Simulation and Sensitivity Analysis of Remote Sensing Reflectance for Optically Shallow Water Bathymetry

Abstract

Optically shallow waters (OSW) enable water depth retrieval through passive optical remote sensing, benefiting from the relatively low attenuation of the water column. However, in OSW environments, remote sensing reflectance is influenced by multiple optical parameters beyond water depth. Comprehensive quantification of these influences remains necessary. This study utilizes numerical simulations to systematically categorize optical parameter ranges and perform variance-based sensitivity analysis. The results indicate that OSW parameter characteristics differ between extremely shallow and moderately shallow waters. In extremely shallow waters, water depth and bottom reflectance are the primary determinants of remote sensing reflectance, with longer wavelengths exhibiting greater sensitivity to depth variations. Bathymetric models utilizing long wavelength combinations demonstrate high accuracy in extremely shallow waters. In moderately shallow waters, sensitivity contributions for shallow water features are concentrated in the blue-green bands, whereas reflectance above 600 nm is primarily influenced by deep-water reflectance and noise, limiting the retrieval of depth and bottom reflectance. Simulations across OSW parameter ranges further reveal that the absolute error in depth estimation increases with depth, whereas relative error is more pronounced in extremely shallow waters compared to moderately shallow waters.

1. Introduction

Optically shallow waters (OSW) are a special type of water body, characterized by the significant influence of the bottom reflectance on the total reflectance, distinguishing them from optically deep waters with negligible bottom contributions [1]. Although OSW regions constitute merely a fraction of the global water surface area, on the order of a thousandth, they support marine biodiversity at a scale of millions of species [2]. Typical OSW environments include coral reefs and seagrass beds, two highly productive regions closely linked to global climate change and the carbon cycle.

The penetration of visible light in water enables cost-effective and efficient exploration of OSWs using optical passive remote sensing techniques. The presence of a reflective bottom boundary in OSWs facilitates the retrieval of parameters along the optical path. The fundamental parameter in early remote sensing studies of OSW regions was water depth [3,4] due to the distinct decrease in reflectance with increasing depth across shallow waters from the shoreline to optically deep waters. The negative relationship between water depth and remote sensing reflectance provides the theoretical basis for using passive optical sensors for OSW depth retrieval and mapping. These bathymetric models typically operate under two key assumptions: (1) water quality conditions are spatially uniform and consistent across the study area, and (2) bottom reflectance can be eliminated using the ratio of different wavelengths. Variability in water quality and bottom types in the OSWs represents the primary source of errors associated with these assumptions.

To address the limitations of assumptions in statistical methods, semi-analytical models based on the radiative transfer equation (RTE) consider water depth, water optical properties, and bottom reflectance [5,6]. These models employ optimization methods to search for suitable water quality and bottom parameters, often resulting in substantial computational demands [7,8]. However, the contribution of each parameter to remote sensing reflectance varies. Quantifying the sensitivity contribution of each parameter enables adaptive adjustment of iteration step sizes, leading to improved computational efficiency.

This study aims to quantify the impact of underwater environmental variations on reflectance and investigate how these variations contribute to the error patterns in passive optical bathymetry. To achieve this, the study employs the SALib library for variance-based global sensitivity analysis (GSA), analyzing how different OSW environment parameters impact remote sensing reflectance [9]. Meanwhile, to evaluate the errors of statistical models based on analytical equations, this study uses the Sobol sequence to generate corresponding remote sensing reflectance based on fixed water depth ground truth and optical parameters. The remote sensing reflectance is then regressed using a statistical model, and the error between the water depth predicted by the model and the fixed water depth used to generate the simulated reflectance is compared.

2. Model and Methods

GSA methods are extensively applied in remote sensing for both optical and microwave research. In optical remote sensing, GSA has been utilized to analyze radiative transfer among canopies [10] and it extended to aquatic vegetation. Aquatic vegetation typically consists of emergent and submerged plants. Submerged plants, being fully or partially underwater, are particularly influenced by water optical properties, which significantly impact optical remote sensing reflectance [11]. Expanding upon submerged plant studies, remote sensing parameterization is also necessary for coral reef ecosystems and other shallow waters dominated by inorganic bottom types, which lack typical characteristics of vegetation.

In OSW regions, the primary parameters influencing reflectance are derived from the RTE and include bottom reflectance, water optical properties, and water depth. These parameters are compiled into spectral datasets, including substrate spectral libraries, simulated water optical properties, and segmented depth intervals. Subsequently, sensitivity analysis is performed to assess the influence of these parameters across wavelengths and depths. Finally, depth inversion simulations are performed using the band ratio bathymetric model, and retrieval accuracy is evaluated by root mean square error (RMSE) statistics. This workflow, which integrates parameter selection, sensitivity analysis, and simulation, is illustrated in Figure 1.

