering densities
p(xj+1t|y1:j, Θ) have been
approximated for all sampling times (tj with j= 1, …,
Nt), these can be used to approximate the likelihood in
Eq. (), since
p(y|Θ)=∏j=1Ntpyj|y1:j-1,Θ=∏j=1Nt∫pyj|xjt,y1:j-1,Θ⋅pxjt|y1:j-1,Θdxjt=∏j=1Nt∫pyj|xjt,Θ⋅pxjt|y1:(j-1),Θdxjt.
For j= 1 in Eq. () we have a set of zero
members and p(yj|y1:j-1,
Θ)=p(y1|Θ). The third line of Eq. () again
assumes conditional independence of the observations and the final integral
can in general be approximated using the predictive ensembles (see
). This procedure can be
repeated for different values of Θ and combined with Eq. ()
to assess posterior probability.
Alternatively, p(Θ|y) can be calculated from a single
application of the filter using a “state augmentation” approach whereby the
parameters Θ are appended to the vector x as additional state
variables with zero dynamics. In practice, random parameter noise may need to
be added to avoid filter degeneracy, such that this approach may be
considered a separate estimation method . However, if such ad
hoc noise can be avoided, or if the parameters are in fact assumed to vary
stochastically, then the augmented-state filter at the end of the
assimilation interval should approximate the theoretical Bayesian posterior
for this time. For other times, a “smoother” algorithm would be required. A
further benefit of the augmented-state filter is that the parameter estimates
for intermediate time periods may show temporal patterns that expose
deficiencies in the model formulation and provide useful information for
model development e.g..
The various types of filter differ essentially in terms of how the integrals
in Eqs. () and () are approximated.
Particle filters use Monte Carlo sampling for both
steps, while the ensemble Kalman filter uses
Gaussian and linear approximations for the analysis step, enabling the use of
smaller ensembles but at the cost of lower accuracy in strongly
nonlinear/non-Gaussian problems. The (extended) Kalman filter applies when
the model dynamics are (quasi-) linear and both model and observational
errors are Gaussian. These conditions allow both integrals to be evaluated
analytically, but appear to be rarely applicable to parameter estimation in
marine ecosystem models. For reviews of sequential approaches the reader is
referred to for marine biogeochemical modelling and to
for oceanography in general.
Variational methods
At present there appears to be some ambiguity regarding the term
“variational” in the context of DA. It is sometimes used to describe
approaches explicitly based on control theory or “inverse methods” that may
not include explicit assumptions about error distributions and where cost
functions are defined a priori, rather than being derived from statistical or
probabilistic models. However, a distribution-free approach seems difficult
to recommend in general for marine ecosystem model parameter estimation,
given the strong nonlinearity, non-Gaussianity, and relatively weak data
constraint often encountered in such problems. Within the marine ecosystem
modelling community, the term “variational DA” is often used more broadly
to refer to all non-sequential methods that involve the minimization of a
cost function, whether or not this is based on a probability model.
In any case, there are some powerful mathematical tools developed for
variational DA that can be applied to minimize cost functions. Adjoint
methods allow the gradient of the cost function with respect to all fitted
parameters to be computed in an extremely efficient manner; see
, and Appendix . This is
particularly useful when dealing with a large number of fitted parameters
(high-dimensional Θ) of computationally expensive models
e.g.. The application of the adjoint method
helps to reduce the number of model runs to provide access to joint posterior
mode and maximum likelihood estimates.
provide useful theoretical background for different
4DVar approaches (four-dimensional, in space and time, variational
approaches) and show how this adjoint method can be used to estimate
ecosystem model parameters jointly with a large number of initial condition
parameters. See also for an introduction to variational
DA and adjoint methods in physical oceanography. However, it can be
disadvantageous to employ a search algorithm that relies too much on local
gradients (e.g. from an adjoint model) to minimize the cost function, because
this may result in finding a local minimum rather than the global minimum
that defines the MAP or ML estimate . This issue appears
to be frequently encountered in marine ecosystem modelling applications, and
should be expected as a product of strong nonlinearity and weak data/prior
constraint. For such cases, a non-local approach such as simulated annealing,
following , or a microgenetic
algorithm, following , may be preferable, at
least during an initial period of the search before the broader region of the
global minimum is located . The main drawback of these
non-local search algorithms is that they tend to require a larger number of
model runs (of at least order 103) to have a good chance of accurately
locating the global minimum, although they may yet provide meaningful
improvements to prior parameter estimates for an order of 100 runs
.
Recent approaches
Much recent interest has focused on combined state and parameter estimation,
whereby model parameters Θ are estimated together with a true
state xte.g.. In the
Bayesian approach, model parameters and system state are both random
variables. We can therefore apply Bayes' theorem to the composite random
variable Ψ= (Θ, xt) and decompose the
prior as p(Ψ)=p(xt|Θ) ⋅p(Θ)
to obtain an expression for the joint posterior:
pxt,Θ|y∝py|xt,Θ⋅pxt|Θ⋅p(Θ).
This equation has so far been applied to stochastic dynamic models with no
kinematic model error cf..
Equation () can be recovered from Eq. () by
integrating (marginalizing) both sides over xt.
In some other recent studies emphasis is put on “hierarchical” error models
. Here, the
traditional model parameters are replaced with stochastic processes over time
and/or space, and parameter identification focuses on the
hyperparameters that describe the stochastic processes (e.g. means,
variances, autocorrelation parameters). This is essentially similar to the
case of parameter estimation for a stochastic dynamical model
(Sect. ) and fits into the general formulation in
Sect. , if we treat the stochastic parameters as
additional state variables with dynamical model errors η.
The hyperparameters could in principle be estimated by ML, sometimes referred
to as an “empirical Bayesian” approach , but it
appears that computational tractability may favour the “hierarchical
Bayesian” approaches e.g., which may also
make use of sequential Monte Carlo methods e.g..
Another important initiative is the estimation of hyperparameters of the
kinematic error model along with the ecosystem parameters
. The posterior of the kinematic model error
provides an estimate of the model discrepancy, introduced by
and originally referred to as model inadequacy. The
model discrepancy is defined as the model error for the “true” values of
the model parameters, i.e. the unknown values of the parameters for which
the model best represents xt. Estimates of model discrepancies may
thus provide useful diagnostics for model skill assessment and development.
From statistical model to cost function
The choice of a suitable estimation method for marine ecosystem model
parameters should be mainly based on the availability of relevant prior
information, as well as on the basic error assumptions (Eqs. –). Once the error model
and estimation method have been chosen, we can derive the probability
densities and cost functions that can be used for parameter estimation.
As a simple but common example, consider a deterministic model with no model
error and data with additive Gaussian observational errors, Eq. (),
with known covariance matrix R. We
wish to use a total of Ny data, summing over all data types, to
estimate NΘ parameters by Bayesian estimation. A survey of the literature
might lead us to model the prior distribution of Θ as Gaussian with a
mean Θb and covariance matrix B. From
Eq. () the posterior density is proportional to a product of the
likelihood and the prior density:
p(Θ|y)∝1(2π)NydetR⋅exp-12dTR-1d⋅1(2π)NΘdetB⋅exp-12ΔΘTB-1ΔΘ,
where the data–model residual d is defined by
d=y-H(x) (see ϵ in
Eq. ). The deviation from the prior
is ΔΘ=Θ-Θb. A MAP or
joint posterior mode estimate of Θ can then be obtained by minimizing
the cost function
J(Θ)=-2logp(Θ|y)+ constant, given by
J(Θ)=dTR-1d+ΔΘTB-1ΔΘ,
where constant terms (since independent of Θ) have been dropped.
Alternatively, nonnegativity constraints on the variables and parameters may
lead us to prefer the lognormal observational error model. Likewise, we can
assume lognormal priors for the parameters. In this case the posterior
density becomes
p(Θ∣y)∝1(2π)NydetR̃∏jyj⋅exp-12d̃TR̃-1d̃⋅1(2π)NΘdetB̃∏lΘl⋅exp-12Δ̃ΘTB̃-1Δ̃Θ,
where the data–model residuals and parameter corrections on the transformed
scale are defined by
d̃=log(y)-log(H(x))+σ̃22
and
Δ̃Θ=log(Θ)-log(Θb)+(σ̃b)22.
A MAP estimator of Θ is then obtained by minimizing
J(Θ)=d̃TR̃-1d̃+2∑l=1NΘlogΘl+Δ̃ΘTB̃-1Δ̃Θ.
The MAP or posterior mode estimator of log(Θ) is equivalent here to
the posterior median estimate and is obtained by maximizing
p(log(Θ)|y). This leads to a cost function given by
Eq. () without the second term,
2∑l=1NΘlog(Θl)cf.. Due to the noninvariance property of Bayesian
estimates, the exponent of the MAP estimator of log(Θ) will generally
differ from the MAP estimator of Θ. By contrast, ML estimates are
obtained by minimizing the cost functions without any of the prior terms
(second term in Eq. , second and third terms in
Eq. ). In each case the same ML estimator for Θ is
obtained whether we use Θ or log(Θ), as expected from the
invariance property of ML estimates.
Remarks on data assimilation terminology
We close this section with some cautionary remarks about different
terminology that the reader may encounter in the literature. First, many DA
papers and textbooks start by assuming a certain cost function, based on
variational or optimal control theory, rather than deriving it from a
probabilistic treatment as herein e.g.. These studies tend to refer to MAP estimates
obtained by minimizing cost functions such as Eq. () as
“weighted least squares estimates”. However, any analogy with regression
analysis is stretched because these estimates are fundamentally dependent on,
and potentially biased by, the assumed prior distributions. Second, many DA
papers and textbooks use the term “likelihood” to refer to the posterior
probability p(Θ|y) in Eq. (), and the term
“maximum likelihood estimators” although modifiers such as “(Bayesian)”
p. 156 or “(posterior)”
p. 40 are sometimes added. This obscures the fact
that posterior mode estimators, like all BEs, are dependent on assumed prior
distributions. Maximum likelihood avoids this dependence, but in doing so
tends to be unsuitable for high-dimensional parameter estimation in the
partially observed systems typically encountered in oceanography and
geophysics.
Typical parameterizations of plankton models and their parameters
Deviant parameter estimates of a model may point towards a deficiency in
model structure, forcing, or boundary conditions. Estimates of the
effectively same parameters may turn out to be different within dissimilar
plankton ecosystem models, even if those models may have been calibrated with
the same data and although they possibly share an identical physical
(environmental) set-up. To understand why parameter estimates can be
different it is helpful to unravel some of the basic differences between
major parameterizations that describe growth and loss rates of phytoplankton.
A crucial element of most plankton ecosystem models is the description of
phytoplankton growth as a function of light, temperature, and nutrient
availability. How growth of algae is parameterized is relevant and the
associated parameter values affect the timing and intensity e.g. of a
phytoplankton bloom in model solutions.
Differences between maximum carbon fixation and maximum growth rate
The build-up of phytoplankton biomass depends on how much of the available
nutrients can be utilized and how much energy can be absorbed from sunlight.
Under nutrient-replete and light-saturated conditions, the carbon fixation
(gross primary production, GPP) reaches a (temperature-dependent) maximum
rate, described as a parameter (PmC) with
unit day-1. For models that do not resolve mass flux of carbon
explicitly, PmC is substituted by a maximum growth
rate (μm) to express the phytoplankton's maximum assimilation
rate of nitrogen (N) or of phosphorus (P). The maximum GPP and the maximum
growth rate are interrelated and in principle one can be derived from the
other . In reality, maximum C-fixation, maximum N- or
P-assimilation, and cell doubling rates are highly variable. This requires at
least cellular C, N, and Chl a to be explicitly resolved,
linking for example intracellular nutrient allocation to
photo-acclimation.
In practice an analogy between PmC and
μm is often assumed in N- or P-based biogeochemical models
(assuming fixed stoichiometric elemental C : N : P ratios for algal
growth). The parameter PmC or μm is
typically multiplied by a dimensionless temperature function (fT)
e.g.,, allowing for temperature-induced
changes in metabolic rates. The actual potential maximum rate
(PmC⋅fT or
μm⋅fT) is then reached at some prefixed
reference or optimum temperature accordingly. In early N-based plankton
modelling studies e.g. the maximum growth rate was mainly adopted from
. In subsequent DA studies this maximum rate was either
subject to optimization e.g. or
it was kept fixed because then parameter values of the limitation functions
could be better identified .
Combining parameterizations of light and nutrient limitation
In many marine ecosystem models two separate limitation functions are
combined: one that expresses the photosynthesis vs. light relationship
(P–I curve) and another that describes the dependence between ambient
nutrient concentrations and nutrient uptake. The two functions are similar in
their characteristics, starting from zero (no light or no nutrients) and
approaching saturation at some high light and at replete nutrient
concentration. Three approaches are generally found in marine ecosystem
models to limit algal growth by photosynthesis and nutrient uptake. The first
is to apply Blackman's law , assuming that growth is
reduced by the most limiting factor, either by light or by nutrient
availability e.g.. The second is to multiply both limitation functions
e.g.. The third
approach involves combinations of light and nutrient limitation that resolve
interrelations between cell quota, N-uptake, and the photo-acclimation state
of the algae e.g.see Sect. .
Whether the first, second, or third approach is considered can be expected to
affect estimates of the associated parameter values.
Photosynthesis as a function of light (P–I curve)
In a P–I curve the level of increase from low to high irradiance is
specified by the initial slope parameter (the maximum of the first derivative
of the P–I curve with respect to light), also referred to as photosynthetic
efficiency (αphot)
.
Photosynthetic efficiencies were derived from P–I measurements, for example
by , , and
, and their mean values were used for many N-based
models e.g.. Published measurements of αphot were
typically normalized to Chl a concentrations. In case of N- or
P-based models careful considerations are then needed with respect to the
phytoplankton's cellular Chl a content, which can vary by a factor
of 10 and more. Values of αphot were found to vary by a
factor of 3 during a 3-month period, which can be
attributed to changes in phytoplankton community structure as well as to
photo-acclimation. reported an even larger
variational range over a 1-year period, from
αphot= 0.03 to
0.63 mg C (mg Chl a)-1 h-1 W-1 m2 within
the upper 10 m.
Algal growth and nutrient limitation
Typical parameterizations of growth limitation by nutrient availability
(ambient nutrient concentrations) are expressed with the half-saturation
constant (Ks) of a classical Monod equation . Another approach is to parameterize limitations of the nutrient
uptake rate, described with a parameter referred to as nutrient affinity
(αaff) . The affinity-based
parameterization may also be applied to describe nutrient-limited growth,
assuming that the rates of nutrient uptake and growth are balanced. In this
case both parameters (Ks and αaff) can be
interpreted as being interrelated:
αaff=μm⋅fT/Ks.
However, αaff is derived from mechanistic considerations
that are fundamentally different from former interpretations
of Ks of a Monod equation . For comparison between estimates
of αaff it is important to know whether this parameter
describes limitation of growth or of nutrient uptake. The description of
nutrient limited growth with the Monod equation, thereby retrieving values
for Ks from measurements, had been discussed in the past
e.g..
This discussion regained attention during recent years and the sole
application of the Monod equation is currently viewed as a considerable
drawback when simulating plankton growth under transient (unbalanced growth)
conditions .
Algal growth and intracellular acclimation
More complex growth dependencies are described with models that consider
intracellular acclimation dynamics e.g.. In these models,
photoautotrophic growth rates become dependent on cell quota, e.g. usually
normalized to carbon biomass (N : C), and the amount of synthesized
Chl a per cell. With such approaches, the changes in the mass
distribution of phytoplankton C and N, as well as the cellular Chl a
content, have to be explicitly resolved in the model. One advantage is that
these models are more sensitive to variations in light conditions and
nutrient availability. The respective equations involve physiological
parameters that are related but not identical to those of classical N- or
P-based growth models, which impedes a direct comparison of older estimates
of growth parameters with values currently used in models with acclimation
processes resolved.
Losses of phytoplankton biomass
Parameterizations of phytoplankton cell losses involve lysis (starvation
and/or viral infection), the aggregation of cells together with all other
suspended matter, and grazing by zooplankton. Exudation and leakage are
processes of organic matter loss that occur while the physiology of the algae
is functional. Cell lysis, exudation, and leakage are usually expressed as a
single rate parameter and this loss of organic matter is assumed to be
proportional to the phytoplankton biomass.
Parameterizations of phytoplankton losses due to the process of coagulation
and sinking of phytoplankton and detrital aggregates are basically derived
from the principle theory of coagulation. The application of coagulation
theory to simulate phytoplankton aggregation is well established for models
that resolve size classes of particles (of phytoplankton cells and detritus)
explicitly . But the representativeness of simplifications
(e.g. reduction to two size classes) assumed for model simulations remains an
open task e.g.. Aggregation
parameters in marine ecosystem models are often assumed to represent the
combination of a collision rate and the probability of two particles sticking
together after collision (e.g. stickiness of algal cells). These two
parameters, collision rate and stickiness, are multiplied by each other to
yield a final aggregation rate. They are therefore difficult to estimate
separately. Unless prior information can be used their estimates are always
collinear, which suggests estimation of their product instead (as done in the
example in Sect. ).
A common problem is to find constraints that allow for a clear distinction
between phytoplankton losses due to the export of aggregated cells and the
loss because of grazing. Both processes can be responsible for the drawdown
of phytoplankton biomass, and data that cover the onset, peak, and decline of
a bloom are needed for a possible distinction. How the complex nature of
predator–prey interaction is parameterized remains a critical element of
plankton ecosystem models. Compared to the approaches that describe algal
growth, an even larger number of different parameterizations exist for
grazing . Experimental data of grazing rates and
collections of field data of zooplankton abundance are therefore of great
value.
