# Expected Improvements in the Quantitative Remote Sensing of Optically Complex Waters with the Use of an Optically Fast Hyperspectral Spectrometer—A Modeling Study

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. Reflectance at the Water Surface

_{rs}) at the water surface. Ecolight is a fast version of Hydrolight [9,10]. It generates radiometric quantities by solving the azimuthally averaged radiative transfer equation, which is suitable for the purpose of this study as we are interested only in the water-leaving R

_{rs}in the nadir-viewing direction, whereas Hydrolight solves the complete radiative transfer equation and generates radiance distribution as a function of depth below water and polar and azimuthal angles, which is computationally intensive. R

_{rs}spectra were generated for a wide range of concentrations of chl-a, CDOM, and SPM, typically encountered in coastal waters. Only low-to-moderate concentrations of SPM were considered because the atmospheric correction program used in this study assumes zero water-leaving radiance in the near-infrared (NIR) wavelengths used for retrieving aerosol parameters. Fourteen different levels of chl-a concentration, seven levels of CDOM concentration, and five levels of SPM concentration were used (Table 1), leading to 490 combinations of constituent concentrations, for each of which an R

_{rs}spectrum was generated using Ecolight. The absorption coefficient of CDOM at 440 nm (a

_{CDOM}(440)) was used as a measure of the CDOM concentration. The water was assumed to be optically deep to avoid interferences from a reflective bottom, which could affect the retrievals. All other relevant parameters, such as the sediment type, phase function of chl-a and sediment particles, etc., were kept the same for all 490 R

_{rs}spectra generated through Ecolight simulations. The R

_{rs}spectra were generated at the wavelength locations and spectral resolution of HICO, which collects data in the 350–1080 nm range at a 5.7 nm spectral resolution [11]. Only data within the 400–725 nm range were used for retrieving water quality parameters.

**Table 1.**Concentrations of chl-a and SPM, and absorption coefficient of CDOM at 440 nm, for which reflectance spectra were generated using Ecolight.

Chl-a (mg m^{−3}) | SPM (g m^{−3}) | a_{CDOM}(440) (m^{−1}) |
---|---|---|

1, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 60, 70, 80 | 1, 2, 4, 6, 8 | 0.1, 0.5, 1, 1.5, 2, 5, 10 |

#### 2.2. Propagation to Top-of-Atmosphere (TOA) Radiance

_{rs}spectra. The location of the target and the date/time used in the simulation resulted in a solar zenith angle of 37 degrees. Atmospheric parameters, such as the atmospheric model, the types of atmospheric gases, the ozone amount, the column water vapor amount, the relative humidity, the aerosol model, the aerosol optical depth at 550 nm, and the wind speed, were also kept the same (Table 2).

**Table 2.**Geographic, illumination, and atmospheric parameter values used for the upward propagation of the at-surface reflectances.

