# Spectral and Radiometric Measurement Requirements for Inland, Coastal and Reef Waters

^{1}

^{2}

^{*}

## Abstract

**:**

_{rs}) spectra of optically deep water for twelve inland water scenarios representing typical and extreme concentration ranges of phytoplankton, colored dissolved organic matter and non-algal particles. For optically shallow waters, R

_{rs}changes induced by variable water depth are simulated for fourteen bottom substrate types, from lakes to coastal waters and coral reefs. The required radiometric sensitivity is derived for the conditions that the spectral shape of R

_{rs}should be resolvable with a quantization of 100 levels and that measurable reflection differences at at least one wavelength must occur at concentration changes in water constituents of 10% and depth differences of 20 cm. These simulations are also used to derive the optimal spectral resolution and the most sensitive wavelengths. Finally, the R

_{rs}spectra and their changes are converted to radiances and radiance differences in order to derive sensor (noise-equivalent radiance) and measurement requirements (signal-to-noise ratio) at the water surface and at the top of the atmosphere for a range of solar zenith angles.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Scenarios

#### 2.1.1. Optically Deep Water

^{−1}for humic acid dominated waters and 0.020 nm

^{−1}when fulvic acids prevail, with a value of 0.014 nm

^{−1}being representative of a great variety of water types [14,15].

#### 2.1.2. Optically Shallow Water

^{−3}, X = 1 g m

^{−3}, Y = 0.2 m

^{−1}) are used to specify the water layer. These concentrations are close to the lower end for the standard scenarios; hence, the water is clear for inland waters compared to most other optically deep water scenarios. For coastal waters, these low concentration values are more representative of an average coastal water.

#### 2.2. Model

^{−3}of chlorophyll-a, $X$ is the total suspended matter concentration in units of g m

^{−3}and $Y$ is the CDOM absorption at 440 nm in units of m

^{−1}. While these wavelength-independent parameters are used to model the concentrations of the water constituents, their optical properties are simulated using the wavelength-dependent SIOPs shown in Figure 2. The specific absorption coefficients of phytoplankton (${a}_{phy}^{*}\left(\lambda \right)$) and non-algal particles (${a}_{NAP}^{*}\left(\lambda \right)$) and the specific backscattering coefficient of phytoplankton (${b}_{b,phy}^{*}\left(\lambda \right)$) are taken from measurements, while CDOM absorption and NAP backscattering are approximated using analytical equations. The parameters of these empirical equations are S, the spectral slope of CDOM absorption in units of nm

^{−1}, $n$, the Angström exponent of NAP backscattering and ${b}_{b,NAP}^{*}\left(555\right)$, the specific backscattering coefficient of NAP at 555 nm in units of m

^{2}g

^{−1}.

^{−1}), ${K}_{d}$ the diffuse attenuation coefficient of downwelling irradiance, ${k}_{uW}$ the attenuation coefficient for upwelling radiance originating from the water layer, ${k}_{uB}$ the attenuation coefficient for upwelling radiance from the bottom, ${R}_{rs}^{b}$ the bottom substrate albedo (irradiance reflectance) and ${A}_{rs,1}$ and ${A}_{rs,2}$ are empirical coefficients close to one. For the equations of the attenuation coefficients ${K}_{d}\left(\lambda \right)$, ${k}_{uW}\left(\lambda \right)$ and ${k}_{uB}\left(\lambda \right)$, see [23,24].

- ${a}_{phy}^{*}\left(\lambda \right)$ is the specific absorption coefficient of green algae from the database of the software WASI [27]. It is based on an absorption measurement of the green algae Mougeotia sp., grown as pure culture in the laboratory [28], which was later fitted for extension to the near infrared and rescaled to 0.023 m
^{2}mg^{−1}at 674 nm to match field measurements from two German lakes [29]. - ${b}_{b,phy}^{*}\left(\lambda \right)$ is the specific backscattering coefficient from normal clear water in Lake Garda, dominated by green algae (provided by C. Giardino, personal communication).
- ${b}_{b,NAP}^{*}$ (555) = 0.011 m
^{2}g^{−1}and $n$ = 0.75 were calculated by averaging measurements from lakes in Italy, Estonia, the Netherlands and Finland using Table 3 of [32].