Elaborate analyses of mesozooplankton and microzooplankton biomass, grazing,
and mortality rates were done by . For their two studies they compiled an extensive
database with laboratory and field measurements. With their data syntheses
they could derive parameter values for simulations with a global ocean
biogeochemical model. Furthermore, independent field data, not used to derive
the mesozooplankton and microzooplankton parameter values, were considered
for assessing the performance of their model on a global scale. Their work
reflects the large effort that can be dedicated to this topic for achieving
reliable simulation results of zooplankton grazing.
The explicit distinction between zooplankton size classes, like
mesozooplankton and microzooplankton, was bypassed in
. Their model allows for omnivory within the
zooplankton community, which is resolved by introducing adaptive food
preferences. These preferences are treated as trait (property) state
variables that adapt to the relative availability of different prey. This
reduces the number of parameters needed to describe a variety of different
behaviours in grazing responses. Field data from three ocean sites in the
North Atlantic were used by for calibrating their
plankton model. They conducted a two-step approach for parameter
optimization. First they optimized parameter values so that depths and dates
of minimum and maximum observed values become well represented by their model
at all three sites. In a second step they refined their parameter estimates
by minimizing weighted data–model residuals. After parameter optimization
they identified distinctive complex patterns between zooplankton grazing and
plankton composition for the three simulated ocean sites. Besides their
phytoplankton grazing losses it turned out that their optimal estimates of
photo-acclimation and maximum C-fixation (αphot,
PmC) agree with those values derived from model
calibrations with laboratory data.
Constraining simulations of algal growth with laboratory and mesocosm data
Parameter values of acclimation models have typically been adjusted to
explain laboratory measurements . So far, there is a limited number of experimental studies
whose data were used to calibrate these acclimation models
. Model calibrations were usually done by
tuning parameter values so that model solutions provide a qualitative good
fit to the laboratory data. In many cases the parameter adjustments relied on
the researchers' experience and intuition, sometimes accounting for prior
parameter values obtained from preceding model analyses
e.g.. Analyses of parameter uncertainties of recent
acclimation models are often lacking. Most laboratory modelling studies had
put emphasis on the physiological mechanistic model behaviour, while error
assumptions for quantitative data–model comparison were hardly considered.
Explicit error assumptions for parameter optimizations and for comparisons of
acclimation model results with laboratory data were introduced by
and by . In both studies
additive uncorrelated Gaussian observational errors were assumed and
optimized results of different model versions had been compared.
applied a simulated annealing algorithm
to fit his optimality-based model version to the
data of . The same data were used to also fit the
model of , and he evaluated the likelihood ratio of
the two ML estimates, to discuss and underpin the improved performance of
his refined acclimation parameterizations. also
compared the performance of two acclimation models, of
and respectively. Optimal
parameter values for the two model versions were obtained with the MCMC
method, minimizing the misfit between model results and data of the
experiment. Apart from mechanistic considerations,
concluded that the models of
and described the assimilated data equally well,
since both cost function minimum values were comparable. However, the
simulated N : C and Chl a : N ratios of the model proposed by
were in much better agreement with observations during the
exponential growth phase, which remained undifferentiated by their error
model (assuming C, N, and Chl a data to be independent). Different
considerations for error models will be addressed hereafter in
Sect. .
To collect diverse data that fully resolve onset, peak and decline of an
algal bloom at ocean sites is difficult to achieve. Data derived from remote
sensing, e.g. Chl a concentration and primary production rates,
provide limited information to explain relevant differences between processes
described before, like N-utilization, fixation and release of C, and
synthesis and degradation of Chl a. Mesocosm experiments that
enclose a large volume of a natural plankton and microbial community can be
helpful in this respect, if they provide a good temporal resolution of the
exponential growth phase as well as of the post-bloom period.
highlighted the benefits of using mesocosm data to test
plankton ecosystem models, as done before by . One advantage is that mesocosms are, apart from the
surface, closed systems and measurements of inorganic nutrients, dissolved
and particulate organic matter should, in principle, add up to approximately
constant concentrations of total nitrogen and total phosphorus. Total carbon
concentrations may only vary due to air–sea gas exchange. By design these
experiments often integrate valuable series of joint and parallel
measurements, yielding detailed data from various scientist with different
expertise
e.g..
Drawbacks are uncertainties in initial conditions and also the
representativeness of mesocosm data to reflect the real dynamics in the ocean
is subject to discussion e.g.. In spite of these
limitations, simulations of mesocosms or of enclosures experiments (e.g. with
large carboys deployed in the field) have helped to identify credible model
parameter values and assess model performance. This is particularly true for
tracing microbial dynamics or for details in the composition and fate of particulate
organic carbon and nitrogen (POC and PON) .
In contrast to laboratory measurements, data from mesocosm experiments
reflect some natural variability of the plankton community, mainly captured
by replicate mesocosms. The availability of measurements from replicate
mesocosms is also helpful when defining error models that specify the
statistical treatment of the data used for parameter estimation.
Error models
Error models define our assumptions about uncertainties and the statistical
relationships between observed data, the true state, model output, model
inputs (forcings and initial/boundary conditions), and model parameters. Here
we review error models that have been applied to address the various sources
of uncertainty in marine ecosystem models and consider their implications for
parameter identification. An explicit treatment of each source of uncertainty
may not be necessary, but we do recommend reflecting on how these
uncertainties can be accounted for when modelling plankton dynamics and
biogeochemical cycles.
Uncertainty in observations
The simplest and most common models for observational error assume that the
observational errors ϵ are (i) additive normal, (ii) constant
variance between samples, and (iii) independent between samples and variable
types. Such models are also commonly used to represent aggregated errors
accounting for both observational and kinematic model error (see
Sect. ); we will refer to these as residual
errors.
The additive normal assumption (i) is straightforward but also restricted, as
it does not capture three common characteristics of some ecosystem data such
as Chl a concentrations: (1) larger values tend to have larger
errors, (2) values cannot be negative, and (3) the error distribution has
positive skew. Characteristic (1) may be captured by scaling the standard
error with modelled values e.g. or with
observed values e.g., while
characteristic (2) can be resolved using truncated error distributions
e.g.. All three characteristics together can be
captured by gamma distributions or power-normal
distributions whereby normality is assumed on a power-transformed scale
. The power-normal family includes
lognormal e.g. and square-root normal models
e.g..
For power-normal, gamma, or proportional error assumptions we have the
difficulty that the variance on the original scale approaches zero at low
values. This may be unrealistic, at least in regard to instrumental noise. In
normal models this problem can be addressed by adding a constant term to the
variance or standard
deviation . Another difficulty is that transform-normal
models may require unbiasing factors when assuming unbiased errors on the
original scale (e.g. exp-σ̃2/2 for the
log-transform). More flexible models may be obtained by e.g. fitting the
power transform parameter , assuming generalized Gaussian
distributions , or using “anamorphic”
transformations . It is yet
unclear whether such extra flexibility is generally necessary, but it has
been demonstrated that the choice of transformation can strongly affect
estimates of plankton ecosystem fluxes and that a good
choice can improve parameter estimation in twin experiments see Fig. 1
and.
Time evolution of parameter estimates in a simulation test of an
ensemble Kalman filter using untransformed data (a–c, top row
panels) and using logarithmic transformed data (d–f, bottom row
panels) (Simon and Bertino, 2012, Fig. 3). Solid lines and shading show
ensemble means and standard deviations averaged over 20 simulation
experiments, while dashed lines show the true parameter values. The data were
generated using Gamma-distributed observational errors with standard
deviation ∼ 30 % (see Simon and Bertino, 2012). A transformation can
significantly reduce the bias of parameter estimates by the end of the
assimilation period. Figure was redrawn from results provided by Ehouarn
Simon, with permission from Elsevier. Copyright of figure content by
Elsevier.
The validity of the constant variance assumption (ii) may be improved by a
scale transformation, although the transformation that best normalizes the
error distribution (see above) may not best promote the homogeneity of
variance. Spatiotemporal variations in the error variance may naturally
occur, for example due to seasonal modulations of the unresolved variability
and hence the representativeness error component. Accounting for this
variation should improve parameter estimates and uncertainty assessment
cf., but in applications this has rarely
been attempted .
In some contexts, e.g. mesocosms, the error covariance matrix might be
estimated from experimental replicates prior to fitting the model
(Sect. ). In problems where sampling is sparse
and/or when the model error contribution is large, the error variances may
not be estimable from data alone . Here the variances may
instead be parameterized and estimated jointly with the ecosystem model by
Bayesian or ML estimation, which has been done in few studies
.
The assumption of independent errors between samples and variable types (iii)
can be invalidated in cases where contributions from representativeness error
or kinematic model error are large, or where the data have been derived by
interpolation or application of a regression model. Neglected correlation may
result in parameter estimates that are less efficient (higher variance) and
more strongly correlated (e.g. see example in Sect. 5.4). Pre-averaging the
data is somewhat helpful to promote independence (and normality, via the
central limit theorem), but might also remove some of the informative
variability. One common ad hoc intervention in the cost function is to scale
the residual error variance with the sample size of each data type, to avoid
biasing the fit in favour of better-sampled variables
e.g.. More formal
treatments have fitted parameterizations of the error correlations jointly
with the ecosystem model e.g..
Whatever the assumptions of the observational/residual error model, it is
possible to test their validity using the assimilated data, either by
analysing the residuals and performing lack-of-fit tests
(, p. 43; )
or by comparing fit statistics with those obtained under alternative error
models (using e.g. likelihood ratio tests and information theoretic or
Bayesian criteria; see Sect. ).
Finally, we caution that certain interpolated or derived data may strictly
invalidate the observational error model, not only due to error correlation
(see above), but also due to the introduction of smoothing bias. Data
interpolated onto a model grid will tend to systematically underestimate true
values where they are high and overestimate them where low; an effect that
will be difficult to account for in the observational error model. In this
situation parameter estimates can become biased towards
values that suppress spatiotemporal variability in plankton dynamics.
Similarly, if the data are derived from a regression model, these estimates
may also “trim the peaks and fill the valleys”, because in a regression
model (e.g. y=a0+a1p+ϵ, where p is some predictor data)
there is always some part of the true variability that is included in the
error term, and therefore subject to smoothing bias. In principle this could
be avoided by including an inverted regression relationship in the operator O
and assimilating the “raw” predictor or proxy data instead of the regression-based estimates.
Prior uncertainty in Θ
Prior uncertainty plays an important role in estimating model parameters.
Typically, there is not enough information in the assimilated data to
constrain all parameters of a biogeochemical model. The results may well be
sensitive to the “error model of prior uncertainty”. Prior uncertainty can
be represented by prior probability densities in Bayesian approaches or
plausible ranges in non-Bayesian approaches. To account for nonnegativity
constraints, prior distributions typically include lognormal
, square-root normal , or beta
distributions , although normal distributions may yet
be applicable for parameters that are well constrained above zero
. To our knowledge no application has yet
incorporated prior correlations between parameters in Θ
(i.e. off-diagonal terms in matrix B introduced in
Sect. ). This is surprising, given the fact that
posterior uncertainty assessments consistently reveal strong correlations
e.g..
Quantifying the prior uncertainty in Θ is often difficult due to
(1) the existing diversity of model structure, functional forms used in the
various parameterizations, and definitions of model state variables, and
(2) the intrinsic variability between assimilated data sets in terms of
taxonomic composition of the plankton community vs. (usually monospecific)
laboratory cultures. As a result, it may not be advantageous to simply set
the prior uncertainty in Θl as the posterior uncertainty from one
previous study. A more common approach is to first gather best estimates
of Θl from a series of previous studies that included
parameterizations and state variable definitions sufficiently consistent with
the present, and then treat these as unbiased data from which a prior
distribution or plausible range can be determined.
When posterior uncertainty becomes unacceptably high, it can be reduced by
reducing the prior uncertainty in Θ, and there are several strategies
for doing this. First, we should incorporate further data, perhaps of a
qualitative nature, into the prior constraints. For example, if it is known a
priori that certain species or functional groups coexist in certain regions
at certain times of the year, then any Θ resulting in competitive
exclusion of one of these groups might be ruled out a priori. Another
possibility within the Bayesian paradigm is to incorporate the subjective
opinion of experts . A second strategy is to model
statistical structure in the prior parameter values, and thereby fill in
missing prior parameter estimates for certain species included in the
modelled species or groups. Examples here include the use of allometric
scaling relationships with cell size e.g. and
phylogenetic relationships derived from stochastic modelling of trait
evolution . Third, we may seek to
reduce the model complexity in terms of the number of free parameters,
thereby removing poorly constrained parameters and parameter correlations
that may act to inflate the posterior uncertainty. This may be achieved using
sensitivity analysis e.g. or model selection criteria e.g..
A risk here is that parameter estimates and uncertainty assessment may be
compromised if model selection uncertainty is not properly accounted for
. Fourth, it may be possible to reformulate the
model in such a way that the prior parameter uncertainty is reduced. For
example, a hierarchical model in which parameters vary randomly over space
or time may enable the
use of stronger prior constraints on the distributional parameters describing
this variability (i.e. the “hyperparameters”). Similarly, a stochastic
trait-based approach e.g. may employ
distributional parameter values that are better known a priori than values
for individual species or functional groups, although such a reduction in
prior uncertainty has not yet been clearly demonstrated in the literature.
Uncertainty in initial conditions (ICs)
Dynamical marine ecosystem models are usually specified by differential
equations that are first-order in time, and therefore require for solution
one initial condition (IC) for each grid cell or spatial location in the
model. These inputs are, in general, uncertain, and liable to impact the
model output, at least during a transient relaxation period, or indefinitely
if the uncertainty spans more than one basin of attraction of the dynamical
system or if the model dynamics are chaotic
e.g..
In some cases it is possible to neglect IC error because of accurate
measurements, or because a steady state (equilibrium or seasonal cycle) that
is only sensitive to Θ can be assumed. Caution is required when
neglecting IC uncertainty because initial concentrations are known to be
small (e.g. in January); small absolute errors may be large relative errors
that can still affect e.g. the timing and magnitude of a spring bloom
.
In non-spatial (0-D) models, IC errors have been modelled as both fixed
parameters e.g. and as random variables (Bayesian
parameters) with specified prior distributions
e.g.. In mesocosm studies, ICs can play a
critical role in determining the model trajectory, and can comprise a large
proportion of the fitted parameters e.g.. For
spatial models, it seems necessary to limit the degrees of freedom of the IC
uncertainty , e.g. by using a Bayesian error model with
spatial covariance in the prior . To
model IC uncertainty, Gaussian distributions are most often employed, often
with a log transform to improve realism of the distributional form (see
Sect. ). For systems with strong physical control, it
may be possible to limit IC uncertainty to only the physical variables,
allowing this to generate biochemical uncertainty over an initial burn-in
period .
Uncertainty in forcings and boundary conditions (BCs)
Marine ecosystem models are usually modulated by time- and space-dependent
environmental drivers (forcings) and boundary conditions that are not
predicted by the model dynamics but are necessary inputs to determine the
evolution of the model state variables. Studies have demonstrated the
sensitivity of biogeochemical variables to errors in bottom-up forcings such
as wind stress and vertical mixing e.g. and top-down forcings
such as fishing e.g.. BC errors may have little impact
on variables strongly controlled by internal dynamics at sufficient distance
from the boundaries, but they may become critical if they affect internal
system constraints such as the supply of limiting nutrients or fluxes of
heat/salinity that drive internal circulation and stratification.
There are basically two approaches to modelling the effects of BC/forcing
error: (1) to consider individual or net impacts on model dynamics as
dynamical model errors (η in Eq. ), thus
requiring a stochastic model, or (2) to consider the net impacts on state
variables as kinematic model errors (ζ in
Eq. ), which may permit a deterministic model. The dynamical
approach (Eq. ) is arguably more realistic, more likely to
generate realistic temporal correlations and cross-correlations, and accounts
for time- and parameter-dependent variation in the form and correlation
structure of the joint state variable probability density. It also allows
individual error sources to be considered separately. However, approaches
based on stochastic models can be computationally intensive and
methodologically complex, and parameterizing all individual sources of
BC/forcing error poses a major challenge. Rather than attempting a
comprehensive treatment, current approaches tend to restrict the dynamical
noise to certain key sources such as the atmospheric forcing
or surface irradiance and
background light attenuation ,
and/or they model the net effect of BC/forcing errors and structural errors
synthetically as additive e.g. or
multiplicative e.g. perturbations.
It may be questioned to what extent the simple parameterizations used to
describe these noise processes accurately describe the net or individual
error sources, and it can be difficult to constrain the distributional
parameters a priori, especially if the structural component is important.
Hierarchical filtering methods may allow these “hyperparameters” to be
estimated jointly with the other parameters but these
may incur a computational cost that is prohibitive for spatial models at
present.
The kinematic approach (Eq. ) offers an immediate
computational saving because the integral over model error configurations
(over xt in Eq. ) can usually be
performed analytically, such that accounting for model error may amount to
simply adding variance and correlation structure to the observational error
covariance matrices. However, this may require a more complex
parameterization of the error covariance that may still not properly capture
seasonal or ecosystem parameter dependence .
demonstrated a Monte Carlo simulation approach to determine the variability
of kinematic error variances due to BC/forcing error, but without accounting
for correlations or θe-dependence. Note that with a
deterministic model, and model error treated kinematically, the ecosystem
parameters θe will likely be optimized to reproduce the
ensemble-mean or ensemble-median behaviour of the true system. This may be
convenient for future simulations, but it may also result in biases when
using previous parameter estimates from laboratory experiments or stochastic
model data assimilations to constrain the prior uncertainty
in θe (see Sect. ).
