Introduction to SDB and GWR

In general, the sea bed shape estimation in Satellite Delivered Bathymetry (SDB) models relies on the assessment of the level of electromagnetic radiation dispersion in the visible (or near visible) spectrum. Based on that level and a specific empirical model, the depth of the sea bed is being calculated for a pixel corresponding to a particular geographic location.

SDB is a survey method that uses satellite or other remote multispectral imagery for depth determination [2] and has become a useful reconnaissance and planning tool for sea and ocean mapping. Especially, the usage of SDB is extremely practical in areas where vessel traffic or the deployment of aids to navigation indicates that charted data is misleading and there are no recent surveys to update the chart [3]. The results obtained from the SDB method could be applied in many hydrographic branches as well as in the marine sector, such as bathymetry, cartography, integrated coastal management, water coastal management, and water quality monitoring.

Although SDB provides bathymetry products at a coarser spatial resolution compared to traditional acoustic surveying or airborne lidar bathymetry (ALB), satellite imagery provides a routine repetitive coverage over the same area with advantages for the study of changes in dynamic seabed areas or areas with little survey data thus it can be also applied for chart adequacy assessment and survey prioritization [3]. It should be mentioned that SDB includes a variety of analytical, semi-analytical, and empirical methods. The first ones, analytical and semi-analytical SDB methods (called inversion models), were built upon the physics of light attenuation and thus require the parameterization of many optical properties of the atmosphere, water surface, water column, and water bottom to derive water depth [4], under the assumption that these properties can vary spatially [5]. Consequently, these methods can generate more accurate estimations of water depth yet at the cost of model complexity and greater computational demand.

As an alternative, empirical methods estimate bathymetry based on statistical regression of depth (derived from in situ depth data collected at locations distributed throughout the study area) on the observed radiance of multiple light-penetrating bands, under the assumption of homogenous water columns (both vertically and spatially) and uniform bottom type across the image [6–8], the latter of which can be compensated through the use of multiband data.

One of the most commonly used algorithms is the log-ratio method developed by Stumpf, Holderied, and Sinclair [9], which uses the ratio of two log-transformed bands, typically the blue and green bands. This method assumes that both bands are similarly affected by the bottom reflectance and thus the ratio of them is insignificantly affected by the bottom type compared to water depth, causing the model to be partially independent of the bottom composition and benthic habitat.

On top of the dispersion and the analytical model, extra factors are taken into account. These might have influence over the satellite measurement of the energy level at a given band. Most common of these factors include technical factors such as sensor’s characteristics and calibration parameters, orbit parameters, spatial resolution and saturation energy. Spatially correlated parameters such as seabed type, atmospheric correction, water turbidity, waving and other environmental factors are also taken into account, especially for locally geographically calibrated models.

Recently, the approaches based on machine learning algorithms, like random forest or different architectures of neural networks, have been introduced to SDB. In the reality, the relation between the electromagnetic radiance registered by satellite sensors and the water depth, utilized in SDB, is influenced by a large number of factors that complicates the physical description of the phenomena, and therefore modeling it by a simple formula may be not efficient. The use of machine learning algorithms for modeling more complicated forms of this relation could be the solution for this problem. One of the first works related to this issue was [10] where the multilayered perceptron network was used for coastal zone depth evaluation from airborne visible/infrared imaging spectrometer measurements. More recently, the random forest machine learning results were presented for SDB with the utilization of several channels: coastal, blue, green, red and near-infrared channels of Landsat 8 imagery along with a large amount of ground truth data [8]. Nowadays, deep learning methods along with several deep neural network architectures: convolutional, encoded-decoder, recurrent, and combinations of them, have been also introduced to SDB [11–13]. In [13], the indirect method of SDB relies on depth estimation from sea wave characteristics derived from a satellite image rather than from a pixel-based radiation level. From the most recent works, in [14] it was presented the application of several machine learning methods for SDB using very high-resolution multispectral Worldview-2 images along with the application of water quality inherent optical parameters obtained using the so-called updated quasi-analytical algorithm, showing that incorporating the latter improves the results. In another work [15], traditional SBD methods based on simple band ratios and their modifications were compared with those based on machine learning, namely, on Random Forest and XGBoost model, with obtaining markedly more accurate results for machine learning methods. In [16] the concept related to GWR was utilized along with machine learning for deriving bathymetry from optical images with a so-called localized neural network algorithm. The extensive and detailed up-to-date review of SBD methods and algorithms, describing and comparing a number of approaches, i.e. analytical, semi-analytical, statistical and empirical, including also those based on machine learning, with finalization of the paper by providing a matrix as an aid for SDB data and technique selection for a needed level of accuracy, is given in [17].

