Characterization of Scale Effects and Determination of Optimal Observation Scales for Bidirectional Reflectance in High-Resolution Remote Sensing of Land Surfaces

Highlights

What are the main findings?

• In-depth characterization of the spatial scale effects inherent in high-resolution surface BRDF.
• Explicit analysis of the linkage between BRDF scale dependence and surface spatial heterogeneity.
• Quantitative determination of optimal observation scales for distinct terrestrial targets.

What are the implications of the main findings?

• Construction of prior BRDF knowledge for typical land features at their optimal scales to fulfill diverse remote sensing applications.
• These findings advance the practical application of remote sensing technology, offering significant implications for enhancing target detection accuracy and the authenticity of remote sensing products.

Abstract

Land surface bidirectional reflectance distribution functions (BRDF) are critical for quantitative remote sensing but are significantly constrained by scale effects, limiting the interoperability of multi-resolution data and the accuracy of quantitative inversion, thereby rendering the investigation of BRDF multi-scale effects increasingly urgent. This study utilized UAV (Unmanned Aerial Vehicle)-based multi-angular observations and the RPV model to retrieve the BRDF of typical land covers, employing the Window Averaging Method to simulate multi-scale responses and systematically investigate the relationship between BRDF characteristics and spatial scale. The results indicate the following key findings: (1) The RPV (Rahman–Pinty–Verstraete) model demonstrated high robustness and inversion accuracy, yielding RMSE (Root Mean Square Error) below 0.06 and RRMSE (Relative RMSE) below 25% across all land covers, with the 840 nm band exhibiting superior performance. (2) Significant spatial scale effects were observed, where BRDF characteristics varied distinctively with scale but eventually stabilized at specific thresholds; specifically, the stabilization scales were identified as 1.3 m for bare soil, 1.5 m for tea plantations, 1 m for rice, and 2 m for forests. (3) The scale evolution of BRDF features exhibited a parallel trend with spatial heterogeneity, a correlation that enables the quantitative identification of optimal observation scales for different land cover types.

1. Introduction

In nature, the vast majority of terrestrial surfaces do not reflect incident radiation uniformly in all directions; rather, they exhibit distinct bidirectional reflectance characteristics. This phenomenon, characterized by significant variations in reflected intensity as a function of solar incidence and View Zenith angle (VZA), constitutes one of the most fundamental macroscopic properties of the natural world. To quantify the intrinsic scattering behavior of surface targets, the BRDF was formally defined [1]. As a pivotal parameter encapsulating both directional spectral signatures and spatial structural information, BRDF has become an indispensable factor in various remote sensing applications, including albedo retrieval [2], atmospheric correction [3], vegetation parameter inversion [4], and land cover classification [5].

Over the past few decades, medium-to-coarse resolution satellite missions, such as MODIS and MISR, have facilitated large-scale BRDF observations, providing foundational support for the investigation of bidirectional reflectance characteristics. However, the application of high-resolution remote sensing data remains constrained by critical challenges: the laws governing BRDF variations across resolution scales remain unclear for diverse land covers, and there is a lack of systematic criteria for determining the optimal observation scale for quantitative retrieval. Consequently, models established at specific scales struggle with cross-scale generalizability, introducing significant uncertainties into the retrieval of key parameters such as Transpiration (T) [6], Leaf Area Index (LAI) [7], and Forest Aboveground Biomass (AGB) [8].

The scale effect has long garnered significant attention within the international remote sensing community and remains a pivotal issue in quantitative remote sensing [9,10,11]. Regarding BRDF scale effects, Liang (2003) [12] investigated variations across scales ranging from 30 m to 1 km, identifying a linear variation pattern. Jiao et al. (2018) [13] compared BRDF discrepancies between MODIS and POLDER data to support the optimized utilization of these products. However, these studies have predominantly focused on coarse- and medium-resolution data, lacking systematic investigations into diverse land covers at high spatial resolutions. In the domain of UAV-based BRDF applications, Deng et al. (2021) [14] utilized UAV oblique multi-angular remote sensing to analyze the impacts of retrieval models, observation counts, and spatial scales on BRDF inversion accuracy. Ye et al. (2025) [15] employed UAV imagery to demonstrate that the NDVI (Normalized Difference Vegetation Index) of natural and artificial grasslands decreases as spatial resolution increases. Xie et al. (2025) [16] verified the dependence of BRDF retrieval accuracy on spatial resolution and the 3D structure of targets using a resampling approach. In summary, existing research is characterized by the following limitations:

• Lack of systematic comparison: A systematic comparison of BRDF scale variation laws for diverse typical land covers under a unified technical framework remains absent at high spatial resolutions.
• Missing mechanistic link: The determination of optimal observation scales has not yet established a linkage mechanism between BRDF characteristics and spatial heterogeneity.
• Insufficient land cover representation: The limited coverage of land cover types fails to fully account for the structural differences between vegetation and bare soil, making it difficult to support diverse remote sensing application requirements.