In either case, BC/forcing error models may fall short in describing
potential errors in phase, like the timing of nutrient depletion.
Model solutions that predict the right sequence of events (e.g. a plankton
bloom) but with slightly wrong timing or spatial location, perhaps due to
phase error in the atmospheric forcing or ocean circulation, may suffer a
double penalty due to changes where none occur in the data and no change
where the data do vary. DA may then “smooth out” the model variability in
order to minimize this double penalty . The problem of phase/timing error has received substantial
attention in numerical weather forecasting and geophysical DA
e.g. and has been highlighted as an issue for
marine ecosystem models .
A simple remedy is to average the data and model over larger spatio-temporal
scales in the data assimilation e.g., but
again this may remove informative variability and result in
a Θ^ that is only suited to those larger scales.
explored a more explicit approach assuming random
time lags between the true state and model state i.e. kinematic model errors
in phase, which can be expressed as ζ(θζ) in
Eq. () (see Appendix ). This may improve the
bias and variance of ecosystem parameter estimates compared to a simpler
approach assuming only additive residual error
Table .
For some problems, in particular for chaotic systems, the phase noise may be
too intense or ill-defined to allow effective use of a parametric phase lag
model. A better approach here might be to use a “synthetic likelihood”
, whereby the raw data and model output are replaced with a
carefully chosen, informative set of phase-insensitive summary statistics
(e.g. means, standard deviations, and lag correlations; cf.
). This approach could incorporate the comparison of
modelled vs. observed Fourier spectra and cross-spectra/coherences
e.g.. Whether the statistics e.g. of spectral
slopes by themselves provide good constraint on ecological parameters should
be tested since it may not be sufficient .
Uncertainty in model formulation and structure
Even with perfectly known parameters, forcings, and initial/boundary
conditions, we would still not expect the modelled fluxes such as primary
productivity and grazing to perfectly reproduce the true fluxes, or the state
variables to perfectly follow the true variability. Aggregation of species
into model functional groups, effects of finite spatial and temporal
resolution, and inherent approximations in the flux parameterizations and
model structure may all contribute to “structural error” in the model
dynamics.
One promising approach to account for structural error is to add stochastic
noise (dynamical model errors) to the ecosystem model
parameters θe (see Sect. ). This
preserves mass conservation and may allow information on the temporal
(e.g. seasonal) variability of species composition within functional groups
to be utilized within the stochastic process parameters
e.g.. However, as with explicit treatments of
BC/forcing error (see Sect. ), a comprehensive treatment
of all sources seems likely to result in an overparameterized error model and
appears to be not yet attempted. An alternative (or complementary) approach
is to treat the structural errors as synthetic dynamical or kinematic model
errors, with one noise process for each state variable. Here it seems the
challenges are to control mass conservation and to find some efficient way to
constrain the distributional parameters a priori or a posteriori.
We note that some structural errors may impose persistent or intermittent
biases in the model output that may not be amenable to a simple statistical
description. For example, a succession in blooming phytoplankton species
might extend or multiply the bloom periods in ways that are not “random”
and that are difficult to reconcile with a single model functional group,
even with stochastic parameters. Limited spatial resolution can also impose
persistent biases that lead to poor extrapolation properties when we try to
correct them by adjusting θe. In
such cases, rather than elaborating the error models, effort might be better
spent improving the explicit biological or spatial resolution of the model,
or exploring implicit resolution techniques
e.g..
An alternative approach might be to employ the tools of multimodel inference
. The idea here is to base
inference of target parameters, states, and fluxes on a family of candidate
models, each differing in structure and parameterization, rather than on a
single model. For example, we might be fairly certain about the form of the
photosynthesis–irradiance (P–I) function in phytoplankton, but much less
certain about the appropriate formulation of zooplankton grazing. Multimodel
inference would allow the P–I parameter values and their uncertainties to be
inferred on the basis of several candidate models, each assuming the same
P–I function but different grazing parameterizations. The resulting
multimodel estimates and uncertainties would be less likely to be biased by a
poor choice of grazing formulation than the inference premised on a single a
priori formulation.
Posterior parameter uncertainties
The determination of parameter uncertainties has many facets, getting to the
core of discussions of Bayesian and frequentist approaches and
interpretations e.g..
Depending on the estimator, uncertainties in the combination of parameter
values may either disclose a credible region of a random distribution of
parameter values (Bayesian interpretation) or they mark a confidence region
that should include the true value with a certain nominal probability of
e.g. 95 % (frequentist interpretation). The latter means that different
data sets would yield different confidence regions and e.g. 95 % of those
regions are expected to include the true “fixed” value.
In general, if we wish to make inference about uncertainties of parameter
estimates (Θ^) we need some knowledge about the
distributional shape of the posterior p(Θ^|y) or of
the likelihood p(y|Θ^). Likewise, we can gather
information about the parameter-cost function manifold in the vicinity of
(Θ^, J(Θ^)). For this we may consider some
threshold offset value ΔJ, which is an upper limit for the
deviation from the minimum value J(Θ^). Such a limit may
identify all cost function values that are insignificantly larger than J(Θ^).
Large deviations from optimal estimates might be
required for some parameters (components of Θ^) before the
corresponding cost function values reach this threshold, while for other
components only small variations are enough. Such tolerance limit defines an
uncertainty region in parameter space:
Θ:J(Θ)-J(Θ^)≤ΔJ.
Typical threshold values are defined as the α quantile of a parametric
or nonparametric probability distribution.
For an unbiased ML estimator, the χ2-distribution with the degree of
freedom (df =Ny-NΘ) has been suggested for deriving
a threshold value χ2(df, α)
e.g..
But for nonlinear models the χ2-distribution might be inappropriate and
the α quantile of the actual distribution,
J(Θ)-J(Θ^), needs to be evaluated by other means
e.g.. Furthermore, the degree of freedom (df) that
specifies location and shape of the χ2-distribution may not be
representative. Only if error correlations have been correctly specified
in J (see Sect. ) and the asymptotic approximation
(for large Ny) is applicable, can the correct degree of freedom be
Ny-NΘ. The effective number of independent observations
can be lower and the considered error correlations can be imprecise, for
example when measurements like Chl a and carbon dioxide
concentrations are negatively correlated during exponential growth but can
then become positively correlated shortly after the peak of an algal bloom.
We therefore expect the effective degree of freedom to be often lower than
(Ny-NΘ) and χ2(df, α) would be
an optimistic threshold, i.e. likely to underestimate the true range of
uncertainty, unless the correct number of degrees of freedom is determined.
Confidence and credible regions
Uncertainty regions in parameter space can be determined basically in two
different ways, based on either Bayesian or frequentist interpretations.
According to the Bayesian interpretation a credible region is specified by
conditional probability distribution of the true value given the data. For
maximizations of the likelihood p(y|Θ) it is often stated that
credible and confidence regions are practically identical. Such
interpretation is imprecise since the methods to confine either region can be
very different with respect to the underlying assumptions, e.g. MCMC
vs. bootstrap approaches.
In case of classical BEs no tolerance limit ΔJ is explicitly
prescribed. Instead, an efficient sampling of (Θ, J(Θ)), or
directly of the posterior p(Θ|y), is applied. Sequential methods
can provide approximations of the posterior parameter distribution once all
data have been assimilated. These approximations differ, depending on how
Eqs. () and () are sampled and
evaluated, as discussed in Sect. . A helpful overview
with some comprehensible examples (of four different methods and three
different ensemble sizes) is given by . BE methods that
do not rely on sequential approaches may also be applied and credible regions
are then simply inferred from selective (acceptance/rejection) sampling
schemes in a MCMC approach, e.g. the Metropolis–Hastings algorithm
. MCMC methods for the derivation of
credible regions are also used for ML estimation problems
e.g.. The main point is that here the data are
assumed fixed.
A fundamentally different approach to the BE methods is to repeat parameter
optimizations many times but with data subsamples or resample data sets.
Large data sets are split up into a series of subsamples that should be as
independent as possible, or many synthetic data sets are created by applying
a random number generator to independently draw bootstrap samples
. This approach accounts for variable
data and it mimics a repetition of an experiment or a repeated sampling at
ocean sites. For each bootstrap data set (y*) a corresponding
optimum estimate Θ^* is obtained. A distribution of
ΔΘ=Θ*‾-Θ^*
can be derived from a series of optimizations with different bootstrap data
sets. Furthermore, nonparametric density estimates of
all J(Θ^*) can be derived and the α quantile can
then be determined from the cumulative distribution of such probability
density. For some situations a bootstrap approach with as few as 10 resample
data sets may suffice to highlight specific uncertainties in some model
parameters e.g.. But to ascertain confidence
regions, much larger bootstrap sample sizes are typically needed
. In the end, both approaches, MCMC and bootstrap
methods, require a large number of model evaluations, typically
of the order of o(102)–o(104). The benefit is that skewed and contorted
posteriors can be better resolved.
Profile likelihoods
An alternative to ensemble-based sequential, MCMC, and bootstrap methods for
determining uncertainties of parameter estimates is the construction of 1-D
or 2-D profile likelihoods . For a 2-D profile
likelihood an array of combinations of two parameters (Θm,
Θn) is constructed. For every combination of parameter values
(elements of the 2-D array) a minimization of J(Θ) is repeated while
varying all other parameters (Θl≠m,n). This is done for all
arrays with possible combinations of two parameters, which requires a large
number of additional optimizations. The advantage is that uncertainty
intervals [Θ^l-ul-,
Θ^l+ul+]) can be well resolved for
each component (l) of Θ, with lower and upper uncertainty limits
possibly being different (ul-≠ul+).
Unfortunately, the evaluation of a profile likelihood is impracticable for
most marine ecosystem model applications, because of the associated
computational costs. Parameter identifiability analyses based on profile
likelihoods have been applied to problems where fast evaluations
of J(Θ) were possible e.g.. evaluated confidence regions for three
parameters (rate constants of production, respiration and water–air gas
exchange) from profile likelihoods and they showed that the error margins of
the parameter estimates can be much larger than those derived with e.g. a
point-wise approximation of a posterior uncertainty covariance matrix,
described in the following.
Point-wise approximations of posterior uncertainty covariance matrix
A single point in parameter space is identified by ML and MAP estimators,
i.e. Θ^ where the posterior p(Θ|y) has its
maximum. Because of the computational costs we often find studies where
parameter uncertainties of ecosystem models had been approximated point-wise
in the immediate vicinity of Θ^. A common theory for deriving
variance information of a ML estimate is based on the inverse of the Fisher
information (; see also e.g.
). The underlying
assumption is that the likelihood p(y|Θ^) is nearly
normal shaped nearby its maximum, which is tantamount to a quadratic increase
of J(Θ) as parameter values are varied around the estimate. Series
expansions, like Taylor power series, around the estimate Θ^
can be applied to derive relevant properties of J(Θ) that are
theoretically attributed to an uncertainty covariance
matrix (UΘ). Confidence regions for Θ^ can
then be expressed in terms of approximations of UΘ. For
example, for some prescribed df an upper critical confidence level can be
specified by the α quantile of a F-distribution
:
Θ:Θ-Θ^TUΘ-1Θ-Θ^≤NΘ⋅Fdf1-α.
Confidence ellipsoids are described with Eq. (),
thus yielding symmetric uncertainty limits around Θ^,
i.e. ul=ul-=ul+. With an approximation
of UΘ a confidence interval for
every single parameter can be described as [Θ^l±ul].
The individual uncertainty limits can be computed as
ul=tdf1-α/2UΘll.
where tdf1-α/2 is the two-tails Student's t
distribution for prescribed α and df . Two
approaches to point-wise approximations of UΘ are found in
ecological and ecosystem modelling studies. One approach uses first derivates
of the model's observation vector with respect to the parameters (Jacobian)
whereas the other requires calculations of second derivatives of J(Θ)
(Hessian).
Uncertainty covariances based on the Jacobian matrix
A first approach considers a linearization (first order power expansion) of
the model's observation vector H(x) around the point
estimate Θ^. As long as H(x(Θ^)) is not
subject to strong nonlinearities, its first derivatives (sensitivity) with
respect to Θ can be used to estimate UΘ. For an
unbiased ML estimator the covariance matrix can be approximated as
UΘ=J(Θ)df⋅HΘTR-1HΘ-1
with the Jacobian matrix HΘ(Θ^), its
transpose (HΘT), and with the observational error
covariance matrix Re.g..
The term J(Θ)/df is added as an approximation of the residual variance
of J, which should be considered unless H(x) is in such good
agreement with data so that the minimum of J(Θ) actually matches the
exact degree of freedom, df. The rows of the Jacobian
HΘ are the first derivatives with respect to the
parameters ∇H(x), with
∇= (∂/∂Θ1,
∂/∂Θ2, …,
∂/∂ΘNΘ) being the Napla operator of first
partial derivatives.
Uncertainty covariances based on the Hessian matrix
Another more common approach for a point-wise approximation of UΘ
is derived from a Taylor expansion around J(Θ^). Since
∇J(Θ^)≈ 0 in the minimum, the first order
term of the Taylor expansion is negligible. The series expansion then approximates the distribution:
J(Θ)-J(Θ^)≈12Θ-Θ^THΘΘ-Θ^.
The matrix HΘ is the Hessian whose elements are second
derivatives of J(Θ) with respect to the parameters
e.g.:
HΘ=∇T∇J(Θ)|Θ=Θ^.
With the Taylor expansion in Eq. () we obtain an
approximation of the local curvature of J(Θ) at point Θ^,
also referred to as the observed Fisher
information. Like in Eq. (), but instead of
using first derivatives of H(x), a posterior uncertainty
covariance of Θ^ is then approximated by computing the
inverse of a Hessian matrix:
UΘ=J(Θ)df⋅2⋅HΘ-1.
Both approximations (Eqs.
and ) yield, in principle, similar results for
accurate ML estimates i.e. when the actual minimum of J(Θ) has been
identified by the optimization algorithm. In practice search algorithms can
terminate at some distance from the actual minimum for numerical reasons,
e.g. when the minimum is located in a flat valley of J and the imposed
convergence criterion makes an algorithm terminate the search in the
periphery of the valley. proposed an approach where
the accuracy of parameter estimates can be improved by minimizing differences
between the results of Eqs. ()
and ().
The Hessian: its approximation and inversion
Hessian matrices have often been approximated with a finite central
differences approach for first and second derivatives of J with respect to
ecosystem model parameters at the point estimate Θ^e.g.. A critical
issue of finite difference calculations of the Hessian's elements is the
choice of an appropriate increment size (δ), which sets
the distance of departure from the optimal parameter point
estimate Θ^. Sometimes a compromise between resolving flat
regions around (Θ^+δ,
J(Θ^+δ)) and numerical precision
has to be found . To approach a high accuracy of
the Hessian approximation it is possible to consider a set of different
increment sizes for the central differences approach, as given in
.
The problem of increment size reduces if first derivatives of J with
respect to the parameters (gradient, ∇J) are readily obtained with
an adjoint model, e.g. as used in a variational DA approach
(Sect. ). Adjoint versions of plankton ecosystem models
have been constructed primarily to compute ∇J for an efficient
search with gradient descent algorithms in the parameter-cost function
manifold e.g.. To elucidate the nature of
adjoint model developments is beyond the scope of this paper, but a brief
summary about adjoint model developments is given in the
Appendix . The advantage is that all elements of the
Hessian can be approximated with finite differences of adjoint model results
e.g..
Computations of the Hessian, Eq. (), provide valuable
identifiability information even if this matrix is not explicitly used to
specify confidence regions of parameter estimates. For example, a
decomposition of the Hessian matrix into its eigenvalues and the
corresponding eigenvectors reveals which parameters are weakly constrained by
the data or it helps to identify structural deficiencies of a model. The
eigenvectors' components (l) represent the components of Θ.
Components of those eigenvectors that belong to small eigenvalues indicate
parameter combinations that are poorly constrained or cannot be estimated. In
contrast, those eigenvectors that correspond with the largest eigenvalues
show parameter combinations that are well constrained. The studies of
and are informative in this
respect, because they provide insight into the range of characteristic
eigenvalues and eigenvectors of 0-D and 1-D marine ecosystem models.
Ideally, every eigenvector would exhibit only one single component, meaning
that values of every parameter can be estimated independently of the other
parameters' values. In practice this is only the case for few parameters of a
planktonic ecosystem model. Eigenvectors with two or more distinct components
disclose those parameters whose estimated values are correlated and for which
correlation coefficients can be explicitly derived e.g.. Correlations between parameter estimates are referred to
as collinearities. A useful collinearity index was introduced by
. Their index expresses how a change in J (or
in H(x)), due to a shift in the value of one parameter, can be
entirely compensated for by adjusting the value of another (correlated)
parameter.
Parameter collinearities: an example with phytoplankton loss parameters
In Sect. we discussed the difficulty of constraining
parameters that determine loss rates of phytoplankton biomass due to grazing,
aggregation or exudation, and leakage or organic matter. With an example we
illustrate typical uncertainties and collinearities in the estimation of
phytoplankton loss parameters in the absence of explicit zooplankton
observations like microzooplankton and mesozooplankton abundance or grazing
rates. Three parameters that affect the loss of phytoplankton biomass have
been optimized together with other parameters. For this we assimilated five
different types of daily mean observations of a mesocosm study
into a plankton ecosystem model
with optimal nutrient allocation and photo-acclimation , as
mentioned in Sect. .
