Uncertainty Evaluation and Compensation for Reservoir’s Bathymetric Patterns Predicted with Radial Basis Function Approaches Based on Conventionally Acquired Water Depth Data

Abstract

Information pertaining to a reservoir’s bathymetry is of utmost significance for water resource sustainability and management. The current study evaluated and compensated the reservoir’s bathymetric patterns established using radial basis function (RBF) approaches. Water depth data were acquired by conventionally rolling out a measuring tape into the water. The water depth data were split into three (3) categories, i.e., training data, validation data, and test dataset. Spatial variations in the field-measured bathymetry were determined through descriptive statistics. The thin-plate spline (TPS), multiquadric function (MQF), inverse multiquadric (IMQF), and Gaussian function (GF) were integrated into RBF to establish bathymetric patterns based on the training data. Spatial variations in bathymetry were assessed using Levene’s k-comparison of equal variance. The coefficient of determination ($R^2$), root mean square error (RMSE) and absolute error of mean (AEM) techniques were used to evaluate the uncertainties in the interpolated bathymetric patterns. The regression of the observed estimated (ROE) was used to compensate for uncertainties in the established bathymetric patterns. The Levene’s k-comparison of equal variance technique revealed variations in the predicted bathymetry, with the standard deviation of 8.94, 6.86, 4.36, and 9.65 for RBF with thin-plate spline, multi quadric function, inverse multiquadric function, and Gaussian function, respectively. The bathymetric patterns predicted with thin-plate spline, multiquadric function, inverse multiquadric function, and Gaussian function revealed varying accuracy, with AEM values of −1.59, −2.7, 2.87, and −0.99, respectively, $R^2$ values of 0.68, 0.62, 0.50, and 0.70, respectively, and RMSE values of 4.15, 5.41, 5.80 and 3.38, respectively. The compensated mean bathymetric values for thin-plate spline, multiquadric function, inverse multiquadric function, and Gaussian-based RBF were noted to be 18.21, 17.82, 17.35, and 18.95, respectively. The study emphasized the ongoing contribution of geospatial technology towards inland water resource monitoring.

1. Introduction

Bathymetry is the component of inland surface water bodies which plays a crucial role in determining a reservoir’s depth and the volume of water [1]. Bathymetric information has been extensively used in hydrology, ecology, water resource management, and geomorphology research [2]. The reservoir’s depth and shape characteristics also affect various water body processes, such as mixing and currents, occurring both on the surface and within the water [3]. Therefore, accurate information regarding a reservoir’s bathymetry is required if sustainability in aquatic resources and developments that require water resources is to be achieved [4]. Several approaches have been proposed for reservoir’s bathymetric pattern retrieval, and these include shipborne sonar or radar technique [5], spatial interpolation techniques based on field-measured depths data [6], topographic data-based approach [7], multispectral remote-sensing-based techniques [8], synthetic aperture radar (SAR) [9], and LiDAR systems [10]. However, all these approaches are not without limitations. Moreover, this approach is expensive, and this poses a serious challenge for bathymetric pattern retrieval in low-income countries.

The application of multispectral images in the surface water body’s bathymetric assessment is also challenging because of complex relationships existing between the image radiance properties and water constituents [11]. For example, increased levels of chlorophyll-a, suspended solids, and organic materials tend to alter the reflectance properties of the reservoir’s bottom [11]. Although LiDAR systems are extensively recognized for their capability to retrieve bathymetry, they are also not effective when scanning cloudy or turbid waters [12]. Moreover, LiDAR systems demonstrated some challenges when scanning bathymetry in the near-shore area of a reservoir [13]. Moreover, bathymetric mapping with LiDAR systems has been supplemented with conventionally measured bathymetric data [14]. The fact that LiDAR-based bathymetry must be supplemented with other bathymetric data, such as sonar-based bathymetry [15], places the reliability of this technique in question. Shipborne sonar techniques can acquire bathymetry data even where shallow waters are subjected to high turbidity levels [13]. However, bathymetric pattern acquisition using this system is not practical for shallow surface water reservoirs as a result of sound saturation, limited swath, and impassable routes for a large ship [16].

Among the cited bathymetric retrieval approaches, spatial interpolation techniques based on field-measured bathymetric data are the most commonly deployed techniques [17]. These techniques provide reliable bathymetric patterns if they are appropriately applied. They consider the spatial relationship existing between measured points to predict values at points where measurements are missing. Interpolation techniques rely on the notion that points that are nearer to each other have a stronger relationship, in contrast to points that are farther apart [18]. In the context of the current study, these methods are commonly applied to predict bathymetry in unmeasured locations. Several studies have used spatial interpolation approaches, such as inverse distance weighting (IDW) [14,19] and kriging [14,20,21], in the prediction of bathymetric patterns. However, these techniques also have their limitations; i.e., these techniques cannot estimate bathymetry below the minimum measured bathymetric value or above the maximum measured bathymetric value [22]. As such, the areas with depths lower or higher than measured end up being predicted within the confines of the minimum and maximum measured bathymetric values.

The introduction of the radial basis function (RBF) interpolation approach by Hardy [23] addresses the limitations associated with conventional interpolation techniques due to its ability to predict values outside the confines of the minimum and maximum measured values. Radial Basis Functions (RBFs) are feedforward artificial neural networks that integrate both supervised and unsupervised learning, known as hybrid learning [24]. Typically, RBF networks have a three-layer structure, i.e., (a) input layer, which sends input information to the hidden layer, (b) hidden layer consisting of non-linear neurons, typically Gaussian, and (c) output layer consisting of linear neurons [25]. The RBF interpolation approaches have widely been acknowledged for their effectiveness in predicting missing data, especially in cases where sample points are sporadically distributed [26]. In the context of the current study, RBF approaches are preferred due to their ability to minimize under- and overestimation of the locations with bathymetric values outside the confines of measured bathymetric minima and maxima. Although RBFs have demonstrated the ability to predict values outside the confines of measured data, any predicted values are subjected to uncertainty; i.e., their accuracy must be evaluated and, subsequently, be compensated if under- or overestimation was detected. If these uncertainties are not properly addressed, they may subsequently be carried along to the spatial modeling stage and ultimately be included in the decision-making stage.

