Directional Applicability Analysis of Albedo Retrieval Using Prior BRDF Knowledge

Abstract

Surface albedo measures the proportion of incoming solar radiation reflected by the Earth’s surface. Accurate albedo retrieval from remote sensing data usually requires sufficient multi-angular observations to account for the surface reflectance anisotropy. However, most middle and high-resolution remote sensing satellites lack the capability to acquire sufficient multi-angular observations. Existing algorithms for retrieving surface albedo from single-direction reflectance typically rely on land cover types and vegetation indices to extract the corresponding prior knowledge of surface anisotropic reflectance from coarse-resolution Bidirectional Reflectance Distribution Function (BRDF) products. This study introduces an algorithm for retrieving albedo from directional reflectance based on a 3 × 3 BRDF archetype database established using the 2015 global time-series Moderate Resolution Imaging Spectro-radiometer (MODIS) BRDF product. For different directions, BRDF archetypes are applied to the simulated MODIS directional reflectance to retrieve albedo. By comparing the retrieved albedos with the MODIS albedo, the BRDF archetype that yields the smallest Root Mean Squared Error (RMSE) is selected as the prior BRDF for the direction. A lookup table (LUT) that contains the optimal BRDF archetypes for albedo retrieval under various observational geometries is established. The impact of the number of BRDF archetypes on the accuracy of albedo is analyzed according to the 2020 MODIS BRDF. The LUT is applied to the MODIS BRDF within specific BRDF archetype classes to validate its applicability under different anisotropic reflectance characteristics. The applicability of the LUT across different data types is further evaluated using simulated reflectance or real multi-angular measurements. The results indicate that (1) for any direction, a specific BRDF archetype can retrieve a high-accuracy albedo from directional reflectance. The optimal BRDF archetype varies with the observation direction. (2) Compared to the prior BRDF knowledge obtained through averaging method, the BRDF archetype LUT based on the 3 × 3 BRDF archetype database can more accurately retrieve the surface albedo. (3) The BRDF archetype LUT effectively eliminates the influence of surface anisotropic reflectance characteristics in albedo retrieval across different scales and types of data.

1. Introduction

Surface albedo is defined as the ratio of reflected solar radiation to incoming solar radiation on the Earth’s surface [1]. It measures the surface’s ability to reflect sunlight. Surface albedo plays a crucial role in climate and energy balance studies, as it directly affects the Earth’s energy budget by determining the amount of solar energy absorbed or reflected back into space. This, in turn, influences global temperature, weather patterns, and climate change dynamics [2,3]. In many applications, it is typically required that the albedo data meet an absolute accuracy of 0.02 to 0.05 [4].

Remote sensing techniques are used to acquire surface albedo data through observations from satellite or airborne sensors, providing information on surface characteristics and changes at large scales and over long time periods [5,6,7]. However, satellites only measure the surface reflectance, which is commonly used to describe the reflective properties of a surface under specific solar and viewing geometries. Since the surface reflection is anisotropic [8,9], the surface reflectance depends on the solar and viewing angles [10,11]. Albedo is the integral of directional reflectance over the entire viewing hemisphere, which provides a more comprehensive representation of a surface’s reflective properties compared to the directional reflectance [10]. The anisotropic reflection characteristics add complexity to albedo retrieval and require modeling and consideration of the surface’s reflection behavior at different angles. Surface anisotropic reflectance characteristics are inherent properties of the surface reflection, commonly described using the BRDF. The definition of the BRDF, given by Nicodemus [12], has been widely used to describe surface reflectance features. Typically, most albedo retrieval algorithms are based on multi-angular data acquired through remote sensing. These algorithms first establish the BRDF of the target and then integrate it over incident and view hemispheres to obtain the albedo [11]. Furthermore, these algorithms require a sufficient number of angles and a dispersed angle distribution [13] to achieve accurate results. Multi-angular observations have been collected from ground measurements [14,15] and several satellite sensors [7,16]. However, ground-based observations are typically limited to specific characteristic points, whereas satellite-based observations, although capable of providing extensive areas, often have a relatively coarse spatial resolution.

Satellites acquire multi-angle data by assuming that the surface anisotropic reflectance characteristics remain constant over a period of time and by combining observations from different swaths within that period. The most commonly used multi-angular products currently are from MODIS [7], Polarization and Directionality of the Earth’s Reflectances (POLDER) [16], Advanced Very High Resolution Radiometer (AVHRR) [17], etc. The MODIS BRDF/albedo product MCD43 [7] has received wide usage from the community. The MODIS instrument operates on both the Terra and Aqua spacecrafts, has a viewing swath width of 2330 km, and views the entire surface of the Earth every one to two days. The MODIS multi-angular observations, which accumulate all cloud-free data within a 16-day time frame, are input into semi-empirical kernel-driven linear BRDF models to invert the BRDF model parameters, surface albedo, and nadir reflectance products [7]. When the observations are sparsely sampled, the typical shape of the underlying BRDF is used as a baseline, and a multiplicative factor adjusts the archetypal BRDF to better fit the observed data [13]. In Version 6, the backup algorithm relies on the most recent high-quality BRDF of the pixel [18]. This approach helps in predicting the probable BRDF of the pixel under observation. The MODIS MCD43 provides daily BRDF/albedo products globally at a 500 m resolution. MODIS BRDF/albedo products are employed for surface energy modeling [19], served as a priori surface anisotropy information for other satellite sensors with limited angular range, facilitating the estimation of sensor-specific albedo [5,20,21,22] and the establishment of sensor inter-comparisons [23]. Moreover, these products help characterize surface structure [24], providing valuable insights for various remote sensing applications.

In contrast to satellite observations at kilometer-level resolutions, the Sentinel [25], Landsat [22], HuanJing (HJ) [26], and GaoFen (GF) [27] satellites provide high-resolution observations, which are crucial for capturing surface details at global or regional scales. The ability of these satellites to acquire surface reflectance is influenced by factors such as cloud cover, atmospheric disturbances, constraints on observation geometry, and the orbit characteristics of the satellite. These high-resolution satellites cannot obtain multi-angle data through swath overlapping. Some satellites, like the Chinese Gaofen-1/6 (GF1/6) Wide Field of View (WFV) images, have a significant range of viewing zenith angles (VZAs), with the maximum reaching 35°, and azimuth angles differing by nearly 180° [28]. Additionally, the solar zenith angle (SZA) varies with the seasons. Studies have proved that the relative errors can reach up to 45% when taking the nadir reflectance as the albedo without considering surface reflectance anisotropy [29]. Therefore, the impact of surface anisotropic reflectance characteristics must be considered when retrieving high-resolution surface albedo through directional reflectance.