^{−3}, $X$ = 1 g m

^{−3}, $Y$ = 0.2 m

^{−1}), the effect on the results of not choosing an additional set of saltwater SIOPs is considered to be minimal.

#### 2.3. Determination of the Optimal Spectral Resolution

^{−6}sr

^{−1}nm

^{−1}. Equation (7) defines the optimal spectral resolution for a measurement with a noise-equivalent reflectance of $\Delta {R}_{rs}$. Equation (8) is used for $\Delta {R}_{rs}$ because it addresses the resolution of the spectral shape of a measurement.

#### 2.4. Determination of the Optimal Radiometric Sensitivity

#### 2.4.1. Rrs. Quantization

#### 2.4.2. Absolute Radiometric Resolution

^{−3}, 10

^{−4}, 10

^{−5}and 10

^{−6}sr

^{−1}.

#### 2.4.3. Signal-to-Noise Ratio

## 3. Results

#### 3.1. Spectral Resolution

^{−1}for all scenarios, hence S was iterated in steps of 0.002 nm

^{−1}. The simulations for shallow water were made for the 14 bottom substrates shown in Figure 1 and altering water depth in 50 steps from 0.01 to 10 m, i.e., $\Delta {R}_{rs}$ in Equation (7) represents the induced changes in ${R}_{rs}$ for depth differences of 20 cm.

^{−6}sr

^{−1}from colors to gray values. This allows us to distinguish between spectral regions in which the induced reflectance changes might be measurable, at least using sensitive field instruments with long integration times (colors), and regions where such differences are below sensor detection limits (gray values).

#### 3.1.1. Optically Deep Water

^{−6}sr

^{−1}. The different colors represent the associated spectral changes, indicating the lowest useful spectral bandwidths according to the color coding in Figure 3. Spectrally finer resolved measurements cannot capture additional information. Similarly, the areas in gray tones represent wavelengths and concentrations where changes in C, X, Y and S above 10% are required in order to induce detectable ${R}_{rs}$ differences, if at all, and the gray values represent the lowest useful spectral bandwidths.

#### 3.1.2. Optically Shallow Water

^{−6}sr

^{−1}are shaded in gray in order to indicate that measuring these differences is very challenging, even for field instruments, and impossible for current satellite sensors.

#### 3.1.3. Optimal Spectral Resolution

^{−6}sr

^{−1}. To reduce the impact of unusually high $\Delta \lambda $ values, the median is used rather than the mean. Figure 7 shows the results.

^{−6}sr

^{−1}were ignored as these are extremely difficult to measure even with sensitive field instruments. This data selection leads to a wavelength dependency of the used number of spectra (Figure 7B). Figure 7B thus illustrates which spectral regions are difficult to measure due to the low dependency of reflectance on water composition, bottom substratum or depth.

^{−1}, changes in S and 20 cm differences in water depth, they do not allow us to derive the upper limits of bandwidth that are still suitable for determining these parameters. All scenarios represent optically complex waters in which ${R}_{rs}$ is usually affected by several water constituents or substratum types simultaneously at each wavelength. These waters therefore do not allow us to determine an unknown parameter using a single wavelength. The ambiguity of single band retrieval algorithms is reduced for algorithms based on several bands; the more bands, the less pronounced are the ambiguities. However, even hyperspectral measurements with a contiguous series of narrow bands can suffer from ambiguities [36], depending on measurement noise and number, type and concentration of water constituents [37] and bottom substrates. The accuracy of the retrieved parameters depends on the used algorithm and its strategy to handle ambiguities [38], as well as on the spectral range, spectral resolution and radiometric resolution of the measurements. Figure 4, Figure 5, Figure 6 and Figure 7 help us to estimate the ideal spectral resolution of measurements and determine the useful spectral range for data analysis in different water types, while the results of Section 3.2 provide information concerning the most sensitive wavelengths and the required radiometric resolution. Which reduced spectral resolutions are still sufficient in practice for accurate parameter retrieval can only be determined for specific retrieval algorithms and given measurement noise. Such analysis is, however, out of scope for this paper.

