Ocean gliders have become ubiquitous observation platforms in the ocean in recent years. They are also increasingly used in coastal environments. The coastal observatory system COSYNA has pioneered the use of gliders in the North Sea, a shallow tidally energetic shelf sea.

For operational reasons, the gliders operated in the North Sea are programmed to resurface every 3–5 h. The glider's dead-reckoning algorithm yields depth-averaged currents, averaged in time over each subsurface interval. Under operational conditions these averaged currents are a poor approximation of the instantaneous tidal current.

In this work an algorithm is developed that estimates the instantaneous
current (tidal and residual) from glider observations only. The algorithm
uses a first-order Butterworth low pass filter to estimate the residual
current component, and a Kalman filter based on the linear shallow water
equations for the tidal component. A comparison of data from a glider
experiment with current data from an acoustic
Doppler current profilers deployed nearby shows that the
standard deviations for the east and north current components are better than
7 cm s

In the near-real-time mode the algorithm provides estimates of the currents that the glider is expected to encounter during its next few dives. Combined with a behavioural and dynamic model of the glider, this yields predicted trajectories, the information of which is incorporated in warning messages issued to ships by the (German) authorities. In delayed mode the algorithm produces useful estimates of the depth-averaged currents, which can be used in (process-based) analyses in case no other source of measured current information is available.

Ocean gliders, or gliders for short, have become ubiquitous observation
platforms in the ocean in recent years. In the Coastal Observing SYstem for
Northern and Arctic seas observatory system (COSYNA;

Gliders have found application in a wide range of research topics; see

Shelf seas are often shallow enough that tidal and wind-driven currents lead
to resuspension and deposition events of sediment

Although currents can be measured from gliders using low power acoustic
Doppler current profilers (ADCPs)

From an operational point of view, any significant variation in the currents
that, from a glider's perspective, appears to have a timescale that is
similar to its subsurface time will cause it to have trouble maintaining the
pre-programmed course. Methods have been developed in order to plan
trajectories for optimal sampling purposes

Safety at sea is in fact a major aspect in glider operations with the
COSYNA coastal observatory. Since glider operations take place mostly in
the German sector of the North Sea, the planning and execution of glider
missions need to comply with the regulations set by the governing German
shipping authority Wasser- und Schifffahrtsamt (WSA). This involves the
application for permission to run gliders in a given region within a certain
time frame. Since there is a risk of a ship–glider collision

In the present work ocean current models are not relied upon to provide information on the water motion. Instead, the aim is to reconstruct the instantaneous currents by recovering (most of) the information in the observed currents lost due to the time averaging.

To that end, an algorithm is proposed that is composed of a simple low pass filter for low frequency variations in the currents due to atmospheric influence, for example, and a Kalman filter based on the shallow water equations to estimate the tidally induced variation in the currents.

The Teledyne Webb Research Slocum electric glider uses a dead-reckoning
algorithm for underwater positioning. The algorithm combines the depth rate
of change from the pressure transducer and heading and pitch from the
attitude sensor to compute the horizontal velocity components. The
dead-reckoned underwater position follows from integrating the current
vectors with respect to time, starting from the latest known GPS position.
The difference between the dead-reckoned resurface position and the actual
GPS position is attributed to a depth- and time-averaged current; see also

In this work, estimates of time-averaged currents are used to reconstruct instantaneous currents. The reconstructed instantaneous time series contain more information than the observed time-averaged currents. Since the extra information required does not come from other measurements or observations, it will have to be provided by a model.

The currents in a coastal sea as considered herein, are dominated by the tide, so that a (simple) model, such as the shallow water equations, can provide this additional information on the tidal motion. Besides tidal currents, non-tidal currents due to atmospheric conditions and fresh water influx, for example, can have significant effects. The non-tidal or residual current, however, is hard to model without resorting to complex numerical models. Therefore the current is decomposed in a tidal current and a slowly varying residual current. The tidal current component is then estimated from a Kalman filter based on the shallow water equations, including at most a few of the main tidal constituents. The slowly varying residual current is estimated by removing the semidiurnal tidal components and their higher harmonics using a low pass filter.

