# Testing of the 4SM Method in the Gulf of California Suggests Field Data Are not Needed to Derive Satellite Bathymetry

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## Abstract

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^{2}(0.90), and a low error (RMSE = 1.47 m). 4SM also showed, over the whole time series, the same average accuracy of the control method (40%). Advantages, limitations and operability under complex atmosphere and water column conditions, and high and low-albedo bottom processing capabilities of 4SM are discussed. In conclusion, the findings suggest that 4SM is as accurate as the commonly used Stumpf’s method, the only difference being the independence of 4SM from previous field data, and the potential to deliver bottom spectral characteristics for further modeling. 4SM thus represents a significant advance in coastal remote sensing potential to obtain bathymetry and optical properties of the marine bottom.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Radiative Transfer Equation

^{−1}, the sum of the down-welling and up-welling terms together (2K = Kdown + Kup) and Z is the actual depth of the seabed. The optical depth 2KZ, is dimensionless. Wavelength dependency subscripts of all L and K terms are omitted for brevity. This will finally allow for the linearization of the bottom contrast:

_{i}< K

_{j}and suitable bottom contrast in band j, water column correction is achieved by increasing Z in Equation (2) until the ratio LB

_{i}/LB

_{j}matches the slope of the soil line (see the optical calibration section below), where LB

_{i}is the average of all bands with K

_{i}< K

_{j}and suitable bottom contrast. This process operates ratios among relative numbers, and therefore does not require radiance terms to be converted into units of calibrated reflectance. All radiance terms in the above equations represent the signal as measured by the sensor, and the terms digital numbers (DN), radiance and reflectance are used interchangeably and remain dimensionless in this article. This is the privilege of a “ratio method”: the 4SM approach does not require formal atmospheric correction.

#### 2.3. Image Resources Preparation

#### 2.4. Optical Calibration of the Simplified RTE in 4SM

_{i}/LB

_{j}of the bottom substrate at null depth for all possible pairs of wavebands i and j. These ratios are then assumed to apply to spectrally non-differentiated shallow water pixels.

_{blue}/K

_{green}of the diffuse attenuation coefficients in the blue and green bands. Then this ratio is used to interpolate spectral K for all visible bands, using Table XXVII of Jerlov [12]; this dataset reports the diffuse attenuation coefficients for downwelling irradiance in oceanic and coastal waters worldwide. For two examples of straightforward optical calibration diagrams over apparently homogeneous waters (one oceanic water type, and one coastal water type), please refer to diagrams in Appendix B. For most scenes, the calibration diagrams exhibit a complex hydrologic situation, which we tentatively explain as follows: the clear deep oceanic waters are usually covered by a layer of slightly turbid coastal waters; the thickness of this upper layer varies across the scene, from just a very few meters to in excess of 30 m (see Appendix B). This vertical structure is illustrated in Figure 2 for a Landsat 8 scene acquired on 1 January 2015: X2, X3 and X5 are linearized Blue, Green and Red bands respectively Equation (4).

- 0–5 m depth range: on that day, the BPL pixels in Figure 2a clearly display along a straight line over the 0–5 m depth range for the pair Blue/Green. The ratio K
_{blue}/K_{green}for this surficial layer is estimated at 0.95; this denotes a water type C1 + 0.17 of Jerlov. The BPL pixels in Figure 2b display along two straight lines for the pairs Blue/Red and Green/Red; these two straight lines have virtually the same slope. Diffuse attenuation coefficients in units of m^{−1}are estimated at K_{blue}= 0.272, K_{green}= 0.285, and K_{red}= 0.774. Please note that 0.272/0.285 = 0.95; - 5–10 m depth range: Figure 2a seems to exhibit a progressive change in water quality over the 5–10 m depth range;
- 10–25 m depth range: on that day, the BPL pixels in Figure 2a clearly display along a straight line over the 10–25 m depth range. The ratio K
_{blue}/K_{green}for this deeper layer is estimated at 0.75; this denotes a water type OII + 0.53 of Jerlov. Diffuse attenuation coefficients in units of m^{−1}are estimated at K_{blue}= 0.173, K_{green}= 0.232. - In case of locally increased attenuation coefficient K (phytoplankton), retrieved depth would be under-estimated accordingly, unless a stratified waters model is specified as shown in Figure 2 in the 0–5 m depth range;
- In case of locally increased sediment turbidity (sediment resuspension), water leaving reflectance would be increased accordingly, therefore retrieved depth would be under-estimated, like shown between 4 and 5 km on Profile A in Figure 3.