#### 3.2. Radiometric Sensitivity

#### 3.2.1. Resolving Spectral Features

^{3}= 1000 spectra were simulated for each scenario, from which $\Delta {R}_{rs,1}$ was derived using Equation (8). The results are shown in Figure 8.

^{−6}sr

^{−1}. $\Delta {R}_{rs,1}$ = 10

^{−5}sr

^{−1}is sufficient for most parameter combinations of the standard scenarios, except scenarios Y+ and C− for X < 1 g m

^{−3}, scenario C− for Y > 0.4 m

^{−1}and scenario X− for Y > 2 m

^{−1}. The extreme scenarios more frequently require $\Delta {R}_{rs,1}$ between 10

^{−5}and 10

^{−6}sr

^{−1}. The median ${R}_{rs}$ differences are 1.1 × 10

^{−4}sr

^{−1}/4.6 × 10

^{−5}sr

^{−1}for the considered C values of the standard/extreme scenarios (upper row of Figure 8), 1.2 × 10

^{−4}sr

^{−1}/2.3 × 10

^{−4}sr

^{−1}for the X values (middle row of Figure 8) and 1.2 × 10

^{−4}sr

^{−1/}2.6 × 10

^{−4}sr

^{−1}for the Y values (lower row of Figure 8). The overall median is 1.5 × 10

^{−4}sr

^{−1}.

#### 3.2.2. Resolving Concentration Changes

^{−3}in 10 steps. In each step, X = 1 g m

^{−3}was set, Y was changed from 0.2 to 2 m

^{−1}and S was changed from 0.010 to 0.020 nm

^{−1}. The reflectance spectrum was calculated for C and 1.1C for each parameter combination to obtain the wavelength of maximum sensitivity and the induced reflectance change. The resulting reflectance differences are the 1000 cyan circles labeled X− in diagram “C” of Figure 9.

^{−5}sr

^{−1}allows the resolution of 10% changes of X, Y and S for all standard scenarios and for the majority of the conditions studied for the extreme scenarios. However, C requires $\Delta {R}_{rs,2}$ as fine as to 3 × 10

^{−6}sr

^{−1}for the standard scenarios and even below 1 × 10

^{−6}sr

^{−1}for the extreme scenarios. High sensitivity is particularly required for the retrieval of low phytoplankton concentrations in dark waters with low X or high Y (scenarios X−, X−−, Y++).

#### 3.2.3. Optimal ${R}_{rs}$ Quantization

^{−1}differences in the spectral slope of CDOM absorption is provided in Figure 11. The left plot shows for each wavelength the medians of $\Delta {R}_{rs,2}$ across all simulations of Figure 9 and Figure 10. The right plot shows a histogram of the wavelengths which are most sensitive to changes in C, X, Y and S. The wavelengths of the local maxima are labeled.

^{−4}and 10

^{−3}sr

^{−1}. The median of 1.5 × 10

^{−4}sr

^{−1}obtained in Section 3.2.1 for resolving the spectral shape of reflectance spectra is inside this range, i.e., the two approaches for estimating the optimal radiometric sensitivity lead to comparable results.

#### 3.2.4. Radiometric Sensor Requirements

^{−3}, 10

^{−4}, 10

^{−5}and 10

^{−6}sr

^{−1}, which represent the four lowermost horizontal lines of Figure 8 to Figure 10 and the left plot of Figure 11. The calculations were made for sun zenith angles of 0°, 20°, 40°, 60° and 70°, using the mid-latitude summer atmospheric model of Modtran-6 [33] for a horizontal visibility of 50 km. Figure 12 shows the resulting radiance differences. To make them comparable with the sensitivity of existing satellite sensors, the noise-equivalent radiances $NEL$ of MODIS and OLCI on Sentinel-3 are also shown, representing instruments optimized for ocean color remote sensing. Sentinel-2 has also been included as an example of a state-of-the art land sensor. $NEL$ is the radiance corresponding to the radiometric sensitivity of a sensor.

^{−3}sr

^{−1}for all considered sun zenith angles, while sensors like OLCI on Sentinel-3 and MODIS can easily measure reflectance differences of $\Delta {R}_{rs}$ = 10

^{−4}sr

^{−1}, and $\Delta {R}_{rs}$ = 10

^{−5}sr

^{−1}can be resolved only above 650 nm. Differences of $\Delta {R}_{rs}$ = 10

^{−6}sr

^{−1}are too small for any of these sensors.