Statistical adjustment methods have been proposed to reduce uncertainties in predicted values [27]. Several statistical adjustment approaches for minimizing uncertainties include empirical distribution matching (EDM) [28], regression of observed on estimated values (ROE) [29], linear transfer function (LTF) [30], linear equation based on Z-score transform (ZZ) [28], second machine learning model used to estimate residuals (ML2-RES) [31] and ROE-Duan [28]. In the current study, the ROE statistical adjustment technique was preferred due to its robustness, simplicity, and ability to be implemented at a point scale [29]. Many studies that sought to map bathymetry through the interpolation approach mainly focused on the performance evaluation of several interpolation techniques without attempting to compensate for the limitations associated with these techniques [3,32,33,34]. The primary objective of this study was to establish, evaluate, and compensate the reservoir’s bathymetric patterns based on bathymetric data collected using a unique traditional approach, i.e., rolling out the measuring tape into water to determine the reservoir’s depth. The advantage of using this approach over other explained approaches is that it provides the actual depth of the reservoir; data acquired using this approach does not need to undergo any processing stage, such as those which are acquired using LiDAR and sonar systems. To the best of our knowledge, studies that surveyed reservoir bathymetry using this approach are extremely scarce. Specifically, this study aims to (a) establish spatial patterns in the reservoir’s bathymetry, (b) evaluate the uncertainties in the established bathymetric patterns, and (c) minimize the uncertainties in the established bathymetric patterns.

2. Materials and Methods

2.1. Description of the Study Area

To carry out this experimental research, we have selected the Kat River Dam as the experimental site for simulating bathymetric patterns using traditionally acquired water depth data. The experimental site is situated in the Seymour area, 40 km north-east of Fort Beaufort Town in the Eastern Cape Province, South Africa. Covering an area of approximately 2.14 km², the Kat River Dam serves as the source of water supply for some household activities, livestock drinking and wildlife sustainability. It is located within 32°33′25.36″ S; 26°45′11.64″ E and 32°35′08.89″ S; 26°47′22.94″ E absolute locations. Figure 1 shows the location of the experimental site in relation to South Africa.

2.2. Methods

The sequence of methods and techniques followed to attain the main objective of this study are provided in this section. Figure 2 presents a schematic flowchart diagram explaining the methods and techniques deployed in this study.

2.3. Data Acquisition

Data on the reservoir’s bathymetry was obtained by employing a traditional approach, using a 100 m Retractable Fiberglass Tape Measure supplied by the Builders, Roodepoort, South Africa. Using an electric inflatable boat to access different parts of the dam, a total of 60 points were randomly surveyed from the dam. The reservoir’s bathymetric data were measured by slowly rolling out the tape, with a 996.3 g metal object appended to the tape hook, into the water. The reason for appending the metal object was to facilitate the gravity of the measuring tape and minimize the resistance of water to tape sinking. Then the dam depth was obtained by recording the tape readings at the water levels. The wind-free weather condition was considered for conducting the field survey, with a view to ensuring boat stagnancy during data collection. At the maximum of 3-m precision, the measurement locations were also recorded using the Garmin eTrex 32x Global Positioning System (GPS) device supplied by Garmin Southern Africa (PTY) Ltd., Johannesburg, South Africa. These points were important in defining the locations at which dam bathymetry data were recorded. The acquired data were divided into three (3) sets, i.e., 36 (60%) points for spatial bathymetric simulation, 15 (25%) points for bathymetry accuracy evaluation, and 9 (15%) points for testing the precision of compensated bathymetry. Moreover, another 30 GPS points were randomly recorded at different sites of the dam edge to mark the locations with 0-m depth.

2.4. Spatial Pattern Analysis of Reservoir’s Bathymetry

The classical one-sample t-test was utilized to assess variations in the measured bathymetry levels across the surveyed sites of the reservoir. Assuming that $\mathrm{n}-\mathrm{th}$ bathymetric data ${\mathrm{x}}_1,{ \mathrm{x}}_2,{ \mathrm{x}}_3,\dots ,{ \mathrm{x}}_{\mathrm{n}}$ is normally distributed, with mean µ and variance (${\mathsf{\sigma }}^2$), i.e., ($\mathrm{N}\left(\mu ,{ \mathsf{\sigma }}^2\right)$), the population mean ($\overline{\mathrm{X}}$) is expected to be normally distributed ($\mathrm{N}\left(\mathsf{\mu },{\displaystyle \frac{{\mathsf{\sigma }}^2}{\mathrm{n}}}\right))$. Under a supposition that ${\mu }_{\mathrm{bathymetry}}={\mathsf{\mu }}_{0\left(\mathrm{bathymetry}\right)}$, the normal bathymetry pattern was computed using Equation (1):

$$Z={\displaystyle \frac{ \overline{\mathrm{X}}-{\mathsf{\mu }}_0}{\mathsf{\sigma }/\sqrt{\mathrm{n}}}}$$

(1)

where $\mathsf{\sigma }$ is the standard deviation, and $\sqrt{\mathrm{n}}$ is the square root of the bathymetric sample size. However, when the sample variance is not established, the $\mathsf{\sigma }$ in Equation (1) is substituted by variance ($\mathrm{S}$), such that

$$\mathrm{Z}={\displaystyle \frac{\mathrm{X}-{\mathsf{\mu }}_0}{\mathrm{S}/\sqrt{\mathrm{n}}}}$$

(2)

Therefore, the bathymetric variance was computed by deploying Equation (3):

$${\mathrm{S}}^2={\displaystyle \frac{1}{\mathrm{n}-1}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{x}}_{\mathrm{i}}- \overline{\mathrm{x}}\right)}^2$$

(3)

where ${\mathrm{x}}_{\mathrm{i}}$ denotes the measured bathymetry value and $\overline{\mathrm{x}}$ denotes the mean of the measured bathymetry data. Subsequently, Equation (2) adheres to the t-population distribution with $\left(\mathrm{n}-1\right)$ degree of freedom, such that

$$\mathrm{t}={\displaystyle \frac{\mathrm{X}-{\mathsf{\mu }}_0}{\mathrm{S}/\sqrt{\mathrm{n}}}}$$

(4)

where t denotes the classical one-sample t-test.