The use of local spatially correlated factors for SDB was first introduced as geographically weighted regression (GWR) in 1996 in the geographical literature drawing from statistical approaches for curve-fitting and smoothing applications. This method works based on the simple, yet powerful idea of estimating local models, using subsets of observations centered on a focal point. Since its introduction, GWR rapidly captured the attention of many in geography and other fields for its potential to investigate nonstationary relations in regression analysis. The basic concepts have also been used to obtain local descriptive statistics and other models such as Poisson regression and probity. The method has been instrumental in highlighting the existence of potentially complex spatial relationships.

Today, GWR is a useful tool for inferring spatial processes, and a relatively simple and effective tool for spatial interpolation [18]. It is also a well-known powerful spatialization approach that enables flexible integration of predictors for systematic analyses of spatially varying relationships [19]. GWR became an established standard GIS routine and presently is applied widely in climate spatialization (e.g., [20–22]).

Although GWR allows for a spatially differentiated analysis of predictor-predictand relationships, one has to keep in mind that these are statistical relations and their physical meaning requires careful examination. Particularly when analyzing, for example, elevation-temperature dependencies in areas of rather low relief energy, GWR may yield physically implausible lapse rates which, in the case of sparse data coverage in adjacent mountainous areas, would serve to bias the spatialization results.

[22] was among the first works where GWR was applied in remote sensing. The technique was demonstrated by examining the normalized difference vegetation index (NDVI) for the rainfall relationship. Some recent examples include modelling the net primary production of Chinese forest ecosystems [23], the estimation of leaf area index (LAI) over a tropical rainforest [24], as well as tree diameter modelling using airborne lidar [25].

For the need of the marine environment, GWR is a reliable tool for estimating the natural variation of salinity in shallow seas. Such a system was used for the southern part of the Baltic Sea as well as for Florida Bay. GWR could be very useful in salinity modelling, because salinity may have strong local variations. It could be rather useful for shallow waters because they have a higher degree of optical complexity than clear deep oceanic waters [26].

GWR is designed to incorporate spatial dependency (non-stationarity) into regression models. To achieve this goal, a local regression model is constructed at every location and different relationships (regression coefficients) are possible at different points [21, 27]. GWR is therefore a truly local approach capable of capturing nonuniform spatial dependencies [28].

Recent GWR approaches are more frequently based on empirical techniques. Frequently, like in SDB and in many other issues of remote sensing at present, the machine learning models are used there to find optimal performance. Usually, different types of machine learning algorithms, e.g. neural networks of several architectures, XGBoost, random forest, ensembles of neural networks, or other learners, are combined with the use of geographically defined weights in the applied learning cost function. The description of this concept may be found for instance in [29]. An example of this approach utilizing machine learning and GWR in SDB is given in [16]. According to other topics, geographically weighted machine learning has been applied, for instance, in forest biomass estimation from satellite imagery [30], in high-resolution spatiotemporal estimations of wind speed from various environmental data [31], and in an investigation of hard coal mining impact on the natural environment [32]. In all these works, a simple, unified form of a radial geographically weighting function is used, depending only on the geographic distance between two points (the example may be the function given by (5) in the next section). It was noticed in [33] that sometimes the geographical properties and spatial dependences describing given phenomena may be more complex and the use of the above approach for modelling them might be insufficient. Therefore the dedicated component of the neural network architecture has been proposed in this work for modelling the fluctuations in GWR, the so-called geographically neural network weighted regression (GNNWR). This concept has been applied for the estimation of the influence of the red tide on several elements of the sea environment [33].

The research objectives

As it has been shown above, there exists a multitude of techniques and algorithms used in SDB, including analytical, empirical and physical approaches. Currently, there is no one universal method that would be considerably better than the others for every use case (see, e.g. [17]). What is more, it is in general difficult to compare directly the results of methods applied in different studies, because, apart from the used method itself, the performance may depend strongly on various conditions, e.g on availability and quality of calibration data sets, quality of acquisition, meteorological conditions, or, in a case of machine learning, also on many details on data preprocessing or used hyperparameters of a given model, or even on the utilized particular software package or library.

In this context, due to there are no many papers focused on performance comparison, with respect to the same measurement dataset, of a number of SDB methods differing not only in used approach but also in many other conditions and details, the objectives of the presented work may be stated as follows: to compare, on the same dataset, the performance of several machine learning methods against polynomial regression algorithms used as the reference approach. The investigated particular SDB models are differing also with respect to several detailed settings, including:

• a particular set of satellite data bands, their ratios etc. used as an input,
• applying or not geographical weighting in the learning process,
• performing some additional preprocessing operations like transforming the data to exponential scale.

The comparison has been made in terms of root-mean-square error, correlation and error distribution.