Many studies have introduced prior BRDF knowledge from coarse-resolution BRDF products into surface albedo retrieval from high-spatial-resolution observations. Cui et al. utilized the POLDER-1 BRDF database to link surface reflectance with broadband albedo using simple linear regression [30]. This method, developed for different SZAs and phase angles, effectively corrects for reflectance anisotropy. Shuai et al. developed an algorithm to retrieve 30 m snow-free albedo from Landsat surface reflectance by leveraging MODIS BRDF data [22]. The process involves classifying the land surface types with Landsat data, aggregating these classifications to match the MODIS resolution, and then extracting BRDF parameters from the concurrent MODIS products. Franch et al. used prior BRDF knowledge to improve Landsat surface albedo retrieval by incorporating the Normalized Difference Vegetation Index (NDVI) to account for surface anisotropy [20]. They segmented the dataset into different NDVI classes, inverted the BRDF model parameters (V and R) for each class, and established linear functions to represent these parameters as a function of NDVI. This is a method that directly estimates broadband surface albedo based on the top-of-atmosphere (TOA) reflectance [5]. This method was initially proposed by Liang et al. [31], where the regression relationship between TOA reflectance and surface albedo was established using simulated data and then applied for surface albedo estimation. The prior BRDF knowledge for this method is based on 1000 samples of randomly selected MODIS BRDF data, including vegetation, soil, water, and snow/ice [5]. Another method involves using the BRDF archetypes to retrieve albedo. The Anisotropic Flat indeX (AFX), computed from BRDF model parameters, is used for the quantitative classification of MODIS BRDF, and BRDF archetypes representing different surface anisotropic reflection characteristics are established [32]. Based on this research, Zhao et al., introduced a new index, named perpendicular anisotropic flat index (PAFX), into the classification process to further refine the BRDF archetypes [33]. These BRDF archetypes are utilized in the retrieval of albedo through airborne multi-angle data [34]. The intermediate-level BRDF archetypes are explored for albedo retrieval from directional reflectance [35]. However, these methods lack an analysis of the applicability of the prior BRDF knowledge under different solar and viewing angles.

Based on the research of the BRDF archetype database and time-series MODIS BRDF products selected globally, this study compares the consistency between albedo retrieved using various BRDF archetypes under different solar and viewing angles and the MODIS albedo products. The BRDF archetype that produces albedo with the highest consistency is selected as the prior BRDF knowledge for that direction, and a LUT is established. The accuracy of the BRDF archetype LUT for retrieving albedo from directional reflectance is validated using various data sources, including MODIS BRDF data from a different year, PROSAIL simulated multi-angle data, POLDER, MODIS, and ground-based multi-angular observations. This study provides a new approach for effectively converting directional reflectance to albedo.

2. Materials and Methods

To explore the method for retrieving surface albedo from directional reflectance, the study utilized a 3 × 3 BRDF archetype database constructed from the 2015 MODIS BRDF product based on AFX and PAFX to describe the reflectance anisotropic characteristics of the surface. Then, using the 2015 MODIS BRDF product, the directional reflectance and albedo under different SZAs were simulated based on the RTLSR BRDF model. In each direction, the nine BRDF archetypes were adjusted according to the directional reflectance, resulting in nine corresponding albedos. These albedos were compared with the reference albedo, and the BRDF archetype with the smallest RMSE was selected as the prior knowledge for that direction. A BRDF archetype LUT was established for any SZA over the entire viewing hemisphere under different SZAs. Finally, this LUT was then sequentially applied to various types of multi-angular reflectance data, and the retrieved albedo was compared with the reference albedo. The accuracy of the BRDF archetype LUT for albedo retrieval was determined based on RMSE. The flowchart for albedo retrieval from directional reflectance based on BRDF archetypes is shown in Figure 1.

2.1. Kernel-Driven BRDF Model and MODIS BRDF

The bidirectional surface reflectance R(θ_i, θ_r, φ, λ) can be approximated as the linear combination of three BRDF kernels (K) weighted by the BRDF model parameter (f).

$$\mathrm{R}\left({\theta }_i,{\theta }_r,\phi ,\lambda \right)=f_{iso}\left(\lambda \right)+f_{vol}\left(\lambda \right)K_{vol}\left({\theta }_i,{\theta }_r,\phi \right)+f_{geo}\left(\lambda \right)K_{geo}\left({\theta }_i,{\theta }_r,\phi \right),$$

(1)

where λ represents wavelength, and θ_i and θ_r are the solar and viewing zenith angles, respectively. The relative azimuth angle (φ) is defined as the difference between viewing azimuth angle (φ_r) and sun azimuth angle (φ_i). The BRDF kernels are approximations made from physically based models, K_iso represents the isotropic scattering, and its value is constant (K_iso = 1). K_vol represents the volume scattering, which accounts for multiple scattering and reflection processes within the scattering medium, influenced by factors such as the shape, density, and arrangement of leaves [36]. K_geo represents the geometric-optical scattering that accounts for the impact of shadows and the geometrical structure caused by the shape and arrangement of objects [37]. This study uses a combination of Ross-Thick (RT) and Li-Sparse-Reciprocal (LSR) kernels, which have been validated favorably over other kernels or combinations. f_iso(λ), f_vol(λ), and f_geo(λ) are the spectrally dependent model parameters that represent the proportion of each of the three different scattering modes in the reflectance R. f_iso accounts for bidirectional reflectance when observing at the solar and viewing zenith angles of 0°.

Black-sky albedo (BSA) is defined as the ratio of reflected radiant flux over the hemisphere to the directionally incoming radiant flux. It is obtained by integrating the BRDF over all viewing angles. White-sky albedo (WSA) is defined as the fraction of the diffuse incoming flux that is reflected over the hemisphere. It is obtained by integrating the BSA over all incident angles. Note that the kernel integrals do not depend on the observations and may, therefore, be precomputed and stored. The kernel integrals at different SZAs are shown in

Table 1. The kernel integrals at different SZAs.

	SZA	0°	15°	30°	45°	60°
Integrals
hk_vol		−0.021079	−0.008762	0.031952	0.114396	0.270480
hk_geo		−1.287889	−1.29717	−1.32476	−1.36929	−1.42528
Hk_vol		0.189184
Hk_geo		−1.377622

. The h_k refers to the integration of volume scattering and geometric-optical scattering kernels over the viewing hemisphere (Equation (2)). H_k is the integration of h_k over the hemisphere of incoming directions (Equation (3)) and remains constant, unaffected by changes in SZA (Table 1). BSA and WSA are then given by Equations (4) and (5).

$$h_k\left({\theta }_i\right)={\displaystyle \frac{1}{\pi }}{{\displaystyle \int }}_0^{2\pi }{{\displaystyle \int }}_0^{{\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{K}_k\left({\theta }_i,{\theta }_r,\phi \right)\sin {\theta }_r\cos {\theta }_{r}d{\theta }_{r}d\phi ,$$