2.5. Construction of Spatial Patterns in Reservoir’s Bathymetry

The spatial patterns in bathymetry were constructed by deploying the ANN-based RBF technique. Given $\mathrm{n}$ water depths data $\mathrm{f}\left({\mathrm{x}}_{\mathrm{n}}\right)$ at ${\mathrm{x}}_{\mathrm{n}}$ locations, the reservoir’s bathymetric patterns were computed using RBF Equation (5) adopted from Bawazeer et al. [35]:

$$\mathrm{f}\left({\mathrm{x}}_{\mathrm{i}}\right)={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\lambda }}_{\mathrm{i}}{\mathsf{\varphi }}_{\mathrm{i}}\left({\mathrm{x}}_{\mathrm{i}}\right)$$

(5)

where $\mathrm{f}\left({\mathrm{x}}_{\mathrm{i}}\right)$ denotes the sum of $\mathrm{n} \mathrm{RBFs}$; $\mathrm{n}$ denotes the number of radial basis functions; ${\mathsf{\lambda }}_{\mathrm{i}}$ are weighting coefficients of linear combination; ${\mathsf{\varphi }}_{\mathrm{i}}$ is a function whose output is determined by the distance from the reference point $\mathrm{c}$; ${\mathrm{x}}_{\mathrm{i}}$ denotes the observed scattered point at i-th location when $\mathrm{i}=1, 2, \dots , \mathrm{n}$, computed using Equation (6) adopted from Yao et al. [26]:

$${\mathsf{\varphi }}_{\mathrm{i}}\left({\mathrm{x}}_{\mathrm{i}}\right)={\mathsf{\varphi }}_{\mathrm{i}}\left(\left|\left|{\mathrm{x}}_{\mathrm{i}}-\mathrm{c}\right|\right|\right)$$

(6)

where $\Vert \Vert$ is the Euclidian norm; ${\mathsf{\varphi }}_{\mathrm{i}}$ is the radial basis function; ${\mathrm{x}}_{\mathrm{i}}$ is the observed data, and $\mathrm{c}$ is the reference point on which the radial basis function ${\mathsf{\varphi }}_{\mathrm{i}}$ is centered at.

Because RBFs are exact interpolation methods, it is important that the constructed surface is passed through each observed point. As such, the RBF algorithm was passed through the spatial interpolation schemes as follows:

2.5.1. Thin-Plate Spline Interpolator

The thin-plate spline interpolator was integrated into RBF to predict spatial patterns in the reservoir’s bathymetry. This was achieved by introducing Equation (7) adopted from Skala [36]:

$${\mathsf{\varphi }}_{\mathrm{i}}\left(\mathrm{x}\right)={\mathrm{x}}_{\mathrm{i}}^2\ln {\mathrm{x}}_{\mathrm{i}}$$

(7)

where ${\mathsf{\varphi }}_{\mathrm{i}}\left(\mathrm{x}\right)$ is the radial basis function, and ${\mathrm{x}}_{\mathrm{i}}$ denotes the observed bathymetry.

Subsequently, the RBF was passed through the thin-plate spline to interpolate the reservoir’s bathymetry using Equation (8) adopted from Keller and Borkowski [37]:

$$\mathrm{f}\left(\mathrm{x}\right)={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{N}}{\mathsf{\lambda }}_{\mathrm{i}}\left({\mathrm{x}}_{\mathrm{i}}^2\ln {\mathrm{x}}_{\mathrm{i}}\right)$$

(8)

where ${\mathrm{x}}_{\mathrm{i}}$ denotes the observed bathymetry and ${\mathsf{\lambda }}_{\mathrm{i}}$ denotes weighting coefficients.

2.5.2. Multiquadric Function Interpolator

The multiquadric function (Equation (9)) used for bathymetric estimation was adopted from Wendland [38] such that:

$${\mathsf{\varphi }}_{\mathrm{i}}\left(\mathrm{x}\right)=\sqrt{\left(1+{\left(\mathsf{\epsilon }\times {\mathrm{x}}_{\mathrm{i}}\right)}^2\right)}$$

(9)

where $\mathsf{\epsilon }$ is the scaling factor, and ${\mathrm{x}}_{\mathrm{i}}$ is the observed bathymetry at the i-th location.

This multiquadric function was integrated into RBF to interpolate the reservoir’s bathymetry such that

$$\mathrm{f}\left(\mathrm{x}\right)={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{N}}{\mathsf{\lambda }}_{\mathrm{i}}\times \left(\sqrt{1+{\left(\mathsf{\epsilon }\times {\mathrm{x}}_{\mathrm{i}}\right)}^2}\right)$$

(10)

where ${\mathsf{\lambda }}_{\mathrm{i}}$ is the weighting coefficients, and ${\mathrm{x}}_{\mathrm{i}}$ denotes the observed bathymetric data.

2.5.3. Inverse Multiquadric Function

Ultimately, the RBF was passed through the inverse multiquadric function to predict bathymetric patterns. In this case, the kernel was defined using the inverse multiquadric function (Equation (11)) adopted from Fornberg and Flyer [39].

$${\mathsf{\varphi }}_{\mathrm{i}}\left(\mathrm{x}\right)={\displaystyle \frac{1}{1+{\left(\mathsf{\epsilon } \times \mathrm{x}\right)}^2}}$$

(11)

where $\mathsf{\epsilon }$ denotes the scaling factor, and ${\mathrm{x}}_{\mathrm{i}}$ represents the observed bathymetry.

Therefore, the RBF was deployed to interpolate the reservoir’s bathymetric pattern by passing it through the inverse multiquadric function such that

$$\mathrm{f}\left(\mathrm{x}\right)={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{N}}{\mathsf{\lambda }}_{\mathrm{i}}\times \left({\displaystyle \frac{1}{\left(1+{\left(\mathsf{\epsilon }.{\mathrm{x}}_{\mathrm{i}}\right)}^2\right)}}\right)$$

(12)

where $\mathsf{\epsilon }$ denotes the scaling factor, and ${\mathrm{x}}_{\mathrm{i}}$ represents the observed bathymetry.