#### 2.5. Depth Retrieval in 4SM

_{j}be the water column corrected radiance in band j with suitable bottom contrast Ls-Lsw; Let LB

_{i}be the average water column corrected radiance of all available wavebands with K

_{i}< K

_{j}and suitable bottom contrast. The inversion of the model is achieved by increasing the Z term in Equation (2) until the ratio LB

_{i}/LB

_{j}matches that of the soil line. As detailed by Morel and Favoretto [23], for a Landsat-8 or WorldView-2 image, several solutions are available: the NIR solution, the Red solution, the Green solution, and the PAN solution. In practice, the PAN solution is preferred, as it carries several distinct advantages. The outputs are: (i) a raster DTM of the shallow water area in units of meter, and (ii) several rasters of water column corrected wavebands in units of relative radiance, ready for bottom typing (a “low-tide” view of the scene). Some seatruth data can be used for fine tuning the estimation of Z and for a tide correction.

#### 2.6. Combining Depths in 4SM

#### 2.7. Groundtruthed DTM

#### 2.8. Depth Retrieval in ENVI-SPEAR Tool

^{2}results (linear, exponential, polynomial model). To compare different results, a linear fitting (Figure 5a) and the model with the highest r

^{2}were chosen for each scene (e.g., Figure 5b).

#### 2.9. Groundtruthing Regressions and Comparisons

^{2}) and a Root Mean Squared Error (RMSE) were calculated for each linear model: see Figure 6 and Table 1. Tide heights were adjusted manually and added to seatruth depths in order to minimize the RMSE result. Two scenes stand out with unrealistic tide heights (1.6 m and 1.7 m): this is possibly the result of a bad estimation of the soil line for these two scenes. In order to compare both 4SM and ENVI-SPEAR tool results versus the interpolated DTM of groundtruthed depth points, an accuracy index was estimated by calculating Depth_residual = (Depth_retrieved−Depth_measured). So a positive depth residual signals an overestimated depth, and vice versa. Depth residuals were classified in 5 classes of accuracy, and the percentage of pixels belonging to each class was calculated (Figure 7).

## 3. Results

#### 3.1. Seatruth Regressions

^{2}(0.99) although it also reports the highest RMSE (2.11 m). The 22 October 2014 scene scored the lowest RMSE. From Figure 6 it is also possible to see how combining depths improved linear fitting results and reduced the noise.

^{2}is plotted. In particular, for eight ENVI-SPEAR scenes regression plots, the highest correlation coefficient r

^{2}was always achieved with a polynomial model.

#### 3.2. Accuracy Assessment

#### 3.3. Depth Residuals

## 4. Discussion

#### 4.1. San Lorenzo Channel’s Conditions

#### 4.2. Models Comparison

#### 4.3. The 4SM Method

_{blue}< K

_{green}< K

_{red}) like in clear oceanic waters. However, in coastal waters, this is not the case and there is hardly any color separation between K

_{blue}and K

_{green}. This actually is a fundamental limitation which precludes a reliable estimation of the model parameters for Coastal water types in methods like Lyzenga’s and Stumpf’s. The PAN solution offers a valuable alternative, as it will always generate enough color contrast, even with Coastal type waters. In the end, 4SM uses the image metadata to convert spectral water column corrected bottom signatures into calibrated reflectance (0−1), ready for bottom typing and time series studies. The method produces a low tide view of the shallow areas along with a DTM; this is a unique feature of 4SM, apart from semi-analytical methods [10].

## 5. Conclusions

- 4SM is independent on field data to achieve the optical calibration;
- its accuracy is equally valid, and in some case better, compared to the accepted and widely used Stumpf’s algorithm;
- it provides a priori important insight on the optical properties of the water column (spectral K, i.e., water quality), and also on the hydrological conditions;
- it uses all bands with significant bottom detection and delivers computed depths, and water column corrected spectral bottom reflectance;
- it provides all valuable information that allows to explore and monitor large coastal areas in a more efficient and cheaper way in terms of resources and time.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Figure A2.**True color composite of the 27 October 2016 image. This scene exhibits a complex atmosphere and distinct blooms of discolored waters.

**Figure A3.**True color composite of the 27 October 2016 image. This scene exhibits distinct blooms of discolored waters (algal blooms).