#### 3.2.5. Signal-to-Noise Requirements

^{−3}, 10

^{−4}, 10

^{−5}and 10

^{−6}sr

^{−1}was simulated using Equation (17). These $\Delta {R}_{rs}$ values represent the four lowermost horizontal lines of Figure 8 to Figure 10 and the left plot of Figure 11. Modtran-6 [33] was used for calculating $L\left(\lambda \right)$ and ${E}_{d}\left(\lambda \right)$ for a mid-latitude summer atmosphere at a horizontal visibility of 50 km and sun zenith angles of 0°, 20°, 40°, 60° and 70°. Figure 13 shows the results. $SN{R}^{path}$ increases strongly from long to short wavelengths and can be very large, particularly for $\Delta {R}_{rs}$< 10

^{−4}sr

^{−1}. Figure 13 can be useful for relating the environmental parameter $\Delta {R}_{rs}$ to the measurement parameter SNR at the top of the atmosphere for given $SN{R}^{BOA}$.

^{−1}changes in the spectral slope of CDOM absorption. $\Delta \mathsf{\lambda}$ indicates the median optimal spectral resolutions of all scenarios (red curve of Figure 7).

^{−6}sr

^{−1}. The 50% quantile represents the median.

^{−4}sr

^{−1}, the average 75% percentile is 2.9 × 10

^{−5}sr

^{−1}and the average 90% percentile is 1.0 × 10

^{−5}sr

^{−1}. From 740 to 900 nm, the corresponding values are 1.5 × 10

^{−5}sr

^{−1}, 4.0 × 10

^{−6}sr

^{−1}and 1.8 × 10

^{−6}sr

^{−1}.

^{−6}sr

^{−1}.

## 4. Summary and Conclusions

_{rs}should be resolvable with a quantization of 100 levels from 400 to 800 nm; (2) concentration changes in water constituents of 10%, spectral slope differences of 0.002 nm

^{−1}of CDOM absorption and depth differences of 20 cm shall produce measurable reflectance differences for at least one wavelength. The results of $\Delta {R}_{rs}$ and $\Delta \mathsf{\lambda}$ are presented in much detail in Section 3.1 and Section 3.2. Both parameters change significantly at around 740 nm. From 400 to 735 nm, the medians are $\Delta {R}_{rs}$ = 1.2 × 10

^{−4}sr

^{−1}and $\Delta \mathsf{\lambda}$ = 2.9 nm; the 90% percentiles are $\Delta {R}_{rs}$ = 1.0 × 10

^{−5}sr

^{−1}and $\Delta \mathsf{\lambda}$ = 1.2 nm. From 740 to 900 nm, the corresponding values are $\Delta {R}_{rs}$ = 1.5 × 10

^{−5}sr

^{−1}and $\Delta \mathsf{\lambda}$ = 13.8 nm for the medians and $\Delta {R}_{rs}$ = 1.8 × 10

^{−6}sr

^{−1}and $\Delta \mathsf{\lambda}$ = 2.3 nm for the 90% percentiles.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Bottom substrate and benthos albedos used for the simulations. See Table 3 for the labeling.

**Figure 2.**Absorption coefficients (left) and backscattering coefficients (right) used for the simulations. The units are given in the legend.

**Figure 3.**Color coding of the spectral resolution plots in Figure 4, Figure 5 and Figure 6. Top: Scheme used for $\Delta {R}_{rs}$ ≥ 10

^{−6}sr

^{−1}. The colors change in steps of 2.5 nm, i.e., dark blue is 0 to 2.5 nm, light blue 2.5 to 5.0 nm, and so on. Bottom: Scheme used for $\Delta {R}_{rs}$ < 10

^{−6}sr

^{−1}. The gray values change gradually from black (0 nm) to white (≥20 nm).

**Figure 6.**Optimal spectral resolution for capturing the spectral details in reflectance measurements for optically shallow waters. For input model parameterization values, see Section 2.2.; for legend of colors, see Figure 3.