2.5.4. Gaussian Function Interpolator

The RBF was also passed through the Gaussian kernel to predict water depths at unmeasured locations. The Gaussian kernel used in this study was computed using Equation (13), adopted from Fasshauer et al. [40]:

$${\mathsf{\varphi }}_{\mathrm{i}}\left(\mathrm{x}\right)={\mathrm{e}}^{-{\left(\mathsf{\epsilon }\times {\mathrm{x}}_{\mathrm{i}}\right)}^2}$$

(13)

where ε denotes the shape parameter of the Gaussian and ${\mathrm{x}}_{\mathrm{i}}$ denotes the observed bathymetry.

Therefore, the RBF was passed through the Gaussian kernel interpolator to estimate bathymetry in unmeasured locations using Equation (14) such that

$$\left(\mathrm{x}\right)={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{N}}{\mathsf{\lambda }}_{\mathrm{i}}\times \left({\mathrm{e}}^{-{\left(\mathsf{\epsilon }\times {\mathrm{x}}_{\mathrm{i}}\right)}^2}\right)$$

(14)

where ε denotes the scaling factor of the Gaussian and ${\mathrm{x}}_{\mathrm{i}}$ denotes the observed bathymetry.

2.6. Spatial Variations in the Interpolated Bathymetry

Upon the successful interpolation of the reservoir’s bathymetry using the RBF algorithm with the selected interpolators, a total of 60 sample points were randomly digitized in the dam. The randomly digitized points were then overlain on each bathymetric map, and the interpolated values on which the randomly digitized samples were superimposed were extracted using the “Extract Multivalues To Points” tool in ArcMap 10.8.2 software package supplied by Esri South Africa, Midrand, South Africa. The attribute table containing the extracted interpolated values was converted to Microsoft Excel format, and Levene’s k-comparison of equal variance test for variance tool embedded in the SigmaXL^® statistical program, supplied by the SigmaXL Inc., Kitchener, ON, Canada, was utilized to assess statistical variance across the established bathymetry maps. The Levene’s test was computed using Equation (15) adopted from Levene [41]:

$$\mathrm{W}={\displaystyle \frac{\left[\left(\mathrm{N}-\mathrm{k}\right){{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{k}}{\mathrm{N}}_{\mathrm{i}}{\left({ \overline{\mathrm{Z}}}_{\mathrm{i}}- \overline{\mathrm{Z}}\right)}^2{\mathrm{n}}_{\mathrm{i}}\right]}{\left[\left(\mathrm{k}-1\right){{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{k}}{{\displaystyle \sum }}_{\mathrm{j}=1}^{{\mathrm{n}}_{\mathrm{i}}}{\left({\mathrm{Z}}_{\mathrm{ij}}-{ \overline{\mathrm{Z}}}_{\mathrm{i}} \right)}^2\right]}}$$

(15)

where $N$ denotes the overall size of combined samples (i.e., $n_1+n_2+\cdots +n_t$); ${\mathrm{Z}}_{\mathrm{ij}}=\left|{\mathrm{Y}}_{\mathrm{ij}}-{\overline{\mathrm{Y}}}_{\mathrm{i}}\right|$, with ${\mathrm{Y}}_{\mathrm{ij}}$ denoting the value of Y for the j-th observation of the i-th subgroup; ${\overline{\mathrm{Y}}}_{\mathrm{i}}$ is the mean of the i-th subgroup; ${ \overline{\mathrm{Z}}}_{\mathrm{i}}$ is the mean of the group ${\mathrm{Z}}_{\mathrm{ij}}$; and $\overline{\mathrm{Z}}$ is the overall mean of the ${\mathrm{Z}}_{\mathrm{ij}}$.

Subsequently, the p-value generated from Levene’s statistical test was analyzed to interpret the variations in the interpolated bathymetric patterns.

2.7. Uncertainty Evaluation of the Established Bathymetric Patterns

The uncertainties in bathymetric patterns established in this study were evaluated. The bathymetric validation samples (15 points), saved as the “.csv” file, were converted into a point shapefile in ArcMap 10.8.2 GIS software package. The created point shapefile was then overlain on each interpolated bathymetric map. Subsequently, an uncertainties evaluation was carried out as follows:

2.7.1. Root Mean Square Error

The root mean square error (RMSE) was deployed to evaluate the accuracy in the interpolated bathymetric patterns using Equation (16) adopted from Chicco et al. [42]:

$$R\mathrm{MSE}=\sqrt{{\displaystyle \frac{1}{\mathrm{N}}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{N}}{\left({\mathrm{X}}_{\mathrm{i}}-{ \overline{\mathrm{X}}}_{\mathrm{i}}\right)}^2}$$

(16)

where $\mathrm{N}$ denotes the number of observations; ${\mathrm{X}}_{\mathrm{i}}$ denotes observed bathymetry and ${\mathrm{X}}_{\mathrm{i}}$ denotes predicted bathymetry. Therefore, the RMSE value closer to 0 indicates accurate bathymetric prediction. As the RMSE value moves away from 0, the bathymetric prediction becomes less accurate.

2.7.2. Uncertainty Evaluation Using Coefficient of Determination

The coefficient of determination $\left({\mathrm{R}}^2\right)$ was utilized to evaluate the accuracy of bathymetric patterns established using RBF with the selected interpolators. This was achieved by employing Equation (17) adopted from Yan et al. [43]:

$${\mathrm{R}}^2=1-{\displaystyle \frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{N}}{\left({\mathrm{Y}}_{\mathrm{i}}-{ \overline{\mathrm{Y}}}_{\mathrm{i}}\right)}^2}{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{N}}{\left(\mathrm{Y}-{ \overline{\mathrm{Y}}}_{\mathrm{i}}\right)}^2}}$$

(17)

where ${\mathrm{Y}}_{\mathrm{i}}$ is the observed bathymetry value, and ${ \overline{\mathrm{Y}}}_{\mathrm{i}}$ is the predicted bathymetry value. Therefore, the bathymetric pattern becomes more accurate as the ${\mathrm{R}}^2$ value grows closer to 1 and more inaccurate as ${\mathrm{R}}^2$ moves closer to −∞.