Generally, low pass filters show a gradual the transition from pass to no-pass at the cut-off frequency (the transition band). In addition, low pass filters introduce a frequency-dependent lag to the filtered signal (the group delay) so that the phase of the signal is not preserved. The design of a low pass filters is a trade-off between how broad the transition band is allowed to be and how much lag is acceptable.

For this purpose, an efficient low pass Butterworth filter is implemented

In order to effectively remove the main tidal signals (semidiurnal components
and their higher harmonics), the cut-off frequency of each filter can be
chosen such that the gain of frequencies with a period of 12 h or smaller
have a gain less than 0.01, yielding cut-off frequencies of

In a practical application the filter can be implemented such that every time a new current measurement becomes available; that is, when the glider resurfaces, the measured value is fed into the low pass filter and yields an estimate of the residual current. This estimate has an error not only because of a time delay but also because of tidal signals that may still be present if the attenuation is insufficient. In the post-processing, when all data are available, the filtering can be improved by running the filter forward and backward: the time delay introduced is compensated, and unwanted signals are further damped.

At this stage it is not clear what filter setting would give the optimum
results, so that, for now, a low pass filter is designed with

Filter responses of Butterworth filters of the first and second order for various cut-off frequencies. Top panel shows the power gain as a function of frequency, and the bottom panel shows the group delay as a function of frequency.

In contrast to the residual currents, the evolution of tidal currents can be captured to a large extent by a simple model. In this section we will cast such a simple model into a Kalman filter to provide an optimal estimate of the tidal current components that the glider will face during its next dive, based on all previous depth- and time-averaged water current data it has collected.

For an introduction to Kalman filters and their derivation, the
reader is referred to e.g.

The Kalman filter is formulated as a dynamical system

The procedure for the Kalman filter is given by the following equations for

The procedure involves sequentially evaluating the set of
Eqs. (

The filter is initialised with estimates for

For a model to capture the main tidal oscillation, it is assumed that the
shallow water equations are a reasonable model for the water dynamics:

Defining the dynamical system (Eq.

The measurements (

The averaged currents are then found by integrating the instant
current components with respect to time and dividing the result by the
subsurface time

The measurement error is assumed to be directionally
uncorrelated, so that

It is less obvious how to quantify the process noise matrix

Below, data of measured currents are used to assess the performance of the current prediction algorithm. Two data sets were used for this purpose. The first data set was obtained from measured currents from a bottom mounted acoustic Doppler current profiler, from which synthetic, but realistic, depth- and time-averaged currents were constructed. These data time series mimic the glider data but have predefined and controllable subsurface times. In addition, the synthetic data set removes any uncertainty introduced by the glider's dead-reckoning algorithm, rendering this a useful data set to assess the performance of the prediction algorithm per se. The second data set uses current estimates from glider data obtained during a field experiment. Analysing the results for both data sets allows us to quantify the effects on accuracy of the subsurface time and the glider's dead-reckoning algorithm.

The data used in this study were collected during a field experiment that
took place in the German Bight, in the German sector of the North Sea in
August 2014 (see Fig.

Eastward and northward currents are shown in the top and bottom panel, respectively. The synthetic 3 h averaged currents are shown in transparent blue, and the forward and forward–backward filtered residual currents are shown in red and green, respectively.