Cost function contours when varying values of a combination of two
parameters J(Θ^m±Δm,
Θ^n±Δn) around the optimum estimate at
(Θ^m, Θ^n, min(J)), while values of
all other parameters remain fixed. Each plot resolves a pairwise combination
out of three parameters that all specify phytoplankton biomass losses. The
two columns reveal differences in error margins due to different cost
functions with same data for the same model: (a) with covariances
explicitly regarded and (b) all data are assumed to be independent.
First row (1a and 1b): combination of maximum grazing rate
(Θ1=gm) and carbon exudation rate
(Θ2=γC). Second row (2a and
2b): combination of the aggregation parameter
(Θ3=Φagg) and γC. Third row
(3a and 3b): combination of gm and
Φagg. Markers show credible regions of parameter estimates
obtained with Markov chain Monte Carlo (MCMC) method (dots for J with covariances, asterisks
for J with variances only). Error ellipses (lines) depict point-wise 95 %
confidence regions derived from an approximated and inverted Hessian matrix,
according Eq. (). The cyan
coloured region embeds all cost
function values that are lower than an upper threshold
ΔJ*(α= 0.05), derived from a distribution of
J(Θ^)-J*(Θ^), where
J*(Θ^) are cost function values at Θ^
using resampled data (Fig. in
Appendix).
Details of the cost functions and the corresponding mapping from model
results x to observations H(x) are given in
Appendix . In our example we consider two cost
functions, with and without covariances respectively (Eqs.
and ). For both cost functions no prior information is
included. As an error model we assume additive Gaussian errors, applying
Eq. () in Sect. . A simulated
annealing algorithm is first used to identify a best parameter estimate
in the vicinity of the global cost function minimum. This point estimate is
then used to derive error ellipses (confidence regions) according
Eq. (). These point-wise approximations of
parameter uncertainties are finally incorporated to initialize the MCMC
method that derives a credible region of posterior parameter uncertainties,
based on an algorithm provided by .
Figure shows contours of
J(Θ^m±Δm,
Θ^n±Δn; m, n= 1, 2, 3) around
the optimum at (Θ^m, Θ^n, min(J)),
while all other parameters are fixed to their optimal
estimates (Θ^l≠m,n). Each plot is thus a combination
of two loss parameters: maximum grazing (Θ1=gm)
and carbon loss rate (Θ2=γC) on top (1a/b in
Fig. ); γC and aggregation
parameter (Θ3=Φagg) in the middle (2a/b);
Φagg and gm on the bottom (3a/b). Results from
MCMC (dots and asterisks) reveal similar collinearities between parameter
combinations that involve gm for the two cost functions (1a/b and
3a/b in Fig. ). It means that gm can
only be estimated in combination with Φagg
and γC. Only if Φagg and
γC were known, then gm could be identified in
this mesocosm model set-up with these available data types. We do not find
such strong collinearity expressed between γC
and Φagg and their estimates seem to be rather independent
(2a/b of Fig. ), given the mesocosm data.
Another peculiarity is that the ranges of the MCMC's posterior indicate
larger uncertainties if the cost function without covariance information is
applied (right side of Fig. ), although model and
data are identical. This behaviour is also resolved by the 95 % confidence
regions that are obtained with a point-wise approximation of error ellipses
(lines). Furthermore, collinearities according to the error ellipses are
smaller for the cost function with covariances compared to the case of
independent data. Here, confidence regions of the error ellipses correspond
well with the credible regions of the MCMC results. We stress that this may
not be the general case and the good correspondence is likely attributable to
the low dimension of the example looked at.
Overall, these results exemplify the uncertainty in constraining major loss
parameters in the presence of grazing, if no explicit prior information about
grazing rates or data of zooplankton biomass are available. Collinearities
between grazing parameters and other phytoplankton biomass losses may be
reduced by testing model performance against independent data, e.g. as done
for the mesozooplankton and microzooplankton grazing in
. In cross-validation studies some combinations of
parameters that produce indistinguishable solutions for one experiment or for
one ocean site are compared with data of another experiment or at another
ocean site, which will be addressed in the following
Sect. .
Cross-validation and model complexity
Good performance should be attributable to a model capturing the predominant
plankton dynamics under varying conditions in different environments.
Parameter values are often optimized for local ocean sites, but ideally,
parameter estimates from one site should improve model performance at other
locations as well. The generality of optimized models can be tested by
cross-validating against independent data, providing a direct and effective
test of predictive skill .
Cross-validation
Parameter optimizations can often improve the fit of a model by selecting
unrepresentative parameter values that serve only to compensate for misfits
between data and model results. It is therefore essential to check whether
the resultant “optimized” model is giving the right answer for the correct
reasons.
, for example, found that while the optimization
of a range of NPZD models to satellite data tended to reduce model–data
misfit, this was often achieved through the adoption of extremely unrealistic
parameter estimates, sometimes being multiple orders of magnitude higher or
lower than their best a priori estimates. The same authors
showed that adding synthetic noise to
assimilated satellite data led to the introduction of similar errors, and a
significant deterioration of one model's predictive skill. The extreme
parameter estimates were not representative for the system and the model
performance turned out to be poor when the model was tested against
independent data that were not used during the optimization procedure.
This is the principle of cross-validation, in which an optimized model is
tested in terms of its ability to reproduce data that were not included in
the calibration phase. This is often achieved by excluding a subset of the
original calibration data set, for later use in model evaluation. For
example, in a variational data assimilation exercise for the Arabian Sea,
repeated their optimization a number of times,
each time excluding data from a particular season. The calibrated models were
then used to predict the system behaviour during the withheld season, with
the resultant model–data misfit labelled the “predictive cost function”.
The cross-validation approach has the advantage of testing one of the key
attributes of marine biogeochemical models, namely their predictive skill.
The technique is, however, not without its difficulties. The first issue is
that it is important to ensure the test data are truly independent of the
training data. In this regard, took advantage of
the highly seasonal nature of the Arabian Sea, but it would perhaps be less
appropriate in regions with a less pronounced seasonal cycle, such as at the
centre of a subtropical gyre. A potentially more serious problem occurs when
researchers simply divide the available data at random, such that highly
correlated data appear in the assimilated and the test data. Under such
circumstances, the cross-validation would give no indication as to the
ability of the model to predict independent data.
The potential to select unrealistic, compensatory, parameter values may not
always be obvious, especially if good estimates of the “true” (or at least
sensible) values of the model parameters are not well known a priori. Such
errors may, nonetheless, strongly impact the ability of a model to reproduce
anything but the assimilated data. This issue appears to be a common theme in
simple marine biogeochemical models calibrated to time-series data, as a
number of studies
have found that parameter optimization resulted in decreased predictive
skill, relative to “off-the-peg”, prior parameterizations. A notable
counterpoint to those studies is given by , who
found that simultaneous optimization of an NPZD model at three time-series
sites led to improved performance when the
model was applied within a 3-D simulation of the North Atlantic. On the one
hand, it seems likely that this improvement was dependent on assimilating
data from three highly dissimilar North Atlantic locations, which prevented
the inclusion of compensatory errors that were highly specific to any one
site see also. On the other hand, in
and in it is
also stressed that the apparent improvement is associated with some ambiguous
rapid nitrogen remineralization pathway in their simple NPZD model, which can
be incorrect in either simulations (1-D and 3-D), but with the same positive
effect on primary production rates in the central North Atlantic.
Model performance as a function of model complexity
Of the many factors that affect the ability of a biogeochemical model to
reproduce and predict observations, the appropriate degree of model
complexity in any given situation is both one of the most important, and one
of the least well defined. This is because there exists a fundamental
trade-off between simplicity and complexity. Simple models have the advantage
of being easier to understand, and with fewer parameters they should also be
better constrained (both before and after optimization). Nonetheless,
simplification requires a degree of abstraction, and it can sometimes be
difficult to draw parallels with the complexities of the observed system.
At the other end of the spectrum, a highly complex model can explicitly
resolve more processes, allowing more detailed comparison with observations.
As models become more complex, the number of degrees of freedom increases,
and the calibrated model will generally be able to match the observations
better than a simpler model. If insufficient observations are available, the
extra degrees of freedom can lead to the introduction of compensatory errors
at the assimilation site, which could then increase uncertainty at other
locations, as illustrated by . Similarly, for
small changes in the assimilated data an extra flexibility may lead to very
different model solutions, also leading to increased uncertainty in model
predictions e.g..
A range of statistical techniques are available to assess this trade off, and
a useful review is given by . One of the most
practical (if not the most general) techniques is cross-validation, as
described in the previous section (see also ,
Sect. 7.10 for an excellent discussion in a general statistical context).
By looking at the effects of adding noise to assimilated remote sensing data,
found that the most complex model they evaluated
was also the most sensitive to the introduction of synthetic errors in the
assimilated data (Fig. ). They attributed this result to
the extra degrees of freedom that could be “fit to noise”. This is consistent
with earlier findings that model predictive skill deteriorates as complex
models can become “overfit” to the data (i.e. too many parameters are fit
to inadequate data) .
Aside from directly assessing a model's predictive skill using
cross-validation, a number of alternative approaches are available to
identify the minimum number of model parameters that are supported by the
available data. One of the simplest techniques (in terms of its
applicability) is the Akaike information criterion (AIC,
). The AIC considers two opposing terms corresponding to
the maximum log-likelihood of the parameters given the data
(ln[L(Θ^|y)], measuring model data misfit) and a
bias-correction factor, which increases with the number of free
parameters (NΘ).
AIC=-2lnLΘ^p|y+2NΘ
Note that for a model fitted by least squares, the log-likelihood can be
approximated by the residual sum of squares (RSS), following
:
ln[L(Θ^p|y)]≈-Ny/2⋅ln(RSS/Ny),
with Ny being the total number of observations. The AIC, and alternative
techniques (weighted AIC, or Bayesian information criterion, BIC), seek to
quantify the trade-off between bias and variance
e.g.. Of a range of competing models, the one
with the lowest AIC has the greatest empirical support.
Predictive skill for five ecosystem models of different complexity,
after assimilation of satellite data (black) and after assimilation of
satellite data with 20 % added noise (grey) .
The most complex model appear to be the most sensitive to errors in the data,
in terms of its cross-validated predictive skill.
A perhaps more intuitive approach is given by the likelihood ratio test (LRT)
for e.g. comparing so-called nested models, in which the simpler model is a
special case of the more complex model, in the sense that
Mp=f1 is a special case of
Mp+1=f1+f2 where f2= 0. Like the AIC,
the LRT aims to account for model complexity in the sense that it compares
log-likelihoods:
LRT=JΘ^p-JΘ^p+q,
with J(Θ^)=-2ln[L(Θ^|y)] and
index p+q indicating the number of free parameters of the full
model. An alternative simpler model (with p parameters) that is not
significantly worse than the full model (with p+q parameters) can
be selected using this ratio. There is a clear analogy to
Eq. () in Sect. . In other
words, although having removed individual parameters (going
from Θp+q to Θp) we may still have an increase in the
data–model misfit that is tolerable or insignificant within some
limit ΔJ. For nested models only, a value for ΔJ can be
derived from a χ2(df =q, α) distribution. The
respective degree of freedom (df) is then assumed to be equal to the
difference in the number of free parameters between the full and the reduced
model, which is q. For LRT with non-nested models an empirical,
non-parametric distribution needs to be derived by other means instead, for
instance using synthetic (or resample) data sets
e.g..
The theory mentioned above is well described by ,
and have already been applied in few ecosystem modelling studies
e.g.. The
techniques for model selection have generally shown that more complex models
are more vulnerable to over-tuning than simpler models. This appears to be
because the number of uniquely identifiable parameters in marine
biogeochemical models is often very low. Studies based on classic NPZD type
models have typically found that the inclusion of as few as three to 15
parameters was supported by the assimilated data . It should however
be noted that these studies made use of only very limited data sets, and a
higher level of complexity would likely be supported with the incorporation
of more comprehensive data sets, especially those describing fluxes.
sequentially removed parameters from a relatively
simple 2NPZD model to show that much of the model structure was redundant,
with respect to the assimilated data, Fig. . They
applied an F-score where the relative change in LRT is related to the
relative change in parsimony (i.e. the difference in the number of free
parameters between the reduced and full model divided by the degrees of
freedom of the full model,
dfp+q=Ny-NΘp+q):
F=LRTJΘ^p+q⋅NΘp+q-NΘpdfp+q-1.
As model complexity was reduced, model predictive skill was initially very
slow to deteriorate, and J remained similarly low. The increased parsimony
of the simpler models led to improved performance in terms of the LRT, and
the AIC and Bayesian information criterion (BIC). Once all of the redundant
components of the model were removed, removal of essential components led to
a rapid increase in J, with an associated increase in the other metrics.
The LRT selects the simplest model with an F score below a variable threshold
value. The AIC and BIC can be used to select a single model with the lowest
score, or preferably to provide individual model weightings for multimodel
inference , although it appears that this latter
has so far seen little application to planktonic ecosystem models.
Space–time variations in model parameters
Theoretical arguments, as well as results from cross-validations, have
revealed problems with the portability of locally calibrated models
e.g. and raise the
question of how representative local estimates are if applied at larger
scales. These limitations encourage the use of estimators that allow spatial and/or
temporal variations of parameter values.
Model selection metrics at sites of the Bermuda Atlantic Time-series
Study (BATS) and the North Atlantic Bloom Experiment (NABE), as a function of
complexity across a suite of nested ocean biogeochemical models
. The least-squares misfit, J (left-hand axis),
increases monotonically with decreasing complexity, as it does not penalize
model complexity. The likelihood ratio test, F (first right-hand axis),
compares each reduced model to the full model, and selects the simplest that
is not significantly worse than the full model (F<F threshold). The
AIC and BIC (second right-hand axis) both contain terms that account for
model data misfit and complexity, and the optimal model is the one with the
lowest score. In each case, the optimal model is indicated by a
dot.
For spatial or temporal variation to be useful we have to make sure that the
corresponding parameter adjustments reflect changes in the actual underlying
(real-world) dynamics. To assess whether this condition is met is a
particularly challenging problem that has yet to be adequately addressed.
Direct comparisons are needed between optimizations that allow variation in
posterior parameter vectors and those that do not. In studies where direct
comparisons are made, a common finding is a reduction in the model misfit to
the assimilated data by allowing these kinds of variations, but this tells us
little. A reduction of the cost function is expected, as a direct consequence
of an effective increase in the number of adjustable parameters. As pointed
out by , “skill assessment using assimilated data
lacks the independence necessary for a comprehensive, objective evaluation”.
Studies where cross-validation is performed to test predictive skill are more
informative.
Switching between different parameter sets in time or for
specific regions may not necessarily be a solution per se but may indicate
where model refinements have to be investigated . From
analyses of spatially and temporally varying parameter estimates that improve
predictive skill we can learn where and when particular model equations are
limited in reproducing changes in plankton dynamics with fixed parameter
values. Such analyses should provide important feedback information on
revising these parameterizations.
Regional differences between parameter estimates
Satellite ocean colour data are widely used to investigate spatial
differences in parameter estimates. In many cases, a local calibration method
is applied where parameters are optimized separately to fit Chl a
data for a number of pre-defined sites or regions spanning a domain of
interest. For example, parameters of a 3D-NPZ model were optimized by
for January and June for two regions, the
North and South Adriatic basins in the Mediterranean Sea. They inferred
comparable parameter vectors for the two regions during bloom conditions in
January but considerable differences between the regionally optimized
parameter sets emerged for June. attributed
this difference to unresolved variations in plankton composition and changes
in biomass concentration between the two basins.
performed a similar assimilation experiment for the Loire and Gironde river
plumes in the Bay of Biscay. On the one hand, they found some similarities
between parameter estimates for the two distinct river plumes for particular
conditions during spring, suggesting the possibility of a common set of
parameter values for both plume areas. On the other hand, the authors
stressed their optimal parameter estimates to be based on data for a specific
period and obtained excessively high Chl a concentrations in the Bay
of Biscay for the entire simulation year when utilizing the mean of parameter
estimates for the two plume regions.
Pronounced regional and seasonal differences are not restricted to adjacent
seas and coastal areas. Large-scale studies for the North Atlantic have shown
comparably strong regional differences between parameter estimates
. A
set of sites representing distinct latitude bands was considered for a 1-year
calibration of an NPZ model in . The
annual cycle at locations on a 5∘ grid was simulated with variable
parameter estimates of a NPZD model in , and individual
parameter estimates for 13 provinces in the North Atlantic, pre-defined
according to , were derived for a six-compartment 3-D
biogeochemical model in .
estimated NPZD model parameters for six 5 × 10∘ regions of
the central North Atlantic. Despite the fact that these studies used
different models, it is possible to compare some optimized parameters that
are equivalent or closely related between all studies. However, little
obvious consistency is seen in the spatial patterns between their estimates,
although suggested some similarity between their
estimates of phytoplankton maximum growth rate and zooplankton maximum
grazing rate with those of .
