Bathymetry Retrieval Algorithm Based on Hyperspectral Features of Pure Water Absorption From 570 to 600 nm

Current efforts for improving the hyperspectral optimization processing exemplar (HOPE) model include further testing of remote sensing reflectance ( $R_{\mathrm {rs}}$ ) features containing useful information for bathymetry retrieval via the minimization of the interference stemming from the variability in inherent optical properties and benthic reflectance. In this article, we detected a novel feature originating from the pure water absorption within the narrow spectral region of 570–600 nm. In most coastal regions of clear water in coral reefs, for example, in a coral reefs environment, pure water accounts for the majority of the total absorption in this spectral range. In addition to the depth variation, the spectral behavior of $R_{\mathrm {rs}}$ (570–600) is primarily dominated by a steep increase in pure water absorption with wavelength, whereas the influence of other optical properties, such as phytoplankton/colored dissolved organic matter (CDOM) absorption, particle backscattering, and benthic reflectance, can be simplified using the spectrally constant shape model. An HOPE pure water (HOPE-PW) algorithm using this feature was developed based on $R_{\mathrm {rs}}$ measurements with a spectral resolution of near 3.5 nm, in which only four uncertainties must be resolved. The validation from light detection and ranging (LiDAR) data and comparison with HOPE-bottom reflectance unmixing computation of the environment (BRUCE) using portable remote imaging radiometer (PRISM) data at 15 sites located in five distinct regions of Palau, Guam, Great Barrier Reef, Hawaiian Islands, and Florida Key confirmed that the HOPW-PW algorithm yielded a considerable performance and provided adequate transferability to other sites with varying bottom and water environments. Furthermore, the sensitivity analysis based on Hydrolight-simulated datasets was carried out and showed that HOPE-PW was less affected by variation of bottom types, but still had some limitations in retrieving water optical properties.


I. INTRODUCTION
S HALLOW water bathymetry data are the key to the development of coastal resources, marine navigation, and other fields [1], [2]. Airborne hyperspectral remote sensing has been verified as a promising approach for obtaining high-spatialresolution (0.1-2 m) bathymetry data in coastal regions, especially in extremely shallow waters. To date, several researchers have proposed solutions for water depth retrieval from remote sensing reflectance (R rs ), including empirical techniques and radiative transfer-based methods [1], [3], [4], [5], [6], [7], [8]. The most widely used radiative transfer-based method is the model-driven spectral optimization technique [hyperspectral optimization process exemplar (HOPE)], first proposed by Lee et al. [9], [10]. Briefly, according to its original formulation, the HOPE model is a semianalytical model that requires seven scalar parameters related to water constituent absorption and scattering, considering bottom reflectivity and bottom depth as input. The low computational complexity of semianalytical models typically enables the application of optimization-based inversion to minimize a given distance criterion (referred to as the cost function) between the modeled and observed subsurface reflectance data by iteratively adjusting the input values of the forward model until convergence [7]. Several studies have discussed the performance of the HOPE model, reporting moderately accurate bathymetry retrieval in water depth shallower than 15 m [11], [12].
Over the past two decades, numerous scholars have attempted to improve the performance of the HOPE model. Among the modified algorithms, one of the vital improvements was modification of the bottom reflectance model. In these models, the bottom reflectance is parameterized as a linear combination of different bottom reflectance spectra of key benthic cover classes [5], [7], [12], [13], [14], [15], [16], [17], [18], [19], [20]. For instance, the bottom reflectance unmixing computation of the environment This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ (BRUCE) model [12] considers three benthic cover classes, including sediment, vegetation, and coral, whereas Brando et al. [13] regarded two benthic classes in the semianalytical model for bathymetry, unmixing, and concentration assessment (SAMBUCA), and Petit et al. [7] considered four classes. Thereafter, Garcia et al. [21] proposed a novel lookup table classification approach, namely, the HOPE-LUT method, aiming to select the potential benthic end-members in a given spectral library that achieves the best spectral fit. Although increasing the number of benthic classes yields a more realistic model, the problem of inversion stability in the bottom reflectance model still remains. This is because numerous benthic classes comprise a notable intraclass spectral variability, and the acquisition of prior knowledge on benthic classes may be challenging in specific coastal regions.
In contrast, when it comes to modifying the bottom reflectance model, at least six model parameters need to be optimized. However, the existence of more parameters inevitably generates an additional degree of freedom during the inversion process. Nonetheless, it potentially introduces more local minima into the multiparameter solution space. These minima add more convergence points, which leads the local optimization algorithms to produce inaccurate retrievals. Several optimization algorithms, such as the Levenberg-Marquardt (L-M) [22] and downhill simplex algorithms [23], have been employed in the inversion process [7], [12], [13], [15], [19], [24]. Recently, Petit et al. [7] synthetically compared the performance of various optimization algorithms corresponding to three cost functions. Their results revealed that the accuracy and robustness of bathymetric estimation were significantly influenced by the optimal selection of the inversion setup.
Petit et al. [7] indicated that future research should further examine the features of reflectance spectra (e.g., overall shape, magnitude, and specific peaks) containing useful information of various quantities (bathymetry, seabed abundance, or water optical properties). Thus, a dedicated inversion setup could be designed for bathymetry retrieval by filtering any noninformative component of the subsurface reflectance spectra. An appropriate method is to utilize the increased spectral resolution to detect the subtle spectral features originating from the narrow spectral regions, which potentially present in water optical properties or bottom albedo. This approach facilitates the reduction of the confounding effects occurring between the optically active parameters and parameter number, which simplifies the assumptions related to the underlying physical or mathematical problem and constrains the inversion problems to nonuniqueness issues [25], [26], [27]. Lee and Carder [28] reported that the utilization of 10-nm-wide contiguous channels provides an almost accurate identical water depth measurement compared to that of 5-nm-wide contiguous channels. However, they reported that a higher spectral resolution may facilitate bathymetry retrieval in more challenging coastal and oceanic regions such as a coral reefs environment and seagrass beds. Nonetheless, most existing studies have focused on the application of HOPE models to multispectral data with a reducing number of bands [29], [30], [31], [32], and only a handful of studies have discussed the contribution of higherspectral resolution data in terms of improving the HOPE algorithm.
In this study, we explored a new hyperspectral feature that could benefit from an increased spectral resolution (∼3.5 nm) for bathymetry retrieval. More specifically, we developed a hyperspectral feature that originates from pure water absorption within the narrow spectral region of 570-600 nm. In this spectral range, the spectral behavior of R rs was observed to be dominated by a steep increase in pure water absorption, with negligible influence of colored dissolved organic matter (CDOM). Accordingly, a spectrally constant shape model can be adopted for other optical substances and bottom reflectance. The application of this feature can reduce the parameter number of the HOPE models from more than 7 to only 4 and notably simplify the benthic cover assumptions and the specific inherent optical properties (SIOPs) of the model. Furthermore, we developed a new HOPE algorithm based on the reflectance from 570 to 600 nm, which is denoted as the HOPE pure water (HOPE-PW) algorithm. Moreover, the proposed approach was applied to airborne hyperspectral data retrieved from three sites, and subsequently, its performance was evaluated via a comparison to light detection and ranging (LiDAR) bathymetry data and results from the existing HOPE models.