2.1. Parameter Selection Based on Radiative Transfer Model

The RTE for OSW environments encompasses parameters associated with underwater topography, water quality, and benthic ecology [5,6]:

$$r_{rs}=W[1-\mathit{exp}(-2KH)]+{\displaystyle \frac{B}{\pi }}\mathit{exp}(-2KH)$$

(1)

$$R_{rs}={\displaystyle \frac{0.52r_{rs}}{\left(1-1.7r_{rs}\right)}}+\mathsf{\Delta }N$$

(2)

where R_rs is the reflectance above the water surface, and r_rs is the refracted reflectance just below the surface. R_rs has two primary sources: bottom reflectance B and deep-water reflectance W. Deep-water reflectance W refers to the signal received by the sensor, which is almost entirely derived from the backscattering of the water column. In the shallow water column, these two sources of reflectance exhibit exponential variation with water column height H and the water attenuation coefficient K. ΔN represents the uncertainty from atmospheric correction and other sensor-related adjustments.

The RTE identifies key parameters that affect reflectance most significantly. Although surface glint, foam, and atmospheric processes also influence reflectance, they are not closely tied to ecological parameters. The parameters affecting OSW remote sensing reflectance are summarized as 3n + 1 [8], where 1 refers to the water depth, and 3 denotes two water optical parameters—water reflectance and water attenuation parameters— and one bottom reflectance parameter, each wavelength dependent. In this context, n refers to the number of bands used in remote sensing or spectral measurements. The 3n + 1 parameterization framework elucidates the structure of OSW reflectance modeling. Each band is associated with an attenuation coefficient, a deep-water reflectance, and a bottom reflectance. Based on the RTE for OSW regions, this study selects water depth, two types of water optical parameters, and bottom reflectance as the primary parameters for subsequent analysis. The water depth parameter is shared across all bands. This allows depth retrieval to leverage information from multiple bands simultaneously. Therefore, early remote sensing research integrated multi-wavelength reflectance trends for depth estimation.

2.2. Bathymetric Methods Based on Passive Optical Remote Sensing

Water depth retrieval methods using passive optical sensors primarily fall into two basic categories: statistical methods and optimization methods [12]. Statistical methods require fitting based on a certain number of sampling points and then extending the fitted model to the entire study area. Among the specific types of models, band ratio models are relatively simple, while more complex models include various basic elementary functions (quadratic functions, power functions, etc.), machine learning, and deep learning. Overall, nonlinear models generally achieve better results. Band ratio models, due to their simplicity, are easier to popularize. Compared to optimization models, it can save computational resources, making large-scale applications feasible. The publicly available Allen Coral Atlas dataset (https://allencoralatlas.org, accessed on 21 January 2025) also processes its bathymetry products based on band ratio models. Therefore, this study selected the simplest and most commonly used statistical model as an example.

The band ratio bathymetric model typically takes the following form:

$$H=m_1{\displaystyle \frac{\mathit{ln}(c{R}_{rs1})}{\mathit{ln}(c{R}_{rs2})}}+m_0$$

(3)

where H represents depth, m₁, and m₀ are fitted coefficients, R_rs1 and R_rs2 are reflectance values at two bands, with c often a large empirical constant, like 1000 or 10,000 [13]. Blue and green bands are typically used for ocean waters [7,14,15], while green and red bands or other optimal band combinations are used empirically in riverine waters [16,17].

Band ratio models exhibit strong statistical regression for depth estimation, simplifying the process by disregarding variations in water quality and bottom optical parameters. The model, although simplified physically, still requires further exploration regarding the principles of its band selection.

Statistical methods require sufficient sampling data for model fitting. When independent data for training bathymetric models is unavailable, more computationally intensive optimization methods are needed to calculate water depth [7,8]. The RTE is often used as the target equation for depth retrieval through optimization methods [18,19]. As another important method, the optimization method typically requires extensive computations to explore the optimal solutions within the parameter space. Sensitivity analysis, as an effective tool for exploring the parameter space, facilitates the investigation of different parameter ranges under various environmental conditions. For example, environmental conditions may vary based on the general depth range (near the shoreline or near the optically deep water) or the general optical properties of the water body. These factors impose different constraints on environmental parameter ranges. These environmental parameter ranges also differ in their spectral response, which in turn affects the performance of using different remote sensing bands. By providing insights into which parameters have the greatest impact on reflectance, the optimization method enables more efficient and targeted exploration of the parameter space.

2.3. Sensitivity Analysis Method

Sensitivity analysis is a modeling approach used to assess the influence of each parameter within a model, identifying key sensitivities while reducing the complexity of less influential variables, to enhance model efficiency. Sensitivity analysis methods range from local sensitivity analysis (LSA) to GSA [20]. LSA is generally insufficient for analyzing high-dimensional and nonlinear parameter spaces. In contrast, GSA evaluates the entire input variable space by simultaneously sampling all parameters, making it the preferred approach for sensitivity analysis [21,22]. A common GSA evaluation metric involves analyzing the variance and covariance of a function, where parameter vectors are input into the target model. A higher variance indicates greater sensitivity to a given parameter, whereas greater covariance indicates interactions among variables. Sobol sequence sampling is an effective method for generating parameter vectors X = (X₁, X₂, …, X_k). Compared to Monte Carlo random sampling, it reduces clustering and enhances convergence [23]. Additionally, it requires fewer samples to achieve lower error levels when estimating statistical data from the model output, making it computationally efficient [24].

