Uncertainty analysis of BRDF Modeling Using 6S Simulations and Monte-Carlo Method

Abstract

This paper presents the method to quantitatively evaluate the uncertainty of the semi-empirical Bidirectional Reflectance Distribution Function (BRDF) model for Himawari-8/AHI. The uncertainty of BRDF modeling was affected by various issues such as assumption of model and number of observations, thus, it is difficult that evaluating the performance of BRDF modeling using simple uncertainty equations. Therefore, in this paper, Monte-Carlo method, which is most dependable method to analyze dynamic complex systems through iterative simulation, was used. The 1,000 input datasets for analyzing the uncertainty of BRDF modeling were generated using the Second Simulation of a Satellite Signal in the Solar Spectrum (6S) Radiative Transfer Model (RTM) simulation with MODerate Resolution Imaging Spectroradiometer (MODIS) BRDF product. Then, we randomly selected data according to the number of observations from 4 to 35 in the input dataset and performed BRDF modeling using them. Finally, the uncertainty was calculated by comparing reproduced surface reflectance through the BRDF model and simulated surface reflectance using 6S RTM and expressed as bias and root-mean-square-error (RMSE). The bias was negative for all observations and channels, but was very small within 0.01. RMSE showed a tendency to decrease as the number of observations increased, and showed a stable value within 0.05 in all channels. In addition, our results show that when the viewing zenith angle is 40° or more, the RMSE tends to increase slightly. This information can be utilized in the uncertainty analysis of subsequently retrieved geophysical variables.

1. Introduction

This reflectance anisotropy, which is fundamental characteristics of surface reflectance, depends on illumination direction, viewing direction, canopy structure such as vegetation density and background characteristic, and wavelengths (Calleja et al., 2016). The surface reflectance measured by satellite sensor may decrease or increase according to surface anisotropy characteristics; and this effects can affect on the quality of satellite-based cloud masking and trace gas retrieval, as well as, retrieval of land products such as vegetation index, land cover and land use, and leaf area index (Roberts, 2001; Lee et al., 2017; Vasilkov et al., 2017; Lorente et al., 2018; Seong et al., 2020; Yeom et al., 2020).

The Bi-directional Reflectance Distribution Function (BRDF) can explain the anisotropy effects of the land surface (Gao et al., 2003; Wen et al., 2018). Among various types of BRDF model, a semi-empirical model, which is consists of the physical kernel and empirical coefficient (BRDF parameter), is widely used in the satellite remote sensing field. The physical kernel is a mathematical equation that expresses the mechanism of geometric and volumetric scattering under the simple assumption for the state of land surface and BRDF parameter means the weight of the physical kernel. This BRDF model enables fast inversion over a wide area while retaining physical interpretation through the kernels, and it is flexible enough to express BRDF characteristics for various land types from dense vegetation to bare soil. In addition, a small number of parameters of the semi-empirical BRDF model are suitable for estimating land surface anisotropic characteristics from satellite observations because there are limitations in the number of available data due to several reasons like clouds and observation cycle (Su et al., 2009).

To describe anisotropy characteristics of land surface, semi-empirical BRDF model requires observations acquired from various sun-target-satellite relationships. However, unfortunately, most multispectral sensors cannot obtain multiangle observation data in one measurement. In order to secure sufficient observations with various geometric conditions, the data over a certain period were used for BRDF modeling under the assumption that the anisotropic characteristic of land surface does not change significantly during a short time (He et al., 2012; He et al., 2019). However, this assumption is not always appropriate. In addition, the state of the land surface in the geometric and volume kernels does not always match the state of the actual land surface. Therefore, due to the issues above mentioned, uncertainties unavoidably occur when performing the BRDF modeling, and it may lead to uncertainty in land modeling, change analysis, and satellite-based land surface variables such as land surface albedo.

In this study, we performed BRDF modeling using the method proposed by Lee et al. (2020) and statistically analyze uncertainty in the simulation of surface reflectance through a semi-empirical BRDF model. For this, Himawari-8/Advanced Himawari Instrument (AHI) was used because it is one of the most suitable sensor to obtain multi-angle observations for performing the BRDF modeling due to its highest temporal resolution (10-min for full-disk domain).

2. Data

1) Spectral Response Function of Himawari-8/AHI

AHI was adapted to Himawari-8 with 16 bands. In this study, the spectral response functions (SRFs) from channel 1 to channel 5, which are corresponding to reflectance channels, were used as input data of the Second Simulation of a Satellite Signal in the Solar Spectrum (6S) Radiative Transfer Model (RTM) to generate input dataset for BRDF modeling. The SRFs of AHI shortwave channels were downloaded from the Meteorological Satellite Center website (https:// www.data.jma.go.jp/mscweb/en), and are shown in Fig. 1. In order to use it as an input data of 6S, it was readjusted at 2.5 nm intervals and used.

Fig. 1. Spectral response function of Himawari-8/AHI shortwave channels.

2) MODIS BRDF parameters

To describe the surface anisotropy characteristics, MODIS BRDF version 6 (MCD43D V006) products were used. These products include empirical coefficients for the isotropic, geometric, and volumetric kernel of the BRDF model, which are mathematical expressions for anisotropic scattering, geometric scattering, and volumetric scattering on the land surface. It covers the entire globe with 30 arc seconds (1000 m) resolution. The V006 product can be provided with improved temporal resolution (daily) compared to the previous version (V005) (Che et al., 2017), and is provided with better quality by using the priori database for the backup algorithm (Roujean et al., 2018).