A. Study Sites and Datasets
The three study sites are located in North Island, Jiajing Island, and Heron Reef, where various coral reef ecosystems exist (see Fig. 1). The first site is situated in the southeastern region of North Island [16.95 • N, 112.31 • E; Fig. 1(a)], which is a component of the Qilian Islets and Cays, belonging to the Xisha Islands, China. In particular, the airborne hyperspectral data were acquired across North Island on October 18, 2018 by the airborne multimodular imaging spectrometer (AMMIS), a newly produced airborne hyperspectral imager designed and manufactured by the Shanghai Institute of Technical Physics, Chinese Academy of Sciences (CAS). The VNIR module can measure 246 bands between 400 and 1000 nm at a 3.5-nm spectral resolution and high signal-to-noise ratio (>500) [33], [34]. We applied an empirical line atmospheric correction approach [35] to the obtained hyperspectral data to derive surface reflectance spectra. The empirical line atmospheric correction approach was primarily based on five standard diffuse boards with Lambertian characteristics and a gradient reflectivity ranging from 5% to 95%, and these boards were placed on the North Island situated in the vicinity of the study area [36], [37]. In addition, a spectral radiometer was equipped atop an aircraft to monitor the downwelling irradiance variation during the survey. Specifically, the Sun and sky glint levels were corrected based on the surface reflectance to generate R rs by attributing the nonzero reflectance at 900 nm to a specular glint, which was considered a uniform additive contribution to the entire spectrum [38], [39]. The LiDAR bathymetry data were acquired on October 12, 2018 using the airborne dual-frequency LiDAR instrument of Mapper 5000, developed by the Shanghai Institute of Optics and Fine Mechanics, CAS. The measured hydrographic vertical and horizontal accuracies of that instrument were 0.23 and 0.26 m, respectively, from 0.25 to 51 m [40]. As these two datasets were acquired on different days, the tidal correction was applied to these two datasets according to published tide tables, which were edited by the National Marine Data and Information Service, Tianjin, China (http://global-tide.nmdis.org.cn/).
The second site [ Fig. 1(b)] is located in the northwestern region of Jiajing Island (18.65 • N, 110.28 • E), which is an uninhabited island in Shimei Bay of Wanning city, Hainan Province. The hyperspectral and LiDAR data were acquired from Jiajing Island on August 8, 2020 using two UAV platforms equipped with imagers and a LiDAR instrument. The airborne hyperspectral data with a spatial resolution of 0.16 m were acquired using an HY-1030 imager, which was designed and manufactured by Hangzhou Hyperspectral Imaging Technology Company Ltd., Hangzhou, China, operating from 400 to 1000 nm with a spectral resolution of 3.5 nm. A standard plaque with a spectrally neutral reflectance of approximately 30% was placed on land near the coastline to derive the surface reflectance spectra. As the flight altitude reached approximately 200 m, the path radiance resulting from the atmospheric scattering by the atmospheric layer between the surface and the imager could be considered negligible. Similar Sun and sky glint corrections were applied on the surface reflectance to generate R rs data. Notably, we identified high-Sun glint and excluded their influence in bathymetry retrieval [41]. In addition, the LiDAR bathymetry data were acquired by the single-photon LiDAR sensor manufactured by the Shanghai Institute of Technical Physics, CAS [42], covering a depth range of 2.0-9.8 m in the second site. Due to the low acquisition period available for the imager and LiDAR sensor (<30 min), the tidal difference was smaller than 0.3 m.
The third study site refers to Florida Keys [25.15 • N, 80.25 • W; Fig. 1(c)], a coral cay located in the southern coast of Florida in the southernmost part of the continental United States. Its hyperspectral R rs imagery was sourced from the NASA website (https://oceancolor.sci.gsfc.nasa.gov/projects/prism-coral/), which has been thoroughly calibrated and atmosphere corrected [21], [43], [44]. On May 28, 2017, we captured this image using the portable remote imaging radiometer (PRISM) instrument at a spatial resolution of 8 m as a part of the Coral Reef Airborne Laboratory (CORAL) project. The PRISM instrument contained 246 spectral bands between 350 and 1046 nm with a full-width at a half-maximum range of 3.27-3.46 nm [21]. The measured depth data were obtained from the NOAA NGS Topobathy LiDAR data downloaded from the NOAA website (https://chs.coast.noaa.gov/htdata/lidar4_z/geoid18/data/9081/). The original LiDAR is sourced from an orthometric vertical datum (North American Vertical Datum, 1988; NAVD88) that uses GEOID18 and converts to tidal datum with the software of "VDatum" (https://coast.noaa.gov/digitalcoast/tools/vdatum.html).
The tidal correction was applied to hyperspectral bathymetry data  Table II. according to the tide tables published by the NOAA, U.S. Department of Commerce (https://tidesandcurrents.noaa.gov/).
To further validate the transferability of the HOPE-PW in other regions different from the three sites, 15 hyperspectral R rs imageries were derived using the instrument of PRISM in five distinct regions of Palau, Guam, Great Barrier Reef, Hawaiian Islands, and Florida Key (Fig. 2) were downloaded from the same NASA website of CORAL Project cited earlier (Table II). In particular, three hyperspectral images were obtained in May 2017 from Palau [ Fig. 2