The glider

The target area for the glider operation was near the buoy NSB3 and the
ADCP. Figure

As a first step, the ADCP measurements were used to evaluate the
performance of the filter. The instantaneous currents measured with the
ADCP are considered as the true currents. Synthetic glider measurement data
were obtained by time and depth averaging the ADCP measurements, followed
by adding white noise. The time averaging was performed over a window
representing the subsurface time of the glider. This interval was set to
3 h, which is a typical value during operations in the North Sea. Below,
however, the influence of the subsurface time on the accuracy on the
prediction algorithm is addressed specifically. The added noise is Gaussian
with zero mean and a standard deviation of 1 cm s

Firstly, the synthetic measurements were low pass filtered using a first-order Butterworth filter as outlined above. The purpose of this filter is to
remove the main semidiurnal and faster tidal signatures from the total
signal. For the present data set the M2 tidal component accounts for 80 and
65 % of the total variance for the eastward and northward currents,
respectively. The result is shown in Fig.

Standard deviations of the errors of the eastward (blue) and
northward (red) current components, as a function of the variance parameter

Errors in the estimated currents. Upper left panel: histogram for eastward direction; bottom left panel: histogram for northward direction; upper right panel: estimated probability density function for eastward current; and bottom right panel: estimated probability density function for northward current. The dashed lines indicate the 50 and 95 % levels.

Secondly, the synthetic current measurements, corrected for the residual
current using the low pass filtered currents, were subjected to the Kalman
filter. The initial conditions were set by the state vector

In order to find the optimal value for the variance parameter

Seventeen-day time series of measured currents and Kalman filter estimated currents for the eastward component (top panel) and the northward panel (bottom panel).

Probability density function estimates of the errors for the
eastward current component (left panel) and the northward component (right
panel). The black and red curves are derived from the results of the Kalman
filter run in near-real-time mode for the glider data and ADCP data,
respectively. The blue and green curves are derived from the Kalman filter
modified for delayed mode (post-processing); see
Sect.

The Kalman filter (Eqs.

In practice, the errors due to the dead-reckoning algorithm that get absorbed
into the current measurements by the glider will degrade the performance of
the current prediction algorithm. To quantify this degradation, the algorithm
is applied to the observed current measurements from the glider and compared
with the currents as measured by the ADCP, which are considered the ground
truth. Since the glider operated within 10 km of the ADCP during most of
the mission time (see Fig.

In contrast to the synthetic data, the glider data do not have a fixed
interval. For the present glider data set, most of the subsurface times were
between 2.6 and 3.0 h, as during most of the mission the glider was
programmed to resurface at 3 h intervals, interpreted as resurface
time-to-resurface time. The reason for the mean subsurface time to be less
than 3 h is due to the fact that at resurfacing the glider spent about
10–15 min afloat to transmit data. Leaving all parameter settings of the
low pass filter and the Kalman filter unchanged, the depth- and time-averaged
current estimates are compared with those computed from the ADCP current
measurements, where time intervals for averaging the ADCP data were matched
to the factual subsurface times of the glider. The results are summarised in
Fig.

Mean (

The

The near-real-time instantaneous current estimates are potentially useful for
assimilation into circulation models; see for example

Comparing the estimates of the instantaneous (depth-averaged) current
components with the instantaneous currents measured with the ADCP, the
standard deviations of the differences amount to 6.5 cm s

It is noted that since for the synthetic data set the “measurement data”
and “observations” are constructed from the same source, namely the ADCP
currents, any bias in ADCP currents will go unnoticed. Indeed, the mean
value of the difference between estimated and observed current for the
synthetic data set amounts to 0; see Table

The approach proposed herein can also be used to reprocess the glider data to
obtain estimates of the instantaneous barotropic currents once the glider
mission has been completed. In delayed mode, a number of improvements can be
applied. First, the depth- and time-averaged current estimates can be improved
by recalculating the dead-reckoned position. The glider's dead-reckoning
algorithm computes the horizontal velocity component from the pitch and the
pressure rate, ignoring the angle of attack. Although the angle of attack is
generally small, the glider algorithm may overestimate its horizontal speed
by a few cm s

Another source of error in the estimated currents is the phase lag introduced
by the Butterworth filter; see Sect.