2.7.3. Uncertainty Evaluation Using Absolute Error of Mean

The absolute error of mean (AEM) was also deployed to evaluate the uncertainties in the interpolated bathymetric patterns. This was achieved by employing Equation (18) adopted from Pantoja-Pacheco and Yáñez-Mendiola [44]:

$$\mathrm{AEM}={\displaystyle \frac{1}{\mathrm{N}}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{N}}\left|{\mathrm{X}}_{\mathrm{i}}-{ \overline{\mathrm{X}}}_{\mathrm{i}}\right|$$

(18)

Best value = 0 worst value = +∞

Where $\mathrm{N}$ denotes the number of observed bathymetry samples; ${\mathrm{X}}_{\mathrm{i}}$ denotes observed bathymetry values and ${ \overline{\mathrm{X}}}_{\mathrm{i}}$ denotes predicted bathymetry values. The AEM values range between 0 and +∞, with a value closer to 0 indicating higher accuracy of the predicted bathymetry and a value towards +∞ denoting low accuracy of the predicted bathymetry.

2.8. Uncertainty Minimization

The uncertainties associated with the simulated reservoir’s bathymetric patterns were compensated for by employing the ROE statistical technique. This approach was implemented at the point scale to adjust the reservoir’s depth at the estimated locations. The compensated values were attained by deploying Equation (19) adopted from Belitz and Stackelberg [28]:

$${\mathrm{Y}}_{\mathrm{ROE}}={\mathrm{m}}_{\mathrm{ROE}}\times {\mathrm{Y}}_{\mathrm{EST}}+{\mathrm{b}}_{\mathrm{ROE}}$$

(19)

where ${\mathrm{Y}}_{\mathrm{ROE}}$ denotes the bias-corrected value; ${\mathrm{m}}_{\mathrm{ROE}}$ is the regression coefficient computed by the regression model and is obtained using Equation (19); ${\mathrm{Y}}_{\mathrm{EST}}$ represents the explanatory variable (interpolated bathymetry data), and ${\mathrm{b}}_{\mathrm{ROE}}$ denotes the estimated value of ${\mathrm{Y}}_{\mathrm{ROE}}$ when ${\mathrm{Y}}_{\mathrm{OBS}}=0$, and is computed using Equation (20):

$${\mathrm{m}}_{\mathrm{ROE}}={\displaystyle \frac{\mathrm{Cov}\left({\mathrm{Y}}_{\mathrm{EST}},{\mathrm{Y}}_{\mathrm{OBS}}\right)}{\mathrm{Var}\left({\mathrm{Y}}_{\mathrm{EST}}\right)}}$$

(20)

where $\mathrm{Cov}\left({\mathrm{Y}}_{\mathrm{EST}},{\mathrm{Y}}_{\mathrm{OBS}}\right)$ denotes the covariance of ${\mathrm{Y}}_{\mathrm{EST}}$ and ${\mathrm{Y}}_{\mathrm{OBS}}$; $\mathrm{Var}\left({\mathrm{Y}}_{\mathrm{EST}}\right)$ denotes the variance of ${\mathrm{Y}}_{\mathrm{EST}}$, which was achieved by deploying Equation (3); ${\mathrm{Y}}_{\mathrm{OBS}}$ denotes the observed bathymetry data and ${\mathrm{Y}}_{\mathrm{EST}}$ denotes the estimated bathymetry data. The covariance of the ${\mathrm{m}}_{\mathrm{ROE}}$ was obtained using Equation (21) adopted from Belitz and Stackelberg [28]:

$${\mathrm{Cov}}_{\mathrm{EST},\mathrm{OBS}}={\displaystyle \frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{Y}}_{\mathrm{EST}}-{ \overline{\mathrm{Y}}}_{\mathrm{EST}}\right)\left({\mathrm{Y}}_{\mathrm{OBS}}-{ \overline{\mathrm{Y}}}_{\mathrm{OBS}}\right)}{\mathrm{n}-1}}$$

(21)

where $\mathrm{n}$ denotes the number of the data values; ${ \overline{\mathrm{Y}}}_{\mathrm{EST}}$ denotes the estimated mean bathymetry and ${ \overline{\mathrm{Y}}}_{\mathrm{OBS}}$ denotes observed mean bathymetry.

Ultimately, the $b_{ROE}$ was computed using Equation (22) adopted from Pérez-Domínguez et al. [45]:

$${\mathrm{b}}_{\mathrm{ROE}}={\displaystyle \frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{Y}}_{\mathrm{OBS}}-{\mathrm{m}}_{\mathrm{ROE}}\left({{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{Y}}_{\mathrm{EST}}\right)}{\mathrm{n}}}$$

(22)

where $\mathrm{n}$ denotes the number of the data values; ${\mathrm{Y}}_{\mathrm{EST}}$ denotes the estimated bathymetric data; ${\mathrm{m}}_{\mathrm{ROE}}$ is computed by Equation (20), and ${\mathrm{Y}}_{\mathrm{OBS}}$ denotes the observed bathymetric data.

3. Results

3.1. Statistical Variations in Field-Measured Bathymetry

Prior to the data split, the descriptive statistics were computed to assess variations in the observed bathymetry.

Table 1. Descriptive statistics results for field-measured bathymetry.

Statistics Results	Bathymetry
N	64
Min	0.0
Max	50.91
μ	16.47
σ	11.57
SE Mean	1.45
t-test	11.39
p-value	0.0001

presents the descriptive statistics results computed from the overall field-measured bathymetry.

From Table 1, the observed bathymetric values ranged from 0.0 m to 50.91 m. At N of 64, DF of 63, and 0.05 significance alpha (α), the mean bathymetric value was noted to be 16.47, with μ, σ, SE Mean, t-test, and p-value being 16.47, 11.57, 1.45, 11.39 and 0.0001, respectively. This confirms variations in the observed bathymetry across the surveyed sites.