B. Hydrolight Simulated Data
To assess the performance of the HOPE-PW and HOPE models, the simulated R rs datasets were generated representing environments with a wide range of water and bottom types. We utilized Hydrolight 5.0 to generate nadir-viewing R rs data. Applying the simulation setup proposed by Lee et al. [9] and Lee and Carder [28], a wind speed of 5 m/s was considered in all calculations, and the water body was assumed as homogeneous. For convenience, the above-surface downwelling irradiance was set to 1.0 in all wavelengths using a solar zenith angle of 30 • , and we simulated only in condition of a clear sky.
Generally, the total absorption coefficient of seawater (a) is expressed as the sum of the absorption coefficients of pure water (a w ), phytoplankton (a phy ), and CDOM (a g ). Moreover, a w was obtained from [45], whereas a phy and a g can be parameterized as follows: where Chl denotes the chlorophyll concentration and a * phy represents the nondimensional chlorophyll-specific absorption spectrum reported in [46], which can be user-defined. The total scattering coefficient (b) is expressed as the sum of the scattering coefficients of pure water (b w ) and suspended particles (b p ). b w was retrieved by Zhang et al. [47] and Hu et al. [48], [49], whereas the scattering coefficients of particles adopted in Hydrolight 5.0 can be parameterized as follows:  In addition, the simulations employed the Petzoldtype average-particle-phase function described by Mobley et al. [50]. As shown in Fig. 5, several benthic environments were simulated here, in which the bottom was assumed as a Lambertian reflector. The simulated R rs data included Chl ranging from 0.1 to 5.0 mg/m 3 , a g (440) ranging from 0.01 to 0.2 m −1 , and the water depth varying from 1.0 to 30.0 m. Furthermore, the Gaussian noise was taken into consideration for the random noise stemming from the instrument electronics and environmental variations. These simulations were intended to test the hyperspectral algorithm in various situations, where all derived values could be compared to the exact input values for Hydrolight.
The comparison between the R rs spectra was measured by the airborne imager and was simulated in Hydrolight for the three sites of North Island, Jiajing Island, and Great Barrier Reef, which is shown in Fig. 3. All these three sites we selected contained various bottom types and diverse water optical properties. The bottom types of "avg ooid sand," "brown algae," and "coral sand" were selected for simulation from the three sites. As observed, the measured and simulated spectral data were consistent throughout the entire spectral range. The good agreement for the results of 570-600 nm can be achieved, while the parameters of water optical properties were manually selected. Thus, Hydrolight-simulated R rs provided numerous beneficial datasets to facilitate the evaluation of hyperspectral algorithm performance.

C. Algorithm Performance Assessment
The matchup comparison analysis is based on the linear regression between any two datasets under assessment. The coefficient of correlation (R 2 ) and the root-mean-square error (RMSE) were computed to evaluate the degree of correspondence between the compared datasets. In addition, the average absolute percent difference (APD) and the average relative percent difference (RPD) of total number N of matchups were considered to assess the retrieval uncertainty as follows: and where x i and y i denote the input and retrieval data, respectively, for matchup i of N total matchups.