In GSA, given a model Y = f (X), where Y denotes the model output, the variance of V(Y), can be decomposed as the sum of contributions from individual variables and their interactions:

$$V\left(Y\right)=\sum _{i=1}^k{V}_i+\sum _{i=1}^k\sum _{j=i+1}^k{V}_{ij}\dots +V_{12\dots k}$$

(4)

where V_i is the variance contribution of individual input variable X_i, the interaction term contributing variance from X_i and X_j.

The first-order sensitivity index S_i and total sensitivity index S_Ti are defined as follows [25]:

$$S_i={\displaystyle \frac{V_i}{V\left(Y\right)}}={\displaystyle \frac{V\left[E(Y\left|X_i)\right.\right]}{V\left(Y\right)}}$$

(5)

$$S_{Ti}=1-{\displaystyle \frac{V\left[E(Y\left|~X_i)\right.\right]}{V\left(Y\right)}}$$

(6)

where E(Y|Xi) represents the expected value of Y given X_i, and E(Y|~Xi) represents the expectation of Y excluding X_i. S_i and S_Ti range between 0 and 1, with higher values indicating greater influence of X_i on Y.

The calculation of S_Ti utilizes the Sobol sequence sampling method from the SALib library. This method constructs a multidimensional Sobol sample set and simulates the combination variations of all parameters in sequence to estimate the expected values and variances. Given the relatively low dimensionality of the corresponding target equation (Equation (1)), the primary computational complexity stems from compiling the spectral information. This study determined the convergence of results by increasing or decreasing the sample size based on the sample scale example provided in the SALib documentation [9]. The final sample size was set to 1000, with second-order sensitivity calculations enabled using the default settings to generate the samples. This sample size falls within the recommended lower range for related studies [24,25]. This computational scale is applied to each band within the visible range to obtain spectral sensitivity contributions.

Previous research considers parameters with normalized S_Ti values below 1% as non-sensitive variables [10]. This study follows this standard, viewing parameters with over 1% contribution to S_Ti, sufficiently sensitive to be retrieved from remote sensing reflectance.

3. Optical Shallow Water Model Parameter Compilation

In the previous section, key parameters were selected based on the RTE for OSW regions, with the optical properties exhibiting wavelength-dependent variations. This study focuses on the variation of optical parameters within the visible spectrum; meanwhile, the parameter range directly affects the results of the sensitivity analysis. The parameters of the OSW–RTE are constrained within a finite range in natural environments, with particular emphasis on the spectral range of optical properties. This study compiled the following ranges for OSW–RTE parameters.

3.1. Water Depth

The OSW criteria require that the water depth be less than the water transparency. Maximum detectable water depths for remote sensing are equivalent to the transparency in the clearest natural waters, while for areas of research, the detection limit in oceanic coral reef areas is typically around 30 m [26,27,28]. In waters with lower transparency, depths of about 10 m are detectable [8,29,30,31,32], while rivers with depths under 3 m also offer a spatial scale suitable for remote sensing mapping [33,34,35]. When performing statistical analysis on remote sensing-derived water depth across the full depth range (e.g., 0–15 m or 0–30 m) in oceanic OSW areas, the RMSE is generally on the order of approximately 2 m. Further subdividing the depth range into smaller intervals reveals additional statistical characteristics of RMSE [7,14,36]. The following analysis first considers the error magnitude across the full depth range.

Further statistical analysis of related studies reveals that depth inversion errors vary across different depth ranges. Therefore, this study further divides the OSW range into extremely shallow and moderately shallow waters. Extremely shallow water represents the transition from the shoreline to deeper areas, while moderately shallow water represents the transition from optically deep water to shallower regions. This classification provides a basic binary division of the OSWs. Further exploration of the extremely shallow division and different characteristics is also part of this study.

3.2. Water Optical Parameters

Beyond depth, water transparency is a key determinant of OSW. Transparency, as measured by Secchi disk depth, is optically approximated as the reciprocal of the attenuation coefficient of the most penetrating wavelength in the water [37]. The attenuation coefficient is typically positively correlated with the concentrations of three components of water color, chlorophyll, colored dissolved organic matter (CDOM), and suspended matter.

Additionally, the software Hydrolight version 5.3.0 (Sequoia Scientific, Inc., Bellevue, WA, USA), which employs the Monte Carlo method, can simulate the optical properties of water bodies based on the parameters of these three components of water color [38].