**Figure 7.**(

**A**): Medians of the optimal spectral resolutions for capturing the details of ${R}_{rs}$ spectra. (

**B**): Number of spectra used for calculating the medians.

**Figure 8.**${R}_{rs}$ differences corresponding to 1% of the dynamic range of ${R}_{rs}$. Left column: standard scenarios; right column: extreme scenarios of optically deep water.

**Figure 9.**Maximum change in ${R}_{rs}$ for a 10% change in the parameter indicated top right for the standard scenarios. Parameter changes of f × 10% alter the shown $\Delta {R}_{rs}$ values approximately by the factor f.

**Figure 10.**Maximum change in ${R}_{rs}$ for a 10% change in the parameter indicated top right for the extreme scenarios. Parameter changes of f × 10% alter the shown $\Delta {R}_{rs}$ values approximately by the factor f.

**Figure 11.**Left: Medians of the maximum changes in ${R}_{rs}$ induced by 10% concentration changes. Right: Number of spectra used for calculating the medians and wavelengths of local maxima.

**Figure 12.**Radiance differences induced by ${R}_{rs}$ changes of 10

^{−3}, 10

^{−4}, 10

^{−5}, 10

^{−6}sr

^{−1}for clear atmospheric conditions and sun zenith angles of 0°, 20°, 40°, 60°, 70°; noise-equivalent radiances of Sentinel-2, Sentinel-3 OLCI and MODIS for comparison.

**Figure 13.**Required signal-to-noise ratio of the path radiance for resolving ${R}_{rs}$ differences of 10

^{−3}, 10

^{−4}, 10

^{−5}and 10

^{−6}sr

^{−1}under clear atmospheric conditions for sun zenith angles of 0°, 20°, 40°, 60° and 70°.

**Figure 14.**Reflectance differences (

**A**) and radiance differences (

**B**) that must be resolved in order to capture the spectral shape of reflectance spectra with a quantization of 100 levels for 50%, 75% and 90% of all simulated cases. The radiance differences are shown for a sun zenith angle of 40° and compared with the noise-equivalent radiances of the sensors OLCI on Sentinel-3 and MODIS.

**Figure 15.**Signal-to-noise ratio at the water surface (

**A**) and at the top of the atmosphere (B) required for capturing the spectral shape of reflectance spectra with a quantization of 100 levels for 50%, 75% and 90% of all simulated cases. The simulations are shown for a sun zenith angle of 40°. The numbers in (

**B**) are the averages for the spectral ranges 400–450 nm, 560–580 nm and 600–900 nm.

**Table 1.**Standard scenarios for optically deep water. A scenario is defined by a constant parameter marked as bold. The scenario name consists of the parameter acronym and a sign indicating a low (−) or high (+) value. The variable parameters are specified by a typical value and a range in the notation typical (min-max).

Scenario | C− | C+ | X− | X+ | Y− | Y+ |
---|---|---|---|---|---|---|

Represents | Low phy | High phy | Low NAP | High NAP | Low CDOM | High CDOM |

Example | Lake Garda | 2 Finnish l. | L. Constance | The Netherlands | L. Maggiore | Lake Peipsi |

C (mg m^{−3}) | 1 | 40 | 2 (0.5–15) | 25 (10–50) | 1 (0.2–5) | 5 (1–20) |

X (g m^{−3}) | 1 (0.2–20) | 10 (5–15) | 1 | 15 | 1 (0.2–10) | 5 (1–10) |

Y (m^{−1}) | 0.1 (0.04–2) | 2.5 (1.5–4) | 0.5 (0.2–2) | 1 (0.5–1.5) | 0.2 | 2.5 |

S (nm^{−1}) | 0.014 (0.01–0.02) | 0.014 (0.01–0.02) | 0.014 (0.01–0.02) | 0.014 (0.01–0.02) | 0.014 (0.01–0.02) | 0.014 (0.01–0.02) |

**Table 2.**Extreme scenarios for optically deep water. The notation is the same as in Table 1, except the label −−of a scenario name indicating an extremely low parameter value and ++ an extremely high value.