III. HOPE-PW ALGORITHM A. Principles of Hyperspectral Method
Similar to the earlier discussed various semianalytical inversion algorithms, the HOPE-PW algorithm follows the work outlined in the study conducted by Lee et al. [9] comprising three components: 1) a forward reflectance model; 2) spectral IOP models; and 3) an inverse solution method [51]. The major difference between the HOPE-PW and standard HOPE algorithms is their spectral IOP and bottom reflectance models, whereas their forward model and inverse solution method are similar. All these methods rely on a form of spectral matching: the modeled reflectance spectrum retrieved from the forward model using a set of bathymetry, bottom reflectance model, and water column IOP values, which was optimized to match the measured R rs spectral data.
For the forward model, the subsurface remote sensing reflectance r rs can be expressed as a function of the absorption and backscattering properties of the water column (a, b b ), bottom reflectance (ρ), and water depth (H ) as follows: where θ v and θ w represent the subsurface sensor-viewing zenith and solar zenith angles, respectively. Subsequently, the model of the above-water remote sensing reflectance R mod rs could be approximately expressed by r rs as where ζ is the diversion across the air-water interface and the 1 − r rs term is the internal reflection [9]. In near-nadirviewing applications, ζ = 0.5 and = 1.5, as determined by Lee et al. [10]. In case R rs (λ) is obtained from any spectrometer, (4) can be solved for retrieving the subsurface properties from the measured R rs (λ) data through the parameterization of a(λ), b b (λ), and ρ(λ) with well-defined relationships. In the HOPE model (Table III), seven independent variables (P, G, X , B, S, Y , and H ) represent the properties of the water column and bottom. P, G, X , and B denote the scalar values. P represents the absorption coefficients of phytoplankton, G represents the CDOM, X represents the backscattering coefficient of suspended particles, and B represents the bottom albedo in the reference wavelength. S represents the spectral slope of CDOM, Y denotes the spectral dependence parameter of particle backscattering, and H indicates the bottom depth [9], [28]. As discussed earlier, an essential improvement among the modified HOPE algorithms (such as BRUCE and SAMBUCA) involved the bottom reflectance of more than a single benthic class. Comparing these modified algorithms [5] displays that the BRUCE obtained the highest overall benthic classification accuracy. Therefore, we selected the BRUCE as the reference HOPE algorithm for comparative analysis with HOPE-PW. In the BRUCE model, the bottom reflectance ρ is parameterized by the linear combination of three bottomreflectance spectra representative of three key benthic cover classes, including sediment (clean ooid sand), vegetation (sea grass), and coral (brown algae). The spectra of all these classes are shown in Fig. 5. However, in the HOPE-PW model, the spectral models for a(λ), b b (λ), and ρ(λ) are sufficiently simplified in Table III, and the number of independent variables is reduced to only four (P, X-band H ), which is expressed as follows: Subsequently, the inversion is performed with the L-M retrieval scheme. The basic concept of these methods includes the minimization of the deviation between the modeled and measured R rs spectra by adjusting the model parameters.
To retrieve the solutions of scalar parameters P, X-band H , the L-M scheme minimized the cost function, err, representing the residual between the measured and modeled R rs curves, which is calculated as err = The detailed parameterizations of the HOPE-PW and HOPE algorithms are comparatively presented in Table III, and the initial setups for their nonlinear optimization are presented in Table IV. Comparatively, the HOPE-PW method is simpler and not related to the site-specific datasets and, thus, has the potential for a broader range of environmental applications. A detailed spectral IOP model description of the HOPE-PW algorithm is presented in the following.

B. Simplified Absorption Model From 570 to 600 nm
One of the greatest advantages of the HOPE-PW algorithm is the simple absorption model yielding from the unique feature of a w . As shown in Fig. 4, the complex spectral behavior of a w , a phy , and a g within 400-750 nm produced the modeling of a in a specific coastal region to demand specific absorption measurements of each component. However, from 570 to 600 nm, the prominent spectral feature encompasses a steep increase in only a w [ Fig. 4(a)]. This feature is considered universal in sea surface water worldwide, as a w values within this spectral range are less influenced by the temperature and salinity of sea water [52]. In addition, in most coastal regions suitable for hyperspectral bathymetry retrieval, the water is extremely clear and is generally associated with the type-1 case. Primarily, spectral values of the absorption coefficient are determined by phytoplankton. The absorption spectra of each component in a chlorophyll concentration (Chla) of 2.0 mg/m 3 are shown in Fig. 4(a) based on the conventional case 1 model proposed by Morel and Maritorena [53]. Both a phy and a g impose a major influence on the blue region. From 570 to 600 nm, a phy attains almost minimum values, and a g is an order of magnitude smaller than a w and a phy due to its exponential reduction with increased wavelength. More importantly, their influence is limited and yields only a minor upward offset in a. As shown in Fig. 4(b), at wavelengths greater than 570 nm, a w accounts for the majority of a, and even if Chla and a g at 440 nm [a g (440)] attain 2.0 mg/m 3 and 0.05 m −1 , respectively, the contribution of a w toward a remains greater than 80%. Furthermore, the smallest CDOM contribution was observed in 570-600 nm, which can be neglected in most clear water bodies. Even if a g (440) attains 0.1 m −1 , a g from 570 to 600 nm is less than 0.016 m −1 . Although a g (440) can increase up to ∼1.0 m −1 in certain coastal regions such as the Moreton Bay-Rainbow Channel [5], the maximum a g deviation from 570 to 600 nm amounts to only ∼0.025 m −1 , which is almost an order of magnitude less than that of a w . Thus, the contribution of CDOM can be incorporated into a phy . For instance, a g (440) is typically less than 0.03 m −1 in most coastal areas, including the South China Sea [54]. Therefore, the absorption model in the HOPE-PW algorithm is simplified as follows: a(λ) = a w (λ) + a phy (λ) + a g (λ) ≈ a w (λ) + a phy (λ). (11) Thus, the unknown CDOM-associated parameters G and Y did not require to be solved in the HOPE-PW model. Therefore, the degrees of freedom during the inversion process are reduced.