Patterns of spatial variation in parameters are not easily validated as most parameters do not
have well-observed equivalents in nature. Nevertheless,
were able to document the plausibility of their posterior photosynthesis
parameter values for the maximum phytoplankton growth rate (μm
in Sect. ) and initial slope of the P–I curve
(αphot in Sect. ) by comparison with
observational estimates of . Six parameters were
optimized in all and the posterior parameter fields were cross-validated in a
3-D version of their model by comparing the output with independent SeaWiFS
chlorophyll data from 1997 to 2003 . The spatially
varying parameter set of , obtained by assimilating
Coastal Zone Color Scanner (CZCS) data for the period 1979–1985, was
interpolated and extrapolated onto the spatial grid of the 3-D model as shown
for the two parameters relevant for phytoplankton growth, μm
and αphot respectively (Fig. ). This
enabled the model to simulate the seasonal patterns in SeaWiFS data much
better than with a fixed prior parameter vector. An important caveat is that
the calibration and validation data sets are essentially two realizations of
the same emerging spatio-temporal patterns. To demonstrate improved
predictive skill attributable to its dynamics the model would be expected to
resolve differences between the two independent data sets, given physical
forcing data specific to each period.
Combining sites or regions
The presence of parameter variation between sites or regions for which a
model was calibrated independently does not refute the existence of a common
parameter vector with which the model could achieve similar results.
and performed
alternative experiments in which regions were combined under a uniform
parameter vector constraint, but did not include predictive skill tests for
direct comparisons of the performance of spatially varying and uniform
parameter solutions. In other studies, sites have been combined without
considering the alternative of allowing parameters to vary spatially. By
optimizing a 13-parameter model for locations of the Ocean Weather Ship
India (OWSI) and of the Bermuda Atlantic Time-series Study (BATS)
simultaneously found that it could capture the
primary observed characteristics of the annual cycle at both sites, despite
being unable to reproduce the cycle at BATS when calibrated at OWSI. As
mentioned in the previous section, the approach of data assimilation over
multiple sites has since been used by with
some success in improving predictive skill of a 3-D North Atlantic simulation
based on a simultaneous three-site calibration.
A relatively complex global model with 45 adjustable parameters was similarly
demonstrated to improve the predictive skill after assimilating time series
data at five different calibration sites .
Spatially varying estimates for the phytoplankton maximum growth
rate (μm in unit day-1) and photosynthetic efficiency
(αphot, in m2 W-1 day-1) used in a 3-D
modelling study of the North Atlantic (Losa et al., 2006). The parameter
estimates are based on those obtained in a previous assimilation of satellite
chlorophyll data (Losa et al., 2004). Permission to include Fig. 2 from Losa
et al. (2006) was granted by the authors. Figure is used with permission from
Elsevier. Copyright of original figure by Elsevier.
There is a clear advantage of combining sites or regions, in that it makes
more data available to constrain parameters. It also creates a representative
sample for the domain of interest, reducing the risk of over-fitting. In
contrast, when assimilating data at a single site,
found it necessary to limit the number of adjustable parameters (to four or
even less) to avoid portability problems. Use of a larger data set
representing a wider diversity of ecosystem behaviour should support a
greater number of parameters to be constrained, which would allow a model's
true flexibility to be more fully exploited. However, there is a potential
disadvantage of combining sites or regions, particularly over large spatial
scales, in that the resultant parameter vectors may be less suitable for
either region than parameter vectors obtained by local calibration.
introduced the idea of allowing provinces that are
in a sense optimal for calibration to emerge during the data assimilation
process. A sample of sites from the domain of interest is divided into two
similarly distributed sets, one for calibration and the other for
cross-validation. The objective is to find “the number and geographic scope
of parameter vectors which allow the lowest possible cost of the calibrated
model, with respect to the stations in the validation set, to be obtained”.
The method involves first performing a whole-domain calibration
where parameters are optimized for all calibration sites, then recursively
splitting the domain into two geographic provinces to investigate whether a
better calibration can be achieved by optimizing parameters for each one
separately, a procedure referred to as split-domain calibration. The
relative merits of the calibration procedures are assessed by
cross-validating the posterior parameter vector or vectors against sites from
the validation set.
Application of the method to the North Atlantic data set used by
, with the same NPZ model and 12 adjustable
parameters, resulted in the discovery of a two-parameter vector solution
having a cross-validation misfit cost 25 % lower than that for the
single-vector solution obtained for all calibration sites. The two
sub-domains are shown in Fig. . The validation cost was
also 24 % lower than that obtained when the model was calibrated locally
using individual sites. This is consistent with subsequent findings of
, where combining sites tends to reduce
validation costs. Note that the validation scheme used by
may not be able to discriminate well between skill
associated with the model dynamics and that associated with the ability of
the model to interpolate spatio-temporal patterns between the calibration
sites shown in Fig. . This could be resolved by
comparison with interpolated output from some purely empirical model fitted
to the calibration data.
Geographic extent of the two sub-domains giving the optimal
calibration in the split-domain calibration study of
, shown here in yellow and green. Also shown are the
distributions of the sites used from the calibration set to obtain the
parameter vectors for each sub-domain and the sites used for
cross-validation. Biogeochemical provinces defined by
are shown for reference. ARCT: Atlantic Arctic Province; SARC: Atlantic
Subarctic Province; NADR: North Atlantic Drift Province; GFST: Gulf Stream
Province; NAST: North Atlantic Subtropical Gyral Province. Figure 6a of
Hemmings et al. (2004) is shown with permission from Elsevier. Copyright of
original figure by Elsevier.
Spatially varying parameter estimates derived with Bayesian hierarchical modelling
proposed a Bayesian hierarchical formulation for
calibrating aquatic biogeochemical models at multiple sites. In this
framework, posterior parameter distributions can vary between sites but the
sites share common prior distributions. used this
approach to estimate parameter distributions for a 1-D NPZD-iron model at two
sites in the Gulf of Alaska. Non-informative prior distributions were
employed for each parameter so the influence of the priors on the solution
for each site was fairly weak. In a parallel Bayes' hierarchical modelling
study for the same model, assimilated satellite
chlorophyll data at nine sites using a spatial Gaussian process model for the
parameters with an anisotropic correlation matrix to allow for differences
between along-shelf and cross-shelf dependence. The methods employed by
and seem promising because of
their potential for rigorous treatment of uncertainty. However, in the
absence of cross-validation experiments, their potential for improving the
predictive skill of the models is not well evaluated at present.
Time-varying parameters
The idea of representing seasonal variation in part by temporal variations in
the parameters has been examined in various studies
.
In some cases, parameters are allowed to vary in space and in time .
Cross-validation tests comparing the merits of varying and
non-varying parameter solutions are mostly lacking, which prevents inferences
being drawn about the superiority of these parameter variations for improving
predictive skill. Temporal variation is handled naturally by adapting widely
used sequential state estimation techniques to obtain parameter values along
with state estimates.
applied a SIR particle filter to a model with
15 time-varying parameters in an assimilation of multi-year time series at
the BATS site. The model was treated as a weak constraint with an additive
system noise term that was uncorrelated between state variables.
instead added noise to their two parameters in a
seven-compartment 3-D biogeochemical model of the Middle Atlantic Bight, with
the advantage that the state evolution over each forecast step was true to
the model and correlated errors between state variables were represented. In
both cases, the error model is highly subjective, yet it can have a major
impact on the results. For instance, found the level of
noise to be a critical factor affecting their solution. This motivated
subsequent experiments in which additional time-varying parameters
representing the noise level for each state variable were optimized
. The posterior parameter trajectories thus obtained
were not consistent with the earlier results. Despite the subjective
characteristics of the system noise, the solution of
improved the model prediction of unassimilated bacteria data. The necessity
of time variation in the parameters for achieving this is unclear, since no
alternative results for static parameter solutions were analysed.
In a more recent BATS assimilation study with a simpler NPZD model,
did compare the performance of time-varying and
static parameter solutions. Rather than employing a sequential method, they
opted to solve the optimal control problem, i.e. to find parameter
trajectories that minimize a cost function for the complete time period. An
annual periodicity constraint on posterior parameter trajectories was
introduced to allow the calibrated model to be also applied for time periods
beyond the range of observations. Optimal periodic parameters were obtained
using a 2-year data set and validated against independent data for the
following 3-year period. In cross-validation tests, this solution was shown
to improve predictive skill over the static parameter solution of
. Their results suggest that the time-varying
parameter model may capture some aspects of the inter-annual variability,
which would indicate dynamical skill.
compared the predictive skill of versions of their
two-parameter model with time-varying and static parameter solutions. Here,
the time-varying solution was obtained using an alternative,
emulator-assisted sequential data assimilation scheme. Their cross-validation
experiments show a modest improvement in the ability to predict the annual
cycle with time-varying parameters. Ability to predict the inter-annual
variability was not tested and the achievability of similar predictive skill
by purely empirical representations of the annual cycle derived from the
observational data is not ruled out.
An experiment allowing both time and space variation in biogeochemical
parameters that includes cross-validation is presented by
. Performance is compared against that of a model with
constant spatially uniform parameters specified a priori but not against
static and/or uniform parameter solutions to the DA problem. The study
employed an ensemble Kalman filter approach for combined parameter and state
estimation in a coupled model of the North Atlantic and Arctic oceans.
Estimates for four model parameters that varied spatially and seasonally over
the domain were obtained by assimilating satellite Chl a data for
2008 and 2009 and applied to the estimation of Chl a in 2010. A
slight improvement was seen in 2010 Chl a relative to that for the
prior parameter simulation. This suggests a small improvement in predictive
skill, perhaps attributable in part to a better representation of persistent
patterns in the annual cycle. A comparison of the assimilating run against
independent nutrient data at Station “M” was generally inconclusive with
regard to the potential of the final parameter estimates to improve
predictive skill for the nutrient fields
Learning from space and time variation in parameter estimates
As shown in this section, a variety of approaches have been explored for DA
with parameters varying in space or time or both. We conclude the section by
considering what might be learnt from these types of studies. A common
finding is that the posterior misfit cost with respect to the assimilated
data is reduced by allowing variation, but this provides no evidence in
itself to support the case for parameter variation. Allowing parameter
variation increases the number of parameter values to be optimized, making it
easier to fit a given data set.
Goodness-of-fit statistics that penalize model complexity in terms of number
of parameters (e.g. the F-score of , described in
Sect. ) could prove more informative, but are not
used. Cross-validation can be used to provide a direct demonstration of
differences in predictive skill. In the few studies which do use
cross-validation to compare uniform and varying parameter solutions
, some evidence
of predictive skill is seen but the cross-validation schemes are not shown to
discriminate reliably between predictive skill associated with model dynamics
and that due to interpolation of patterns in space or persistence of an
annual cycle. Better cross-validation schemes will be needed before we can
convincingly demonstrate real improvements in the models as a result of
introducing spatial and/or temporal variation in parameters.
Allowing parameters to vary reduces the extent to which their values can be
constrained by a given set of observations, making an already
under-determined problem worse. It could therefore be argued that parameter
variation is justified only when there is good evidence to infer that a given
model cannot adequately represent the observed variability under the uniform
parameter vector constraint. The evidence should be statistically robust,
taking into account all relevant sources of uncertainty. The consideration of
these additional uncertainties, motivated by its potential for improving
parameter estimates , may tend to weaken data
constraints further and make the introduction of parameter variation less
practical, as well as affecting the strength of the evidence in support of it.
Heterogeneity in the parameter vector is most likely to be useful for
structurally simple models. Those models may lack the required flexibility to
capture some distinct spatial features observed within large domains or they
may fail to resolve specific events during a complete annual cycle. Its
introduction may be a sensible alternative to increasing structural
complexity as it does not increase the computational demands of 3-D
simulations. From an ecological point of view, the need to introduce space
and time variations in parameter values reflects limitations in resolving
physical environmental changes, or deficiencies in physiological or
ecological processes, or all of these factors together. For example,
variations in plankton elemental stoichiometry, e.g. variable
Chl a : C and C : N ratios, induce variations in photosynthetic
rates that may not be well described by a model's parameterization of
Chl a synthesis and assimilation of nutrients (as discussed in
Sect. ). It is helpful to consider biological or
environmental reasons why space or time variations of parameter values are
expected to improve model performance.
If good reasons are found to support the use of parameter variation for model
improvement, then the issue of how to benefit from this spatio-temporal
information must be addressed. Spatially varying parameters can be applied
directly in 3-D models e.g.. This should work well
for hindcasts and short-term forecasts where the application is not
compromised by large-scale ecological changes. For forecasting,
climatological trajectories such as those estimated by
are likely to be of advantage, although their
direct application to long-term prediction in the context of global change
would be difficult to justify. Application of spatially varying parameters to
long-term predictions of global change is possible but will be more
complicated than their use in short-term forecasting and it may be necessary
to find ways of allowing spatial patterns in biogeochemical parameters to
evolve with predicted changes in the physical regimes.
Emulator approaches
Systematic approaches for parameter optimization that were successfully
applied in 0-D or 1-D set-ups, may become too costly as resolution in space
is increased and if the time period for integration is prolonged. This is the
case when spatially 3-D models with high resolution or steady annual cycles
(i.e. periodic solutions) are considered. For the computation of a steady
annual cycle (or fixed point) typically thousands of years of model time are
necessary, which may result in a number of time steps in the order
of o(107). Since DA usually involves an iterative optimization process,
typically hundreds or more model evaluations are necessary to obtain a
satisfactory parameter set. Thus the necessary time steps during procedures
of parameter identification can even reach o(1010). Recent attempts aim
at replacing computationally costly models with approximations that are less
expensive; i.e. emulators have the goal to provide an approximation of the
model output trajectory x:=(xi)i=0Nt,
recalling Eq. () of Sect. :
xi+1=Mxi,Θ,fi,ηi,i=0,…,Nt-1,
by substituting the original model M by a simpler one, the emulator (M̃).
Here we disregard a stochastic model approach and consider ηi= 0 for simplicity.
The application of emulators has emerged in many different fields of science
and thus the theoretical background is relatively well developed
e.g.. Two distinct approaches to emulation
exist, which we refer to as dynamic emulation and statistical emulation. Both
approaches are outlined in the following. Note that the terminology in
literature may vary somewhat depending on the respective research field.
Dynamic emulators
A dynamic emulator (or reduced order or surrogate model) is a substitute for
the original model M. It makes use of the original model equations but is a
simpler representation in terms of resolution or details resolved in the
dynamics. The term “simple model” refers here to the computational effort
needed to evaluate a solution that is a useful approximation of the solution
obtained with the full model. A typical number of model evaluations needed
for an automized optimization process can easily reach the order
of 1010. In this case an emulator becomes particularly valuable, because
its application should be much faster than the original model, while as much
as possible main properties of the original model are retained. Only then an
emulator-based DA approach will give satisfactory results.
Dynamical or physical emulators are based on a simplified model
version (M̃), which might be additionally aligned with interim
evaluations of the original model. The term “dynamic” refers to the fact
that the emulator is still based on dynamical physical or biogeochemical
equations. These can be similar to the ones in the original model but might
have some reduced complexity, either by neglecting some processes or by
simplifying e.g. the forcing f̃. Another option is the
reduction of accuracy in model output by coarsening the spatial or temporal
discretization. For instance, the Transport Matrix (TM) method
can be interpreted as an emulator approach with a kind
of coarse model. The TM is an emulator that simplifies the original model M
by using an approximated and averaged forcing f̃ in
Eq. () and a linear approximation of the spatial
discretization, compared to nonlinear advection schemes typically used in
ocean models. For the case of a spin-up, as mentioned above, a reduction of
accuracy can be achieved by introducing a different criterion that specifies
when a tolerable steady periodic solution as been approached.
When using dynamic emulators, it is often insufficient to take the output of
the faster but less accurate coarse model during optimization, because the
accuracy of the coarse model M̃ might be too low to effectively
support parameter search process. It can be worthwhile or even necessary to
gradually enhance (or update) the emulator's accuracy during the optimization
procedure by introducing special alignment or correction operators. To
explain their definition, let us assume we have computed state vectors of the
original and of the coarse model with a current set of values for the
parameter vector Θℓ in the ℓth step of the
optimization run, i.e.
xi+1=Mxi,Θℓ,fi,x̃i+1=M̃x̃i,Θℓ,fĩ,i=0,…,Nt-1.
We recall that the model state vector xi consists of the values of
the Nx state variables. Thus, in a spatially distributed model,
xi is a vector where every element represents the values at a
certain spatial grid point. We here assume that the same numbering is used
for the coarse model state x̃i.
The alignment operator in optimization step ℓ is then defined
element-wise for xi and point-wise in time by
AℓiM̃x̃i,Θℓ,fĩ=Mxi,Θℓ,fi.
Thus, every Aℓi is a diagonal matrix. At the current
iterate Θℓ, the emulator's output equals the output of the
original model. For a parameter vector Θ close
to Θℓ, the emulator uses the correction of
Eq. () – being exact at Θℓ –
for the coarse model evaluated at Θ, thus giving only an
approximation of the original model. The idea of this response correction method is that the deviation between both model outputs remains
uncritically similar in a vicinity of Θℓ. The emulator is
thus not just the coarse model M̃, but an aligned one,
AℓiM̃, that is now locally optimized. The local
optimization process does not require any additional evaluations of the
original model, but only of the cheaper, coarse one. When this inner
optimization gives some new parameter vector Θℓ+1, the
original model is evaluated once again, and the procedure in
Eq. () is repeated, defining the new emulator for
the (ℓ+ 1)th outer optimization step. In the inner optimization
loop no runs of the original model are needed, and the total number of outer
iterations is expected to be lower than in an classical direct optimization
using M. This type of optimization procedure fits in the framework of trust
region methods, a class of state-of-the-art algorithms for which a
mathematical convergence analysis is shown in .