Water optical parameters are typically expressed as apparent optical properties (AOPs) or inherent optical properties (IOPs). IOPs characterize fundamental light transmission processes, whereas AOPs are more practical for instrument-based measurements.

Two key IOPs, the absorption coefficient, and scattering coefficient, can be further decomposed into contributions from different components of water color. Absorption primarily results from pure water and pigments, whereas scattering is predominantly caused by pure water and particulates, as expressed in the following optical coefficients:

$$a_t=a_w+a_{phy}+a_g$$

(7)

$$b=b_f+b_b$$

(8)

$$b_b=b_{bw}+b_{bp}$$

(9)

where a_t is the total absorption coefficient, a_phy is the absorption coefficient due to chlorophyll, and a_g is the CDOM absorption coefficient. The scattering coefficient b includes directional components; b_f represents forward scattering, and b_b represents backscatter, with units of m⁻¹. Figure 2 shows IOP limits for typical clear oceanic waters over many years [39].

Given the spectral attenuation characteristics of the three components of water color, clear waters typically exhibit high reflectance in the blue band, whereas turbid waters display elevated reflectance in the green and yellow bands.

Absorption and scattering can often be transformed into two types of AOPs: the attenuation coefficient K and the deep-water reflectance W. The conversion relationship between AOPs and IOPs is given as follows [5]:

$$K\approx a_t+b_b$$

(10)

$$W=0.084{\displaystyle \frac{b_b}{a_t+b_b}}+0.17{\left({\displaystyle \frac{b_b}{a_t+b_b}}\right)}^2$$

(11)

where K exhibits a positive correlation with both a_t and b_b, where b_b is significantly smaller in magnitude than a_t, making K primarily influenced by a_t. Meanwhile, W is positively correlated with b_b and negatively correlated with a_t.

In solving the RTE, at least two parameters are needed to represent the absorption and scattering processes of radiation. These two parameters can be described using AOP or IOP, such as K and W [19,28,40], a_t and b_b [39], or K and b_b [8]. This study chooses the combination of K and W to calculate sensitivity as AOP is more suitable for field surveys using profiling and spectral instruments.

In water quality studies, waters are often classified as Case 1 or Case 2 waters based on the influence levels of the three key components of water color [41,42]. The optical coefficients of Case 1 waters are mainly influenced by chlorophyll, and most oceanic waters can be considered Case 1 waters. In Case 2 waters, the optical coefficients cannot ignore CDOM and suspended solids, resulting in more complex optical properties, typical of most coastal and inland water areas.

This study utilizes long-term water color remote sensing data from the E.U. Copernicus Marine Service Information to determine the range of three key water color components across several typical water areas [43]. Chlorophyll and suspended solids are typically recorded by concentration, while CDOM is typically characterized in water optical modeling by its absorption coefficient at 440 nm, a_g(440).

Representative regions include OSW areas adjacent to continents, those near large islands, and those in open oceans. The typical OSW region locations and time series information of ocean color products referenced in this study are presented in Appendix A as screenshots from the interactive webpage of the Copernicus Marine Service. Hydrolight was used to perform optical simulations of waters in OSW to determine the range of optical parameters for highly transparent waters during detectable OSW periods based on the non-outlier range of the three components of water color in these areas. Specific values are shown in

Table 1. Values used for the three simulated components of water color.

Component	Unit	Class
Chlorophyll (concentration)	mg/m3	0.1, 0.2, 0.5, 1, 2, 5
CDOM (ag(440 nm))	m−1	0.01, 0.1
Suspended solids (concentration)	g/m3	0.3, 3

, where six chlorophyll levels were selected, and two representative levels for CDOM and suspended solids were chosen. For Case 2 water models, 24 sets of water optical parameters were generated, covering three components of water color. Additionally, optical parameters for Case 1 waters with chlorophyll concentrations of 0.1, 0.2, and 0.5 mg/m³ were added to supplement the scenarios with higher transparency. In total, 27 sets of water optical parameters were generated for both water types. Figure 3 and Figure 4 show the AOP spectra for the selected range of the components of water color, including the attenuation coefficient spectrum and deep-water reflectance spectrum. The fixed parameters used in the Hydrolight simulations are listed in Appendix B.

The relationship between Secchi depth and K_min is used to cover water transparency between 3 m and 30 m in Figure 3, with attenuation coefficients increasing as transparency decreases across the spectrum.

Figure 4 illustrates non-monotonic trends in deep-water reflectance across different wavelength bands as transparency decreases. Blue-green reflectance does not exhibit a monotonic relationship with transparency, whereas long wavelength reflectance increases as transparency decreases. In short wavelengths, the deep-water reflectance spectra present two typical clusters: one with decreasing reflectance across wavelengths and the other showing a peak in the yellow-green band. These two patterns correspond to typical Case 1 and Case 2 waters, where overall reflectance increases in more turbid Case 2 waters. These spectra closely align with findings from relevant water reflectance studies [44,45,46,47].