C. Spectrally Constant Shape Approximation
Another modification in the HOPE-PW model adopted a spectrally constant shape approximation to model the additional optical properties of a phy , b bp , and ρ. Excluding a w , the additional optical properties exhibited a low spectral variation in the narrow range from 570 to 600 nm. As shown in Fig. 5(a), the a phy spectra revealed notable spectral variation within the range from 400 to 750 nm with two peaks near 440 and 675 nm, and large differences between the species due to varying pigment compositions. However, all the normalized a phy spectra from 570 to 600 nm, normalized by their mean values in the 570-600-nm range, displayed almost flat shapes [ Fig. 5(b)]. The variation in each a phy spectrum did not exceed ±10%, and while the spectrum was within 580-590 nm, the variation was even lower, namely, at only ±5%. In terms of the backscattering coefficient of suspended particles (b bp ), the representative particle backscattering spectra retrieved from models [55], [56] and measurements [55], [57] are presented in Fig. 5(c). b bp evaluated from any of the models or measurements exhibited a spectral variation within 400-750 nm, wherein in 570-600 nm, the spectral dependence of b bp became extremely weak, particularly for the phytoplankton spectra [ Fig. 5(d)]. Overall, similar features were observed in benthic reflectance spectra. We acquired a series of representative bottom reflectance spectra from the studies of Dekker et al. [5], Petit et al. [7], and Mobley et al. [58] with notable spectral differences [ Fig. 5(e)]. Based on the normalized ρ spectra from 570 to 600 nm shown in Fig. 5(f), we can clearly see that the variation in sediment-or sand-dominated types was only ±5%. Although a comparatively high variation can be observed in coral-dominated types, the variation remained within ±10%, which can still be considered low.
Accordingly, we considered a phy , b bp , and ρ from 570 to 600 nm as a wavelength-independent parameter and, thereby, applied a single scalar value to simulate these parameters as follows: where the scalar parameters P, X , and B can be considered as the average values of a phy , b bp , and ρ, respectively, within the range of 570-600 nm, as well as b bw is the backscattering coefficients of pure water [47], [48], [49]. Thus, in the HOPE-PW model, the IOP and bottom reflectance models were considerably simplified, which did not necessitate any prior knowledge of specific optical properties derived from field measurements.

A. Characteristics of R rs From 570 to 600 nm
The representative R rs spectra measured along a given depth gradient in the three study regions are plotted in Fig. 6(a)-(c). The common spectral features can be determined among the spectra retrieved at all three sites. Generally, the overall magnitude of R rs in the spectral range of 400-750 nm gradually decreased with the increasing water depth, as the signal contribution of the bottom reduced due to the water attenuation. The variation in magnitude can be more conveniently determined in the range from 400 to 550 nm, and a relatively low reflectance is observed in the spectral range beyond 600 nm. Although multiple peaks or troughs can be observed in the blue-green band, the spectral variation in this range is typically low. Therefore, the spectral shape can be considered smooth. In principle, the most predominant hyperspectral feature exhibits a drastic decline from 570 to 600 nm, which was observed at all sites, regardless of the bottom type and optical properties of water. Primarily, this spectral feature is associated with a dramatic increase in the spectral absorption of pure water. The pure water absorption results in two evident declines in R rs within the spectral ranges from 650 to 670 nm and from 700 to 720 nm, particularly at Heron Reef [ Fig. 6(c)]. Regardless of a slight increase in the water depth, these two declining features could not be prominently observed because of high absorption. Thus, the features are unreliable for bathymetry retrieval.
To further illustrate the influence of the water depth on the spectral shape ranging from 570 to 600 nm, the R rs spectra The first five spectra were redrawn from the Ocean Optics Book (https://www.oceanopticsbook.info) [59]. The following two spectra are retrieved from [57], and the ultimate spectra are acquired from [60]. (c) and (d) Normalized particle backscattering spectra retrieved from the model of Morel [56], the detritus model of Stramski et al. [55], and various algal species obtained from [55] and [57], which are normalized by their mean values ranging from 400 to 700 nm and from 570 to 600 nm. (e) Bottom reflectance spectra of various benthic cover types redrawn from [5], [7], and [50]. (f) Normalized bottom reflectance spectra from 570 to 600 nm. In Fig. 3(b), (d), and (f), the black and gray dotted lines indicate ±10% and ±5% ranges, respectively.
were obtained from North Island [ Fig. 6(a)] covering a large bathymetry range from 0 to 35 m were normalized through R rs at 585 nm, as shown in Fig. 6(d). The reduction in R rs increases with the water depth. According to the model proposed by Lee et al. [10], R rs or r rs is attributed to the water column and bottom. As shown in Fig. 5, the spectral shape of the bottom reflectance is overall flat in this spectral range, and the contribution of the seabed in shallow-depth water dominates the spectral behavior from 570 to 600 nm and yields a gentler decline. In contrast, the reduction steepens with the increasing depth and decreasing reflectance signal from the bottom. These results implied that the reflectivity from 570 to 600 nm varied with the depth and the diminishing slope within this range is closely related to the water depth, indicating that this feature from 570 to 600 nm can be utilized for bathymetry retrieval. More importantly, hyperspectral measurements with higher resolution are indispensable for acquiring this spectral feature in such a narrow range.

