# Effects of Atmospheric Correction on Remote Sensing Statistical Inference in an Aquatic Environment

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## Abstract

**:**

_{rs}), which requires AC. This raises the question of whether AC is necessary for remote sensing statistical inference. We conducted a theoretical analysis and image validations by testing 24 water bodies observed by Landsat-8 and compared their spectral probability distributions (SPDs) calculated from R

_{rs}before and after AC (using the ACOLITE model). Additionally, we tested and found that, if we use remote sensing inference as a tool to quantitatively infer statistical parameters of a specific waterbody, it is better to perform atmospheric correction. However, if the quantitative inference is applied to a large number of water bodies and high inference accuracy is not required, atmospheric correction may not be necessary, and a quick calculation based on the strong correlations between R

_{rs}at the surface and sensor-observed reflectance can be used as a substitute.

## 1. Introduction

_{rs}), which is widely used in water color remote sensing. Because SPDs used in remote sensing inference are normalized linearly by their own scale and shift factors, it is possible that the normalized SPDs are not significantly affected by the atmosphere. However, it has not been investigated if AC is necessary for using R

_{rs}-based normalized SPDs in remote sensing inference, as there are many other remotely sensed variables available for making SPDs, such as digital number (DN) and radiance at the top of the atmosphere (TOA), which can be obtained without AC. Retrieving R

_{rs}from satellite images is a complicated process involving not only AC but also removing the reflective effect of the water surface. If the SPD of R

_{rs}is the same as the SPD of DN, meaning AC is not necessary, then a lot of work can be saved in remote sensing inference applications.

## 2. Theoretical Analysis

_{rs}of different waters into the same range 0–1 [1,16], using the following equation:

_{rs_norm}is the linearly normalized R

_{rs}and max(R

_{rs}) and min(R

_{rs}) are the maximum and minimum R

_{rs}, respectively, of all ROI pixels. Using Equation (1), the maximum and minimum R

_{rs}are normalized to 1 and 0, respectively, and the other R

_{rs}are normalized to values between 0 and 1.

_{TOA}or reflectance R

_{TOA}and the DN measured by the satellite (e.g., Landsat-8) is given by

_{TOA}can be calculated using the same Equation (2) but with different gain and offset.

_{TOA}can be further decomposed into three components, such that

_{path}is the atmospheric path radiance, L

_{reflected}is the radiance reflected by the ground surface, and L

_{adjacency}is the radiance reflected by the adjacent pixels. If we assume (A1) that the signals from adjacent pixels are quite small and can be ignored, and (A2) that the atmosphere is horizontally homogenous within the water of interest (WOI), which typically covers tens or hundreds of km

^{2}, then Equation (3) becomes

_{surf}is the surface reflectance, E

_{d}is the total downwelling irradiance, T is the total transmittance from the surface to the sensor, and S is the spherical albedo of the atmosphere [17]. By estimating the parameters L

_{path}, E

_{d}, T, and S, atmospheric corrections can predict R

_{surf}based on the sensor-observed L

_{TOA}. If we further assume (A3) that R

_{surf}× S is much smaller than 1, then Equation (4) becomes

_{surf}is contributed by the signals reflected only from the water surface and the signals leaving the water, which are reflected by in-water components as well as by the water itself, so, to obtain R

_{rs}, the water-surface signals must be removed using the equation

_{surf}is the radiance reflected by the water surface [18].

_{TOA}and R

_{rs}are in different units, to compare them, we use the variable R

_{rs_TOA}, defined by

_{rs}and R

_{rs_TOA}are both in the unit sr

^{−1}.

_{TOA}/R

_{TOA}, R

_{surf}, and R

_{rs}. Therefore, the results of their linear normalization should be invariant, as the scale and shift factors in Equations (2), (6) and (7) will be canceled out when using the normalization Equation (1).

_{rs}should be approximately equal to those of the linearly normalized DN. However, this approximation depends on the above three key assumptions, (A1), (A2), and (A3), which transform Equation (3) into Equation (6). If these three assumptions—namely, (A1) the effects of adjacent pixels can be ignored, (A2) the atmosphere is horizontally homogenous, and (A3) R

_{surf}× S is much smaller than 1—are not true, then the SPDs of R

_{rs_norm}may deviate significantly from the SPDs of the linearly normalized L

_{TOA}to some extent. Therefore, in the following section, we compare the SPDs of normalized R

_{rs}and R

_{rs_TOA}retrieved from real images to determine whether the above assumptions hold true and whether any differences can be ignored.