(2)

$$H_k=2{{\displaystyle \int }}_0^{{\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{h}_k\left({\theta }_i\right)\sin {\theta }_i\cos {\theta }_{i}d{\theta }_i,$$

(3)

$$BSA\left({\theta }_i, \lambda \right)={\displaystyle \sum }_{k=1}^3{f}_k\left(\lambda \right)h_k\left({\theta }_i\right),$$

(4)

$$WSA\left(\lambda \right)={\displaystyle \sum }_{k=1}^3{f}_k\left(\lambda \right)H_k,$$

(5)

Based on the kernel-driven, Ross-Thick/Li-Sparse-Reciprocal (RTLSR) semi-empirical BRDF model, the MODIS has been providing operational BRDF, albedo, and NBAR products (MCD43) since 2000 [7]. The global time-series MODIS BRDF dataset in 2015 and 2020 in the red and near-infrared (NIR) bands were used in this study. A uniform sampling was conducted at a 2° interval globally, resulting in approximately 15,000 sampling points that encompass most of the IGPB land-cover types. High-quality MODIS BRDF products for these points were selected for the years 2015 and 2020. Each year’s sample size is approximately 2 million, and these samples include a wide variety of surface reflectance anisotropy characteristics. The 2015 MODIS BRDF data have been used to establish the BRDF archetype database [33], please refer to Figure 5 of the previous study for more information about the distribution of the normalized MODIS BRDF in the red and the NIR bands. The distribution of the 2020 MODIS BRDF shows a similar pattern and is not further detailed here. In this study, we further developed a BRDF archetype LUT based on the data of 2015 to retrieve albedo from directional reflectance. The 2020 MODIS BRDF dataset was used to validate the accuracy of the LUT. The MODIS BSA under different SZAs is recalculated by the kernel-driven model based on the BRDF model parameter products and kernel integrals under various SZA (Table 1). The recalculated BSA and the MODIS WSA products are considered the true values of albedo.

2.2. BRDF Archetype Database

When establishing the BRDF archetype database based on the MODIS BRDF, firstly, AFX and PAFX are calculated according to the MODIS BRDF parameters. The calculation formulas are as shown in Equations (6) and (11). Then, using the iterative self-organizing algorithm, AFX and PAFX are separately classified, and the intersection classes of the two classifications are obtained. Finally, the mean of the corresponding normalized MODIS BRDF model parameters within each intersection class is taken as the BRDF feature of that class, which represents the corresponding BRDF archetype. The BRDF archetype database used in this study was established by Zhao [33], with nine BRDF archetypes in each band. Figure 2 illustrates the shapes of the MODIS BRDFs in each class and the corresponding BRDF archetype on the principal plane (PP) for the NIR band.

The BRDF normalization is the ratio of BRDF model parameters to the isotropic scattering parameter (f_iso), which can eliminate the influence of differences in reflectance magnitude on the BRDF [32]. The calculation formula is as shown in Equation (7), where F represents the normalized BRDF model parameters. The formula for calculating the AFX based on F is shown in Equation (8). By considering F_vol as the independent variable and F_geo as the dependent variable, Equation (8) can be rewritten as Equation (9). To more finely divide the two-dimensional plane space formed by F_vol and F_geo, we present Equation (10), which represents a line perpendicular to the line indicated by Equation (9). Based on this, the formula for calculating the PAFX from F (Equation (11)) is derived. For more information on constructing the BRDF archetype database, please refer to Zhao’s article [33].

$$\mathrm{AFX}\left(\lambda \right)={\displaystyle \frac{WSA\left(\lambda \right)}{f_{iso}\left(\lambda \right)}}=1+{\displaystyle \frac{f_{vol}\left(\lambda \right)}{f_{iso}\left(\lambda \right)}}{H}_v+{\displaystyle \frac{f_{geo}\left(\lambda \right)}{f_{iso}\left(\lambda \right)}}{H}_g,$$

(6)

$$F_{iso}\left(\lambda \right)=0.5,\hspace{1em}{F}_{vol}\left(\lambda \right)={\displaystyle \frac{f_{vol}\left(\lambda \right)}{2f_{iso}\left(\lambda \right)}}, F_{geo}\left(\lambda \right)={\displaystyle \frac{f_{geo}\left(\lambda \right)}{2f_{iso}\left(\lambda \right)}},$$

(7)

$$\mathrm{AFX}\left(\lambda \right)=1+2F_{vol}\left(\lambda \right)H_v+2F_{geo}\left(\lambda \right)H_g,$$

(8)

$$F_{geo}\left(\lambda \right)=-{\displaystyle \frac{H_v}{H_g}}{F}_{vol}\left(\lambda \right)+\left({\displaystyle \frac{AFX\left(\lambda \right)-1}{2H_g}}\right),$$

(9)

$$F_{geo}\left(\lambda \right)={\displaystyle \frac{H_g}{H_v}}{F}_{vol}\left(\lambda \right)+{\displaystyle \frac{\mathrm{PAFX}\left(\lambda \right)}{2}},$$

(10)

$$\mathrm{PAFX}\left(\lambda \right)=-{\displaystyle \frac{2H_g}{H_v}}{F}_{vol}\left(\lambda \right)+2F_{geo}\left(\lambda \right).$$

(11)

2.3. BRDF Archetype LUT for Albedo Retrieval from Directional Reflectance

Given the prior BRDF knowledge of surface anisotropic reflectance (F), the albedo (α) can be retrieved from directional reflectance ρ(θ_i, θ_r, φ) by first using a kernel-driven BRDF model (Equation (1)) and the F to simulate prior BRDF-based directional reflectance ρ′(θ_i, θ_r, φ) for the same solar and viewing angles as ρ. The prior BRDF-based albedo (α′) corresponding to F can be calculated using Equations (4) and (5). The ratio of the simulated directional reflectance (ρ′) to the actual observed reflectance (ρ) is then calculated. Finally, based on the assumption that the ratio of the prior BRDF-based albedo (α′) to the reference albedo (α) is equivalent to the ratio of the prior BRDF-based directional reflectance ρ′(θ_i, θ_r, φ) to the actual reflectance ρ(θ_i, θ_r, φ), the prior BRDF-based albedo (α′) is adjusted to obtain the sought albedo (α_l), expressed as Equation (12).

$${\mathsf{\alpha }}_l={\mathsf{\alpha }}^{\prime }\times \left(\rho \left({\theta }_i,{\theta }_r,\phi \right)/{\rho }^{\prime }\left({\theta }_i,{\theta }_r,\phi \right)\right).$$

(12)