B. Validation of HOPE-PW With LiDAR Depth
The HOPE-PW algorithm was applied to hyperspectral imagery retrieved from all three sites on North Island, Jiajing Island, and Florida Key for depth retrieval. To demonstrate the HOPE-PW model performance, the bathymetry output is presented, followed by a comparison of HOPE-BRUCE algorithm. These algorithms function pixel-by-pixel. Therefore, the spatial resolution of the depth results is the same as the image resolution at various study sites. As retrieved by the HOPE-PW model, the maps of phytoplankton absorption and backscattering coefficients are neither presented nor discussed here for the sake of brevity.
Bathymetry maps were computed with the HOPE-PW and HOPE-BRUCE models for the area near North Island. The bathymetric surfaces derived from these two algorithms, including that derived from the LiDAR survey, are visually compared in Fig. 7(a)-(c), which revealed an overall consistency. Nevertheless, the quality of the two hyperspectral bathymetries indicated spatial discrepancies between these two algorithms. The scatter plots of the hyperspectral bathymetry were generated as a function of the LiDAR bathymetry, revealing further details of various behaviors [ Fig. 7(d) and (e)]. In the 0-5-m depth range (Circle A), the HOPE-BRUCE model yielded superimposed point clouds that slightly overestimated the bathymetry. In the moderate depth range from 5 to 20 m (Circle B), the highest linearity of depth prediction was obtained with both the HOPE-BRUCE and HOPE-PW models, where the error dispersion was the lowest. At greater depths, the HOPE-BRUCE bathymetry progressively exhibited asymptotic behavior from ∼20 m. Especially in the deep sandy area at the bottom-left corner of Fig. 7(b), the HOPE-BRUCE bathymetry was visibly underestimated, and therefore, more points were observed below the 1:1 line, as shown in Fig. 7(d). Comparably, the HOPE-PW algorithm yielded an accurate result, as its depth estimation was similar to the 1:1 line, and a high coefficient of determination (R 2 = 0.92) and low RMSE (= 2.3 m) was attained. At greater depths (>20 m) (Circle C), a relatively high error dispersion was detected in the HOPE-PW model results [ Fig. 7(e)]. This is probably attributed to the relatively low R rs values from 570 to 600 nm, which may largely be influenced by the measurement noise stemming from the instrument. The error dispersion could be reduced by incorporating the spatial correlation between the neighboring pixels, as proposed by Jay and Guillaume [18]. Another assessment conducted in the area of Jiajing Island indicated that the two algorithms provided identical accurate results, despite the spatial variations in the accuracy of bathymetry retrieval between these algorithms (Fig. 8). Due to the limited LiDAR bathymetry data, the depth retrieval approach was assessed from the hyperspectral imagery only for shallow depths near 0.0 and 7 m [ Fig. 8(a)]. Both depths yielded suitable R 2 values greater than 0.7, RMSE values less than 0.9 m, and APD values approximating to 21% [ Fig. 8(d) and (e)]. In the RGB image, broadly two bottom types could be visually observed, i.e., bright sand and dark algae. In both algorithms, the bottom depth tended to be underestimated in certain dark seabed pixels. This could be linked to the spectral variation in the bottom reflectance if a high bottom signal contribution occurs in shallow waters, but this underestimation remained within an acceptable range. In addition, the depth estimation remained dispersed around the 1:1 line. These discrepancies may originate from the extremely high spatial resolution (0.16 m) caused by the low flight height during the measurements, where minor surface roughness or surface slope differences can exert a considerable impact on R rs .
At Florida Keys, another hyperspectral dataset measured by the PRISM is reported by the COREL project, including the LiDAR bathymetry data available from NOAA. As shown in Fig. 9(a)-(c), the three bathymetric maps were consistent in terms of their spatial patterns. A quantitative analysis of the statistics of the matchup of the hyperspectral retrievals and LiDAR depths is presented in Fig. 9(d) and (e). Both HOPE-PW and HOPE-BRUCE algorithms yielded high R 2 values of 0.97 and 0.98, respectively, and their RMSE was less than 0.8 m. Based on these comparisons, the transferability of the HOPE-PW method onto the hyperspectral sensors can be confirmed with spectral resolution wider than 3.5 nm.
More notably, an assessment of the processing period between the HOPE-PW and HOPE-BRUCE models was performed for the AMMIS swath shown in Fig. 1(a). Based on a single-core computer, the above image was processed by the HOPE-PW model simulated with the C programming language, containing 201 210 optically shallow water pixels, in 12 006 versus 50 096 s required by the HOPE model. This improvement in computational speed is attributed to fewer independent variables that must be retrieved and fewer bands used in the HOPE-PW model.

C. Comparison Between HOPE-PW and HOPE-BRUCE
Insightfully, to further explore the transferability of the HOPE-PW method to other sites with distinct bottom and water environments, 15 hyperspectral imageries derived using the PRISM instrument were selected from five distinguished regions-Palau/Guam/Great Barrier Reef/Hawaiian Islands/Florida Keys-all of which have <5% cloud coverage. According to the in situ optical property data acquisition that occurred between June 2016 and May 2017 during six discrete field campaigns, Russell et al. [61] indicated that benthic environments across these sites exhibited various types, including varying mixtures of live and dead coral, coralline rubble and pavement, multiple types of benthic micro-and macro-algae, seagrasses, and carbonate and basaltic sediments. Furthermore, large variability in the measured optical properties was observed across geomorphic zones, including fore, back, fringing, and patch reef zones, including lagoons, river mouths, and near-shore coastal areas. Although the validation of depths retrieved by HOPE-PW at each site presented a challenge, these images were representative and deemed sufficiently diverse to conduct statistical comparisons between the HOPE-PW and HOPE-BRUCE methods due to the difficulty in obtaining LiDAR data. As observed, the HOPE-PW and HOPE-BRUCE retrievals were consistent in all sites. For brevity, the bathymetry maps of four sites retrieved from the two methods are shown in Fig. 10(a)-(h), which were selected from Palau, Great Barrier Reef, Hawaiian Islands, and Florida Keys. Evidently, the two methods yielded almost the same spatial patterns. Scatter plots and multivariant statistics for HOPE-PW and HOPE-BRUCE bathymetry [ Fig. 10(i)-(l)] were presented as well. Notably, the results for the depth beyond 15 m were not provided because the depth retrievals from both these methods exhibited large uncertainties that could be attributed to the decreasing bottom reflection contribution associated with depth. As observed in Fig. 10(i)-(l), all the points distributed around the 1:1 line indicated a high linear correlation between the two methods. High R 2 was greater than 0.9 at all four sites, whereas the RMSE was relatively low and ranged within 0.9-1.5 m. However, more dispersion was observed beyond 10 m depth, particularly in Palau [ Fig. 10(i)], and thus, the two algorithms were compared in the depth range of 0-10 and 0-15 m. The comprehensive comparison results for the 15 sites are presented in Table V. In the depth range below 10 m, the R 2 values were high for all sites, ranging from 0.89 to 0.98, whereas the RMSE was below 1.0 m at most sites with a maximum of only 1.2 m. The total R 2 and RMSE of all 15 sites were 1.0 and 0.8 m, respectively. However, the total R 2 is reduced to 0.9 and RMSE increased to 1.3 m when the data were included for depths greater than 10 m. The larger dispersion observed at depths greater than 10 m was potentially caused by most coastal regions suitable for hyperspectral bathymetry. R rs , at 570-600 nm is generally less than that at blue-green bands (Fig. 3). Similarly, the bottom reflectance contribution at 570-600 nm decreased more rapidly with depth compared to that at blue bands, which is caused by the high absorption coefficients at 570-600 nm due to the high pure water absorption [ Fig. 4(a)]. Thus, the HOPE-PW performance at greater depth regions was more likely to be influenced by noise from both the instrument and the environments. Comparatively, according to the cost function in Table III, the HOPE-BRUCE relies more on R rs at blue-green bands and, therefore, may yield improved results in regions of greater depth. According to the RPD results of linear fitting in Table V, the HOPE-PW estimated depths were slightly smaller than the data done by HOPE-BRUCE in most sites. Such underestimation in HOPE-PW may be attributed to the variations in the scattering phase functions between 550 and 570-600 nm, which should form the scope of future study. Generally, we conclude that HOPE-PW provides comparable results to both HOPE-BRUCE and LiDAR.

