# HYDROPT: An Open-Source Framework for Fast Inverse Modelling of Multi- and Hyperspectral Observations from Oceans, Coastal and Inland Waters

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. HydroLight Simulations

#### 2.2. Polynomial Forward Model

#### 2.3. Optimization

#### 2.4. IOCCG Dataset

#### 2.5. Hyperspectral Phytoplankton Size Class Dataset

#### 2.6. Ocean-Color Instruments

## 3. Results

#### 3.1. Forward Model Validation

#### 3.2. Hyperspectral Inversion

#### 3.3. Retrieval of Phytoplankton Size Classes

#### 3.4. Comparison of Multi- and Hyperspectral Retrievals

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RT | Radiative transfer |

HYDROPT | HydroLight Optimization |

IOP | Inherent optical properties |

CDOM | Colored dissolved organic matter |

VSF | Volume scattering function |

FF | Fournier–Forand phase function |

PACE | Plankton, Aerosol, Cloud, ocean Ecosystem |

MERIS | MEdium Resolution Imaging Spectrometer |

CZCS | Coastal Zone Color Scanner |

OLCI | Ocean Land Color Instrument |

SeaWiFS | Sea-viewing Wide Field-of-view Sensor |

${R}_{rs}$ | Observed remote sensing reflectance-either from HydroLight simulations or synthetic dataset |

$\widehat{{R}_{rs}}$ | Predicted remote sensing reflectance by the HYDROPT forward model |

${R}_{rs}^{\prime}$ | Natural log-transformed remote sensing reflectance |

a | Absorption coefficient |

${b}_{b}$ | Backscatter coefficient |

$\tilde{b}$ | Backscatter ratio |

${a}^{*}$, ${b}_{b}^{*}$ | chlorophyll-a- or mass specific absorption and backscatter coefficient |

${\rho}_{i}$ | Concentration of constituent i or absorption by CDOM at reference wavelength |

$\widehat{{\rho}_{i}}$ | Estimated concentration or absorption of optical constituent i |

RMSRE | Root mean squared relative error |

MAE | Mean absolute error |

## Appendix A

**Figure A1.**Spectral absorption for CDOM and detrital matter. (

**a**) CDOM absorption. (

**b**) mass specific detrital absorption. CDOM and detrital matter absorption are shown for a range of realistic spectral slopes (blue lines) [26]. Red dashed lines indicate averaged spectral absorption used for the inversion.

**Figure A2.**Mass specific detrital backscatter. Spectral slopes varies randomly between −0.2 and 2.2 [27]. Red dashed line indicates spectral backscatter with averaged spectral slope (${\gamma}_{2}=1$) used for the inversion.

Parameter | Value | Units | Notes | References |
---|---|---|---|---|

Case-II bio-optical model | ||||

Sea-water | ||||

Absorption | - | m${}^{-1}$ | See references | Pope and Fry [67] for >550 nm Mason et al. [20] <550 nm |

Phase function | - | sr${}^{-1}$ | See reference | Equation 3.30 in Mobley [68] |

Elastic scattering | - | m${}^{-1}$ | See reference | Equation. 3.31 in Mobley [68] |

Inelastic (Raman) scattering | - | m${}^{-1}$ | No inelastic scattering | |

Phytoplankton | ||||

Absorption | - | m${}^{-1}$ | See reference for spectral absorption | Prieur and Sathyendranath [69] |

Phase function | - | sr${}^{-1}$ | Fournier-Forand (1.4% backscatter ratio) | |

Scattering | - | m${}^{-1}$ | Spectral backscatter according to: $b{b}_{phyto}=0.00255\ast {\left[Chl\right]}^{0.471}$ | |

Fluorescence | - | - | No chlorophyll fluorescence | |

Concentration | 0.01–31.62 | mg m${}^{-3}$ | ||

Colored dissolved organic matter | ||||

${a}_{CDOM}\left(440\right)$ | 0.005–1 | m${}^{-1}$ | ||

Absorption | - | m${}^{-1}$ | See reference for spectral absorption | Babin et al. [26] |

Slope | 0.017 | nm${}^{-1}$ | Exponential decay function with reference at 440 nm | Babin et al. [26] |