3.2. Spatial Prediction of Reservoir’s Bathymetry

Upon data split, bathymetric patterns were established based on the training data. Figure 3 displays the spatial distributions in bathymetry in the study area.

As shown in Figure 3, the RBF passed through thin-plate spline, multiquadric function, inverse multiquadric function, and Gaussian function predicted maximum bathymetric value of 54.96 m (Figure 3a), 50.85 m (Figure 3b), 55.64 m (Figure 3c), and 51.86 m (Figure 3d), respectively. The highest bathymetric values were observed closer to the dam wall and gradually declined as the distance increased from the dam wall.

3.3. Spatial Variations in the Predicted Bathymetry

Levene’s k-comparison of equal variance technique revealed spatial variations in bathymetric patterns generated using integrating RBF with the selected interpolators.

Table 2. The detailed Levene’s statistical results explaining variations in predicted bathymetry.

Results	N	DF	α	µ	σ	Levene’s Test Stats	p-Value
TPS	60	59	0.05	16.614	8.935		0.0001
MQF	60	59	0.05	13.277	6.863	7.159
IMQF	60	59	0.05	10.563	4.357
GAU	60	59	0.05	17.740	9.651

provides detailed Leven’s statistical results explaining variations in the predicted bathymetry.

From Table 2, the bathymetric mean value for RBF with thin-plate spline, multiquadric function, inverse multiquadric function, and Gaussian function was noted to be 16.614, 13.277, 10.563, 17.740, respectively. Overall, Levene’s test statistics value and p-value were noted to be 0.0001. Since the obtained p-value is less than the 0.05 α value, confirming significant variability in the simulated mean bathymetric values.

3.4. Uncertainty Evaluation in the Predicted Bathymetric Patterns

The uncertainties in the predicted bathymetric patterns were assessed by evaluating the predicted bathymetry against the observed bathymetry based on the validation dataset as follows:

3.4.1. Absolute Error of Mean Results

The uncertainties in the predicted bathymetric patterns were evaluated using the absolute error of mean (AEM). The AEM results are presented in

Table 3. The absolute error of mean results.

Interpolator	${\mathit{\Sigma }}_{\mathit{o}\mathit{b}\mathit{s}\mathit{e}\mathit{r}\mathit{v}\mathit{e}\mathit{d}}$	${\mathit{\Sigma }}_{\mathit{p}\mathit{r}\mathit{e}\mathit{d}\mathit{i}\mathit{c}\mathit{t}\mathit{e}\mathit{d}}$	AEM
Thin-plate spline	304.470	328.267	−1.586
Multiquadric function	304.470	344.954	−2.699
Inverse multiquadric function	304.470	347.447	2.865
Gaussian function	304.47	312.56	−0.993

.

Table 2 showed that the predicted bathymetric patterns deviated from the observed patterns, with the AEM value of −1.586, −2.699, 2.865, and −0.993 for RBF with thin-plate spline, multiquadric function, inverse multiquadric function, and Gaussian function, respectively. Whereas RBF with thin-plate spline, multiquadric function, and Gaussian function overestimated bathymetry at the validation sites, underestimation was observed with RBF with inverse multiquadric function.

3.4.2. Coefficient of Determination Results

The coefficient of determination was used to evaluate the uncertainties in the interpolated bathymetry. Figure 4 shows the coefficient of determination results.

The R² values generated from the relationship between the observed bathymetry and the bathymetry predicted using RBF with thin-plate spline, multiquadric function, inverse multiquadric function, and Gaussian function were noted to be 0.68 (Figure 4a), 0.62 (Figure 4b), 0.5 (Figure 4c) and 0.70 (Figure 4d), respectively. By inference, RBF with the Gaussian function revealed better accuracy against RBF with other selected interpolators.

3.4.3. Root Mean Square Error (RMSE) Results

The uncertainties in the predicted bathymetric patterns were also assessed using the root mean square error technique. The RMSE results for the predicted bathymetric patterns are provided in

Table 4. The RMS results for predicted bathymetric patterns.

Interpolator	Total Error	RMSE
Thin-plate spline	−23.797	4.146
Multiquadric function	−40.484	5.407
Inverse multiquadric function	−42.977	5.798
Gaussian function	31.910	3.381

.

Table 4 shows the difference between predicted and measured bathymetric values, with RMSE values of deviations of 4.146, 5.407, 5.798, and 3.381 for RBF with thin-plate spline, multi quadric function, inverse multiquadric function, and Gaussian function, respectively. The total error values show underestimation in bathymetric patterns predicted by RBF with thin-plate spline, multiquadric function, and inverse multiquadric function. However, there was also bathymetric overestimation by RBF with the Gaussian function. Therefore, the discrepancies between predicted and observed bathymetry signified the importance of minimizing uncertainties in the predicted bathymetric patterns.

3.5. Uncertainty Compensation of the Predicted Bathymetry

The uncertainties in the predicted bathymetric patterns were compensated at the validation point locations.

Table 5. Computed coefficients for determining $m_{ROE}$ and $b_{ROE}$.

Interpolator	$${\mathit{\sigma }}^2$$	Covariance	$${\mathit{m}}_{\mathit{R}\mathit{O}\mathit{E}}$$	$${\mathit{b}}_{\mathit{R}\mathit{O}\mathit{E}}$$
Thin-plate spline	76.756	81.159	1.057	−2.834
Multi quadric function	79.313	78.408	0.989	−2.446
Inverse multiquadric function	116.599	92.934	0.797	1.837
Gaussian function	54.065	68.836	1.273	−1.313

presents the variance and covariance results that served in the coefficient ($m_{ROE}$) and $b_{ROE}$ computation.

Whereas the variance values for RBF with thin-plate spline, multiquadric, inverse multiquadric, and Gaussian functions were found to be 76.756, 79.313, 116.599, and 54.065, respectively, the covariance values were noted to be 81.159, 78.408, 92.934 and 68.836, respectively. These parameter values were used to compute $m_{ROE}$ and $b_{ROE}$ coefficients, which were 1.057, 0.989, 0.797, and 1.273, respectively, and −2.834, −2.446, 1.837, and −1.313, respectively. The computed $m_{ROE}$ and $b_{ROE}$ coefficients were used as input values into the $Y_{ROE}$.