A. Effect of Bottom Reflectance Model
The bottom type considerably impacts the bathymetric retrieval results of HOPE models. For instance, in certain areas of the Hawaiian Islands where the turf algae is distributed at the bottom [ Fig. 2(l)], both the original HOPE model [9], [10] and the HOPE-BRUCE model with the three original types of sand, sea grass, and brown algae [12] yield unpredicted high depth retrievals in the relatively shallow region  [ Fig. 11(a) and (b)], where the influence of bottom reflectance is more significant. The reflectance spectra of bottom types used in the two models are presented in Fig. 11(e) and (f). As shown in Fig. 11(c), when we only used the special bottom type of "turf algae" instead of "brown algae" in HOPE-BLUCE, the water depth could more accurately be retrieved. More importantly, HOPE-PW can obtain satisfactory results without considering the actual situation of the bottom type [ Fig. 11(d)].
The impact of benthic-type variability on the effectiveness of depth retrieval was further assessed based on the simulated datasets. In total, 15 distinct types of bottom reflectance spectra [ Fig. 5(e)] were chosen for the Hydrolight simulation, which are categorized into five classes: sand, algae, coral, sea  grass, and other types, to facilitate the comparison of various HOPE models. In addition, the simulations were conducted in a wide range of IOPs as Chl ranging from 0.1 to 2.0 mg/m 3 and a g (440) ranging from 0.01 to 0.2 m −1 . As shown in Fig. 12, the original HOPE model considering only a single bottom type (sand or coral) yielded the least R 2 value of 0.89 and the highest APD of 18.5%. Overall, the HOPE-BRUCE provided an improved result with an R 2 value of 0.90, an RMSE of 2.6 m, and an APD of 10.7%. Comparatively, the HOPE-PW model retrievals [ Fig. 12(c)] achieved a similar adequate performance with its R 2 value reaching up to 0.96 and RMSE and APD to only 1.6 m and 9.1%, respectively. Generally, the assumption of the bottom reflectance model with an unchanging spectrum of 570-600 nm is sufficient for the depth retrieval of HOPE-PW model in regions with various bottom types, whereas comparatively, current HOPE models are more sensitive toward the variations in seabed types, especially in shallow water.

B. Effect of CDOM Absorption
Another advantage of the HOPE-PW model is that it does not need to solve the CDOM-associated unknowns S and G. The depth retrievals from HOPE-BRUCE models with varying inversion setup of CDOM in North Island are presented in Fig. 13, implying that when the variation of S is neglected, i.e., being set as a fixed value of 0.015 nm −1 , the HOPE-BRUCE depth retrievals are significantly underestimated with increasing depths, and the underestimation will be more serious when CDOM is not considered in the inversion procedure (G = 0.0 m −1 ). Similar results were obtained in the simulation datasets in which the input of a g (440) ranging from 0.01 to 1.0 m −1 [ Fig. 14(a)-(c)]. These underestimations probably associated with the significant influence of CDOM on R rs of blue bands. In addition, compared to the depth retrievals corresponding to an input range of a g (440) from 0.01 to 0.2 m −1 shown in Fig. 12, the increase of CDOM concentration reduced the performance of HOPE-BRUCE [ Fig. 14(a)] as well as that of HOPE-PW [ Fig. 14(d)], especially in the depth range below 10 m. This is because a g (440) increasing above 0.2 m −1 will contribute to CDOM on the absorption coefficients at 570-600 nm, which cannot be neglected and the spectrally constant shape approximation was not appropriate for the specific absorption model. Despite that, such a high a g (440) (>0.2 m −1 ) seldom occurs in the region suitable for hyperspectral bathymetry [61]. Therefore, HOPE-PW is a more simplified model than the current HOPE.