## 3. Image Data Analysis

#### 3.1. Data and Method

_{rs}and R

_{TOA}in the five visible and near-infrared bands of Landsat-8, i.e., B1 (443 nm), B2 (483 nm), B3 (555 nm), B4 (665 nm), and B5 (865 nm), were obtained from 24 images (the years of their observation range from 2013 to 2019, and the months range from January to November). These images cover 24 water bodies around the world, including lakes, bays, lagoons, and estuaries (Figure 1). Note that water bodies studied by remote sensing inference should be closed and connected, so they are labelled as closed-connected water bodies [1]. While most of the lakes are closed-connected, bays, lagoons, and estuaries with narrow channels to the open sea can also be considered as approximately closed-connected.

_{rs}were obtained using the ACOLITE atmospheric correction processor, which is designed for use in coastal and inland water applications. ACOLITE employs the ‘dark spectrum fitting’ approach, which automatically selects the most relevant spectral band and aerosol model [19,20,21,22]. ACOLITE supports Landsat-8. It can mask all non-water pixels in an image and calculate R

_{rs}of all water pixels. Note that many of the known AC methods (such as iCOR, FLAASH, and 6S) were not compared in this study as they are not specifically developed for water applications and their results are only slightly different.

_{rs_norm}and thus the SPD. The maximum 2% and minimum 2% R

_{rs}within the WOI pixels were excluded from the SPD calculations, and the remaining 96% were used to construct the SPD’s histograms with bin number = 20 [1]; see SPD examples in Figure 2. As the number of WOI pixels in each water body varied, we calculated the pixel probability (the percentage of WOI pixels in each bin) rather than the raw pixel count.

_{TOA}was calculated directly from the Landsat-8 DN using Equation (2), and the gains and offsets were retrieved from the image metadata files (_MTL.txt). To calculate the SPDs of R

_{TOA}, we used the exact same WOI pixels that were used to calculate the SPDs of R

_{rs}. R

_{TOA}were also linearly normalized using the same method in Equation (1), and then SPDs of R

_{rs_TOA}= $\frac{1}{\pi}$R

_{TOA}were also calculated accordingly.

_{rs}and R

_{rs_TOA}was expressed as MAPE (mean absolute percentage error) using the following formula:

_{rs}, i) and SPD(R

_{rs_TOA}, i) are the spectral probabilities (percentage counts) of R

_{rs}and R

_{rs_TOA}, respectively, in the i-th bin. In addition, we also calculated the RMSE (root mean squared error) and bias between the two variables to provide more information about their difference.

#### 3.2. Results and Discussion

_{rs}and R

_{rs_TOA}at five Landsat-8 bands is only 0.84% (Figure 3). This error is quite small, indicating that the SPDs of R

_{rs}and R

_{rs_TOA}are almost the same. Their differences in RMSE and bias were also quite small (overall average RMSE = 0.0012% and bias = 0.000059%). From the two examples in Figure 2b,d, the five respective SPD curves of R

_{rs}and R

_{rs_TOA}in Kainji Lake and Lake Geneva both matched perfectly well. Good agreement also occurred in the other 22 water bodies, proving the correctness of the above theoretical analysis in Section 2. The three assumptions, (A1), (A2), and (A3), are typically true, i.e., the effects of the adjacent radiance, atmospheric horizontal inhomogeneity, and multiple reflections can be ignored. When using SPDs of R

_{rs_norm}for water quality or classification analysis, we can directly use the SPDs of linearly normalized R

_{rs_TOA}or even DN, and it is not necessary to make atmospheric corrections to first calculate R

_{rs}.

_{rs}and R

_{rs_TOA}. The results—see examples in Figure 2a,c—show that the shapes of the corresponding curves are similar but fall in different ranges (in unit sr

^{−1}) because of the effects of atmospheric path radiance. It seems that, if we stretch/squeeze and shift the SPDs of R

_{rs_TOA}, then we get the SPDs of R

_{rs}, which why their normalized SPDs match well; see Figure 2b,d. As the unnormalized SPDs of R

_{rs}and R

_{rs_TOA}were of different magnitudes, the quantitative statistical information derived from their SPDs will also be different. For example, in Lake Geneva and for band B3 (555 nm), the mean R

_{rs}is 0.0075 sr

^{−1}, while the mean R

_{rs_TOA}is 0.0487 sr

^{−1}and, obviously, if we use mean R

_{rs_TOA}to replace mean R

_{rs}in inference models, it will bring large errors. Therefore, in order to quantitatively calculate some statistical parameters, such as the mean and median of R

_{rs}, we need to make atmospheric corrections to remove the noise signals caused by atmospheric path radiance. It is the same as in remote sensing inversion, that is, atmospheric correction is necessary to apply inversion models to images.