Definition of the problem in the context of machine learning

Classical approaches to SBD rely on the phenomenon of light passing through a water column of a certain depth described by the Beer-Lambert law: (1) where I–light energy after passing the water column of depth (height) z and light attenuation coefficient k, I₀ –light energy before passing the water column. From satellite remote sensing data, we can derive the reflectance for a shallow water area for more than one electromagnetic band, e.g. for visible blue, visible green and even more bands. However, due to strong variability of several factors influencing the radiance value registered by the satellite sensor, i.e. air-water interface reflectance, wave attenuation in the water column and water-seabed reflectance, it is not easy to obtain a simple, universal formula for efficient sea depth estimation from satellite imagery. On the other hand, usually we may have at our disposal a set of sea depth data acquired by more accurate methods than satellite imaging, i.e. in-situ echosounder measurements. These data may be used for calibration of methods and models used in SDB.

In general, we may define the problem of SBD as an optimization of the function: (2) where – sea depth of a given location (x, y) estimated by SDB, B₁, B₂, …, B_Nb−reflectance (pixel) values acquired by satellite sensor for this location for a set of bands, Nb–number of used bands. The optimization is done using the calibration depth measurement data z₁, z₂, …, z_n, n–number of the used calibration data points, what denotes the minimization of the cost function: (3) where E[·] is an expected value operator, n–number of training (calibration) points, z_i−depth of i-th calibration point, – depth of i-th point estimated by SDB. Due to expected geographical variability of the optimal form of f(·) function from (2) [1], the optimization may be performed locally instead of globally, assuming an adequately, geographically weighted influence of particular z_i values on estimation for a given location (x, y).

Instead of assuming the closed, analytical form of a function from (2) and estimating, for instance, only some of its parameters, the dependence of depth z on bands values B_i may be derived by means of machine learning, e.g. using an artificial neural network with appropriately designed architecture, or other machine learning methods. This approach is applied in the presented work. The investigated machine learning models, accompanied by the reference not AI-based models implemented for results comparison, are presented in the following subsections.

Proposed machine learning models for SBD

For the presented study, we have prepared 12 different machine learning models optimizing the f(·) from (2), described below. Each model was calibrated on the basis of the same data calibration set and observations. Works on the development of the algorithm for bathymetry estimation focused on multitude approaches, mostly based on shallow neural networks and regression trees that use localized geographical models for calibration. On top of that, four models which do not rely on AI were included for reference purposes.

All models were implemented in Python using numpy and scipy libraries (the latter was selected for its ability to deal with sparse matrices’ calculations). Neural network local models were prepared using Keras and Tensorflow libraries. This allows for a flexible description of the algorithm as a graph operating on tensors. The graph itself contains parameters, which are being calculated during the algorithm’s learning process. In the case of neural networks, the parameters represent the weights of specific neurons. The networks were finally composed to obtain a specific geographically weighted model (GWM) using Keras library.

The general model training schema is presented in Fig 1. Input data tuples are combined from Sentinel-2 MSI data and bathymetry reference data. Entire set is divided randomly to Test data (20%) and base training data (80%). If a given model supports validation data, base training data set is divided randomly into validation data (16%) and training data (64%). Training data and validation data (if supported) is used to train model. This process produces trained model and training results and metrics. Test data is used on trained model to perform final verification.

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Fig 1. General model training and testing scheme.

https://doi.org/10.1371/journal.pone.0291595.g001

RMSE

Below, the RMSE obtained for particular models (subsequent rows in Table 1) will be compared with that corresponding to the reference model ref.

gwmk and gwmk_ref As expected, the reimplementation of the well-known GWR algorithm in Keras (gwmk) did not improve the performance. On the contrary, the error increased by 12%. Even more discouragingly, applying localized data to Keras models as inputs in model gwmk_ref (as in the reference work) performed even worse, increasing the error by 22%.

The introduction of artificial intelligence was a step in the right direction. With the linear regression layers replaced by neural networks in the model gwnk_nn, the error was reduced by 17%.

The selection of specific data as inputs proved to be of great importance. While using inputs as per the original work of Chybicki [1] yields improved performance, it performs worse than inputs composed of direct band data from Sentinel 2 satellites in model gwmk_nn_bands with RMSE of 0.96.

Having that said, it is noted that when an approach incorporates together the direct band data and the quotient of bands’ logarithms, like in gwmk_nn_bands_qlog, the error is reduced even further to 0.76 m.

Using the exponent at the output neuron of the network, as in gwmk_nn_exp occurs as a suboptimal approach since the performance is worse than that of the algorithm with linear output.