Third, a Kalman filter can be formulated that uses both “historic”
and “future” observations. To that end the Kalman filter described
above is run forward and backward, whereas the final estimate of the
vector

Figure

Developed in the 1990s, the AIS, which is based on VHF radio communications, allows ships to both see and be seen by other marine traffic in their area. The system augments radar and has increased the safety at sea. Since AIS instrumentation is generally bulky and would take a substantial cut from the glider's energy resources, and AIS signals do not penetrate water, it is for technical reasons not feasible to equip a glider with an AIS transmitter. Being able to broadcast its position to surrounding ships would, however, reduce the probability of a collision between a glider and a ship drastically. An alternative to AIS is virtual AIS, whereby the position of an object (glider) is broadcasted from an AIS transmitter elsewhere (a land station). In Germany the authority Wasser-und-Schifffahrtsverbund (WSV) regulates the use of virtual AIS and has shown interest in this approach.

The principle of operation of a virtual AIS system is as follows. Two situations are discerned, namely the period when the glider is at the surface and when it is underwater. When the glider is at the surface and has established a (satellite) communication link with a server on shore, its actual GPS position is known. This information is instantly and automatically relayed to an operator room of WSV, from which the positional information of the glider is broadcasted as an AIS message. When the glider is underwater, and no actual GPS position is available, an estimated position can be broadcasted. To estimate a position, information is required on the local current field (drift) and the behaviour of the glider in terms of hardware behaviour (how it is programmed and, consequently, how it reacts to the environment) and dynamic behaviour (how and how fast it flies through the water). The modelling of the glider behaviour is considered beyond the scope of this study and therefore not discussed.

Assuming that an adequate model of the glider behaviour is available, it is
furthermore required to quantify the drift due to the current whilst the
glider is underwater. The drift can be estimated from integrating the
estimated instantaneous (depth-averaged) currents over the period of the
dive. The instantaneous current is computed as outlined in the previous
section, except for some modifications. Since no new information can be taken
into account until the glider resurfaces again, the residual current
component cannot be computed from a linear interpolation during the dive.
Instead, the residual current component is taken equal to its estimate at the
time of diving and held constant during the dive. For the same reason, using
Eqs. (

It is, in fact, possible to use interpolated amplitudes, based
on the a posteriori estimates at the time of diving and the a priori
estimates at the time of resurfacing; however, this brings no benefit because
of Eq. (

It is expected that the uncertainty in the underwater glider position grows
the longer it is underwater. The synthetic data set can be used to quantify
the effect of subsurface time on the uncertainty in position, as this data set
can easily be divided in predefined subsurface times. Running the (forward)
low pass and Kalman filters repeatedly for subsurface times, spanning 12 h
with 10 min intervals, ensembles of six consecutive runs are formed.
Figure

As the position error increases from zero at the time of diving, the errors
shown in Fig.

Presently, this system is not implemented yet. The authority WSV has expressed its interest and also indicated that the errors in the prediction for 3-hourly dives are acceptable. Technical limitations of the AIS system in use by WSV prevents a (semi-)automatic implementation. Furthermore, the range of the land stations to broadcast the AIS messages is limited to about 70 km offshore and would not reach far enough to cover the outer parts of the German sector of the German Bight.

Error in estimated resurfacing position as a function of subsurface time.

The approach presented herein comes with a number of advantages. First, with
a focus on glider path prediction, previous experience has shown that an
unjustifiable amount of effort is required to guarantee current model output
to be available at all times. Using glider estimated currents removes this
vulnerability, as this information is always available, assuming a glider
operates normally. Second, the proposed algorithm provides independent
estimates of the instantaneous currents. In near-real time these estimated
currents can be assimilated into the COSYNA-run ocean current models of the
German Bight in a fashion similar to how radar observations of surface currents
are assimilated

In delayed mode, when all data are available, the accuracy of the current
estimates can be further improved. Still, the accuracy would remain inferior
to the accuracy that can be achieved with direct measurements from devices
such as ADCPs. However, as often, for practical and logistical reasons, few,
if any, independent current data are available that co-locate with
glider data, so that a third advantage is that for many applications the
improved glider based current estimates may be the only information on
instantaneous currents available. This can facilitate the data analysis in
studies involving gliders in tidal waters, similar to those published on
phytoplankton blooms

We chose to decompose the currents into a tidally driven part and a residual current. Lacking a realistic model for the residual currents, this component was quantified by a simple low pass filter, whereas the tidally driven currents were estimated using a Kalman filter based on the shallow water equations. Instead of this approach, a variety of other formulations could have been considered.