_{rs}from the same parameters of R

_{rs_TOA}. We found that there were good correlations between the corresponding parameters of the two variables in the 24 water bodies in the study; see the results in Table 1. The overall correlation coefficient (r) of all bands and parameters is r = 0.92. For bands B3, B4, and B5, the correlations (r = 0.94, 0.95, and 0.96, respectively) were even better than the correlations for bands B1 (r = 0.86) and B2 (r = 0.89). Among the five statistical parameters, the mean, median, and std show better correlations (r = 0.95, 0.95, and 0.97, respectively) than the other two parameters, min (r = 0.87) and max (r = 0.87). These good correlations imply that we can calculate the statistical parameters of R

_{rs}directly from the corresponding parameters of R

_{rs_TOA}, without using the atmospheric correction to know R

_{rs}at each pixel. We have performed the linear regressions between the respective parameters of R

_{rs}and R

_{rs_TOA}, so their relationships are easily expressed and the statistical parameters of R

_{rs}can be quickly calculated using equations such as those shown in Figure 4. We suggest that, when using remote sensing inference to quantitatively study a large number of water bodies or images, we can use the above quick calculations, but when focusing on only one specific water body, atmospheric correction is needed to obtain R

_{rs}and thus calculate its more accurate statistical parameters. Note that the empirical relationships shown in Figure 4 were only obtained from the 24 water bodies in the study, so in future work, we suggest further checking whether they apply to more water bodies or whether they need to be modified.

## 4. Conclusions

_{rs}and R

_{rs_TOA}is less than 1%, which means that, in remote sensing inference, if we only use linearly normalized SPDs to qualitatively classify and analyze the water types or qualities, it is not necessary to make atmospheric corrections, and SPDs can be calculated directly from the DN values. If we use statistical parameters, such as the mean, median, std, min, max, or statistical distributions of surface spectra, to quantitatively infer statistical parameters of non-optical properties of a particular water body, it is better to make atmospheric corrections. However, if the quantitative inference has been applied to a large number of water bodies for comparative analysis and inference accuracy is not a strict requirement, then atmospheric correction is sometimes not necessary. Instead, we can use the simple correlations between the corresponding statistical parameters of R

_{rs}and R

_{rs_TOA}to approximate the statistical parameters of R

_{rs}—this would be much simpler and faster than using atmospheric correction.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Normalized and unnormalized R

_{rs}and R

_{rs_TOA}of Lake Geneva and Kainji Lake: (

**a**,

**c**) unnormalized R

_{rs}and R

_{rs_TOA}and (

**b**,

**d**) normalized R

_{rs}and R

_{rs_TOA}.

**Figure 3.**Errors (MAPEs) between linearly normalized R

_{rs}and R

_{rs_TOA}for the 24 water bodies in five Landsat-8 bands.

**Figure 4.**Relationships between the statistical parameters ((

**a**) mean, (

**b**) median, (

**c**) std, (

**d**) min, and (

**e**) max) of unnormalized R

_{rs}and R

_{rs_TOA}for the 24 water bodies at the B3 band (555 nm) of Landsat-8.

**Table 1.**Correlation coefficients between the statistical parameters of unnormalized R

_{rs}and R

_{rs_TOA}for the 24 water bodies at five Landsat-8 bands and the averages for all five bands and five parameters. The p-values of their correlations, with the orders of 10

^{−12}to 10

^{−14}, are much smaller than 0.05.

Stat. Para. | B1 | B2 | B3 | B4 | B5 | Average |
---|---|---|---|---|---|---|

Mean | 0.88 | 0.93 | 0.98 | 0.98 | 0.97 | 0.95 |

Median | 0.88 | 0.93 | 0.98 | 0.99 | 0.97 | 0.95 |

Std | 0.95 | 0.95 | 0.97 | 0.98 | 0.98 | 0.97 |

Min | 0.75 | 0.79 | 0.89 | 0.89 | 0.98 | 0.87 |

Max | 0.81 | 0.85 | 0.89 | 0.92 | 0.89 | 0.87 |

Average | 0.86 | 0.89 | 0.94 | 0.95 | 0.96 | 0.92 |

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**MDPI and ACS Style**

Zhu, W.; Xia, W.
Effects of Atmospheric Correction on Remote Sensing Statistical Inference in an Aquatic Environment. *Remote Sens.* **2023**, *15*, 1907.
https://doi.org/10.3390/rs15071907

**AMA Style**

Zhu W, Xia W.
Effects of Atmospheric Correction on Remote Sensing Statistical Inference in an Aquatic Environment. *Remote Sensing*. 2023; 15(7):1907.
https://doi.org/10.3390/rs15071907

**Chicago/Turabian Style**

Zhu, Weining, and Wei Xia.
2023. "Effects of Atmospheric Correction on Remote Sensing Statistical Inference in an Aquatic Environment" *Remote Sensing* 15, no. 7: 1907.
https://doi.org/10.3390/rs15071907