The method was successfully applied for parameter identification of a
transient 1-D configuration with an NPZD ecosystem model and for periodic
states with climatological forcing in a 3-D setting in a N-based model with
dissolved organic phosphorus (DOP) .
Therein, a coarser time-stepping and a less accurate computation of the fixed
point (i.e. a shorter spin-up) respectively were used to construct the simple
model M̃. For this computationally very costly 3-D model, it
turns out that the most efficient way is to start the optimization using the
emulator- or surrogate-based optimization procedure (with a very coarse
model), and then increase its accuracy during the outer optimization
.
Statistical emulators
In contrast to a dynamical emulator, statistical emulators relate the input
parameters statistically to the model output and thus to H(x),
regardless of the dynamical model structure. Generally, statistical emulators
interpolate the results of a numerical model from a set of training runs with
differing parameters. The aim is to approximate the unknown model output for
other input parameters, not included in the training parameter set. Common
approaches are based on a polynomial fit (of varying degree). Typically, such
interpolations are extended by Bayesian techniques to also obtain uncertainty
estimates. For this purpose it is commonly assumed that the model outcome can
be represented by a Gaussian process and also that the model output changes
smoothly as parameter values are varied. A priori assumptions about reliable
parameter ranges and their distribution are required. Another prior choice
needed is to determine the respective model output of interest, e.g. results
required for H(x) to determine p(Θ|y) or
L(y|Θ), Sect. . Although there
are methods available to reduce the dimensionality for multi-dimensional
model output e.g., it remains
practically infeasible to capture the complete output of a 3-D coupled ocean
ecosystem model. While the theory for statistical emulation is relatively
well described e.g., statistical emulators are so far rarely applied in
biogeochemical ocean modelling.
Simulated (a, c) and emulated (b, d)
rms (root mean square) error depending on the
maximum growth rate of phytoplankton and the maximum grazing rate. Simulated
and emulated rms errors are provided relative to “synthetic observations”,
based on a simulation for a given parameter set (HI = 15 W m-2;
m= 0.06 day-1; μmax= 0.51 day-1;
Hn= 0.8 mmol N m-3; mPD= 0.1 day-1;
mDN= 0.1 day-1;
HZ= 0.9 mmol N m-6;
mZN= 0.01 day-1;
mZD= 0.01 day-1;
gmax= 0.21 day-1), which is disrupted by reddish noise
(AR(3) process) with a standard deviation of 0.09 mmol N m-3.
(Notation after .) Sub panels (a, b)
are based on all prognostic variables, while the rms error in (c, d)
is based on nitrate (NO3-) only (c, d). Red crosses mark the
training data.
In Fig. an example of a statistical emulator is provided
based on a simple NPZD-type box model. The model set-up is adopted from
, thereby resolving seasonal variations in
photosynthetically available radiation. Since computational costs are low,
the chosen example set-up would not necessarily require emulation. However,
the model is well suited for testing an emulator approach, because it allows
us to evaluate a wide range of model solutions. Figure
depicts simulated and emulated root mean square (rms) errors relative to a
set of synthetic observations (i.e. with noise added to model results that
are obtained for a prescribed set of parameter values). For our example we
use the maximum growth rate of phytoplankton and the maximum grazing rate as
free model parameters, while all other model parameters remain fixed. The
emulation is based on a second order polynomial, following the approach of
. The training runs comprise 25 model simulations
in a Latin hypercube design, according to .
Figure shows very similar results for the emulator and
for the full model. In particular, the location of the minimum can be well
reproduced by the emulator. Thus, the agreement between emulated and
simulated model–data misfit is satisfactory and the emulator could be
applied for parameter optimization. The precision might be further enhanced
by considering higher order polynomials and/or more training data sets. Note,
however, that the complexity of the problem increases with the number of free
parameters. In particular, the numerous parameter collinearities in
biogeochemical models e.g. can complicate emulation. Increasing the dimension of
the model introduces additional difficulties. One suggestion on how to reduce
the dimension of a complex model output is given by .
The authors decomposed modelled surface Chl a concentrations of a
suite of training runs into singular vectors and predicted the leading modes
in dependence of a suite of biological and physical model parameters. During
a subsequent parameter optimization with respect to satellite chlorophyll,
they identified zooplankton grazing rate and the light response of
phytoplankton to be the most influential parameters. In contrast to most
other approaches, where variances are estimated based on Bayesian techniques,
used a Bayesian approach to estimate the mean values.
The study of applied a similar technique for DA.
Another example for statistical emulation in biogeochemical modelling is
presented by . Their emulator approach was based on
polynomial chaos expansion e.g.. emulated simulation results of
Chl a concentrations as a function of “maximum zooplankton grazing
rate” and the Chl a : C ratio in the Middle Atlantic Bight in the
year 2006. The authors used an emulator instead of the model to minimize the
model–data misfit with respect to daily Chl a concentrations
observed from remote sensing. They optimized time-constant as well as
time-varying parameter estimates. Both approaches improved the overall model
performance with respect to Chl a. While the original time-varying
estimates disregard the actual state of the system, the use of the polynomial
chaos method formed the basis of an updated, more reliable method in the
study of previously discussed in
Sect. .
Another study of analysed the uncertainty of
modelled hypoxia for the Texas–Louisiana shelf based on statistical
emulators. The authors investigated the uncertainty due to initial and
boundary conditions of biological variables as well as river nutrient loads
and phytoplankton growth rate. Additionally, physical factors like river
runoff, wind forcing, and ocean mixing coefficients were taken into account.
The authors revealed considerable uncertainties as their estimates for the
hypoxic area varied by more than 40 % when considering reasonable
uncertainties in freshwater runoff. Such an extensive analysis would not have
been possible without taking advantage of emulators. Furthermore, the use of
emulators opens up the possibility of new approaches to exploring the
parameter space. One emulator-based technique referred to as “history
matching” , now well established in other fields and
recently applied to the constraint of coupled ocean–atmosphere model
parameters , seems a particularly promising
approach for parameter identification in marine ecosystem modelling. This
relatively simple method uses Bayesian inference to rule out areas of
parameter space as implausible, given some set of observations. Estimated
uncertainties in both the observations (with respect to the truth) and the
emulator (with respect to the model) can be taken into account. The method
can be applied iteratively with different observation sets to reduce the size
of the plausible region at each stage, either as a precursor to more formal
model calibration or as a parameter identification method in its own right.
Combining dynamical and statistical approaches
While emulations based on statistical approaches are comparatively fast, such
methods rely on sufficiently large sets of training data (i.e. full model
simulations). To generate such training data can be costly, especially for 3-D
models with high spatial resolutions. To overcome this problem one might
consider a combination of statistical and dynamical emulators.
A two stage emulation process is suggested by .
Their idea is to use a set of 1-D models as a dynamical emulator that
describes the evolution of the 3-D model at representative sites. This Stage
1 emulator allows large ensemble simulations to be run, providing output that
could be used as training data for construction of a statistical emulator
(Stage 2). The dynamical emulator of is not used in
an inner optimization loop but is used instead to predict 3-D model output
for arbitrary parameter vectors. It is thus used more like a statistical
emulator. In fact, a particular innovation in their study was to quantify
uncertainty in the emulator outputs for inference purposes. Another
innovation was the inclusion of biogeochemical perturbations associated with
lateral advection that are typically ignored in 1-D calibration studies.
These were derived by averaging 3-D model diagnostics over a 10-member
ensemble simulation based on a sample of parameter vectors from the search
space. Accounting for the lateral flux information was helpful, contributing
strongly to the emulator accuracy. The emulator with uncertainty estimates
gave robust results for the surface Chl a concentration of an
ecosystem model of intermediate complexity, considering variation in eight
parameters.
The ultimate aim of the two-stage procedure would be to use a sufficiently large
number of state estimates of the model, based on a (sufficiently precise) dynamical emulator,
for the construction of a statistical emulator for a cost function or similar metric. The dynamical
emulator would effectively bridge the gap between a small reference ensemble
that is practical to generate with the full 3-D model and the statistical
emulator that requires a relatively large training set. The respective metric
must incorporate an error model that takes into account all sources of
uncertainty in the statistical emulation of the full model. Thus, the
uncertainty estimates obtained when training the statistical emulator must be
inflated by combining them with the dynamical emulator's own uncertainty
estimates. Stage 1 emulation results suggest that it may be important to
first extend the latter to include temporal covariance estimates for the
parametric uncertainty associated with the averaged 3-D model output used.
Another important consideration is that global 3-D models require long spin-up
times to overcome an initial model drift (see Sect. ). The application of dynamical emulation techniques
for accelerated spin-up, such as the TM method
mentioned in Sect. , could help to provide a better
representation of the parametric variation by increasing the practical length
of the spin-up period.
Parameter estimation of large-scale and global biogeochemical ocean circulation models
Global biogeochemical ocean models are commonly used to investigate the
mutual interactions between ocean biota and climate change, a famous example
being coupled Earth system models (ESMs) applied in the fifth assessment of
the Intergovernmental Panel on Climate Change and those
models that are evaluated as part of the Coupled Model Intercomparison
Project CMIP5;. Besides individual evaluations of
biogeochemical ocean model components e.g., global ocean biogeochemical simulation results are often
specifically evaluated in terms of their representations of the carbon cycle
e.g.. More recent studies also focus on
analysing the spread of oxygen minimum zones e.g..
Consistency between tracer distribution and ocean circulation field
A major challenge in calibrating biogeochemical models on global scale is
that the simulations require many millennia until tracer distributions are in
equilibrium with the given circulation field and the biogeochemical processes
. Equilibrium solutions are usually achieved by
integrating tracer fields for several thousand years in a so-called model
spin-up, based on some seasonally cycling climatological circulation fields.
Convergence to steady state conditions depends on the region, tracer type,
and form of boundary condition . It also depends on the
values assigned to the parameters of the biogeochemical model, and it is not
necessarily a monotonic function of time, but can exhibit inflection points
that reflect the interaction of diverse processes happening on different
timescales . For parameter optimization it is
meaningful to exclude from a cost function those transient model solutions
that involve continuing trends in the redistribution of tracers see
also.
To attain some equilibrated biogeochemical cycling requires considerable
computational time, which makes it particularly difficult to employ methods
that exploit the parameter-cost function manifold with a large ensemble of
model runs like the MCMC method. The derivation and application of emulators,
as described in Sect. , is therefore of great
value for parameter optimization of global biogeochemical ocean models. An
alternative approach to accelerate the spin-up time is to apply
Newton–Krylov methods, by iteratively solving the dynamical system for
steady state e.g..
Some speed-up of long-term model simulations can also be achieved with an
appropriate balance between a model's spatial resolution and the complexity
of biogeochemical tracer dynamics, as approached by
. Using a coarse grid and a time step of 0.05 years
(≈ 18 days), they could apply an ensemble Kalman filter for
estimating parameters of their relatively “abstract” biogeochemical
component of an ESM of intermediate complexity, building on a DA set-up of
. Another option is to decrease the number of model
runs by applying the variational adjoint method for parameter optimization
(Sect. ). Results of an adjoint global biogeochemical
model were used by to determine first derivatives
of a cost function with respect to the parameters; see also
Appendix . However, because of local minima or flat
regions in the cost function, optimal estimates may then depend on the
initial guess of parameter values, as discussed in
Sect. .
Some DA applications may not require equilibrated tracer dynamics to maintain
steady seasonal cycles, e.g. when applying sequential DA approaches with
recurrent analyses steps and corrections of the simulated state variables. An
example is the study of , who introduced an
ensemble-based DA method for a large-scale biogeochemical model of the North
Atlantic and Arctic oceans. The focus of their study was to estimate spatial
and temporal variations of phytoplankton and zooplankton loss rate parameters
as well as model states, in order to establish an operational system for
hindcasts and forecasts of Chl a concentrations. Their model was
initialized with climatological data of nutrients and oxygen and initial
values of the other biogeochemical state variables were set to low constant
values. Prior to the DA period (2007–2010) their model was integrated for a
6-year period, starting in the year 2000. This simulation period is much
shorter than the few hundreds of years typically needed to equilibrate tracer
distribution and ocean circulation in the North Atlantic and Arctic oceans
e.g. and the optimized hindcast simulations
may therefore not be expected to represent detrended seasonal cycles of
biogeochemical tracer distributions and mass flux.
In summary, various procedures for calibrating large-scale and global
biogeochemical ocean circulation models exist, but are presently challenged
by overcoming limitations in computational time to approach equilibrated
steady cycles in biogeochemical tracer distributions. Data availability on
global scale introduces additional limitations to act as constraints for
parameter identification of global biogeochemical models.
Data for parameter estimation and calibration in global ocean biogeochemical models
With regard to the ocean's key role in global carbon cycling and hence for
the climate system, four different types of data are typically considered for
assessing and calibrating global biogeochemical ocean models: (i) data of
dissolved inorganic tracers, e.g. distributions of nutrients, oxygen,
alkalinity, and dissolved inorganic carbon, (ii) data products derived from
remote sensing measurements, e.g. of chlorophyll a, or plankton
primary or net community production, (iii) in situ measurements or composite
data of organic and inorganic matter concentrations, fluxes, and rates
e.g. at different time-series stations, and (iv) observations of the
gravitational flux of organic particles to the ocean interior, transporting
particulate organic matter through the water column.
For the calibration and assessment of large-scale or global biogeochemical
models, many studies resort to using climatological data sets, e.g. of
nutrients and oxygen, components of the carbonate system
e.g.. Also common is the
additional or exclusive use of observational estimates that were derived from
remote sensing measurements, like primary production rates and surface
concentrations of Chl ae.g.. Given the often high
level of structural complexity of ocean biogeochemical models we find only
few studies that involved more elaborate data such as organism groups or
fluxes of organic matter. Examples can be found in ,
who compared simulated and observed particle fluxes, or
, who compared simulated and observed dissolved iron
concentrations and nitrogen fixation rates. Likewise,
considered satellite-based estimates of surface Chl a concentrations
of different taxonomic groups as specified in .
One reason for the fallback to rather basic data types such as climatological
nutrient concentrations for global model evaluation is the sparse
distribution of open ocean, in situ observations. One example is the scarcity
of global microzooplankton biomass observations in the ocean, as depicted in
. Direct, in situ, open ocean ship-based
observations are sparse in space and time mainly for logistic reasons (and
costs) and we therefore find available sets of situ data to be noticeably
biased towards certain areas and periods e.g. towards coastal areas,
summer season in the high latitudes, and the northern hemisphere,.
Ocean measurements of rates are particularly valuable, but these may not be
straightforward to accomplish, e.g. isotopic measurements on a research
vessel. Some rate measurements may also suffer from large methodological
uncertainties, e.g. measurements of nitrogen fixation. Of similar value,
comparable to rate measurements, are observations of oceanic particle flux,
as obtained from sediment traps or from optical methods
e.g.. These data provide only patchy
information about the particle flux in the world ocean. Their analysis and
interpretation are also difficult, since particles produced at the surface
are subject to horizontal transport by advection, hampering the establishment
of correlations between surface and deep fluxes, particularly for slowly
sinking particles (e.g. a metre per day) in energetic current fields (e.g. a
meter per second) e.g..
Attempts to calibrate global models against individual observations of
particle flux have not yet revealed any unique “best” model solution
. To establish a consistent
linkage between surface primary production rates, e.g. as derived from remote
sensing, and observed in situ measurements of particle flux remains a major
challenge. This requires a close look at parameters that link production the
euphotic zone to deep carbon export. Parameters that specify vertical flux
and remineralization of organic matter ultimately determine carbon storage
.
Parameters relevant for global ocean biogeochemical modelling
The joint effect of particle flux and remineralization is often described by
one or two parameters in global models. Early models referred to an
exponential function of remineralization with depth
, which – in equilibrium – would
correspond to a constant particle sinking velocity and constant
remineralization. Another, common description of particle flux (and hence of
subsequent remineralization) is the consideration of a power law of depth:
F(z)∝z-b, where b is usually set to b= 0.858, representing the
open-ocean composite value derived by from sediment
traps e.g.. Empirical fits to various
observations of particle flux suggest that b may vary between 0.3 and 1.4
.
This typical range of variation of b has been used and tested in global
biogeochemical models e.g. analysing how its value affects dissolved tracer
concentrations in the ocean . coupled a simple global
biogeochemical model with a one-box atmosphere and found a large effect of
this parameter on atmospheric pCO2, highlighting the relevance of this
parameterization in ESM simulations. Since this parameterization is widely
used e.g. we will have a closer look at its implicit assumptions in
the following and discuss potential constraints for the estimation of
respective parameters.
Under steady state conditions b can be interpreted as being equal to a
constant remineralization rate r divided by a particle sinking speed a
that increases with depth: b=r/a. The
associated potential mechanisms that may lead to a vertical increase in
sinking speed are selective export of large and fast particles to deeper
layers, or repackaging of small particles into larger ones by zooplankton
egestion. An alternative interpretation is to assume the sinking speed to be
constant while the remineralization rate decreases with depth. This implies
that particles may become more refractory and less susceptible to bacterial
degradation, or that bacterial activity is reduced by the decrease in
temperature at depth. Other parameterizations of particle flux profiles have
been applied in global models, e.g. constant sinking and remineralization
leading to an exponential flux curve;
e.g., or models that explicitly simulate
different groups of particles with different size and properties
e.g..
provide an excellent overview about different parameterizations for models
applied in CMIP5.