3.3. Bottom Reflectance

Opaque water bottoms, unlike atmospheric and water columns with certain transparency, exhibit simpler optical processes. Bottom reflectance represents the contribution of the substrate to the total remote sensing reflectance. Since direct measurements are challenging due to water coverage and attenuation [48], organizing a spectral library under controlled laboratory conditions can provide valuable references for related studies. Hydrolight, an optical water simulation software, includes a finite bathymetric model and a spectral library of measured bottom reflectance, as shown in Figure 5. In recording practices, bottom spectra are typically documented as surface reflectance, theoretically ranging from zero to one, and dimensionless. The water reflectance in Figure 4 is typically recorded as remote sensing reflectance, whose spherical integral corresponds to surface reflectance ρ. For Lambertian reflection at the bottom, the relationship between surface reflectance and remote sensing reflectance is as follows:

$$Rrs={\displaystyle \frac{\rho }{\pi }}$$

(12)

Spectral signatures of similar bottom types across different regions exhibit some variation, yet their overall trends remain consistent [49,50,51]. Common patterns in bottom reflectance include an upward trend with increasing wavelength, a minimum reflectance around 650~700 nm, and typically lower reflectance for organic compared to inorganic bottoms. Organic bottom types exhibit spectral characteristics similar to those of terrestrial vegetation.

In terms of satellite remote sensing resolution, a single pixel typically does not cover an area composed of a single bottom type. Unlike qualitative classifications that assign a single bottom type, quantitative bottom reflectance enables the assessment of mixed coverage habitats within a pixel. As the proportion of a specific bottom type within a pixel, its spectral signature more closely resembles that of the pure type [52]. Higher spectral resolution and more remote sensing bands assist in identifying multiple mixed benthic habitats. Therefore, this study incorporates the visible spectral range of common bottom types as continuous values into the sensitivity analysis framework to match the mixed benthic types found on the bottom of actual OSW areas. The water optical simulation results of the above cases can be found in the Supplementary Materials.

3.4. Noise from Above-Surface Processes

The previously discussed environmental parameters influence radiative transfer from below the surface. Above the surface, reflectance signals are influenced by sea surface roughness induced by wind, sun glint, scattering of atmospheric molecules and aerosols, as well as sensor noise. Since terrestrial targets exhibit higher reflectance than deep-water reflectance, uncertainty in remote sensing reflectance has a relatively smaller impact on land-based remote sensing. In oceanic remote sensing, the absolute errors in short wavelength reflectance are typically greater than those at longer wavelengths [53,54,55,56]. Although long wavelength reflectance exhibits lower absolute uncertainty, its inherently low magnitude in water results in a relatively higher error impact. Reflectance products in the red and infrared bands of water frequently show negative values [43]. This is primarily due to the extremely low reflectance of long wavelength bands in water, which makes these signals highly susceptible to noise. Consequently, atmospheric and sun glint corrections are more likely to affect these weak signals. Appendix A presents the reflectance values of common bands in the Copernicus dataset, highlighting the challenges associated with using long wavelength data in clear waters. In OSWs, the combined influence of water columns and bottom reflectance results in values that fall between those of terrestrial surfaces and deep water. As the environment transitions toward deeper waters, the impact of remote sensing reflectance uncertainty becomes increasingly significant.

Remote sensing reflectance uncertainty is strongly influenced by atmospheric factors, which vary regionally and seasonally. Based on the reference, this study treats above-surface processes as secondary influences, combining them into a noise function added to the reflectance signal. The standard deviation of remote sensing reflectance uncertainty varies from approximately 0.001 sr⁻¹ in the blue-violet range, decreasing in wavelength to approximately 0.0001 sr⁻¹ in the red range. The noise function was simply fitted as an exponentially decaying signal with wavelength to match the above uncertainty levels [53,54,55,56]:

$$\mathsf{\Delta }N(\lambda )=0.001·{10}^{{\displaystyle \frac{400-\lambda }{300}}}$$

(13)

This noise function characterizes the wavelength-dependent sensor noise in remote sensing systems. It indicates that the noise level decreases exponentially as the wavelength λ increases. The coefficient 0.001 represents the baseline noise level at 400 nm and decreases to 0.0001 at 700 nm.

4. Results

4.1. Parameter Sensitivity Across Different Parameter Ranges

This section classifies different water quality conditions and depth ranges to derive corresponding GSA results. Water depth and water quality conditions contribute comparably to the formation of OSW. Clearer waters enable detection at deeper depths, whereas turbid waters restrict detection to shallower depths.