Table 6. Uncertainty compensation equations for correcting the predicted bathymetric patterns.

Interpolator	Uncertainty Compensation Equations
Thin-plate spline	$Y_{ROE}=1.057\ast Y_{EST}-2.834$
Multiquadric function	$Y_{ROE}=0.989\ast Y_{EST}-2.446$
Inverse multiquadric function	$Y_{ROE}=0.797\ast Y_{EST}+1.837$
Gaussian function	$Y_{ROE}=1.273\ast Y_{EST}-1.313$

provides the $Y_{ROE}$ equations generated for compensating uncertainties in the predicted bathymetric patterns.

The established compensation equations presented in Table 6 were then used to adjust bathymetric values on the predicted bathymetric maps. Figure 5 shows spatial distributions in compensated bathymetric patterns.

3.6. Spatial Comparison Between Predicted and Compensated Bathymetric Patterns

In this study, the compensated bathymetric patterns were compared with the predicted bathymetric patterns (

Table 7. Descriptive statistics results for predicted and compensated bathymetry.

	Thin-Plate Spline		Multi Quad. Func.		Inv. Mult. Quad. Func.		Gauss. Func.
	Pred.	Comp.	Pred.	Comp.	Pred.	Comp.	Pred.	Comp.
N	50	50	50	50	50	50	50	50
μ	19.339	17.608	20.089	17.422	15.374	18.258	18.733	19.618
σ	9.910	10.475	9.987	9.877	7.354	9.362	9.582	8.449
SEM	1.401	1.481	1.412	1.397	1.04	1.324	0.789	0.629
${\sigma }^2$	98.208	109.72	99.742	97.560	54.086	87.648	31.159	19.792
Coeff. Var.	51.243	59.491	49.715	56.695	47.837	51.277	52.905	43.420

).

From the computed descriptive statistics, variations in the mean value between the sampled predicted and sampled compensated bathymetry were observed. Whereas the mean values for the predicted bathymetry by RBF with thin-plate spline, multiquadric function, and inverse multiquadric function were noted to be 15.374, 19.339, 20.089, and 18.733, respectively, the mean values for compensated bathymetry were found to be 17.422, 18.258, 19.618 and 17.608, respectively. The mean profile for the predicted and compensated bathymetry was subsequently plotted (Figure 6).

The differences between the predicted and compensated mean bathymetry were noted with all interpolators. Whereas the compensation process reduced the bathymetry predicted using RBF with the thin-plate spline and multiquadric interpolators, there was an increase in the bathymetry predicted using inverse multiquadric and Gaussian functions during the compensation process. However, the Gaussian function showed a pattern closer to the compensated bathymetry.

3.7. Accuracy Evaluation of the Compensated Bathymetric Patterns

In this study, the compensated bathymetry patterns were evaluated against the test datasets.

Table 8. Descriptive statistics explaining variations between compensated and measured bathymetry.

	TPS	MQF	IMQF	GAUS	Testing Data
N	9	9	9	9	9
μ	18.211	17.822	17.351	18.949	19.819
σ	12.100	11.163	10.661	11.224	10.590
SEM	4.033	3.721	3.554	3.741	30.910
${\sigma }^2$	146.42	124.61	113.66	125.98	5.910
Coeff. Var.	66.445	62.635	61.443	59.233	10.150

presents the descriptive statistics used to explain bathymetric variations and evaluate their accuracy by comparing them with the test dataset.

The discrepancies between the compensated and measured bathymetric values are explained by differences in the mean values in Table 8. Therefore, the compensated mean bathymetric values for thin-plate spline, multiquadric function, inverse multiquadric function, and Gaussian function were noted to be 18.211, 17.822, 17.351, and 18.949, respectively. These mean values were then plotted against the mean bathymetric value of the test dataset (19.819), as shown in Figure 7.

The comparison of the compensated mean bathymetry against the observed bathymetry revealed the effectiveness of the RBF with the Gaussian function and ROE in accurately mapping the reservoir’s bathymetry. The RBF with thin-plate spline, multiquadric function, and inverse multiquadric function were noted to be the least effective approaches in bathymetric mapping.

4. Discussion

Monitoring the changes in the reservoir’s bathymetry is crucial for understanding inland aquatic ecosystem functioning, aquatic life risk reduction, and water resource management. This study sought to evaluate and compensate for the inland reservoir’s bathymetry by integrating RBF with thin-plate spline, multiquadric, inverse multiquadric, and Gaussian functions to estimate bathymetric patterns and by employing ROE to compensate for under- and overestimation of the bathymetry during the interpolation process. The bathymetric models, derived from depth measurements, are the most effective approach for determining the bottom surface of reservoirs [46]. In this study, the water depth measurements were acquired using the conventional approach, i.e., rolling out the measuring tape deep into the water. This bathymetric data acquisition approach is not very popular, and studies that employ this approach are very scarce. Currently, the proposed techniques for bathymetric mapping include methods relying on topographic data [7], remote sensing [47], and spatial interpolation using field-based depth data [6], as well as methods based on collected depth samples [8]. However, topographic-based bathymetric information is very scarce since it is required prior to filling a reservoir with water [3].