## Appendix B

**Figure A4.**Plot of linearized calibration data (Equation (4)) for the whole of the 11 October 2016 scene. Bands 2, 3 and 5 are Blue, Green and Red, respectively. (

**a**) is the calibration diagram of linearized Blue (X

_{2}) and Green (X

_{3}) and reports estimated K

_{2}/K

_{3}values with Jerlov water types. (

**b**) is the calibration diagram for Blue (X

_{2}), Green (X

_{3}) and Red (X

_{5}) bands respectively. The backdrop in shades of gray represents the bi-dimensional histogram for all image pixels (white ROI in Figure 1). Blue dots are the actual pixels used for BPL calibration, each referenced to its row/line position in the image. Small white circles represent the scatter of averaged bare land pixels, as a proxy for the Soil Line.

_{blue}/K

_{green}= 0.47. In Figure A4b, the BPL pixels (blue and green dots) display along two straight lines over a depth range of ~5 m, with slopes K

_{blue}/K

_{red}= 0.103 and K

_{green}/K

_{red}= 0.219. Please note that 0.103/0.219 = 0.47. Using this value to interpolate spectral K from Jerlov’s data yields the following values in units of m

^{−1}: K

_{1}= 0.075, K

_{2}= 0.083, K

_{3}= 0.174 and K

_{5}= 0.651. This case is misleading though, as the optical properties of the waters in this area exhibit much variability, both horizontally and vertically.

**Figure A5.**Plot of linearized calibration data (Equation (4)) for the whole of the 29 January 2016 scene. Bands 2, 3 and 5 are Blue, Green and Red, respectively. (

**a**) is the calibration diagram of linearized Blue (X

_{2}) and Green (X

_{3}) and reports estimated K

_{2}/K

_{3}values with Jerlov water types. (

**b**) is the calibration diagram for Blue (X

_{2}), Green (X

_{3}) and Red (X

_{5}) bands respectively. The backdrop in shades of gray represents the bi-dimensional histogram for all image pixels (white ROI in Figure 1). Blue dots are the actual pixels used for BPL calibration, each referenced to its row/line position in the image. Small white circles represent the scatter of averaged bare land pixels, as a proxy for the Soil Line.

_{blue}/K

_{green}= 0.79. Using this value to interpolate spectral K from Jerlov’s data yields the following values in units of m

^{−1}: K

_{1}= 0.191, K

_{2}= 0.186, K

_{3}= 0.234 and K

_{5}= 0.878. Please note that, if K

_{blue}= K

_{green}, the ratio K

_{blue}/K

_{green}= 1.0 and there is no color separation at all by diffuse attenuation between these two bands: many shallow pixels would display as a function of noise. Therefore, unlike the Pan solution, which achieves good color separation [23], we can expect that the Green solution would yield bad depth results for coastal waters. In Figure A5b, the ratio K

_{blue}/K

_{red}= 0.33 and the ratio K

_{green}/K

_{red}= 0.37. Please note that 0.33/0.37 = 0.89: the upper layer in the 0–5 m depth range might be slightly less clear.

## Appendix C

^{2}.

## Appendix D

**Figure A14.**Underwater picture of SLC relevance of cyanobacteria mats over sandy bottoms (Favoretto et al., unpublished data).

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**Figure 1.**Landsat scene: LC80340432013292LGN00. (

**a**) whole image with study area (orange rectangle), and Espiritu Santo island and the SLC (white rectangle); (

**b**) red square is the area used for parameter calibration, and green rectangle is the SLC.

**Figure 2.**Example of calibration diagram linearized data in logarithmic values for the white ROI of Figure 1. (

**a**) is the calibration diagram of linearized Blue (X

_{2}) and Green (X

_{3}) and reports estimated K

_{2}/K

_{3}values with Jerlov water types. (

**b**) is the calibration diagram for Blue (X

_{2}), Green (X

_{3}) and Red (X

_{5}) bands respectively. The backdrop in shades of gray represents the bi-dimensional histogram for all image pixels (white ROI in Figure 1). Blue dots are the actual pixels used for BPL calibration, each referenced to its row/line position in the image. Small white circles represent the scatter of averaged bare land pixels, as a proxy for the Soil Line.

**Figure 3.**Plot of the depth combination process along four profiles across the SLC. All profiles run from North to South. Thick black profiles are combined depths.