Section 3.2 outlined the optical water parameters for three typical OSW environments: near continental coasts, near large islands, and in open ocean regions. Based on the distinctions among these three types, the optical water parameters from Figure 3 are divided into three categories: clear, moderate, and turbid. The clear condition refers to a water body with high transparency (20–30 m) and a relatively small range of environmental variations. The moderate condition is characterized by relatively high transparency but greater environmental variations. The turbid water is defined as a water body where the green band has a more dominant influence than the blue band. The parameter ranges for each environmental condition are set as shown in

Table 2. Parameter ranges for different water environments.

Condition	AOP Range	Bottom Type
Clear	Ranking 1~3 in Figure 3 and Figure 4	From Kelp to Sand in Figure 5
Moderate	Ranking 1~12 in Figure 3 and Figure 4	From Kelp to Sand in Figure 5
Turbid	Ranking 13~27 in Figure 3 and Figure 4	From Kelp to Sand in Figure 5

.

This section categorizes depths into three levels starting from the shoreline: 0~10 m, the transitional range of 10~20 m, and 20~30 m, which approaches optically deep water. Shallow-water environments contain various substrates, ranging from low-reflectance organic substrate to high-reflectance inorganic substrate. Accordingly, Table 2 includes the reflectance range from aquatic vegetation to sand in the GSA.

Figure 6 illustrates the full-order sensitivity index contribution (S_Ti) ratio of each shallow water parameter across the 400~700 nm visible light band within the 0~10 m depth range under different water conditions. In the 0~10 m depth range, water depth and bottom reflectance are the primary contributors to the sensitivity of remote sensing reflectance. Remote sensing reflectance in short wavelengths exhibits higher sensitivity to bottom reflectance. In contrast, longer wavelengths, due to higher attenuation coefficients, exhibit higher sensitivity to variations within this depth range. Depth and bottom reflectance variations contribute approximately 40~70% and 20~30%, respectively, to the overall sensitivity.

The water attenuation coefficient primarily affects short wavelengths, contributing up to 20% to sensitivity, depending on the water quality. Long-wavelength optical attenuation is mainly influenced by pure water, so variations in the components of water color have a minimal impact. An increase in suspended matter concentration enhances the backscattering, resulting in higher reflectance in turbid waters, as described by Equation (11). In clear waters, the attenuation coefficient slightly contributes to total reflectance sensitivity in short wavelengths and is negligible in long wavelengths. In moderate water quality, the attenuation coefficient has a more significant sensitivity contribution in the short wavelength range. In turbid water, short-wavelength deep-water reflectance may contribute up to 10% to sensitivity.

In the 0~10 m OSWs, remote sensing reflectance is generally higher than in optically deep water and is less affected by noise. With increasing depth, remote sensing reflectance decreases, and each parameter’s sensitivity contribution to remote sensing reflectance changes significantly.

Figure 7 illustrates the wavelength-dependent sensitivity of remote sensing reflectance in the 10~20 m and 20~30 m depth ranges for clear and moderately clear waters. At wavelengths above 600 nm, remote sensing reflectance is primarily influenced by deep-water reflectance and noise, with minimal impact from depth, attenuation coefficient, and bottom reflectance in moderate depth. In clear water, long wavelength reflectance is lower than in moderately turbid water, thereby increasing the relative impact of noise. At longer wavelengths, deep-water reflectance and noise contribute approximately 30% and 70% to the sensitivity in clear water, whereas in moderate water, their contributions are about 70% and 30%, respectively. With increasing depth and turbidity, the influence of deep-water reflectance becomes more pronounced, while shallow water features are primarily evident at wavelengths below 600 nm but gradually weaken with depth. Within the 20~30 m depth range, the sensitivity to shallow water feature peaks around 490 nm, with contributions varying based on water transparency—clear waters are primarily influenced by bottom reflectance, whereas turbid waters are more affected by attenuation coefficients.

Turbid waters exhibit minimal shallow-water characteristic peaks between 10~30 m; hence, turbid water parameters are not included in Figure 7 for these depth ranges.

Figure 6 and Figure 7 illustrate depth-specific parameter sensitivity contributions as functions of wavelength. Such results can be used for hyperspectral selection, while the more widely used remote sensing applications are multispectral data with fixed bands. Based on the depth measurement error range mentioned in Section 3.1, assuming that water depth in the clear condition from Table 2 is preliminarily estimated with true depth varying within a 4 m range, Figure 8 shows the sensitivity contributions of each environmental parameter in commonly used satellite remote sensing bands as functions of depth. For a given depth variation, depth sensitivity decreases with increasing depth. ±2 m variation within the 0~5 m shallow water is also considered a substantial change. The sensitivity contribution of bottom reflectance to total reflectance decreases as depth increases, while the influence of deep-water reflectance and noise increases. The peak sensitivity of the attenuation coefficient occurs at intermediate depths.