Parameter | Value |
---|---|

Date | 9 April 2011 |

Time | 15:30:00 (GMT) |

Latitude | 37°30′ |

Longitude | −76°10′ |

Ground Elevation | 0 km |

Sensor Altitude | 400 km |

Sensor Zenith Angle | 0° |

Sensor Azimuth Angle | 0° |

Atmospheric Model | Mid-Latitude Summer |

Atmospheric Gases | H_{2}O, CO_{2}, O_{2}, N_{2}O, CO, CH_{4}, O_{3} |

Ozone Amount | 0.34 atm-cm |

Column Water Vapor | 2.50 cm |

Relative Humidity | 50% |

Aerosol Model | Maritime |

Aerosol Optical Depth at 550 nm | 0.225581 |

Wind Speed | 2 m s^{−1} |

#### 2.3. Addition of Sensor Noise

#### 2.3.1. Considerations for the Sensor Configuration

#### 2.3.2. Sensor Noise Model

- where, λ is the wavelength of the incident radiation (in units of μm)
- h is the Planck’s constant
- c is the velocity of electromagnetic radiation (in units of m s
^{−1}) - ${L}_{\text{\Delta}\lambda}$ is the incoming radiance at the sensor in the waveband Δλ (in units of Wm
^{−2}Sr^{−1}) - D is the diameter of the aperture (in units of m)
- f is the focal length of the imaging system (in units of m)
- p is the spatial width of the detector pixel (in units of m)
- T is the exposure time (in units of s)
- ${\eta}_{\text{sys}}$ is the overall system efficiency, which is given by,
- ${\eta}_{\text{sys}}={\eta}_{\text{op}}\times {\eta}_{\text{QE}}\times {\eta}_{\text{g}}$ , where,
- ${\eta}_{\text{op}}$ is the optical transmittance of the system
- ${\eta}_{\text{QE}}$ is the quantum efficiency of the detector
- ${\eta}_{\text{g}}$ is the grating efficiency, which is given by,
- ${\eta}_{\text{g}}(\lambda )={\eta}_{\text{g}0}{\text{sinc}}^{\text{2}}\left[{f}_{g}\left(1-\frac{{\lambda}_{b}}{\lambda}\right)\right]$ , where,
- ${\eta}_{\text{g}0}$ is the grating efficiency at the blaze wavelength, ${\lambda}_{b}$ ,
- $\mathrm{sin}\text{c}x=\left(\mathrm{sin}\pi x\right)/\pi x$
- ${f}_{g}$ is the fraction of a grating groove that is at the blaze angle.

^{−34}Js) and c (= 3 × 10

^{8}ms

^{−1}) are constants. Nominal values for the relevant instrument-related quantities in Equation (3) were taken from the corresponding values for HICO. Lucke et al. [11] have provided a detailed description of the HICO instrument. For the high-SNR (i.e., F-number = 1.0) system, the quantum efficiency of the commercially available FBX-2K256 CMOS array from Brandywine Photonics (Exton, PA, USA), which has the required electron well depth to remain unsaturated when used in an f/1.0 system over shallow coastal waters, was used in the noise calculation. The quantum efficiency of this detector was slightly higher than that of the HICO detector at wavelengths below 450 nm and above 750 nm and lower than that of the HICO detector at wavelengths between 450 nm and 750 nm.

**Figure 2.**The average SNR calculated from the at-sensor radiances (of all 490 images) for the low- and high-SNR systems. The ratio of the average SNR for the high-SNR system to that for the low-SNR system is plotted on the secondary axis.

#### 2.3.3. Improving Effective SNR through Post-Processing

#### 2.4. Atmospheric Correction of Noisy TOA Radiance Images

_{obs}is the observed radiance at the sensor, L

_{atm+sfc}is the combination of atmospheric path radiance, the radiance specularly reflected from the water surface, and the radiance from whitecaps that is transmitted through the atmosphere to the sensor, L

_{w}is the water-leaving radiance, and t

_{u}is the upward transmittance of L

_{w}to the sensor. Converting the radiance quantities into reflectance quantities and taking into account the absorption and scattering processes in the ocean-atmosphere system along the sun-surface-sensor path, Equation (4) can be rewritten as (see [13] for details),

_{g}is total atmospheric gaseous transmittance in the sun-surface-sensor path. The term $1-s{\rho}_{\text{w}}$ accounts for the loss of photons due to the downward reflection of a part of the water-leaving radiance by the atmosphere back to the water surface. The effects of subsequent multiple reflections of photons between the water surface and the atmosphere are neglected. Rearranging terms,

_{rs}is calculated by division by π) can be determined if the quantities on the right-hand-side (RHS) of Equation (6) are determined through measurements or theoretical modeling. Tafkaa uses pre-computed values (stored in a look-up-table) of the quantities on the RHS of Equation (6), calculated using a slightly modified version of the vector radiative transfer model developed by Ahmad and Fraser [21] for a variety of solar illumination and sensor viewing geometries and five aerosol models (each at five levels of relative humidity and 10 levels of optical thickness). For illumination/viewing geometries and atmospheric conditions not explicitly contained in the look-up-table, Tafkaa interpolates using adjacent values to determine the radiometric quantities.