So far few attempts have been made to systematically calibrate
parameterizations of particle export and remineralization in global
biogeochemical models. assimilated annual mean
phosphate data into a simple global ocean biogeochemical circulation model to
optimize globally uniform b. Their study shows that the value of of
b≈ 1 can be well identified for their model when using global
climatological data. According to their approach, the tracer distributions
are dynamically consistent with their solution of ocean circulation. Such
consistency is relevant and b may not be derivable by applying any
simulated circulation field to climatological data, e.g. of phosphate
. Furthermore, also discussed
how the identification of b is affected by uncertainties in the transport
and remineralization of dissolved organic matter.
Projections from the parameter–cost function manifold
(Θ^l, J(Θ)) as obtained during the optimization of
six biogeochemical parameters. Parameters shown are quadratic zooplankton
mortality κzoo (left panels) and rate of vertical increase
in particle sinking speed, a, expressed as quotient b=r/a, where
r is the particle remineralization rate (right panels). Upper panels: cost
function (volume-weighted root-mean square error, divided by the global mean
concentration of each tracer) expressed as its deviation from the minimum.
Parameters of all model simulations in the optimization trajectory were
grouped into 50 classes. Grey bars show the minimum cost within each class.
Red and black horizontal lines indicate deviations from the minimum cost of
1 % and 0.1 % respectively. Squares show the cost of each individual.
Note that the y axis only extends to 5 % above the minimum cost at
(Θ^, J(Θ^). Lower panels: parameter
distribution (PDF) of all model simulations whose costs do not exceed a
threshold limit of ΔJ= 1.01 ⋅J(Θ^)
(1 %, red bars) or ΔJ= 1.001 ⋅J(Θ^)
(0.1 %, open bars): 0.1% (open bars) of the minimum cost; see
Eq. () and the text.
In a recent study of the export parameter b turned
out to be well identifiable, with an optimal value of ≈ 1.3, based
on annual mean climatologies of dissolved nutrients and oxygen. As in
their biogeochemical model explicitly resolves
seasonal cycles. Plankton parameters that act on seasonal scale within the
upper, near surface layers are more difficult to identify, if annual mean
climatological data are used. Figure exemplifies
this difficulty, based on results from , who optimized
six biogeochemical parameters in total. The example reveals differences in
the sensitivity of the cost function with respect to variations of two
contrasting parameters, the zooplankton mortality (κzoo) and
b respectively. These differences can be visualized from projections of the
parameter-cost function manifold (Θ,
J(Θ)), as obtained during parameter optimization
. To better illustrate the
discrepancy between the two parameters in Fig. we
defined two arbitrary cost function threshold limits
ΔJ=J(Θ)/J(Θ^)- 1 and
ΔJ= 0.01 and ΔJ= 0.001 (see
Eq. in Sect. ). The projected
pattern of the zooplankton mortality reveals a much smaller sensitivity of
the cost function (larger uncertainty), compared to the robust (nearly
quadratic) pattern of the export parameter b. Furthermore,
for κzoo some bimodal structure exists within
ΔJ≤ 0.01, which impedes parameter identification. Clearly,
annual mean climatologies of dissolved inorganic tracers provide only little
information on plankton dynamics in the upper layers, while particle export
dynamics (which integrate over large spatial and temporal scales) are well
constrained by the large-scale distribution of dissolved inorganic tracers.
Thus, simulated tracer concentrations at great depth do not critically depend
on every parameter that specifies growth and mortality of the plankton.
In the presence of very diverse timescales and space scales, which is typical
in global biogeochemical ocean modelling, the selection of data sets and the
definition of the error model strongly affect parameter identification. We
also stress that parameter estimates of global biogeochemical modelling
studies are conditioned by the applied circulation, which can have a large
impact on simulated tracer fields , and by the
boundary conditions of e.g. of organic matter burial at the sea floor
. To date, it remains unclear whether parameters
optimized for a given circulation field will improve model simulations in a
different setting, e.g. with a different circulation or forcing, as induced
by climate change scenarios.
Impact of parameter uncertainties on climate model projections into the future
A typical large-scale application of marine biogeochemical models is their
use in ESMs from which projections of future climate change can be derived
for different emission and land-use scenarios. Output of such models helps to
inform scientists, but also society and policymakers about possible
consequences of human action on the climate system. A key example is the most
recent assessment report of the IPCC that featured ESMs with fully
interactive carbon cycles . An appropriate treatment of the
uncertainties contained in the applied scenarios and employed models is
crucial for correctly interpreting model projections, informing the societal
debate about climate policies and thus strengthening the base for developing
relevant measures. A full treatment of uncertainties in the projections of
ESM is beyond the scope of our review and we can only address this topic here briefly.
A comprehensive attempt to account for uncertainties in the models when
determining likelihoods of reaching certain climate goals, like the
politically widely accepted 2 ∘C warming goal, was presented by
and . Employing a
somewhat simplified ESM of intermediate complexity, they ran perturbed
parameter ensembles with some ad hoc assumptions about prior probability
distributions of the model parameters. The skill of individual ensemble
members was then measured by comparison of model hindcasts with available
observations of the current state of the Earth system. A single, pragmatic
skill score was used in the assessment and led to an improved posterior
estimate of parameter probability distributions. The model dynamics then
mapped the parametric uncertainty onto the model projections. From the large
ensemble of model solutions that were, in hindcast mode, not inconsistent
with the observational constraints, the authors could then successfully
derive likelihoods of reaching various climate goals.
Note that reproducing the current climate state is merely a necessary
condition for model skill, but may not constrain the model's ability to
correctly simulate the sensitivity to natural or anthropogenic environmental
change. Observational information on past climate change, such as
glacial–interglacial changes may help to better constrain the models'
sensitivity to changing environmental conditions, even though no historical
analog of the current anthropogenic perturbation is known in terms of the
rapid rate of change. Still, any information about model sensitivities to
applied perturbations is extremely valuable, be it derived from lab or
mesocosm experiments or from historical information. DA is a promising tool
to combine such information on very different space scales and timescales and
to develop an improved understanding of how the earth system works and may
respond to ongoing environmental change.
Summary and perspectives
The survey of revealed that relatively few
aquatic biogeochemical modelling studies (a) considered parameter
optimization (8.5 %), (b) provided values of data–model misfit (30 %),
or (c) performed quantitative parameter sensitivity analyses (28 %). Since
then there has been a vast increase in the number of those studies where the
assimilation of biological and chemical data into planktonic ecosystem models
is described. Likewise, we now find a wide field of different studies that
address problems of parameter identification. Although positive, this
development has also brought up diverse approaches whose contexts and
connections are sometimes difficult to understand. Furthermore, we face a
variety of terminology and notation, which makes it even more arduous to
comprehend the various studies and the significance of their findings. With
this review we aim to provide support to readers.
The theoretical backbone for studies of parameter estimation and uncertainty
builds first of all on how model errors and observational errors are treated.
Specifying the error model is an essential first step in the workflow of
parameter identification, enabling the subsequent derivation of conditional
probabilities and cost functions. Our review shows that there is no ultimate
standard error model or procedure but a meaningful practice is to become
explicit about these errors and to reconsider the underlying assumptions for
discussions of parameter estimates and model results. Whether the DA approach
conserves mass and/or energy is relevant in this respect, depending on the
scientific problem addressed. Some ecosystem model applications may not
critically depend on mass conservation, e.g. when simulating plankton growth
to act as food source in regional simulations of fish stock size and
recruitment. In biogeochemical applications the conservation of mass can be
essential, in particular for large-scale or global ocean applications.
As in many other fields of science, the basic estimation methods considered
in plankton ecosystem DA studies are Bayesian estimation and maximum
likelihood. Their major differences are how prior information enters the DA
approach and how estimates and uncertainties are evaluated. The consideration
of prior parameter values from preceding studies is meaningful and likely
alleviates parameter identification problems. A drawback then is that
asymptotic (point-wise) approximations of posterior uncertainty covariance
matrices, as described herein, may not apply. But when the model parameters
in question have been estimated before in a number of comparable settings, it
may seem a tragic waste of effort and information to pursue an ML approach
without prior information. A similar issue arises in specifying an
“ignorance” prior, and the choice of using BEs when no prior information is
available can also be questioned.
We included a section on typical basic parameterizations of plankton models,
mainly to stress that the treatment of light- and nutrient limitation may
differ between modelling studies. Furthermore, we touched on the problem of
resolving phytoplankton losses specified by e.g. grazing and aggregation
parameters. Latest plankton growth models account for physiological
acclimation effects, responsible for variations between carbon fixation,
cellular allocation of nitrogen and phosphorus, and Chl a synthesis.
Those variations are relevant for DA, in particular if flux estimates of
carbon (e.g. CO2 utilization and respiration) are of primary concern. It
is thus worthwhile to discuss some of the underlying dynamics that can be
resolved with the plankton ecosystem model rather than treating it as a
“black box” for simulating Chl a concentrations.
Many acclimation- or optimality-based models have been qualitatively
calibrated with data from laboratory experiments. DA approaches for parameter
estimation were only done in a few of these studies. Going from laboratory
data to the assimilation of data from mesocosm experiments can be a useful
intermediate step for testing e.g. acclimation or adaptive models and for
assessing uncertainty ranges of parameter values. In this respect, parameter
estimates of one experiment can be used for cross-validation with data of
another independent mesocosm experiment. On the one hand, simulations of the
physical environment of mesocosms are easier to implement, compared e.g. to
setting up a 1-D model for an ocean site. On the other hand, parameter
estimates obtained from the assimilation of mesocosm data might not be
representative for ocean simulations. Although more difficult, model
cross-validations between different ocean sites or regions provide valuable
insight, eventually specifying a model's predictive skill under oceanic
conditions.
Some studies have shown that an increase in model complexity may not
automatically improve predictive skill. This can be partially attributed to
over-fitting, which can yield parameter estimates that improve model–data
misfits at one site but induce unreasonable model results at other ocean
sites. Such results illustrate the vital role played by well-designed
cross-validation experiments. A critical element of cross-validation is
whether the assimilated data are truly independent from the data used for
testing model skill. This is, for instance, not typically the case if
observations from different years but of the same characteristic region are
used unless inter-annual variability dominates over the repeating seasonal
dynamics. Regional differences between parameter estimates are informative
and have the potential to reveal a model's limitations in a way that can
suggest improvements.
Parameter identification becomes more difficult as we go from local and
regional-scale to large-scale and global model simulations. Algorithms for
parameter optimization require multiple model evaluations, which can be
computationally expensive for global biogeochemical models. The procedure for
optimizing parameter values can be accelerated with the application of an
emulator. We discussed the use of dynamical and statistical emulators. The
dynamical emulator is a simpler representation of a full model operator that
is computationally expensive, thereby approximating the underlying model
dynamics. A statistical emulator interpolates model output from a set of
training runs with different values assigned to the parameter vector. Based
on the derived statistics it can be applied to approximate unknown model
output for other input parameters. Both emulator approaches have been shown
to efficiently support the search for optimal parameter values. The
development and use of emulators of biogeochemical models will likely gain in
importance along with improved computer performance. A promising approach is
to apply models with coarser resolution or a series of 1-D models
(distributed over ocean regions) as dynamical emulators for 3-D global
biogeochemical model simulations. Studies have shown that sufficient accuracy
of the emulator can be achieved with repeated intermediate alignments of the
dynamical emulator. Alternatively, differences between 1-D and 3-D results
can be statistically quantified as emulator uncertainty, impacting on the
parameter search process and used to modify the emulator-based cost function.
Parameter identification in global marine biogeochemical circulation models
is still in its infancy, due to the high computational requirements, the huge
range of spatial and temporal scales to be covered, and the comparatively
sparse spatial-temporal distribution of data in the ocean. In contrast to
local optimizations, the consideration of all relevant spatial and temporal
scales has one major advantage in that it provides the opportunity to
rigorously test and benchmark biogeochemical models. In addition to tasks and
complications mentioned in our review, care must be taken in the selection of
appropriate data sets, assuring their relevance (or potential) for answering
the questions posed. Moreover a critical evaluation of the respective roles
of physics, biogeochemistry, exchanges across the model's boundaries and,
possibly, ecology is an as yet unresolved task.
A recurring problem associated with parameter optimization is that marine
biogeochemical models are often unrealistically simplified, while at the same
time remaining unconstrained by data. Ideally, models should be developed to
minimize the number of uncertain parameters yet maintain a level of
complexity that is suited to their intended use in answering specific
questions e.g.. To accomplish this we may not only
think of new model approaches, but also of collecting respective data that
can help to constrain solutions of these models.
Modelling prospects
A commonality of new model formulations is to focus on principles, e.g. by
considering the adaptation of traits towards optimal trade-offs
e.g., or by
accounting for allometric relationships in growth and plankton interaction
e.g., or by using microbial
traits in a functional gene approach . Recent studies
have begun to simulate ecosystem complexity and allow the model to
“self-organize” according to a relatively simple set of ecological and
physiological rules or “trade-offs” . A major advantage of this approach is that the models are
able to resolve greater ecological diversity with fewer specified parameters
whose values can be assumed to be spatially invariant. This diversity allows
the simulated plankton community to reorganize across broad environmental
(e.g. spatial) gradients. But the identification of the most important
trade-offs governing competition between organisms remains a major challenge
.
Perhaps one of the most remarkable developments is the revival of
thermodynamically inspired ecosystem theories for modelling biogeochemical
cycling in the oceans e.g.. In the review of
the concept and potential of the maximum entropy
production principle are addressed. In this modelling approach life in the
ocean is perceived as units of e.g. covalent bonded chains of carbon atoms
that create disequilibria of energy and mass between organisms. These
disequilibria lead to different functional pathways in biogeochemical
cycling, accompanied by a flexible evolution of structural dependencies
between nutrient or substrate availability, plankton and other organisms.
Such novel or revised approaches are expedient and help to create new ideas
in terms of how to design models and measurement strategies that may
alleviate the problems of parameter identification.
Examples of recent advances in data availability
The use of previously underexploited data sets for example those
linking organism size to key ecophysiological rates; have the potential to bring new constraints on
model behaviour, and may go some way to alleviating the degree of
underdetermination that is typically associated with parameter estimation.
New data sources, such as the Bio-Argo profiling floats, should also advance
our understanding, e.g. by documenting seasonal variations of deep
Chl a maxima in remote oligotrophic regions .
These Bio-Argo profile data have the advantage that they resolve
biogeochemical properties with a relatively high frequency of 5 to 10 days
over a sampling period of up to 2 years.
A substantial fraction of recent fluorescence measurements from Bio-Argo
platforms has already been included in a new global Chl a database
described and provided by . Their quality-controlled
data comprise profiles of total Chl a concentration together with
some additional estimates of the relative contributions from pico, nano, and
micro phytoplankton. The employed relationship between the relative size
distributions and total Chl a concentration was derived from an
extensive analysis of high-performance liquid chromatography pigment data in
combination with Chl a fluorescence measurements
. The consideration of these profile data will
possibly facilitate the estimation of photo-acclimation parameters in
particular, and of phytoplankton growth parameters in general.
Data products from remote sensing measurements are continuously improved and
new empirical relationships between photosynthesis and respiration are
derived to estimate net community production (NCP) on the global scale
e.g.. These spatially
resolved estimates may help to constrain parameters of plankton respiration
and remineralization rates. In spite of large uncertainties, the assimilation
of NCP estimates from remote sensing into biogeochemical models may impose
additional constraints on parameters that affect solutions of air–sea
exchange of CO2 and of organic matter export. In this respect we also
stress that upgrades and analyses of time-series data are more then ever
essential to make inference about organic matter flux and ecosystem
functioning e.g., which may introduce additional
constraints for identifying values of a larger number of parameters of
plankton ecosystem models. Finally, we point to latest products from
compilations and syntheses of oceanic and atmospheric CO2 data collected
by a large international community . Data products like air–sea CO2 flux of specified
ocean regions (biomes), as derived in , in
combination with data of nutrient concentrations and O2 will likely put
new light on those parameters that determine variations of the elemental
stoichiometry (C : N: P : O2) in model results of inorganic and
organic matter cycling.
Harmonizing research foci in marine ecosystem modelling and data assimilation
The application of DA methods has become standard for calibrating marine
ecosystem and biogeochemical models. But scientific insight can differ
between DA studies considerably. In the literature we find that there is
often an imbalance between the level of sophistication of the ecosystem model
used and the DA method employed. This is likely due to the fact that marine
ecosystem/biogeochemical modelling studies integrate knowledge from different
scientific fields, of which each has its own foci, objectives, and expertise,
i.e. plankton ecology, physical oceanography, marine geochemistry, and
mathematics and statistics. It is difficult to track major advancements in
marine ecosystem modelling when considering the different views from each of
these research fields. Furthermore, the design of experimental studies and
the collection of field data are often achieved without harmonizing the needs
of biologists with the modellers' exigencies .
Facets of parameter identification in biological modelling disclose major
commonalities and disparities between the objectives expressed in the
different research fields. Discussions on parameter identification are
therefore helpful to achieve a common understanding and to promote
communication between observers, modellers, and statisticians. Problems of
parameter identification may thus be well addressed by pooling expertise
across multiple disciplines, without losing sight of scientific objectives.
Such joint efforts should help planktonic ecosystem models to fulfil their
potential as quantitative tools for aquatic sciences.
Results presented in Figs. 2, 7, and 8 and Figs. A1, A2,
and B1 are made available by the respective authors. The results are centrally stored.
Please send requests to mschartau@geomar.de.
The variable lag fit with unknown error variances (Sect. )
In a variable lag fit (VLF), we assume that the truth at time ti is
related to the model output by a kinematic model error (ζ) in phase or
time lag τi. Equation () becomes
xtti=xti+τi.