For the four commonly used remote sensing bands selected, the sensitivity contribution of each parameter exhibits a smooth variation with depth. The sensitivity profiles of 450 nm and 490 nm are comparable. The sensitivity analysis of water depth further indicates that deep-water reflectance increases with depth. However, reflectance at 490 nm exhibits lower sensitivity to noise. In contrast, shallow water features become indistinct at relatively shallow depths in the 560 nm band. In the 0~15 m range, the sensitivity contributions in the 560 nm band are ranked as bottom reflectance, depth, and water attenuation coefficient. The 650 nm band is capable of detecting depth uncertainty and bottom signals within the 0~7 m depth range. At greater depths, red band reflectance is primarily influenced by uncertainties in deep-water reflectance and noise, as discussed in Section 3.4, regarding its instability in water color products. In OSW waters, the red band attenuation coefficient is not significantly higher than pure waters’, so slight changes in the water attenuation coefficient do not yield substantial sensitivity contributions.

Overall, the sensitivity characteristics of OSW remote sensing parameters can be divided into extremely shallow and moderately shallow waters. In extremely shallow depths, variations in depth parameters induce significant changes in reflectance statistics, with long wavelength reflectance being more affected by shallow-water characteristics. In moderately shallow waters, long wavelength reflectance behaves similarly to that in optically deep water.

In the preceding paragraphs, Figure 6 and Figure 7 analyzed a water depth variation range of 10 m. Figure 8 illustrates the sensitivity contributions of various parameters across commonly used bands as water depth increases, based on a 4 m variation range. Figure 9 focuses on the commonly used 490 nm band, which exhibits relatively high penetration in clear water, and illustrates the sensitivity contribution of the depth parameter across variation ranges of 2, 4, 6, and 8 m. The optical parameters align with the clear water conditions listed in Table 2.

Figure 9 indicates that reflectance exhibits greater sensitivity to depth changes in shallower regions. With increasing depth, a greater depth variation range is required to induce noticeable changes in reflectance. In Figure 9, remote sensing reflectance of clear water over the (29 m, 33 m) depth range still contributes more than 1% to sensitivity, indicating that depth inversion generally has a detection accuracy of better than 4 m in common ocean depth ranges. In a 2 m depth range, the (21 m, 23 m) range or deeper may not exceed 1% in sensitivity contribution. Depth detection with ±1 m accuracy is only feasible in shallower regions. Overall, obtaining depth from differences in remote sensing reflectance has certain accuracy limitations.

4.2. Simulation Based on the Band-Ratio Bathymetric Model

To explore the limitations in accuracy and compare the differences between analytical and empirical models, this study generates water depth points and corresponding optical parameters in the visible wavelength range using the Sobol sequence and conducts numerical simulations based on clear water optical conditions.

The simulation process entails inputting optical parameters and water depth into the RTE to derive remote sensing reflectance in two bands. The simulated reflectance of these two bands is then used in the band ratio model to calculate the fitting results.

Figure 10 selects a sandy bottom and the clearest water conditions simulated earlier. Two typical cases are designed, one with a spectral variation of 2.5% considered a uniform case, where water quality and bottom variations are minimal. The other with a spectral variation of 25% considered a typical case, where a certain amount of dark biological bottom is present within the sandy environment, leading to noticeable changes in water color. The chosen band combinations are the commonly used blue and green bands in remote sensing, and the reflectance of the two bands is fitted using the band ratio formula in Equation (3) to derive the predicted water depth from the band ratio bathymetric model. Figure 10 also shows the depth variation range corresponding to 1% STi and the relative error at different depths. The absolute error of the bathymetric model increases with depth, rendering precise depth estimation more challenging as it nears optically deep water.

The sensitivity contribution of different parameters at different depth ranges significantly impacts long wavelength band reflectance sensitivity across depth ranges. This sensitivity affects the optimal band selection for the band ratio bathymetric model. The results in Figure 10 represent a specific band combination selection. To identify optimal band combinations within the visible spectrum, this study simulates various band combinations at 10 nm intervals based on the water optical properties and bottom spectra of the corresponding typical case and evaluates the resulting R². Figure 11 presents the R² values derived from water depth inversion simulations across different depth intervals and band combinations. Results indicate that, with increasing depth, the high-correlation divisor band in the bathymetric model gradually shifts from the red band to the blue-green band. For the detectable depth range of 0~30 m in OSW using remote sensing, the band combinations within 400~560 nm and 490~570 nm show a good fit with depth.

As depth increases, the band combinations with high R² shift from red bands to blue-green bands. This trend aligns with the depth sensitivity patterns across different depth intervals in Section 4.1, where bands with greater sensitivity contributions to depth are more suitable for empirical bathymetric models. Long wavelength bands can be used for model fitting in extremely shallow waters, while in moderately shallow waters, the reflectance of long bands barely changes with depth, rendering the band ratio model less effective in correcting for bottom reflectance and water quality parameter variations.

5. Discussion

This study summarizes the sensitivity contributions of various parameters in OSW remote sensing and compiles the optimal band combinations for water depth inversion within different depth ranges through simulations. These band combinations are systematically selected based on high sensitivity. The simulation results demonstrate a similar regression trend and inversion error magnitude to those observed in actual environments. Figure 12 summarizes the RMSE from recent bathymetric studies with similar depth ranges for statistical analysis [14,27,36].