#### 2.5. Estimation of Water Quality Parameters

_{CDOM}(440) simultaneously and (ii) a semi-analytical NIR-red algorithm to estimate chl-a concentration.

_{CDOM}(440) were estimated from the atmospherically corrected noisy at-surface reflectances by using the Levenberg-Marquardt method [22,23], which is a non-linear least squares curve fitting procedure that estimates parameters by minimizing the squared difference between measured data and modeled data. In this case, the estimated parameters were the concentrations of chl-a and SPM and a

_{CDOM}(440); the atmospherically corrected noisy at-surface reflectances were treated as the measured data; and the modeled data were obtained from fresh runs of the radiative transfer model Ecolight for a given set of parameters. The parameter estimation was done through MPFIT [24], which is an enhancement of the FORTRAN-based software MINPACK [25,26], and is written for use with IDL (Interactive Data Language). For a given noisy spectrum, MPFIT takes in an initial set of parameter values, which were set equal to the corresponding original constituent concentrations (Table 1), and through an iterative process estimates the optimal concentrations of chl-a and SPM and a

_{CDOM}(440) by minimizing the squared difference between the noisy R

_{rs}spectrum and the R

_{rs}spectrum generated by Ecolight for a given set of parameters.

_{CDOM}(440) values were omitted from this subset because the spectral overlap of the absorption features of chl-a and CDOM in the blue spectral region introduces additional retrieval errors in algorithms that use reflectances in the blue region. The omission of very high a

_{CDOM}(440) values was considered acceptable because this study is not a test of the robustness of the algorithm but rather its sensitivity to uncertainties caused by sensor noise, which meant that the uncertainties due to inherent errors in the bio-optical algorithm had to be kept at a minimum so that the uncertainties observed in the retrievals can be confidently attributed largely to the effects of sensor noise.

**Table 3.**Constituent concentrations considered for the analysis of the effect of sensor noise on the water quality parameters retrieved through the non-linear least squares error minimization approach.

S. No. | Chl-a (mg m^{−3}) | a_{CDOM}(440) (m^{−1}) | SPM (g m^{−3}) |
---|---|---|---|

1 | 1 | 0.1 | 1 |

2 | 5 | 0.1 | 2 |

3 | 5 | 0.5 | 1 |

4 | 10 | 0.1 | 4 |

5 | 10 | 1 | 6 |

6 | 15 | 0.5 | 4 |

7 | 20 | 0.1 | 2 |

8 | 20 | 0.5 | 4 |

9 | 25 | 2 | 8 |

10 | 30 | 0.1 | 2 |

11 | 30 | 1 | 6 |

12 | 35 | 1.5 | 8 |

13 | 40 | 1 | 2 |

14 | 45 | 1.5 | 4 |

15 | 50 | 1 | 8 |

16 | 50 | 2 | 4 |

17 | 60 | 1.5 | 8 |

18 | 70 | 1.5 | 4 |

19 | 70 | 0.5 | 6 |

20 | 80 | 2 | 8 |

_{NIR}and R

_{red}are reflectances in the NIR and red regions, respectively. The two-band NIR-red model has been previously shown to yield accurate estimates of chl-a concentration (e.g., [28]) when applied to data from MERIS, which has spectral channels centered at 665 nm (red) and 708 nm (NIR). Because HICO does not have a spectral channel centered at 665 nm, the average of the reflectances at 662 nm and 668 nm were used instead.