Scenario | C−− | C++ | X−− | X++ | Y−− | Y++ |
---|---|---|---|---|---|---|

Extreme for | Low phy | High phy | Low NAP | High NAP | Low CDOM | High CDOM |

Example | Italian lakes | Lake Taihu | Lake Garda | Lake Taihu | Lake Garda | Finnish lakes |

C (mg m^{−3}) | 0.2 | 1000 | 1 (0.1–10) | 20 (1–1000) | 1 (0.1–10) | 5 (1–10) |

X (g m^{−3}) | 1 (0.2–20) | 50 (10–300) | 0.1 | 300 | 1 (0.2–20) | 2 (0.5–5) |

Y (m^{−1}) | 0.1 (0.04–2) | 1 (0.2–3) | 0.1 (0.04–2) | 1 (0.2–3) | 0.04 | 10 |

S (nm^{−1}) | 0.014 (0.01–0.02) | 0.014 (0.01–0.02) | 0.014 (0.01–0.02) | 0.014 (0.01–0.02) | 0.014 (0.01–0.02) | 0.014 (0.01–0.02) |

No. | Substrate Type | Reference |
---|---|---|

0 | Chara contraria (macrophyte) | [16] |

1 | Potamogeton perfoliatus (macrophyte) | [16] |

2 | Rock | [17] |

3 | Bleached coral | [18] |

4 | Dark silt | [17] |

5 | Bright sand | [18] |

6 | Yellow porites sp. (coral) | [18] |

7 | Purple encrusting coralline algae | [18] |

8 | Brown porites sp. (coral) | [18] |

9 | Posidonia australia (seagrass) | [19] |

10 | Detritus (sea-grass wrack) | [17] |

11 | Ecklonia radiata (kelp) | [17] |

12 | Coarse coral rubble | [18] |

13 | Dark sand | [20] |

**Table 4.**Wavelengths most sensitive to concentration changes, their medians of the optimal spectral resolution $\Delta \mathsf{\lambda}$, their median $\Delta {\mathrm{R}}_{\mathrm{rs}}$ of the induced ${R}_{rs}$ differences and their medians of the signal-to-noise ratio at the water surface (${\mathrm{SNR}}^{BOA}$), for the atmospheric path radiance at a sun zenith angle of 40° (${\mathrm{SNR}}^{path}$) and at the top of the atmosphere (${\mathrm{SNR}}^{TOA}$).

Wavelength (nm) | $\Delta \mathsf{\lambda}$ (nm) | $\Delta {\mathbf{R}}_{rs}$ (sr ^{−1})
| $SN{R}^{\mathit{B}\mathit{O}\mathit{A}}$ | $SN{R}^{\mathit{p}\mathit{a}\mathit{t}\mathit{h}}$ | $SN{R}^{\mathit{T}\mathit{O}\mathit{A}}$ |
---|---|---|---|---|---|

488 | 1.8 | 3.3 × 10^{−4} | 62 | 151 | 213 |

501 | 1.6 | 4.0 × 10^{−4} | 21 | 108 | 129 |

543 | 1.9 | 2.5 × 10^{−4} | 26 | 118 | 144 |

566 | 2.1 | 6.8 × 10^{−4} | 13 | 36 | 49 |

580 | 0.8 | 8.0 × 10^{−4} | 16 | 28 | 44 |

588 | 0.6 | 2.0 × 10^{−3} | 14 | 11 | 25 |

638 | 4.3 | 6.8 × 10^{−4} | 13 | 21 | 34 |

676 | 6.7 | 5.6 × 10^{−5} | 56 | 203 | 259 |

705 | 1.2 | 1.9 × 10^{−3} | 19 | 5 | 24 |

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**MDPI and ACS Style**

Gege, P.; Dekker, A.G.
Spectral and Radiometric Measurement Requirements for Inland, Coastal and Reef Waters. *Remote Sens.* **2020**, *12*, 2247.
https://doi.org/10.3390/rs12142247

**AMA Style**

Gege P, Dekker AG.
Spectral and Radiometric Measurement Requirements for Inland, Coastal and Reef Waters. *Remote Sensing*. 2020; 12(14):2247.
https://doi.org/10.3390/rs12142247

**Chicago/Turabian Style**

Gege, Peter, and Arnold G. Dekker.
2020. "Spectral and Radiometric Measurement Requirements for Inland, Coastal and Reef Waters" *Remote Sensing* 12, no. 14: 2247.
https://doi.org/10.3390/rs12142247