It also shows the segmented RMSE results for the simulated water depths from Figure 8. In typical cases, the RMSE closely matches values observed in actual waters, whereas in uniform water environments, the simulated RMSE is generally lower than that in actual waters. The uniform case results can be considered an approximation of the lower error bound for water depth inversion. In the same study area, an overall trend of absolute error increases with depth, while extremely shallow waters exhibit higher relative error.

In benthic ecological research, it is possible to classify bottom types in OSW using remote sensing imagery, even without accounting for the water column [34]. This highlights the significant influence of bottom reflectance on remote sensing reflectance. Traditional benthic ecological studies primarily rely on visual analysis to identify major bottom types. However, when analyzing mixed pixel composition ratios or studying benthic health, bottom reflectance serves as a more accurate descriptor than bottom types. Obtaining bottom reflectance presents challenges due to the limited utility of long wavelength information.

In water color research, the attenuation coefficient at 490 nm (Kd490) is one of the most popular water color products and can often be solved through empirical formulas using reflectance at specific wavelengths in optically deep water. According to numerical modeling and sensitivity analysis of water optics, the attenuation coefficient is challenging to obtain effectively in extremely shallow waters. However, bands near 490 nm demonstrate high sensitivity to attenuation coefficient in regions with moderately deep waters, enabling feasible retrieval in specific OSW regions. Deep-water reflectance is correlated with particle backscattering in IOP and thus strongly correlates with suspended solids. Despite significant noise affecting long wavelength reflectance measurements, wavelengths above 600 nm have been successfully used to detect turbidity or particle concentration [57,58,59].

The main difference between sensitivity modeling and practical studies lies in the need for modeling to fully explore the parameter space, whereas large water quality gradients and substrate diversity may not be present in actual OSW environments. Empirical algorithms can treat non-research parameters of low sensitivity as constants, yielding good results in ideal real waters. The three water body conditions in Table 1 serve as case studies rather than general representations, illustrating sensitivity trends with increasing depth range and decreasing attenuation coefficients.

This study did not treat the solar zenith angle as a variable. For Hydrolight simulations, it was fixed at 45°, as both satellite and aerial remote sensing can adjust for this by controlling sampling time. Therefore, the solar zenith angle is excluded as an environmental parameter. Since sunlight is treated as parallel light, the solar altitude angle affects its propagation direction within the water column, effectively increasing the actual optical path length. Consequently, OSW transparency decreases when the solar altitude angle is low. Hydrolight can simulate non-uniformity in water column parameters. However, highly stratified or heterogeneous water conditions were not considered further, as conventional passive optical remote sensing has limited capability for retrieving water quality parameters across depth profiles. In OSW regions, this limitation is mitigated because the sensor primarily detects sunlight that has traveled through the entire water column. However, the optical signal integrated over a two-dimensional pixel does not linearly correspond to the actual distribution of water constituents at different depths.

6. Conclusions

This study summarizes the ranges of environmental parameters involved in OSWs within radiative transfer models and employs numerical simulations to convert water color component concentrations to optical parameters. Using spectral data and water depth profiles, this study evaluates the GSA of key parameters, leading to the following conclusions within the compiled parameter ranges:

The sensitivity of OSW remote sensing reflectance to environmental parameters varies across different parameter ranges. Two distinct sensitivity patterns emerge: extremely shallow waters and moderately shallow waters. In extremely shallow waters, long wavelength reflectance is highly sensitive to depth variations, whereas deep-water reflectance and noise have a minimal effect. In moderately shallow waters, bands above 600 nm are mainly influenced by deep-water reflectance and noise. Reflectance in blue-green bands is primarily sensitive to depth, bottom reflectance, and attenuation coefficient.

With increasing depth, the influence of depth and bottom reflectance on OSW reflectance gradually decreases, while the impact of deep-water reflectance and sensor noise on OSW reflectance gradually increases. The influence of the attenuation coefficient increases initially and then decreases.

Extremely shallow waters are unsuitable for water quality parameter detection. In moderately deep waters, blue wavelengths are effective for quantitative estimates of absorption and attenuation coefficients, whereas long wavelengths are suitable for estimating backscattering.

Simulation of empirical bathymetric models shows that band combinations achieving the best inversion performance are related to the depth range. Depth-sensitive bands produce more accurate inversion results. In addition, using remote sensing reflectance for depth inversion introduces a characteristic error trend—absolute depth inversion error increases with depth, while relative error is higher in extremely shallow waters and relatively stable in moderately shallow waters.

Overall, sensitivity analysis of environmental parameters provides a theoretical basis for assessing the feasibility of retrieving specific parameters using reflectance-based remote sensing within the corresponding environmental parameter ranges.