_{CDOM}(440) values at seven discrete levels for each combination of chl-a and SPM concentrations. This meant that for a given pair of chl-a and SPM concentrations, there were seven different reflectance spectra, each varying based on the a

_{CDOM}(440) value, resulting in seven different values for the two-band NIR-red algorithm, which made it not possible to derive a single robust regression relationship between the two-band NIR-red ratio and chl-a concentration for the whole dataset. Therefore, the two-band model was parameterized separately (Table 4) at each level of a

_{CDOM}(440) instead of using a single regression model for the entire dataset. This is acceptable because the goal here is not to develop a universally applicable chl-a algorithm but to simply test the sensitivity of the two-band NIR-red model to uncertainties arising from the sensor noise. The regression equations (Table 4) were applied to the original, noiseless R

_{rs}spectra at each a

_{CDOM}(440) level to estimate the noiseless, “true” chl-a concentration according to the two-band NIR-red algorithm. The same equations were also applied to atmospherically corrected, noisy reflectance data, to estimate the “noisy” chl-a concentrations, which were compared with the “true” chl-a concentrations to assess the impact of sensor noise on the estimated chl-a concentration.

**Table 4.**Coefficients of the second order polynomial regression $\text{Chl}-a=\text{A}{x}^{2}+\text{B}x+\text{C}$, where $x={R}_{708}/\text{avg}.({R}_{662},\text{}{R}_{668})$, at various levels of a

_{CDOM}(440).

a_{CDOM}(440) (m^{−1}) | A | B | C |
---|---|---|---|

0.1 | 264.16 | −200.21 | 34.499 |

0.5 | 271.61 | −215.54 | 39.562 |

1 | 279.74 | −232.86 | 45.419 |

1.5 | 286.88 | −248.64 | 50.882 |

2 | 294.82 | −265.68 | 56.956 |

5 | 341.05 | −367.56 | 96.38 |

10 | 429.34 | −567.56 | 186.18 |

#### 2.6. Measures of Uncertainty

^{th}pixel in the j

^{th}noisy atmospherically corrected 1000-pixel image, and ${X}_{original\_j}$ is the parameter value for the original reflectance spectrum used to generate the j

^{th}image. The average percent error for each parameter of interest was calculated as follows:

_{rs}spectrum from which the noisy image was generated.

## 3. Results and Discussion

#### 3.1. Effects of Sensor Noise on Atmospheric Correction

_{550}) and the atmospheric column water vapor amount (C

_{wvap}), in addition to the variations in the atmospherically corrected R

_{rs}spectra. For the atmospherically corrected R

_{rs}spectra, the average percent errors and PNRMSEs (Figure 3) were calculated for each spectral band.

**Figure 3.**The average PNRMSEs of the atmospherically corrected R

_{rs}spectra from the 490 images for the low- and high-SNR systems and the percent improvement (on the secondary axis) in the variations in R

_{rs}when the F-number is changed from 3.5 to 1.0.

_{wvap}decreased from 85.3% to 5.1%. Significant reductions were also obtained in the variations of the retrieved τ

_{550}and reflectances across the whole visible-NIR spectral range (Figure 4 and Figure 5).

**Figure 4.**Average (

**a**) PNRMSE and (

**b**) percent error of C

_{wvap}and τ

_{550}retrieved from all 490 images for the low- and high-SNR systems.

**Figure 5.**Average (

**a**) PNRMSE and (

**b**) percent error of the reflectances retrieved from all 490 images for the low- and high-SNR systems.

_{wvap}, τ

_{550}, and R

_{rs}when reflectances in the NIR spectral bands are spatially averaged for both systems; The results show that the retrieval uncertainty for the high-SNR system (without spatial averaging) is significantly lower than the uncertainty for the low-SNR system with spatial averaging at the NIR spectral bands. It must be noted that spatial variations in the atmosphere within the area covered by the pixels that are spatially averaged have been ignored in this study. Therefore, in a real scenario, especially in coastal regions near urban areas, where the atmosphere often changes on a much smaller spatial scale than over the open ocean, the retrieval uncertainty might be higher for the case of spatial averaging at the NIR spectral bands. Averaging over a higher number of pixels, say, 5 × 5, 7 × 7, etc. will produce different results, with increased smoothing out of noise but at the risk of removing real spatial variations in the data.