Non-algal particles | ||||

concentration | 0.01–100 | g m${}^{-3}$ | ||

Absorption | - | m${}^{-1}$ | See reference for spectral absorption | Babin et al. [26] |

Slope (spectral absorption) | 0.0123 | nm${}^{-1}$ | Exponential decay function with reference at 443 nm | Babin et al. [26] |

Phase function | - | sr${}^{-1}$ | Fournier-Forand (1.4% backscatter ratio) | |

Backscatter | - | m${}^{-1}$ | See reference for spectral backscatter | Babin et al. [70] |

Slope (spectral backscatter) | −1 | nm${}^{-1}$ | power-law with reference wavelength at 550 nm | Babin et al. [70] |

Sea-surface boundary model | ||||

Wind speed | 5 | m s${}^{-1}$ | ||

Real index of refraction of water | 1.34 | - | Wavelength indepedent | |

Atmospheric model (RADTRAN-X) | ||||

Solar zenith angle | 0–80 | degrees | 10 degree intervals | |

Cloud cover | 0 | percent | Clear sky | |

Earth-sun distance | - | - | Yearly average | |

24-h averaged wind speed | 5 | m s${}^{-1}$ | ||

Horizontal visibility | 15 | km | ||

Relative humidity | 80 | percent | ||

Precipitable water content | 2.5 | cm | ||

Total ozone | 300 | Dobson units | Yearly average | |

Airmass type | 1 | - | Marine | |

Bottom reflection model | ||||

Depth | - | m | Infinitely deep (no bottom reflection) | |

Output | ||||

Wavebands | 400–710 | nm | 5-nm resolution | |

Radiance (upwelling) | - | W sr${}^{-1}$ m${}^{-2}$ nm${}^{-1}$ | Radiance distribution for all HydroLight quads | |

Irradiance (downwelling) | - | W m${}^{-2}$ nm${}^{-1}$ |

**Table A2.**Definition of retrieval statistics. x are the $lo{g}_{10}$-transformed forward modelled values, y are the estimated $lo{g}_{10}$-transformed values from the inversion. Total number of successful retrievals are indicated by n (with $\delta \le 2$; see Equation (13)) and total number of ${R}_{rs}$ spectra in the dataset by ${n}^{*}$ (=430).

Abbreviation | Definition | Formula |
---|---|---|

MAE | Mean absolute error | ${10}^{\widehat{}}\left(\frac{{\displaystyle \sum _{i=1}^{n}|{y}_{i}-{x}_{i}|}}{{\displaystyle n}}\right)$ |

MB | Mean bias | ${10}^{\widehat{}}\left(\frac{{\displaystyle \sum _{i=1}^{n}{y}_{i}-{x}_{i}}}{{\displaystyle n}}\right)$ |

${R}^{2}$ | Coefficient of determination | $1-\frac{{\displaystyle \sum _{i=1}^{n}{[{y}_{i}-{x}_{i}]}^{2}}}{{\displaystyle \sum _{i=1}^{n}{[{x}_{i}-\overline{x}]}^{2}}}$ |

slope | slope of linear regression model | $\frac{{\displaystyle \sum _{i=1}^{n}[{x}_{i}-\overline{x}][{y}_{i}-\overline{y}]}}{{\displaystyle \sum _{i=1}^{n}{[{x}_{i}-\overline{x}]}^{2}}}$ |

f | fraction of successful retrievals | $n/{n}^{*}$ |

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**Figure 1.**Chlorophyll-a specific absorption for (

**a**) micro- (

**b**) nano- and (

**c**) pico-phytoplankton. A hundred absorption spectra are randomly sampled to visualize variability within and across size classes (blue lines). Mean absorption spectra are used as for the inversion with HYDROPT (red dashed lines). Absorption coefficients and standard errors are obtained from Uitz et al. [11].

**Figure 2.**Chlorophyll-a specific backscatter for (

**a**) micro- (

**b**) nano- and pico-phytoplankton. A hundred backscatter spectra are randomly sampled to visualize variability within and across size classes (blue lines). Mean backscatter spectra are used for the inversion with HYDROPT (red dashed lines). Backscatter coefficients and standard errors are obtained from Table 3-database D in Brewin et al. [12].