3. Method and results

It was well known that the uncertainty of BRDF modeling depends on the number of observations that means the number of times the land surface has been observed from the satellite-mounted sensor during the composite period of BRDF modeling. In general, the accuracy of the BRDF model increases as the number of observations increases (Che et al., 2017). However, the uncertainty of BRDF modeling varies depending on the sampled surface reflectance (e.g. geometric and atmospheric conditions), thus, it is difficult that quantifying the uncertainty of BRDF modeling using simple uncertainty propagation law. Therefore, in this thesis, the Monte-Carlo Method (MCM) was used to estimate the uncertainty that occurred when performing BRDF modeling. The MCM is a mathematical method for modeling the uncertainty of a system and solution by generating random variables. This method is known as the most dependable method to analyze dynamic complex systems (Boudina et al., 2020). It provides a probabilistic estimate of the uncertainty in a system or solution through iterative simulation and was utilized for quantifying the uncertainty of satellite-derived variables (Lobell et al., 2003; Białek et al., 2020).

To estimates the uncertainty of BRDF modeling, first, the hourly surface reflectances for 5days (June 22, 00:00 UTC to June 27, 23:00 UTC) were simulated using 6S RTM. In simulation, the actual geometric conditions (solar zenith angle; SZA, viewing zenith angle; VZA, and relative azimuth angle; RAA) for randomly selected 1, 000 pixels were used, and one of the following five land types was randomly assigned to each pixel: forest, croplands, grasslands, shrublands, urban, and barren. The surface anisotropy of each land type was considered using the mean value of MODIS BRDF parameters over 2017. The examples of simulated surface reflectance using 6S RTM were shown in Fig. 2. The simulated surface reflectances well present typical reflectance according to land type. Since this study aims at the uncertainty arising from BRDF modeling, it is assumed that there is no uncertainty in the surface reflectance. These 1, 000 datasets were used not only as an input for BRDF modeling, but also as a reference for uncertainty evaluation. Then, for each dataset, the surface reflectances to be used for BRDF modeling was randomly selected for each number of observations (4, 7, 10, 15, 20, 25, 30, and over 35), and 1, 000 datasets of surface reflectance for each number of observations were constructed. Finally, BRDF modeling was performed using each dataset and the bias and Root Mean Square Error (RMSE) between reproduced surface reflectance and the surface reflectance simulated using 6S RTM were calculated.

Fig. 2. Examples of simulated surface reflectance using 6S RTM for (a) forest, (b) croplands, (c) grasslands, (d) shrublands, (e) urban, and (f) barren.

Fig. 3 shows overall uncertainty, which is expressed as bias± RMSE, in reproduced surface reflectance according to the number of observations (4, 7, 10, 15, 20, 25, 30, and over 35). In all cases, the reproduced surface reflectance tends to be underestimated. This negative bias tends to become more pronounced as the number of observations increases, however, it is negligible because it is a very small uncertainty of less than 0.01 in all channels. The RMSE tends to decrease as the number of observations increases because the accuracy of the statistical calculation was improved as the number of input data increases. It means that as the number of observations increases, the BRDF modeling is performed more stably. In addition, the higher RMSEs were found in the channel 4 and 5 spectrum than in the channel 1~3. This may have occurred because the surface reflectance of NIR and SWIR has a relatively high signal amplitude in all land types as shown in Fig. 2.

Fig. 3. Uncertainty of BRDF modeling according to number of observations.

Fig. 4 shows RMSE of BRDF modeling in BRDF modeling according to VZA range and number of observations for channel 1~3. Channel 4 and 5 shows similar results with other channels, thus, these are excluded for brevity. In general, as shown in Fig. 3, in the same VZA section, the smaller the number of observations, the higher the RMSE. In addition, in the case where the VZA is above 40°, the RMSE is slightly higher than that of the case below 40°. This result means that the performance of BRDF modeling is slightly reduced in VZAs above 40°, but still shows a convincing uncertainty within 0.05. It means that the BRDF modeling well describes surface anisotropy characteristics from satellite measurements for most cases.

Fig. 4. RMSE of BRDF modeling according to number of observations and VZA range: (a) channel1, (b) channel2, and (c) channel3.

4. Conclusion

In this study, the uncertainty of BRDF modeling was analyzed according to a number of observations and VZA conditions through iterative simulation. For this, 1, 000 datasets that include the hourly simulated surface reflectance using 6S RTM and MODIS BRDF parameters were used. The uncertainty of BRDF modeling was varied according to the VZA and the number of observations, but in most cases, it showed a reasonable uncertainty within 0.05. This means that the semi-empirical BRDF model well simulates the anisotropic characteristics in clear-sky surface reflectance. The method presented in this study can be used for uncertainty analysis of geophysical variables calculated by complex calculation systems as well as BRDF modeling. In addition, the results of this study can be used to understand the products that are subsequently applied and to analyze the spread of uncertainty by understanding the uncertainty of BRDF modeling. This study focused on the analysis of the uncertainty that occurs when performing the BRDF modeling, thus, the uncertainty in surface reflectance was not considered. However, the surface reflectance may include uncertainty depending on various atmospheric conditions and observation conditions, and this may cause uncertainty in BRDF modeling, therefore, it is necessary to consider this in future studies.