**Figure 6.**Improvements in (

**a**) the average PNRMSE and (

**b**) the average percent error of the retrieved C

_{wvap}and τ

_{550}for the low-SNR system with spatial averaging in the NIR spectral bands, the high-SNR system, and the high-SNR system with spatial averaging in the NIR spectral bands over what was obtained for the low-SNR system.

**Figure 7.**Improvements in (

**a**) the average PNRMSE and (

**b**) the average percent error of the retrieved R

_{rs}for the low-SNR system with spatial averaging in the NIR spectral bands, the high-SNR system, and the high-SNR system with spatial averaging in the NIR spectral bands over what was obtained for the low-SNR system.

**Figure 8.**Average percentage of pixels with negative reflectances in the 400–450 nm region for the low- and high-SNR systems with and without spatial averaging in the NIR spectral bands.

#### 3.2. Effects of Sensor Noise on the Estimated Water Quality Parameters:

_{CDOM}(440) were simultaneously estimated from the 20 (Table 3) atmospherically corrected images. In a few rare cases, the numerical method converged to extremely high, unrealistic values for the water quality parameters, due to noisy spikes in the data. Such pixels were considered invalid and omitted from the error calculations.

_{CDOM}(440), and almost one-third for SPM concentration. The error in chl-a concentration for the high-SNR system is within the 30% threshold that is generally accepted as the minimum accuracy required for estimating chl-a concentration in open ocean waters [2]. Spatially averaging the NIR reflectances from the low-SNR system resulted in an increase in the number of valid pixels for water quality parameter retrieval (from 68.2% to 82%); however, the retrieval error in the valid pixels was no better than that for the low-SNR system without spatial averaging for chl-a and SPM concentrations and better by 35% for a

_{CDOM}(440). When the reflectances from the high-SNR system were spatially averaged at the NIR wavelengths, the retrieval error reduced to 7.01% for chl-a concentration, 3.43% for SPM concentration, and 4.88% for a

_{CDOM}(440).

**Figure 9.**The average percent errors for the low- and high-SNR systems in the concentrations of chl-a and SPM and a

_{CDOM}(440) retrieved from the 20 atmospherically corrected noisy images using the non-linear least squares error minimization approach

**Figure 10.**Average percent errors in the chl-a concentrations retrieved using the two-band NIR-red algorithm for the low- and high-SNR systems with and without spatial averaging in the NIR spectral bands.

## 4. Conclusions

_{CDOM}(440) and 65% for SPM concentration; it reduced the uncertainty in the chl-a concentration by the semi-analytical NIR-red algorithm by 92%.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Moses, W.J.; Bowles, J.H.; Corson, M.R.
Expected Improvements in the Quantitative Remote Sensing of Optically Complex Waters with the Use of an Optically Fast Hyperspectral Spectrometer—A Modeling Study. *Sensors* **2015**, *15*, 6152-6173.
https://doi.org/10.3390/s150306152

**AMA Style**

Moses WJ, Bowles JH, Corson MR.
Expected Improvements in the Quantitative Remote Sensing of Optically Complex Waters with the Use of an Optically Fast Hyperspectral Spectrometer—A Modeling Study. *Sensors*. 2015; 15(3):6152-6173.
https://doi.org/10.3390/s150306152

**Chicago/Turabian Style**

Moses, Wesley J., Jeffrey H. Bowles, and Michael R. Corson.
2015. "Expected Improvements in the Quantitative Remote Sensing of Optically Complex Waters with the Use of an Optically Fast Hyperspectral Spectrometer—A Modeling Study" *Sensors* 15, no. 3: 6152-6173.
https://doi.org/10.3390/s150306152