A notable feature of this model error representation is that it introduces
unknowns τ that can be conditionally optimized by searching forwards and
backwards in time within saved model output, i.e. without rerunning the dynamical model. For the demonstration in Fig. we
assumed that the time lag errors are normal and independent:
τi∼N(0, στ2). This independence assumption
may seem restrictive; for example, a misplaced eddy might be expected to
impose some correlation between the τi for a set of cruise data.
Nevertheless, we find that the method is somewhat robust to neglected lag
correlation. Moreover, this formal neglect enables a large computational
simplification since the lags can then be optimized one by one; see
.
For the observational error in Fig. we assumed lognormal
errors with no interpolation or conversion factors, and that all measured
variables were sampled simultaneously. Equation () becomes
yij=xijt⋅expϵij-σj22
at each measurement time ti and for each measured variable j
(nutrient, phytoplankton, and zooplankton). For simplicity we further assumed
that the observational errors were independent between measurements and data
types, hence ϵij∼N(0, σj2). Note that the
ϵ may be considered to include a component of kinematic model
error (ζ) without affecting the parameter estimation, hence we refer to
them as residual errors below. Assuming that the ecosystem
parameters θe, time lags τ, time lag
variance στ and observational error variances σ are all
unknown, a joint posterior mode estimate of
Θ= (θe, τ, στ,
σ) is obtained by maximizing the posterior density
p(Θ|y), equivalent to minimizing the following cost function:
J(Θ)=nlogστ2+∑iτi2στ2+n∑jlogσj2+∑ijlogyij-logxjti+τi+0.5σj22σj2.
To test this cost function, we simulated data from the NPZD model of
in a 0-D setting using the parameter values and
sine-squared forcing function from . Three years of
simultaneous weekly samples of N, P, and Z were simulated assuming
independent normal time lag errors with standard deviation
στ= 10 days and independent normal residual errors
σlogN= 0.1, σlogP= 0.2,
σlogZ= 0.3. The data were assimilated into the same
NPZD model by one of two methods. In the “standard fit”, no time lag error
was assumed and search parameters Θ={θe,
σlogN, σlogP,
σlogZ} were estimated by minimizing only the final two
terms in Eq. () with τi= 0 for all i. In the VLF,
Θ={θe, τ, στ,
σlogN, σlogP,
σlogZ} was estimated by minimizing Eq. (). In
both cases, we assume uncertainty in only two of the 15 biological
parameters, namely the phytoplankton maximum uptake rate Vm and
the zooplankton maximum grazing rate g (hence
θe= (Vm, g)). For all search parameters,
allowed ranges were ±50 % about the true values, equivalent to unbiased
uniform priors with 29 % prior uncertainty. Initial values of the search
parameters were chosen at random from this prior, and optimizations were
repeated over 10 random restarts to avoid local minima. The experiment was
repeated over 20 simulated data sets to obtain the statistics in
Table .
Caution must be exercised here regarding the estimation of στ.
If the prior for στ permits very low or zero values, then the
MAP estimation will push the estimate of στ towards zero
irrespective of its true value. This is because, unlike the fourth term in
Eq. (), the second term can be made exactly zero with
τ=0 as long as στ2> 0,
in which case the negative contribution of nlogστ2 may
produce a spurious, deeper minimum of J near to στ= 0. We
have found that this spurious minimum need not influence estimation as long
as the sample size and the lower limit of the allowed range or rectangular
prior for στ are sufficiently large,
Fig. . An alternative solution may be to assume a
prior that drops smoothly to zero as στ2→ 0,
such as an inverse gamma distribution cf..
To investigate estimation of the time lag variance parameter στ
we obtained cost function profiles by fitting the same data set using a range
of fixed values of στ, Fig. . We see
that with 3 years of weekly NPZ sampling the cost function function has a
strong minimum close to the true value of 10 days, and this minimum should be
approached even if the allowed range (prior uncertainty) for στ
reaches as low as 1 day. However, if we decrease the number of sampled years,
or especially the number of sampled variables, the minimum becomes weaker and
a spurious minimum close to στ= 0 starts to encroach on the
profile. A sufficiently low minimum allowed
value στ(min) may then lead to estimates converging
to this spurious minimum.
Demonstration of the variable lag fit (VLF) applied to a simulated
data set. (a, d) show the system trajectory with the true parameter
values (solid lines), the data (dots) simulated assuming normal and
independent time lag errors (στ= 10 days) and residual
errors (σlogN,P,Z= 0.1, 0.2, 0.3; see
Table ), and the system trajectory with the VLF parameter
estimates (dashed lines) from the standard fit (a) and from the VLF (d).
(b, e) compare the true time lags (solid) with the
modelled time lags (dashed) for the standard fit (b) and the VLF (e).
(c, f) compare the true residual errors (solid) with the model estimates
(dashed) for the standard fit (c) and the VLF (f, same colour
code as in a, c). Three years of data were assimilated, but only the initial and
post-bloom period of the first year is shown for clarity.
Profiles of the variable lag fit cost function (-2 ×
posterior density) relative to the minimum value for a range of assumed
values of the time lag error standard deviation στ. For each
στ, Eq. () was minimized over (θ, τ,
σlogN,P,Z) for the same data set. Different curves correspond to
different scenarios for the number of sampled years (at weekly sampling
frequency) and the number of simultaneously sampled variables (black:
3 years; blue: 2 years; solid lines with circles:
nutrient–phytoplankton–zooplankton sampling; dashed lines with triangles:
nutrient–phytoplankton sampling). The extent to which each curve has a deep
minimum close to the true value στ(true) = 10 days indicates
the feasibility of estimating στ for the corresponding sampling
plan.
True parameter values and means ±1 SD of estimates over
20 simulated data sets, using a standard fit method and a variable lag fit
method (see Eq. ). Three years of weekly NPZ data were simulated
using the true values (first row) for the maximum nutrient uptake
rate Vm, zooplankton grazing rate g, residual standard
deviations σlogN,P,Z, and time lag standard deviation
στ (for experiments with lags imposed). With no time lags, the
standard fit accurately recovers the true parameter values (third row), but
with time lags (fourth row) the standard grazing rate estimates are biased
and imprecise, while the residual variances have strong positive bias as they
are forced to account for the time lag errors. The variable lag fit avoids
these biases and accurately partitions the variance between residual error
and time lag error (fifth row).
For our example we account for six different types of measurements from
mesocosms of the Pelagic Ecosystem CO2 Enrichment Study
PeECE I,: (1) dissolved
inorganic carbon (DIC, mmol m-3), (2) nitrate (NO3-,
mmol m-3), (3) nitrite (NO2-, mmol m-3),
(4) Chl a (mg m-3), (5) PON (mmol m-3), and (6) POC
(mmol m-3). Concentrations of NO3- and NO2- are not
explicitly resolved by the model and therefore these measurements are
combined. We refer to their sum as dissolved inorganic nitrogen (DIN). Thus,
the number of components of the observation vector y is
Ny= 5. Observations are available on a daily basis over a period of
23 days (Nt= 23). The vector includes daily means of nine mesocosms
at ti, i= 1, …, Nt. The dynamical model equations
determine 12 state variables (Nx= 12). The corresponding vector of
model counterparts to observations is Hi(x), with carbon and
nitrogen biomass concentrations of phytoplankton (PhyN and PhyC), of
zooplankton (ZooN and ZooC), of detritus (DetN and DetC), and carbon
concentration of (particulate) macrogels (GelC). The data–model residual
vector is
di=yi-Hi(x)=DICiDINiChlaiPONiPOCi︸obs-DICiDINiθiChl:C⋅PhyCi(PhyN+ZooN+DetN)i(PhyC+ZooC+DetC+GelC)i︸model.
As an error model we assume additive Gaussian errors applying
Eq. () in Sect. . The standard
errors (σi) include the observed variability between
the nine mesocosms, based on daily measurements. Residual error covariance
matrices can thus be derived for every sampling day:
Ri=SiC(y)Si. The
matrices Si include diagonal elements with
σi at date ti, while off-diagonal elements are zero.
The elements of matrix C(y) represent correlations between the
different types of observations, which were determined for two time
intervals: exponential growth and post-bloom period. The distinction between
periods of bloom build-up and post-bloom can be particularly meaningful when
C and N (or P) data are assimilated. Correlations can switch sign and thus
the sign of the data–model residual
di=yi-Hi(x) matters. For
example, PON and dissolved inorganic carbon (DIC) are strongly negatively
correlated during the exponential growth phase. During the post-bloom period
DIC may still decrease at times when PON concentration declines as well,
which yields a weak but positive correlation.
The standard errors (σi) can be written in matrix notation with
off-diagonal elements being zero:
Si=σi(DIC)0⋯00σi(DIN)⋯⋮⋮⋮⋱00⋯0σi(POC).
Correlations during exponential growth
(ti; i= 1, …, 13) are
C(y)=DICDINChlaPONPOC10.96-0.95-0.97-0.97.1-0.96-0.95-0.95..10.960.92...10.94....1
and during the post-bloom period (ti;
i= 14, …, 22):
C(y)=DICDINChlaPONPOC10.2-0.220.20-0.64.1-0.37-0.260.16..10.63-0.26...1-0.55....1.
For days with some missing observations (e.g. no PON measurements), the
dimension of the vectors Hi(x) and yi and matrices
S(yi) and C(y) have to be adjusted for that
date accordingly. We disregard any prior information and the cost function
(Eq. in Sect. ) reduces to
J(Θ)=∑i=1Ntyi-Hi(x)TRi-1yi-Hi(x).
For our second cost function we assume all data to be independent (i.e. all
off-diagonals of C(y) are zero) and Eq. ()
can be further simplified to a sum over all individual vector components (indexed with j):
J(Θ)=∑i=1Nt∑j=1Nyyij-Hij(x)2σij2.
The mesocosm model environment was coded in FORTRAN and compiled as shared
library so that we could use R as free software environment for statistical
computations. For parameter optimization (simulated
annealing) and for the analysis of the
posterior (Markov chain Monte Carlo method) we applied R package FME of
.
Observations of nine mesocosms (red asterisks), resampled data (grey
markers), and optimized simulation results (blue lines): dissolved inorganic
nitrogen and carbon (DIN and DIC), particulate nitrogen and carbon (PON and
POC), and chlorophyll a concentration (CHLa).
Development of an adjoint model (Sect. )
Adjoint models can be used to efficiently compute the derivative (or
gradient) of the cost function J. In a parameter identification problem,
J depends on Θ both indirectly via the state variable x and
also directly if prior information is incorporated. The optimization problem
can thus be written as
minΘJ(x(Θ),Θ),
where x=(xi)i=0Nt summarize all time instances of the
model variables. To evaluate the derivative of the cost w.r.t. the
parameters Θ, we may apply the chain rule and obtain
dJdΘ=∑i=0Nt∂J∂xidxidΘ+∂J∂Θ,
where we omitted the arguments x(Θ) and Θ for brevity.
The needed derivatives of the model variables xi w.r.t. the
parameters Θ can be obtained by taking the total derivative w.r.t. Θ
of the equations of the dynamical model, Eq. ():
dxi+1dΘ=∂M∂xidxidΘ+∂M∂Θ,i=0,…,Nt-1.
This time propagation scheme for the derivatives is often called the tangent
linear model.
The idea behind adjoint models is to avoid this direct computation, whose
effort grows linear with the number of parameters Θ. For this purpose,
we re-formulate Eq. (), treat both arguments of J independently
and use the model equation as a constraint in the optimization process. This
can be expressed as
min(x,Θ)J(x,Θ)s.t.xi+1=Mxi,θe,f,i=0,…,Nt-1.
A useful overview of adjoint model construction and applications is given in
. An established approach to construct an adjoint model
is to generate adjoint code directly from the numerical code of a model,
based on algorithms that implement the chain rule for automatic
differentiation . According to the
description of , a numerical model can be treated
as a composition of differentiable functions, where each function represents
a statement in the numerical code. The differentiation of such composition
can be automated by highly sophisticated tools that yield tangent linear and
adjoint FORTRAN code e.g..
The application of adjoint construction tools (e.g. Tangent linear and
Adjoint Model compiler, TAMC, of ) have been
shown to perform well for studies with large-scale ocean general circulation
models that include even complicated boundary conditions
e.g..
Another approach is based on a discretized extended Lagrange equation. Under
certain mathematical assumptions, a solution of Eq. ()
corresponds to a saddle point (x, Θ, λ) of the
Lagrangian
L(x,Θ,λ)=J(x,Θ)+∑i=0Nt-1λi⊤Mxi,θe,f-xi+1.
The vector λ=(λi)i=0Nt-1
contains the Lagrange multipliers λi, each of which
corresponds to one time step in the model. A saddle point of L
satisfies the conditions
0=∂L∂xi=∂J∂xi+λi⊤∂M∂xi-λi-1⊤,i=1,…,Nt,0=∂L∂Θ=∂J∂Θ+∑i=0Nt-1λi⊤∂M∂Θ,0=∂L∂λ.
Here, we again omitted the arguments, and set λNt= 0 in the
first equation to keep the compact notation. Note that all derivatives are
partial ones since the idea is to decouple x and Θ and realize
their dependency by implying the constraint in Eq. (). For
simplicity we neglect additional parameter bounds which otherwise would
affect Eq. (). Taking the derivative in
Eq. () for each λi separately
results in the model equations (Eq. ) again. From
Eq. () we deduce
∂J∂xidxidΘ=λi-1⊤dxidΘ-λi⊤∂M∂xidxidΘ,i=1,…,Nt
and apply Eq. () to obtain
∂J∂xidxidΘ=λi-1⊤dxidΘ-λi⊤dxi+1dΘ-∂M∂Θ,i=1,…,Nt
where λNt= 0 as above. Summing up gives
∑i=1Nt∂J∂xidxidΘ=λ0⊤dx1dΘ+∑i=1Ntλi⊤∂M∂Θ=λ0⊤∂M∂x0dx0dΘ+∑i=0Ntλi⊤∂M∂Θ
where we used again Eq. () for i= 1. The first term
includes the derivative of the initial values x0 w.r.t. the
parameters and in many cases will be zero. As result, the derivative of the
cost can be computed from Eq. () using the multiplier
vector λ, but without the tangent linear model. Note that the
derivative of the model w.r.t. Θ in the sum is a
partial/derivative only; thus, it does not include the derivative of
the model variables, but only those of the model equations w.r.t. Θ.
The multipliers λi satisfy a time-stepping scheme
themselves, but in the reverse direction. Using the transposed form of
Eq. (), we obtain
λi-1=∂M∂xi⊤λi+∂J∂xi⊤,i=Nt,…,1,
with λNt= 0 (see above) as the starting point of the
computation. Since here the transpose (or adjoint) of the linearization of the model
operator M occurs, these equations are referred to as the adjoint equations
or the adjoint model. Accordingly, the multipliers λ are also
referred to as adjoint variables or adjoints. Given a model
trajectory x and using Eq. (), the trajectory of the
adjoints λ can be computed. It is crucial to note that both
time-stepping schemes, for the variables x and the
adjoints λ, have opposite directions. This requires – except
for the case of a linear model M – the complete model trajectory to be
stored or recomputed in order to compute λ.
The adjoint model construction starting from a discretized extended Lagrange
equation, Eq. (), can easily become extensive, in particular
when discretizations of advection and mixing are included in the model
dynamics. Furthermore, even small changes in the equations can entail
considerable additional efforts in updating the adjoint model equations. The
application of automatic differentiation tools may therefore be better suited
for cases where the ecosystem dynamical model is subject to regular
modifications.
Individual sections of our review were written by one or more lead author(s),
with contributions from the other authors (Phil Wallhead, PW;
John Hemmings, JH; Ben Ward, BW; Ulrike Löptien, UL; Thomas Slawig, TS;
Iris Kriest, IK; Andreas Oschlies, AO; and Markus Schartau, MS). All authors
were involved in mutual revisions of the individual sections. The sections'
lead authors are (1) Introduction (MS), (2) Theoretical background (PW, MS,
and JH), (3) Typical parameterizations of plankton models (MS), (4) Error
models (PW), (5) Parameter uncertainties (MS), (6) Cross-validation and model
complexity (BW), (7) Space–time variations in model parameters (JH),
(8) Emulator approaches (UL and TS), (9) Parameter estimation of large-scale
biogeochemical ocean circulation models (IK, AO, and MS), (10) Summary and
perspectives (MS), Appendix A (PW), Appendix B (MS), and Appendix C (MS
and TS). Shubham Krishna performed parameter optimizations, MCMC computations
of the mesocosm modelling example, as well as calculations of the 2-D
parameter arrays.
The authors declare that they have no conflict of interest.
Acknowledgements
We gratefully acknowledge the support from the International Space Science
Institute (ISSI). This publication is an outcome of the ISSI's Working Group
on “Carbon Cycle Data Assimilation: How to consistently assimilate multiple
data streams”. We would like to thank four anonymous referees who provided
constructive and helpful comments. The time and effort they spend on our
manuscript is much appreciated. The examples of mesocosm data assimilation
are based on the mesocosm modelling environment designed for large integrated
projects Surface Ocean Processes in the Anthropocene (SOPRAN, 03F0662A) and
BIOACID (03F0728A), both funded by the German Federal Ministry of Education
and Research (BMBF). Contributions from Iris Kriest, Ulrike Löptien, and
Thomas Slawig were supported by the BMBF-funded PalMod – Paleo Modelling: A
national paleo climate modelling initiative.
Edited by: M. Scholze Reviewed by: four anonymous referees
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