Instead of using a low pass filter, a Kalman filter for the residual
current could be formulated based on the model

A different approach could be to incorporate the residual current Kalman
filter based on Eq. (

The robust solution, therefore, is to apply a low pass filter to the measured
currents to separate the residual and tidal current components. Butterworth
low pass filters of various orders and cut-off frequencies were considered. A
first-order filter with a cut-off frequency of

Figure

Mean error in estimated resurfacing position as a function of
subsurface time for various Butterworth low pass filters with order

Although the navigational algorithm implemented on board the (Slocum) glider yields depth- and time-averaged currents, the time resolution, set by the subsurface time, is often too coarse for the purpose of data analysis. This is particularly the case in regions where the currents are dominated by the tides. In this work an algorithm, tailored to coastal seas with strong tidal currents, was presented that can be used to estimate the instantaneous currents from the time-averaged current measurements obtained by the glider. The algorithm considers a current component driven by the tides and a residual current.

During a glider mission, the algorithm can be used to predict the currents,
which is essential to make a projection of the glider trajectory up to 12 h
or so ahead. Run as a predictive tool, both the low pass filter and the
Kalman filter are run forward in time only, which inevitably leads to lagging
effects. This can particularly be apparent in the residual currents resulting
from the low pass filter. For a typical application of a glider run in the
North Sea, a first-order Butterworth filter with a cut-off frequency of

To assess its performance, the algorithm was first applied to depth- and time-averaged currents, constructed from instantaneous currents measured with an ADCP, with known noise levels added (synthetic data). By averaging over time, information is lost, so that the measurements presented to the Kalman filter contain less information than the original ADCP measurements. The loss of this information is mostly compensated by information provided by the shallow water model. For an anticipated subsurface time of 3 h, the correlation coefficients calculated for the estimated and ADCP measured instantaneous currents were found to be 0.97 and 0.96 for the eastward and northward components, respectively. This result indicates that the algorithm as such lives up to the expectations and is capable of reconstructing the instantaneous currents to a large extent.

When applied to the glider-derived current measurement, and compared with
ADCP data measured within a radius of about 10 km, the algorithm performs
slightly worse, with correlation coefficients of 0.96 and 0.93 for the
eastward and northward current components, respectively. This regression is
attributed to the additional uncertainty caused by the navigation algorithm.
Still, for subsurface times of 3 h, which is a typical operational
setting, the estimate of the instantaneous current has standard deviation of
6–7 cm s

A further application could be to incorporate the present algorithm in a virtual AIS system to enhance the safety at sea. Herein the glider's position between surfacings can be estimated. The uncertainty in the estimated position grows with the time that the glider is underwater. Quantifying the effect of the uncertainty in the currents on the positional accuracy, it was found that subsurface times up to 3 h would yield a positional accuracy that was still acceptable for the German authority WSV.

In delayed mode, the performance of the algorithm can be increased by running
the low pass filter and the Kalman filter in forward–backward mode. The
backward run in effect counters the lag introduced in the forward sweep. The
standard deviation of the instantaneous current estimate was found to drop
below 6 cm s

The glider data used in this work are available from the CODM, the Cosyna
data portal

This work was jointly financially supported by GROOM of the 7th Framework Programme of the European Union under grant agreement no. 284321 and through the Coastal Observing System for Northern and Arctic Seas (COSYNA). The comments and suggestions of four anonymous reviewers are greatly appreciated. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.Edited by: K. Juniper Reviewed by: four anonymous referees