There are differences between the prior BRDF knowledge (F) and the actual surface BRDF. The ratio of reflectance (ρ) to the simulated directional reflectance (ρ′) based on the BRDF archetype changes with solar and viewing angles. Consequently, the BSA and WSA retrieved using this method will vary with angles. Given the nine types of prior BRDF knowledge, we explored the applicability of each BRDF archetype for albedo retrieval at different solar and viewing angles. Using the 2015 MODIS BRDF product, we simulated directional reflectance for a given solar and viewing angle with the kernel-driven BRDF model. We then obtained nine sets of simulated surface albedo data using each BRDF archetype and Equations (1), (4), (5) and (12). These nine sets of albedo data were compared with the MODIS albedo product, and the RMSE and bias for each set were calculated (Equations (13) and (14)). The BRDF archetype with the least RMSE was considered the optimal BRDF archetype for that solar and viewing angle. Finally, we established a LUT for the optimal BRDF archetypes, corresponding to SZA ranging from 0 to 70° (at 1° intervals) and VZA ranging from 0° to 80° (at 2° intervals), respectively. The viewing zenith and azimuth angles of the LUT are shown in Figure 3, with a total of 5017 observation directions for a specific SZA.

$$\mathrm{RMSE}\left({\theta }_i,{\theta }_r,\phi \right)=\sqrt{{\displaystyle \frac{1}{n}}{{\displaystyle \sum }}_{i=1}^n{\left(\alpha -{\mathsf{\alpha }}_l\right)}^2},$$

(13)

$$\mathrm{Bias}\left({\theta }_i,{\theta }_r,\phi \right)={\displaystyle \frac{1}{n}}{{\displaystyle \sum }}_{i=1}^n\left({\mathsf{\alpha }}_l-\alpha \right).$$

(14)

To evaluate the improvement of the BRDF archetype LUT for albedo retrieval, we also directly compared the directional reflectance (ρ) with the albedo. This comparison assesses the effectiveness of the optimal BRDF archetype for a given direction. In the subsequent analyses, we used RMSE_a to describe the consistency between the LUT-based albedo (${\mathsf{\alpha }}_l$) and the reference albedo (α), while RMSE_r was used to describe the consistency between the directional reflectance (ρ) and the reference albedo (α).

In this study, we set thresholds for RMSE in the red and NIR bands. When the RMSE is less than this threshold, we consider the albedo retrieved through the BRDF archetype LUT to have relatively high accuracy. We then count the number of directions with RMSE below this threshold and divide it by the total number of directions included in the LUT to calculate the proportion of directions with high accuracy.

2.4. Multi-Angular Data

To further validate the accuracy of using the BRDF archetype LUT to convert directional reflectance to albedo, this study incorporates additional multi-angle reflectance and albedo data for verification. The data sources include the PROSAIL model-simulated multi-angle data, MODIS reflectance products, POLDER datasets, and ground- and airborne-based multi-angle observations.

We utilized the multi-angle reflectance data simulated by the robust canopy radiative transfer model, i.e., the PROSAIL model, to validate the ability of the BRDF archetype LUT for retrieving albedo through directional reflectance. The PROSAIL model can simulate canopy reflectance spectral in the optical domain from 400–2500 nm at a 1 nm resolution, as well as reflectance at arbitrary sun and viewing angles [38,39]. This study references the latest 4SAIL canopy BRDF model, which considers the hotspot effect, and the PROSPECT-5 model, which describes leaf optical reflectance and transmittance (http://teledetection.ipgp.jussieu.fr/prosail/, accessed on 10 August 2023) [40,41]. The leaf reflectance and transmittance can be calculated from leaf parameters and PROSPECT model. The input model parameters include leaf structure parameter (N), chlorophyll a and b content (Cab), carotenoid content (Car), brown pigment content (Cbrown), equivalent water thickness (Cw), and leaf mass per unit leaf area (Cm). The leaf reflectance and transmittance are then input into the SAIL model to calculate the canopy reflectance ρ. In this process, additional parameters are used, including leaf area index (LAI), average leaf angle (ALA), hotspot (Hspot), soil coefficient (Psoil), diffuse/direct radiation (SKYL), SZA, VZA, and relative azimuth angle (RAA).

A comprehensive simulation dataset containing 20,000 vegetation parameter combinations was generated using the Saltelli periodic function [42] via the uniform sampling of seven leaf and canopy parameters. This dataset was used to verify the leaf area index inversion from MODIS BRDF data [38,43]. For more details on the parameter ranges and step sizes, refer to the cited studies. Given a set of parameters, the reflectance in any direction can be simulated using the PROSAIL model. The corresponding BSA and WSA albedos can be calculated through integration of the directional reflectance over the viewing hemisphere or over both the illumination and the viewing hemisphere [10], with the following formulas:

$$BSA\_Prosail\left({\theta }_i\right)={\displaystyle \frac{1}{\pi }}{{\displaystyle \int }}_0^{2\pi }{{\displaystyle \int }}_0^{{\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\rho \left({\theta }_i,{\theta }_r,\phi \right)\sin {\theta }_r\cos {\theta }_{r}d{\theta }_{r}d\phi ,$$

(15)

$$WSA\_Prosail=2{{\displaystyle \int }}_0^{{\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}BSA\_Prosail\left({\theta }_i\right)\sin {\theta }_i\cos {\theta }_{i}d{\theta }_i.$$

(16)

MODIS surface reflectance (MOD09GA) at two periods (2021.101–2021.116 and 305–320) in tile h20v11 was also used in this study. This tile is located in the southern part of Africa, with the main surface types of grassland, open shrubland, savanna, and cropland/natural vegetation mosaic [33]. Only the first layer of the surface reflectance with high quality during the 16-day period was used. Figure 4 shows an example of the MODIS angular samplings at the two periods. The observations during the first period (Figure 4a) are close to the cross-principal plane (CPP) with the SZA around 45°, while the observations during the second period (Figure 4b) are located near the PP with the SZA range from 13° to 32°. For validation purposes, the high-quality MODIS albedo product (MCD43A3) from the ninth day within these periods was used as a reference. This reference was instrumental in assessing the accuracy of albedo retrievals derived from directional reflectance using the BRDF archetype LUT.

POLDER BRDF datasets [44] were also used to validate the accuracy of the LUT-based albedo. This dataset provides high-quality reflectance observations collected by the POLDER-3 sensor from January to December 2008, which was the year with the most continuous data collection. The observations were made under various viewing geometries and classified into 16 land cover types according to the International Geosphere–Biosphere Programme (IGBP), excluding water, representing different surface anisotropic characteristics. Figure 4c shows an example of the POLDER angular samplings. The pixels in this dataset were atmospherically corrected and cloud-filtered, containing only homogeneous land cover types. For this study, only the high-quality observations that have low aerosol optical thickness and high consistency with the kernel-driven BRDF model, from the red and NIR bands, were used to validate and analyze the possibility of using the BRDF archetype LUT to retrieval albedo from directional reflectance.