C. Limits in Water Optical Property Retrieval
For additional unknowns (e.g., a phy , b bp , and ρ), a comparison between HOPE-PW and HOPE-BRUCE would provide more details on the model performances, which consequently reveals the limits of the HOPE-PW. Both in terms of water    Fig. 15, which is representative of the present study areas. Upon comparing the hyperspectral depth retrievals in Fig. 10(d) and (h) with LiDAR results in Fig. 15(a), the depth retrieval of both HOPE-PW and HOPE-BRUCE is considered extremely successful in this area because R 2 and RMSE for HOPE-PW were 1.0 and 0.8 m, whereas those for HOPE-BRUCE were 0.98 and 0.5 m, respectively. According to the RGB image [ Fig. 15(b)], two broad seabed types of sand and coral are presented. In particular, ρ(550) derived from HOPE-BRUCE and mean bottom reflectance derived from HOPE-PW [ρ(570-600)] exhibited accurate spatial patterns of sand that could be identified from coral because the detected sand generally achieves a high reflectance, while the reflectance of coral is relatively low. In addition, ρ(550) is generally less than ρ(570-600) for coral bottom, which is reasonable as most coral reflectance spectra exhibited an increase from 500 to 600 nm [ Fig. 15(c) and (d)].
Nevertheless, the inversion of water optical properties shows certain limitations. Although a phy (440) from HOPE-BRUCE is much higher than a phy (570-600) from HOPE-PW, the results in Fig. 15(e) and (f) illustrate that a phy retrievals from both the algorithms were higher on darker seabed (i.e., coral) in terms of the benthic cover. On the contrary, these retrievals were lower on brighter seabed (sand), which can be partially explained by the impact of the reef benthic community on water column a phy because the coral reef produces and resuspends fine carbonate sediments [62], [63]. However, the interference of benthic reflectance variability on the inversion of water optical properties cannot be completely eliminated in both HOPE-PW and HOPE-BRUCE algorithms. For b bp retrievals [ Fig. 15(g) and (h)], HOPE-PW was more sensitive toward the benthic reflectance variability than HOPE-BRUCE. For HOPE-PW, the spatial pattern of b bp was extremely similar to that of a phy . More notably, HOPE-PW and HOPE-BRUCE yield opposite results in certain areas [as the rectangle regions of Fig. 15(g) and (h)], where HOPE-BRUCE provides an extremely high value of b bp . Nevertheless, reflectance from HOPE-PW is unexpectedly low and even proximate to zero. Similarly, the overestimation of HOPE-BRUCE and the underestimation of HOPE-PW often appear in the b bp retrievals of simulated datasets (Fig. 16). The water optical properties, especially b bp , derived for HOPE-PW, still pose significant uncertainties, which should be further validated using in situ data.

VI. CONCLUSION
This study proposed a novel HOPE-PW algorithm that utilized only R rs from 570 to 600 nm to specifically retrieve the bottom depth, which could minimize the interference stemming from the variability in SIOPs and benthic spectral reflectance. This algorithm was developed based on the finding that the rapid reduction in R rs in the narrow spectral region from 570 to 600 nm can be observed in almost all R rs spectra in coastal waters, and this reduction is primarily governed by a steep increase in a w . In this spectral range, the CDOM absorption was extremely low, yet its influence can be neglected in most clear coastal water bodies. All the other optical models of a phy , b bp , and bottom reflectance were effectively simplified to derive spectrally constant shape models. Thus, the feasibility of applying the HOPE-PW method was emphasized with less than four independent variables that required resolution, including the simpler and more versatile IOP models adopted from 570 to 600 nm.
The bathymetry retrieved by applying the HOPE-PW method to obtain airborne hyperspectral data at the sites of North Island, Jiajing Island, and Florida Keys was evaluated with LiDAR bathymetry measurements. The results signified suitable agreements at all three sites, especially on North Island. Based on the excellent signal-to-noise ratio of AMMIS hyperspectral radiometric measurements, a maximum retrieved depth of 35 m could be attained. The comparison of results with HOPE-BRUCE using PRISM data at 15 sites located in five diversifying regions-Palau, Guam, Great Barrier Reef, Hawaiian Islands, and Florida Key-confirmed that the HOPW-PW algorithm yielded a considerable performance and provided adequate transferability to other sites with varying bottom and water environments. In addition, we achieved a considerable improvement in computational speed, and the HOPE-PW processing speed was almost four times greater than that of the HOPE-BRUCE model.
The field measurements indicated that HOPE-PW performance was more potentially influenced by the noise both from the instrument and environments of deeper depth regions. In addition, the detailed sensitivity analysis based on Hydrolight-simulated datasets within a wide range of IOPs and a series of distinct benthic types was conducted for the HOPE-PW and HOPE-BRUCE models. The results suggested that the HOPE-PW model was much less influenced by the spectral variability in the bottom reflectance compared to HOPE-BRUCE, and the presence of CDOM for a g (440) was less than 0.2 m −1 . Nevertheless, both two models still have certain limitations in the inversion of water optical properties. Similar to HOPE-BRUCE, a phy and b bp retrievals from HOPE-PW are sensitive to the type of benthic cover to a certain degree, which was higher on the darker seabed and, conversely, lower on brighter seabed. In addition, HOPE-PW yielded unexpected underestimation of b bp at certain levels that should be further validated with in situ data. Despite all these factors, the prominent spectral feature of the steep increases from 570 to 600 nm in a w that was insensitive toward the variations in the temperature and salinity of sea water, which can be considered universal in surface sea water worldwide. Moreover, the proposed parameterization scheme of the HOPE-PW model is neither image-specific nor sitespecific and provides a more convenient and feasible approach for estimating the depth from hyperspectral images in most coastal waters without prior knowledge.
In future, more in situ R rs spectra considering a wider range of optical water types, especially with high concentrations of CDOM or detritus, should be included for assessing the uncertainties and maximum depth of the HOPE-PW algorithm. In addition, considering the 3.5-nm spectral resolution is quite high and may limit the potential application ability of the proposed model, the influence of spectral resolution on the performance of HOPE-PW should be studied. Despite the drawbacks, the application of the HOPE-PW algorithm in a narrower spectral range from 580 to 590 nm may obtain a higher inversion accuracy with the increasing spectral resolution due to the much lower spectral variation observed in a phy , b bp , and bottom reflectance.