Remote-sensing approaches have evolved to be a powerful tool for bathymetric data acquisition over recent decades [48] due to their efficiency and applicability to various environments. In their study to map bathymetry using a remote-sensing approach, Curtarelli et al. [3] noted that its effectiveness is constrained by water transparency. If the water is transparent, radiation can reach the bottom of the reservoir [49], allowing for the determination of water depth. Therefore, Sichoix and Bonneville [50] noted that the reliability of this technique can be improved by introducing shipboard data directly acquired from the reservoir. Moreover, the presence of optically active water constituents, such as chlorophyll-a, turbidity, and suspended matter, may limit the applicability of this approach. As such, Ji et al. [51] noted that these water constituents must be included in the empirical models for bathymetric mapping based on remotely sensed data. Although numerous studies recommended the LiDAR approach for bathymetric information acquisition [52,53], the actual depth at which radiation reaches is also a function of water constituents and radiation wavelength [54]. Under clear water conditions, Sinclair and Barker [55] found that LiDAR can detect depths up to 70 m. However, the fact that these techniques do not have physical contact with the water depth places their accuracy in question. Although the conventional approach employed in the current study was noted to be limited to shallow bathymetry, it is worth noting that this approach was perfect for reservoirs with at most 100 m bathymetry. The sonar-based echo sounding offers accurate bathymetric information, but it is not suitable for use in shallow water bodies, such as the experimental site of the current study [56]. Moreover, SAR has not extensively been deployed in bathymetric data acquisition due to its susceptibility to wind and low precision [51,57].

In the current study, the RBF passed through thin-plate spline, multiquadric, inverse multiquadric, and Gaussian functions facilitated a successful spatial prediction of bathymetric patterns. This technique was applied to the conventionally acquired water depth data. However, the selection of the interpolation technique can differ based on data type, nature, and modeling objectives [25]. Although numerous research studies have recommended kriging and IDW [57,58,59,60], the RBF was preferred in the current study due to its ability to predict values below the minimum and above the maximum value in the dataset. This interpolation technique has gained interest from researchers for its applications in interpolation, differentiation, and solving partial differential equations [37]. Silveira et al. [61] noted that the accurate prediction of bathymetric patterns can be achieved if the interpolator components are carefully selected. Even though the prediction of bathymetric values below the minimum observed bathymetry and above the maximum observed bathymetry gives the RBF the upper hand when compared with other traditional interpolation techniques such as IDW and kriging, the fact that it can predict negative values in cases where the minimum observed bathymetric value is 0 becomes a concern. In the current study, this limitation was overcome by assigning the value of 0 to the negative bathymetric values.

Through the analysis of the R², RMSE, and AEM results, the Gaussian-based RBF function produced better bathymetric estimates when compared with the thin-plate spline, multiquadric, and inverse multiquadric-based RBF. This was supported by Jasek et al. [62], who noted that Gaussian-based RBF interpolation achieved better interpolation results due to its ease of use, flexibility, and high interpolation accuracy [63]. In the current study, the Gaussian-based RBF interpolation technique revealed better accuracy in predicting the reservoir’s bathymetry. Although Cheng [63] noted that the multiquadric function provided more precise interpolation results compared to other RBF interpolations, the results of the current study are in keeping with Sun et al. [64] and Smola et al. [65], who concur that the Gaussian RBF have proven to provide high overall accuracy in pattern retrieval. In the study to map bathymetry using remote-sensing techniques, Gao [4] revealed that the Gaussian performed better in predicting the bathymetry than other employed interpolators. However, the choice of the shape parameter is critical to the success of the Gaussian-based RBF interpolation, as it directly affects the shape of the basis function [64]. The MQF interpolator showed some discontinuous dots in bathymetry. This could be attributed to the decline in its convergence rates as the order of differentiation increases. However, Madych [66] noted that integrating the smoothing technique into MQF can offset this limitation. The shape of the Gaussian basis function facilitated the smoothing of the interpolated bathymetry in the study area. However, the Gaussian basis function is not without limitations. Although the bathymetric pattern interpolated using the Gaussian function showed better accuracy than other interpolators, it overestimated bathymetry at the reservoir’s shore. This could be attributed to either the smoothing process or the sample size used in training the model since RBF requires a large dataset.

The globally supported radial basis functions such as Gaussians or generalized (inverse) multiquadric have excellent approximation properties [67]. Furthermore, the multiquadric function may obtain more accurate solutions than other RBF interpolations [63]. In this study, the ROE approach enabled the improvement in the accuracy of the interpolated bathymetric patterns. Through a literature search, studies that attempted to compensate for uncertainties associated with bathymetric patterns established using spatial interpolation approaches were found to be lacking. This technique demonstrated that bias emanating from the established bathymetric patterns can indeed be compensated. Although Song [29] noted that the ROE technique may appear counterintuitive, Belitz and Stackelberg [28] recommended it for adjusting the estimated values to match observed values. The employment of the ROE method aimed to ensure that the residuals were independent and had homoscedasticity. The comparison of bathymetric mean values between the predicted and compensated formed the basis for evaluating the performance of the ROE technique. It is imperative to note that, during the bathymetric pattern compensation, the bathymetric values in the observed locations were not excluded from being adjusted. It is also worth noting that the RBF neural networks require a large dataset to establish the reservoir’s bathymetric patterns. In the current study, a total of 36 training points were used, and this might have slightly influenced the results of the study.

5. Conclusions

This research highlighted the significance of mapping bathymetry, evaluating and minimizing uncertainties in bathymetric patterns generated using the RBF interpolation techniques based on conventionally acquired water depth data. The spatial patterns in the reservoir’s bathymetry were computed by deploying thin-plate spline, multiquadric, inverse multiquadric, and Gaussian-based RBF interpolation techniques. The uncertainties in the interpolated patterns were compensated for by employing the ROE technique based on the validation dataset. In general, this research emphasized the importance of employing RBF interpolation techniques in establishing the reservoir’s bathymetric patterns and the significance of ROE in minimizing the bias emanating from the interpolated bathymetric patterns. Nevertheless, it would be intriguing to further investigate the extent to which other bias correction techniques could improve bathymetric patterns established using these RBF interpolation techniques. Moreover, the integration of the smoothing functions and MQF must be evaluated to determine the extent to which their integration would enhance the accuracy of the MQF approach in bathymetric patterns. Overall, the study emphasized the ongoing contribution of geospatial and machine learning techniques towards addressing water security issues in inland surface water bodies. Moreover, it will be interesting to also investigate the efficacy of the employed interpolators with a large bathymetric dataset. Ultimately, we recommend that the findings of this study be evaluated against the bathymetric data acquired using universally employed techniques such as echo sounding and LiDAR. This process may serve to verify the success of the current work.