A total of 73 sets of ground- and airborne- based multi-angular observations were used to evaluate the applicability of the BRDF archetype LUT across different scales of observational datasets. Figure 4d shows the angular samplings of one dataset. The ground-based sensors included a calibrated Barnes Model 12-1000 Modular Multiband Radiometer (MMR) [45], Mark III three-band radiometer [29,46,47], Portable Apparatus for Rapid Acquisition of Bidirectional Observations of the Land and Atmosphere (PARABOLA) [14,48,49,50], and airborne sensors such as the POLDER instrument [51,52] and the Cloud Absorption Radiometer (CAR) [53]. Additionally, there was a set of satellite-based AVHRR data [54]. The land cover types included barren soil with varying roughness, sparsely vegetated grass, grass-like or broadleaf crops, and forests. Twenty-seven datasets were previously utilized for algorithm testing by Hu et al. [55], and sixty eight were employed to develop BRDF archetypes for a backup algorithm for the early operational MODIS BRDF/albedo products [54]. Sixty-nine datasets were used to establish the BRDF archetype database according to the theory of AFX [32].

For the latter two types of multi-angular data, we input these data into the kernel-driven BRDF model to calculate the model parameters, which are then used to determine the BSA and WSA for each dataset, which serve as the reference values. A constraining technique is applied to constrain negative parameters to zero and re-fitting a two-parameter model to meet the least-squares error function through iterative methods [7,13]. When the SZA corresponding to a particular observation data is less than 65 degrees, we use the BRDF archetype LUT to convert the directional reflectance into albedo. The reference values are then compared against the albedo values calculated using the directional reflectance and the BRDF archetype LUT, thereby exploring the applicability of the BRDF archetype LUT.

3. Results

3.1. The BRDF Archetype LUT for Albedo Retrieval from Directional Reflectance

Take the SZA of 45° and the view zenith angle (VZA) located at 55° backward on the PP as an example. Firstly, directional reflectance is simulated based on the 2015 MODIS BRDF product and the kernel-driven BRDF model. Then, nine BRDF archetypes are sequentially employed to the retrieval surface albedo based on the simulated directional reflectance. Figure 5 illustrates the scatter plot of the WSA retrieved from nine BRDF archetypes compared to the high-quality MODIS WSA product, demonstrating the influence of surface reflectance anisotropy on albedo retrieval from single-direction reflectance. Inappropriate prior BRDFs can lead to large errors; for instance, BRDF archetype A1P3 yields significantly lower results than the true values, corresponding to an RMSE of 0.116. Conversely, appropriate prior BRDFs result in most albedo values being similar to the true values. For example, BRDF archetype A3P3 yields an RMSE of only 0.057. Even when the RMSE reaches its minimum value, there are still some scattered samples, which is also due to significant differences between the anisotropic reflectance characteristics of the sample and the selected BRDF archetype. Since the corresponding number of samples is small, the impact on the final results is not significant.

This also indicates that although surface reflectance characteristics exhibit rich anisotropic patterns, for specific directions, using the BRDF archetype with the minimum RMSE can convert the directional reflectance into albedo with higher accuracy. It is also worth mentioning that other BRDF archetypes may yield equally high-precision results. For instance, in the NIR band, A2P2 also achieves high-precision results, with corresponding RMSE and bias values of 0.061 and −0.012, respectively. To achieve the objective of albedo retrieval through a BRDF archetype and directional reflectance, subsequent studies will use the BRDF archetype with the minimum RMSE as the prior BRDF knowledge in that direction.

Figure 6 illustrates the comparison between albedo retrieved from the BRDF archetype with the least RMSE and MODIS albedo in the NIR band. The first three observations are located on the PP with VZA of 55° in the backward hemisphere, 0°, and 45° in the forward hemisphere, respectively. The last observation is located in the CPP, with the VZA 60°. The SZA is set to 45°. To demonstrate the advantages of BRDF archetype inversion for albedo, the comparison between directional reflectance and MODIS albedo is also presented. In these four selected directions, the least RMSE between the albedo derived from the BRDF archetypes and MODIS albedo (RMSE_a) is smaller compared to the directional reflectance (RMSE_r). In directions where there is a significant difference between reflectance and albedo, the improvement of the BRDF archetype is more pronounced. For example, for the VZA of 55° in PP, the RMSE_r is 0.105, while the RMSE_a decreases to 0.057. Similarly, for the direction in the forward of PP with a VZA of 45°, the RMSE decreases from 0.061 to 0.042, with the corresponding bias tending towards 0 as well. Even in the direction with VZA of 60° on the CPP, where the difference between reflectance and albedo is minimal, the RMSE_a continues to decrease further. The results indicate that, at these directions, albedo retrieved based on BRDF archetype with least RMSE exhibits higher consistency with MODIS albedo compared to directional reflectance.

Figure 7 and Figure 8 illustrate the distribution of RMSE_r and RMSE_a over the entire viewing hemisphere under different SZAs. Contour lines of 0.025 or 0.045, which indicate a higher consistency, are also shown for the red and NIR bands. These two thresholds are determined after considering the scatter plot characteristics in Figure 5 and Figure 6, as well as the distribution characteristics of RMSE in the viewing hemisphere. The results demonstrate that the RMSE in the NIR band is slightly higher than in the red band, likely due to the typically higher reflectance in the NIR band. When the SZA is relatively small, directional reflectance exhibits higher consistency with albedo in many directions, except near the hotspot and at large VZAs. As the SZA increases, directions with higher consistency (smaller RMSE) tend to decrease and cluster around the CPP. Compared to directional reflectance, a larger proportion of directions exhibit higher consistency between the BRDF archetype-based albedo and MODIS albedo. Improved directions are primarily observed near the backward hotspot directions and forward dark-spot directions. However, there are still some directions where the RMSE_a is higher, such as those with larger VZAs in the forward and backward hemispheres when the SZA is large. This is related to the fact that, under large SZAs and VZAs, the kernel-driven BRDF model cannot flexibly capture the anisotropic reflectance characteristics of the Earth’s surface. Both of the WSAs and BSAs follow similar patterns.

According to the least RMSE, the corresponding BRDF archetype is taken as the prior BRDF knowledge for albedo retrieval in that direction. The distribution of the optimal BRDF archetypes over the viewing hemisphere under different SZAs for the BSA and WSA are shown in Figure 9 and Figure 10. There are obvious differences in the optimal BRDF archetype across different directions. The same BRDF archetype exhibits a continuous block-like distribution rather than a random one, indicating that the BRDF varies continuously and gradually within the viewing hemisphere. The proportion of different BRDF archetypes varies significantly. For example, the proportion of A1P3 in the LUT is relatively small, while A2P1 and A3P2 have a higher proportion. This is somewhat related to the anisotropic reflectance characteristics of the samples used in the study.