At first, the introduction of regression tree algorithms seemed like not the best approach. An implementation which takes the quotient of logarithms as input performed worse than the analogical neural network approach of gwmk_nn, having an RMSE of 1.74 m versus 1.55 m of the latter. Similarly, approaches taking raw reflectance data and a mix of reflectance and quotient approach for regression tree algorithms (that is rt_bands and rt_qlog_bands) performed roughly 10% worse than their neural network counterparts. It is important to note that regression tree algorithms do not inherently take into account the local character of the data, whereas neural network approaches do. Introducing information regarding input data localization to regression tree algorithms (as per rt_spatial_qlog, rt_spatial_bands, rt_spatial_qlog_bands) significantly improves the performance, leading to the lower error for rt_spatial_qlog_bands approach with RMSE of 0.69 what is the best-obtained result according to RMSE.

Random forest models A natural step when dealing with the class of decision tree algorithms is to test the random forest approach. It was assumed that with the randomization of input data batches it would be possible to reduce the potential overlearning of the models. But unfortunately, the results of using random forest algorithms did not yield satisfactory improvements. For each input data class (quotient of logarithms, raw reflectances and the combination of both), the approaches performed slightly worse than their decision tree equivalents. Having in mind the extra complexity introduced by implementing random forest compared to regression tree algorithms, this approach was abandoned.

Coefficient of determination and overall performance

At this point, it is important to address the overall performance of the studied algorithms regarding depth estimation on test data, with respect to the correlation between the reference and estimated depth and to obtained estimation accuracy for particular ranges of values (Fig 2). In addition to measures such as RMSE, it is also important to take into account the relative error in the analysis of the obtained results. However, it should be borne in mind that in the case of SDB analysis, such a comparison may not be reliable, because the error characteristics are strongly dependent on factors that are difficult to take into account in empirical modeling or measurement data such as high variability of the light scattering pattern from the waves for small depths or the temporal and spatial variability of bathymetry, especially apparent for the sandy bottom. For this reason, as part of the analysis and in addition to the analytical data, we also focused in the discussion of the results on the error characteristics depending on the depth and the algorithms used.

The performance of the original GWR model is presented in detail in Fig 2 (ref). It can be observed that for depths down to 8 m the model tends to overpredict the depth, while for larger depths an underprediction is noticed. The model which takes into account the exponential radiation absorption of water (Fig 2 exp) significantly improves the symmetry of the acquired results (i.e., it was not more overpredicting than underpredicting), although it maintains (or even increases) their spread.

Reimplementation of the GWM algorithm in Keras maintained the shape of the estimated data versus test data. It again demonstrated overprediction for smaller depths, and underprediction for larger ones, as shown in Fig 2 gwmk and Fig 2 gwmk_ref.

The introduction of neural networks slightly reduced the overestimation problem (see Fig 2 gwmk_nn and Fig 2 gwmk_nn_exp). At the same time, it did not result in the reduction of estimation spread. On that note, the inclusion of direct band data to the inputs did eliminate far-over- and far-underestimates (Fig 2 gwmk_nn_bands and Fig 2 gwmk_nn_bands_qlog).

A noticeable feature of the results provided by regression trees and random forest algorithms is the quantization of data. This is expected, as the result space for such algorithms is discrete. According to the performance comparison, as in the case of neural network algorithms, it was observed that using raw bathymetry as input data does not end with a satisfactory outcome (Fig 2 rt_qlog, Fig 2 rf_qlog). The estimations are overpredicted and are laden with big errors. Making use of EM reflectance data from all available bands removes this issue and significantly improves the performance of this class of algorithms (Fig 2 rt_bands, Fig 2 rf_bands) and when these inputs are combined with the raw bathymetry the outcome proves to perform even better (Fig 2 rt_qlog_bands, Fig 2 rf_qlog_bands).

Good performance is obtained by including data localization to the input data for regression tree algorithms. The best performing combination of input data proves to be the one that includes both reflectance data and raw bathymetry (compare Fig 2 rt_spatial_bands, Fig 2 rt_spatial_qlog and Fig 2 rt_spatial_qlog_bands). Not only does this significantly reduce the errors, but also provides correct estimations for the whole range of depths. The errors are still present but are significantly reduced and restricted to the quantified values generated by the regression tree.

Visualizations of bathymetry estimations for two representative example locations are presented in Figs 3 and 4. Results obtained for the port of Gdynia (Fig 3) visualize the local character of the training data sets. The transition of seabed depth estimation from the main port area to the northern local set is discontinuous, which is clearly an artifact resulted by overlearning of the algorithm in a specific area. Bathymetry data calculated for the harbor of Jastarnia (Fig 4) demonstrate the usefulness of satellite-delivered bathymetry methods for seabed modelling by clearly being able to identify the harbor’s approach path. With satellite data being available significantly more frequently than in-situ measurements, numerical models provide an accurate approximation of changes in seabed shape in time, which may allow authorities to react in time in case of any unexpected disturbances.