**Figure 3.**Cross-validation of the model in Equation (2) for different polynomial degrees (n). The model is fitted on ${R}_{rs}$ data at nadir with the sun at zenith angle of 30°. The accuracy metric used is the root mean squared relative error (RMSRE). The blue line and purple line show the validation and training score respectively. The 68% confidence envelope is shown. The dashed line indicates the most parsimonious model chosen in this study.

**Figure 4.**Distribution of the relative error for 8 (out of 63) wavebands in ${R}_{rs}$ between the HydroLight simulations and the predicted values by the 4th degree polynomial model. Validation results are for a nadir viewing angle and solar zenith angle of 30°.

**Figure 5.**Mean relative prediction error in percent (%) in ${R}_{rs}$ at 440 nm for the 4th degree polynomial model for different viewing geometries. Nadir angle ($\theta $) and azimuthal angle ($\varphi $) follow the default HydroLight quad layout with 10° resolution in $\theta $ and 15° resolution in $\varphi $. The sun is positioned at an azimuthal angle of 0° and zenith angle (${\theta}_{s}$) of 30°.

**Figure 6.**Validation of ${R}_{rs}$ predicted by the polynomial model vs. the IOCCG HydroLight simulations. (

**a**–

**c**) validation results, (

**d**–

**f**) distribution of the relative error (%). Results are for a nadir viewing angle with solar zenith angle (${\theta}_{s}$) of 60°. For statistics refer to Table A2. Number of samples indicated by N. Black dashed line is 1:1 line, red dashed line is the linear model. Data point density is indicated by color (yellow = high, blue = low).

**Figure 7.**Comparison between forward- and inverse-modelled hyperspectral data for (

**a**) ${R}_{rs}$ (

**b**) total absorption excluding water (

**c**) total backscatter excluding water (

**d**) absorption by pico-phytoplankton (

**e**) absorption by nano-phytoplankton (

**f**) absorption by micro-phytoplankton. Red lines/dots indicate results of the inversion and blue lines/dots represent the forward modelled values.

**Figure 8.**HYDROPT retrieval of chlorophyll-a concentrations for (

**a**) pico-, (

**b**) nano- and (

**c**) micro-phytoplankton and the sum of the three size classes (

**d**). For statistics refer to Table A2. The f-score indicates the fraction of reflectance spectra that could be successfully inverted. Data point density is indicated by color (yellow = high, blue = low).

**Figure 9.**IOP retrievals for detrital matter and CDOM. (

**a**) Detrital matter absorption at 440 nm and (

**b**) backscatter at 550 nm. (

**c**) CDOM absorption at 440 nm. For statistics refer to Table A2. The f-score indicates the fraction of reflectance spectra that could be successfully inverted. Data point density is indicated by color (yellow = high, blue = low).

**Figure 10.**Inter-comparison of retrieval statistics for total chlorophyll-a (sum chlorophyll contained in pico-, nano- and micro-phytoplankton) between sensors. Mean bias (MB), mean absolute error (MAE) and slope are projected on a [0–1] interval according to Equation (17). A transformed statistic of 1 indicates a perfect score whereas a score of 0 indicates ≥100% deviation from a perfect score. For calculation of statistics see Table A2. Three sensors are compared: the hyperspectral PACE sensor and multispectral OLCI and SeaWiFs sensors.

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**MDPI and ACS Style**

Holtrop, T.; Van Der Woerd, H.J.
HYDROPT: An Open-Source Framework for Fast Inverse Modelling of Multi- and Hyperspectral Observations from Oceans, Coastal and Inland Waters. *Remote Sens.* **2021**, *13*, 3006.
https://doi.org/10.3390/rs13153006

**AMA Style**

Holtrop T, Van Der Woerd HJ.
HYDROPT: An Open-Source Framework for Fast Inverse Modelling of Multi- and Hyperspectral Observations from Oceans, Coastal and Inland Waters. *Remote Sensing*. 2021; 13(15):3006.
https://doi.org/10.3390/rs13153006

**Chicago/Turabian Style**

Holtrop, Tadzio, and Hendrik Jan Van Der Woerd.
2021. "HYDROPT: An Open-Source Framework for Fast Inverse Modelling of Multi- and Hyperspectral Observations from Oceans, Coastal and Inland Waters" *Remote Sensing* 13, no. 15: 3006.
https://doi.org/10.3390/rs13153006