When the SZA is small, there are differences in the BRDF archetype LUT for WSA and BSA albedo, while the patterns of the LUT become more similar as the SZA increases. This is due to the differences in the kernel integrals used for BSA and WSA when the SZA is small (Table 1). The BRDF archetype LUTs across different bands also exhibit significant differences. Due to the spectral basis of the BRDF archetype database used in this study, the BRDF archetypes across different bands are not directly comparable. The BRDF archetype database across different bands exhibits a high degree of similarity. To explore the differences in the BRDF archetype LUTs across spectral bands, it may be considered to establish a new BRDF archetype database applicable to different spectral bands in future studies.

3.2. Analysis of BRDF Archetype LUTs Effectiveness Based on MODIS BRDF

3.2.1. Influence of the Number of BRDF Archetype

The study investigated the influence of the number of BRDF archetype on the accuracy of albedo retrieval from directional reflectance. Similar to many previous studies, we initially explore using the average of 2015 MODIS BRDF data as prior BRDF knowledge. The volume scattering and geometric optical coefficients of the mean BRDF archetype is 0.2276 and 0.0750 for the red band, and 0.2668 and 0.0520 for the NIR band. The 3D BRDF patterns with an SZA of 30° is shown in Figure 11. In the red band, the shape of BRDF resembles more of a dome shape. In the NIR band, there is slightly stronger volume scattering and weaker geometric optical scattering, with a shape more inclined towards bowl-shaped. In comparison, the prior BRDF knowledge is very close to one of the 3 × 3 BRDF archetypes in Figure 2, indicating that the average BRDF represents only specific BRDF styles.

Using this mean BRDF and simulated directional reflectance from MODIS BRDF in 2015, albedo is calculated and compared with MODIS albedo products at various SZAs. The RMSEs between the BRDF-archetype-based albedo and the MODIS albedo in the red and NIR band over the viewing hemisphere are shown in Figure 12. Compared to the results based on 3 × 3 BRDF archetype LUT (Figure 7 and Figure 8), the directions with smaller RMSEs showed a significant decrease. Using the mean BRDF as prior knowledge, the directions consistent with MODIS albedo show a significant reduction in the red band, and the accuracy in the backward hemisphere direction also significantly decreases in the NIR band.

Figure 13 shows the proportion of directions with RMSE_a below 0.025 and 0.045 in the red and NIR bands. The proportion of directions with a high accuracy of albedo based on the mean BRDF is always smaller compared to that of the 3 × 3 BRDF archetype LUT. In the red band, the proportion of high-accuracy directions based on the mean BRDF is even about 30% lower compared to the results from directional reflectance. Using the mean BRDF as prior knowledge may require further surface classification to effectively improve the accuracy of albedo retrieval from directional reflectance. When the SZA is less than 35°, the proportion of directions with smaller RMSEs is relatively large and stable. When the SZA exceeds 35°, this proportion begins to decline, but the proportion of albedo obtained through the 3 × 3 BRDF archetype LUT remains the highest. Furthermore, this study explored the impact of the different numbers of BRDF archetypes on the accuracy of albedo retrieval. Increasing the number of BRDF archetypes leads to a notable increase in the proportion of lower RMSEs. For example, when the LUTs includes 2 × 2 BRDF archetypes, the proportion of directions with smaller RMSEs significantly increases compared to the directional reflectance, especially in the red band. However, this proportion is still 5% smaller compared to using the LUTs based on 3 × 3 BRDF archetypes. The results also show that the percentage of high-accuracy directions for WSA retrieval using LUT based on the 6 × 1 BRDF archetypes (blue lines) is lower than those based on the 2 × 2 BRDF archetypes. This indicates that even though the number of BRDF archetypes is relatively small when incorporating both AFX and PAFX, the accuracy of the albedo obtained from the LUT is higher. This further demonstrates the necessity of refining BRDF characteristics using both AFX and PAFX. When further increasing the number of BRDF archetypes, such as to 5 × 5, the increase in proportion is limited. This suggests that BRDF archetype LUT established based on 3 × 3 BRDF archetypes can be used accurately for albedo retrieval from directional reflectance. Subsequent research will continue to use the BRDF archetype LUT based on 3 × 3 BRDF archetypes.

3.2.2. Validation Based on MODIS BRDF

To further investigate the accuracy of the LUTs based on 3 × 3 BRDF archetype, verification was conducted using 2020 MODIS BRDF data. Initially, directional reflectance under different observation geometries was simulated using the MODIS BRDF and kernel-driven BRDF models. Corresponding BRDF archetypes were then selected from the LUTs based on the observation geometry. The BRDF archetype was adjusted according to the simulated directional reflectance to retrieve albedo. Finally, the results were compared with MODIS albedo products, and the RMSE between them was calculated. The RMSE under different SZAs is illustrated in Figure 14. The results show a similar trend to Figure 7 and Figure 8, where the proportion of directions with higher albedo consistency is relatively large when the SZA is small. Conversely, when the SZA is large, the directions that have higher albedo consistency are mainly concentrated around the CPP, especially in the red band. This further underscores the feasibility of using BRDF archetype LUTs for albedo retrieval from directional reflectance.

The MODIS BRDF data of 2015 and 2020 both exhibit clustering characteristics, with model parameters displaying a discrete distribution pattern with one or more clustering centers.

Table 2. The proportion of each archetype class for the MODIS BRDF of 2020.

	Red (%)			NIR (%)
	P1	P2	P3	P1	P2	P3
A1	19.5	12.2	1.1	15.7	13.4	1.2
A2	11.8	17.5	6.1	15.5	18.2	6.7
A3	6.9	8.6	16.4	4.1	6.5	18.9

shows the proportion of each archetype for the MODIS BRDF of 2020. The results indicate that the archetype classes A1P1, A1P2, A2P1, A2P2, and A3P3 constitute a relatively large proportion, accounting for over 76% of the total, while the A1P3 archetype class has the smallest proportion, at only about 1%.

Therefore, the statistical RMSE in Figure 14 may be constrained by the number of samples and the characteristics of their BRDF features. We further apply the BRDF archetype LUTs to MODIS BRDF data within each BRDF archetype class to explore the applicability of the LUTs under specific anisotropic reflection characteristics. Based on the LUTs and simulated directional reflectance, albedo can be retrieved. The RMSE_a over the viewing hemisphere for each BRDF archetype class can be calculated. The distribution of RMSE_a for the nine BRDF archetype classes in the NIR band over the viewing hemisphere at an SZA of 45° is shown in Figure 15. To further illustrate the improvement of BRDF archetypes in albedo retrieval from directional reflectance, Figure 16 provides the distribution of RMSE_r over the viewing hemisphere within each BRDF class.

Table 3. The proportion of directions where RMSE_a (RMSE_r) is less than 0.025 and 0.045 in the red and NIR bands.