**Figure 4.**(

**a**) is the interpolated DTM of groundtruthed depth points; (

**b**) is 4SM satellite retrieved depth layer from 19 October 2013; (

**c**) is satellite retrieved depth layer processed with ENVI-SPEAR tool on the same scene.

**Figure 5.**Absolute depth calibration plot calculated with ENVI-SPEAR tool with (

**a**) linear model calibration and (

**b**) polynomial curve model calibration.

**Figure 6.**Scatterplots of 4SM retrieved depths vs tide corrected groundtruthed depths for nine scenes, plus combined depth (lower right). Please note the different scale in the combined depths plot.

**Figure 7.**Results of the accuracy assessment on retrieved depth for 4SM and ENVI/SPEAR tool methods. The accuracy index is classified as follow: blue tones for two depth overestimation classes, red tones for two depth underestimation classes, and turquoise tone where the residual is less than 1 meter. Barplots show the percentage of pixels that belong to each of these classes. (

**A**) is a Landsat RGB color composite view of 22 October 2014, white arrow indicates an example of bright shallow bottom, while black arrow represents darker shallow bottoms. (

**A.1**) is 4SM accuracy index; (

**A.2**) is the accuracy index calculated on ENVI-SPEAR with line model calibration; (

**A.3**) is the accuracy index calculated on ENVI-SPEAR with polynomial model calibration. Topright barplot reports % of pixels belongings to each index classes: <−5 m difference class is due to the detection limit constraint of remote sensing in coastal water, since it is represented by pixels in deeper waters. (

**B**) is a Landsat RGB color composite view of 19 October 2013, (

**B.1**) is 4SM accuracy index; (

**B.2**) is the accuracy index calculated on ENVI-SPEAR with line model calibration; (

**B.3**) is the accuracy index calculated on ENVI-SPEAR with polynomial model calibration. Topright barplot reports % of pixels belongings to each index classes calculated on the scenes.

**Figure 8.**Absolute difference between 4SM retrieved depths and interpolated DTM of groundtruthed depths.

**Figure 9.**4SM is capable of calculating bathymetry values without field data and this is the results of satellite derived bathymetry for the whole La Paz bay.

**Table 1.**Landsat 8OLI scenes codes processed in this study, with seatruth regression for 4SM retrieved depths.

Scene File Code | Date | Tide (m) | r^{2} |
---|---|---|---|

LC80340432013292LGN00 | 19 October 2013 | 0.0 | 0.90 |

LC80340432013308LGN00 | 4 November 2013 | 0.6 | 0.91 |

LC80340432014007LGN00 | 7 January 2014 | 0.0 | 0.86 |

LC80340432014039LGN00 | 8 February 2014 | 0.6 | 0.89 |

LC80340432014295LGN00 | 22 October 2014 | 1.6 | 0.91 |

LC80340432016029LGN00 | 29 January 2016 | 1.7 | 0.83 |

LC80340432016061LGN00 | 1 March 2016 | −0.3 | 0.89 |

LC80340432016285LGN00 | 11 October 2016 | −0.2 | 0.95 |

LC80340432016301LGN00 | 27 October 2016 | 0.0 | 0.99 |

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## Share and Cite

**MDPI and ACS Style**

Favoretto, F.; Morel, Y.; Waddington, A.; Lopez-Calderon, J.; Cadena-Roa, M.; Blanco-Jarvio, A.
Testing of the 4SM Method in the Gulf of California Suggests Field Data Are not Needed to Derive Satellite Bathymetry. *Sensors* **2017**, *17*, 2248.
https://doi.org/10.3390/s17102248

**AMA Style**

Favoretto F, Morel Y, Waddington A, Lopez-Calderon J, Cadena-Roa M, Blanco-Jarvio A.
Testing of the 4SM Method in the Gulf of California Suggests Field Data Are not Needed to Derive Satellite Bathymetry. *Sensors*. 2017; 17(10):2248.
https://doi.org/10.3390/s17102248

**Chicago/Turabian Style**

Favoretto, Fabio, Yann Morel, Andrew Waddington, Jorge Lopez-Calderon, Marco Cadena-Roa, and Anidia Blanco-Jarvio.
2017. "Testing of the 4SM Method in the Gulf of California Suggests Field Data Are not Needed to Derive Satellite Bathymetry" *Sensors* 17, no. 10: 2248.
https://doi.org/10.3390/s17102248