	Red (%)			NIR (%)
	P1	P2	P3	P1	P1	P3
A1	55.2 (44.1)	52.0 (41.3)	43.4 (37.5)	68.2 (46.5)	62.7 (36.2)	35.8 (25.2)
A2	64.3 (49.6)	68.0 (39.9)	63.0 (46.1)	65.2 (83.8)	97.6 (47.1)	70.6 (34.3)
A3	57.2 (91.1)	90.6 (58.3)	83.2 (66.7)	72.8 (94.0)	94.8 (70.6)	75.0 (47.7)

gives the proportion of directions where RMSE_r or RMSE_a is less than 0.025 and 0.045 in the red and NIR bands over the viewing hemisphere under an SZA of 45°.

The results indicate that BRDF archetype classes with strong anisotropic reflection characteristics (e.g., A1 and P3 series classes) exhibit a significant impact of surface anisotropic reflection characteristics, resulting in notable differences between directional reflectance and albedo. For instance, in the A1 and P3 series classes in the NIR band, the maximum percentage of directions with RMSE_r less than 0.045 is 47.7%, while the A1P3 class has the smallest proportion at only 25.2%. After correction using the BRDF archetype LUTs, the proportion of directions with lower RMSE significantly increases. Specifically, in the A1P3 class, the proportion slightly improves to 35.8%, with the distribution still concentrated mainly in the direction around the CPP. In the A3P1 class in the red band and the A2P1 and A3P1 classes in the NIR band, the proportion of directions with an RMSE_a smaller than 0.045 is actually smaller compared to that of directional reflectance. The BRDF archetype LUTs have an opposite effect in these classes. The statistical analysis based on all the data indicates that the archetype-based LUT can improve the accuracy of albedo inversion in more directions. This is due to the fact that the reflectance anisotropic characteristics of these BRDF archetype classes are either excessively weak or pronounced, compounded by the insufficient number of samples for these classes. The statistical approach places greater emphasis on the reflectance anisotropic characteristics of BRDF archetype classes with larger sample sizes. Since the number of samples for these classes is relatively small, their impact on the overall statistical results is not significant. In the remaining BRDF archetype classes, there are varying degrees of improvement in the consistency between MODIS albedo and LUT-based albedo. In the absence of a better method for connecting the anisotropic reflectance characteristics to a specific pixel, using BRDF archetype LUT to retrieve albedo from directional reflectance can ensure improved retrieval accuracy over a wide range of observations.

3.3. Validation Based on Multi-Angular Observations

3.3.1. Validation Based on PROSAIL

The PROSAIL model was used to simulate the reflectance in the viewing hemisphere under different SZAs for 20,000 sets of parameter combinations. Figure 17a–c presents the distribution of RMSE_r between the directional reflectance and PROSAIL albedo over the whole viewing hemisphere in the NIR band. The results show that when the SZA is 15°, the high consistency directions are mainly in the backward direction of the viewing hemisphere. When the SZA is 45°, the high consistency directions are only near the CPP. As the SZA further increases, the consistency at larger VZA areas will further decrease. There is a significant difference between the directional reflectance and albedo with the RMSE_r less than 0.045 accounting for 32.2%, 35.9%, and 21.8%, respectively. After being corrected with the BRDF archetype LUTs, the RMSE_a (Figure 17d–f) decreases in more directions, and the proportion of directions with an RMSE_a less than 0.045 is 63.8%, 46.5%, and 30.0%.

It is worth noting that near the nadir direction, the RMSE_r corresponding to the LUT-based albedo remains relatively large. This is mainly due to the significant differences between the PROSAIL model and the kernel-driven model near the nadir direction [56]. On the other hand, the improvement effect of LUTs on the reflectance simulated by the PROSAIL model is not as significant as that on MODIS reflectance, which is related to the PROSAIL model’s more detailed characterization of vegetation’s anisotropic reflectance characteristics. In contrast, the BRDF archetype LUTs are established based on all kinds of surface types. Overall, based on the BRDF archetype LUTs, a greater number of the directions of the PROSAIL model’s simulated reflectance can be accurately converted to albedo. This demonstrates the broad applicability of the BRDF archetype LUTs in the retrieval of albedo from directional reflectance.

3.3.2. Validation Based on Actual Observations

The directional reflectance simulated by the kernel-driven BRDF model and the PROSAIL model is ideal, while actual observations contain various kinds of noise. To fully validate the applicability of the BRDF archetype LUT, we also applied it to various observational multi-angle datasets, including MODIS reflectance datasets, POLDER datasets, and ground- and airborne-based multi-angle datasets. However, these observations are only discretely distributed in the viewing hemisphere; therefore, the effects of solar and viewing zenith and azimuth angles are not distinguished here. Only the overall effects are considered.

The correspondence between the MODIS directional reflectance dataset and MODIS albedo is shown in Figure 18. The results indicate a high similarity in the red band, with an RMSE of only 0.014 (CPP) and 0.031 (PP). By comparison, the differences in observations located on the PP are slightly larger. This may be because the anisotropic reflectance characteristics are more pronounced when observations are on the PP. In the NIR band, however, the differences are more significant, with some reflectance values noticeably greater than reflectance and others noticeably smaller, corresponding to an RMSE of 0.035 (CPP) and 0.042 (PP). This discrepancy arises from observations distributed within both the forward and backward hemispheres, where typically reflectance in the backward hemisphere is higher than that in the forward hemisphere. Converting these reflectance values to albedo using the BRDF archetype LUT significantly improves the consistency with the MODIS albedo. The RMSE in the red and NIR bands is 0.012 and 0.020 for the observations close to CPP (Figure 18b,d), and 0.024 and 0.028 for the observations close to PP (Figure 18f,h), both less than the directional reflectance situation. This indicates that the BRDF archetype LUT significantly eliminates the influence of surface anisotropic reflectance characteristics when retrieving albedo from directional reflectance.

The validation results based on the POLDER dataset and ground measurements are illustrated in Figure 19 and Figure 20. For the POLDER dataset, the comparison between reflectance and albedo reveals a negative bias, indicating that reflectance is generally lower than albedo. In the NIR band, the distribution of the sample is relatively scattered, with observations located both above and below the 1:1 line. This is caused by the different viewing geometries of the observations. The RMSE values for the red and NIR bands are 0.04 and 0.057 (Figure 19a,c), respectively. Following computation using the BRDF archetype LUT, the consistency between the retrieved albedo and the POLDER albedo is higher, with corresponding RMSEs of 0.033 and 0.039 (Figure 19b,d), respectively, and the biases also tend to approach 0. For ground observation data, the number of observations is relatively smaller, the comparison results indicate that the RMSEs between the reflectance and albedo retrieved from multi-angular observations are 0.025 and 0.067 in the red and NIR bands (Figure 20a,c). The albedo calculated using the BRDF archetype LUTs shows higher consistency with the albedo based on multi-angular observations, with more data clustered around the 1:1 line. The corresponding RMSEs are 0.02 and 0.038 (Figure 20b,d), with a noticeable improvement in the NIR band. This demonstrates that the BRDF archetype LUTs can enhance the accuracy of albedo estimation from POLDER and ground directional observations.

Figure 19 and Figure 20 also indicate that for some multi-angular observations of the same target, the range of albedo calculated using the BRDF archetype LUTs may be larger. This could be partly due to the errors in the observations. Additionally, while the BRDF archetype LUT is applicable for statistical analysis across a large dataset, its ability to retrieve albedo from individual, specific multi-angular datasets may be less accurate.

4. Discussion

Satellite remote sensing provides a feasible means for acquiring extensive surface data. However, satellite observations are typically from a single direction. Due to the anisotropic characteristics of surface reflectance, data collected at different times are often not comparable. Surface anisotropic reflectance features significantly impact the inversion of surface albedo from reflectance. Numerous studies have shown that neglecting these features can introduce substantial errors into albedo retrievals [29]. This study also demonstrates that the albedo results derived from the same directional reflectance using different prior BRDF knowledge can vary significantly. Inappropriate BRDF information can lead to considerable errors.

Extracting prior BRDF knowledge from coarse-resolution BRDF products to improve surface albedo inversion under insufficient observational data has been widely applied and has achieved relatively accurate results. This study, based on a BRDF archetype database established using the MODIS BRDF product, explores the applicability of various BRDF archetypes under different sun-viewing angles and proposes a method for directly inverting albedo from directional reflectance. Validation shows that selecting different BRDF archetypes as prior knowledge for different directions can effectively eliminate the impact of surface reflectance anisotropy, thereby improving the accuracy of surface albedo inversion. Compared to using specific prior BRDF knowledge obtained by averaging, the BRDF archetype LUT established from a 3 × 3 BRDF archetype database can fully account for the anisotropic reflectance characteristics of surfaces, significantly enhancing the accuracy of albedo retrievals from directional reflectance. The LUT derived from the MODIS BRDF archetype database successfully converts directional reflectance into albedo with high accuracy when applied to reflectance data simulated by the PROSAIL model, as well as to coarser resolution POLDER and MODIS observations, and ground- and airborne-based multi-angular data.

Previous studies that utilized prior BRDF knowledge to invert albedo from single directional reflectance typically focused on establishing relationships between coarse-resolution prior BRDF knowledge and high-resolution surface data. This was often achieved by constructing relationships between different scale data through surface types or spectral indices like vegetation indices [20,21,22]. However, these approaches have some limitations. First, the relationship between surface anisotropic reflectance characteristics and spectral indices requires further exploration. Second, these studies generally obtained a specific prior BRDF knowledge pattern through statistical means. Third, the data used in the study were close to the nadir direction, with insufficient consideration given to the effects of variations in solar and viewing angles. The prior BRDFs extracted from coarse-resolution BRDF products are always mixed, and our study found that using averaged BRDFs might lead to worse results than the Lambertian assumption. Moreover, studies that classify surface anisotropic reflectance characteristics have mainly focused on extracting different patterns of surface anisotropic reflectance and examining their effects on albedo [32,35]. The challenge remains in how to combine these surface anisotropic reflectance characteristics with actual surface targets. This study builds on the research of classifying surface anisotropic reflectance characteristics and explores the applicability of various BRDF archetypes for albedo inversion under different sun-viewing angles. By fully utilizing multiple surface anisotropic reflectance characteristics, this approach avoids the limitations of surface classification and simplifies the computational process. Overall, the BRDF archetype LUT provided in this study can improve the effects of surface anisotropic reflectance characteristics in most cases and accurately convert directional reflectance into albedo.

There are still many comparisons that need improvement in this study. For instance, the normalization of BRDFs using isotropic parameters effectively eliminates the effects of spectral differences, but a separate BRDF archetype database was still created for different spectral bands. The similarity of these BRDF archetype databases across different bands suggests the possibility of developing a unified BRDF archetype database that applies to multiple bands. Furthermore, the method based on the BRDF archetype LUT, while suitable for inverting albedo from directional reflectance over large areas, assumes that the distribution of the surface anisotropic of the data exhibits clustering characteristics. This assumption may not always hold true, leading to suboptimal inversion results for specific BRDFs. Additionally, the error in the reflectance data is fully delivered to the albedo in the proposed method, necessitating highly accurate surface reflectance measurements.

To address these limitations, several improvements can be considered. Simplifying BRDF archetypes and developing a unified set of surface anisotropic reflectance patterns applicable to different spectral bands could enhance computational efficiency. Additionally, refining the BRDF archetype database by creating specific database for particular surface types and periods could further narrow the range of BRDF variations. This would reduce discrepancies between prior BRDF knowledge and actual surface anisotropic reflectance characteristics, thereby improving the accuracy of surface albedo inversion.

5. Conclusions

Building on the quantitative classification of surface anisotropic reflectance characteristics, this study explored the applicability of various BRDF archetypes under different solar and viewing angles. A BRDF archetype LUT was established to convert directional reflectance to albedo. The feasibility of using the BRDF archetype LUT for albedo inversion from directional reflectance was validated using the 2020 MODIS BRDF products, PROSAIL model-simulated multi-angular data, and various ground and satellite multi-angular measurements. The results are as follows:

(1)

The surface anisotropic reflectance characteristics represented by the BRDF archetypes cause notable differences in albedo retrieval from directional reflectance. In any given direction, there is always one BRDF archetype that provides the highest inversion accuracy. This characteristic can be used to establish a BRDF archetype LUT for albedo inversion under various solar and viewing conditions.

(2)

The improvement achieved using a specific prior BRDF knowledge based on average methods was limited and may even have adverse effects. The BRDF archetype LUT, established using a 3 × 3 BRDF archetype database that fully considers surface anisotropic reflectance characteristics, significantly improved albedo inversion from directional reflectance. Further, increasing the number of BRDF archetypes showed limited additional improvement.

(3)

Validation with the 2020 MODIS BRDF data indicated that the effectiveness of the BRDF archetype LUT is influenced by the BRDF characteristics of the sample data. The BRDF archetypes in the LUT for each direction represent the dominant BRDF characteristics of the statistical data. When applying the BRDF archetype LUT to datasets with significantly different anisotropic characteristics from the prior BRDF, both the degree of improvement and the proportion of directions yielding optimal results decreased.

(4)

When applied to directional reflectance data from the PROSAIL model simulations and ground- or satellite-based measurements, the BRDF archetype LUT showed high consistency between the inverted albedo and the reference albedo. This demonstrates that the BRDF archetype LUT is suitable for albedo inversion from directional reflectance across different scales.

In summary, this study, based on the quantitative classification of surface anisotropic reflectance characteristics, explored the accuracy of albedo inversion from directional reflectance using different BRDF archetypes under various solar and observational geometries. This provides a new approach for albedo inversion from directional reflectance.