Focusing on four typical land covers—bare soil, tea plantations, rice, and forests—this study elucidates the spatial scale variation laws of BRDF and establishes a quantitative method for determining the optimal observation scale. The remainder of this paper is organized as follows: Section 2 details the study area, data acquisition and processing workflows, and core research methodologies. Section 3 analyzes the fitting accuracy of the RPV model, the characteristics of BRDF scale variation, and differences in spatial heterogeneity. Section 4 discusses the driving mechanisms underlying scale variations and the rationality of the determined optimal scales. Section 5 summarizes the research findings, addresses limitations and future perspectives, and provides a scientific reference for enhancing the accuracy of high-resolution quantitative remote sensing retrieval.

2. Data and Methods

2.1. Data Acquisition

The UAV remote sensing system serves as a pivotal instrument for precision remote sensing monitoring, primarily consisting of a UAV flight platform and a sensor payload. In this experiment, the DJI Matrice 350 RTK (Shenzhen Dajiang Innovation Technology Co., Ltd., Shenzhen, China) platform and the MS600 multispectral camera (YUSENSE, Qingdao, China) were employed for field data acquisition, as shown in Figure 1. The system integrates a Global Positioning System (GPS) and an Inertial Measurement Unit (IMU). It utilizes a high-precision servo gimbal (with an angular accuracy of ±0.1°) to dynamically adjust the sensor’s viewing direction, facilitating the acquisition of user-defined angular sequences. Detailed technical specifications are presented in

Table 1. Technical parameter table of the DJI Matrice 350 RTK platform.

Parameter	Index
Hovering Accuracy (Windless or light wind)	Vertical: ±0.1 m Horizontal: ±0.1 m
RTK Positioning Accuracy (RTK FIX)	Vertical: 1.5 cm + 1 ppm Horizontal: 1 cm + 1 ppm
Max Angular Velocity	Pitch: 300°/s Yaw: 100°/s
Max Pitch Angle	30°
GNSS	GPS + GLONASS + BeiDou + Galileo

and

Table 2. Technical parameter table of the MS600 multispectral camera.

Parameter	Index
Spectral Channels	450 nm@35 nm 555 nm@25 nm 660 nm@20 nm 720 nm@10 nm 750 nm@15 nm 840 nm@35 nm
Ground Sample Distance (GSD)	7.5 cm@h120 m
Image Footprint	96 m × 72 m@h120 m
Field of View (FOV)	HFOV: 43.6° VFOV: 33.4°
Image Resolution	1280 × 960
Integration Time	1/200–1/500
Gain	16

.

Implementing multi-angular Earth observation via UAV systems entails addressing multifaceted technical requirements, including flight path planning, sensor parameter control, and adaptation to diverse application scenarios. Given that the vertical extent of the targets in this study was negligible compared to the UAV flight altitude, a unified spiral descent flight path was adopted, as illustrated in Figure 2. This approach constitutes a hemispherical multi-angular observation mode, enabling the effective acquisition of spectral reflectance signatures across diverse VZAs and full View Azimuth angles (VAAs). The observation principles are detailed as follows:

• Full 0° to 360° coverage is achieved with a sampling interval of 45°;
• Five zenith angles are sampled within the 0° to 60° range at 15° intervals, with the nadir observation corresponding to 0°;
• Throughout the observation process, a constant distance is maintained between the UAV and the target center. The sensor inclination is dynamically set based on the relative height, radius, and imaging resolution of the target area; this is achieved by coordinating the UAV’s hovering position with the gimbal system, ensuring the lens remains consistently oriented towards the target.

This study selected four typical land covers (bare soil, rice, tea plantations, and forest) located in Liyang, Changzhou City, Jiangsu Province, as the observation targets. The study area and the spatial distribution of these targets are illustrated in Figure 3. The experiments were conducted under ideal observational conditions characterized by clear skies, low wind speeds, sufficient illumination, and the absence of significant occlusion. Specific observation parameters are detailed in

Table 3. The record table of UAV flight experiments.

Task	Land Cover	Time	SZA(°)	SAA(°)	Spatial Resolution (m)
1	Bare Soil	17 October 2023 14:21:52–14:36:11	55	230	0.17
2	Tea Plantation	16 October 2023 12:26:10–12:38:40	40	200	0.15
3	Rice	16 October 2023 14:12:14–14:25:11	55	230	0.14
4	Forest	18 October 2023 10:12:25–10:23:44	45	150	0.16

. By optimizing the UAV flight path and sensor viewing angles, the onboard multispectral camera acquired a minimum of 33 data collections for each target area, effectively securing multi-angular, high-resolution multispectral imagery. Concurrently, the research team conducted synchronous field measurements of the geometric structural characteristics of the targets. These systematic and comprehensive in situ measurements not only fully characterize the geometric attributes of the different land covers but also provide robust data support for subsequent quantitative analysis, model parameterization, investigation of scale effects, and mechanistic elucidation. The specific contents and parameters of these measurements are listed in

Table 4. Field-measured geometric structural parameters of the target land covers.

Land Cover	Structural and Biophysical Parameters
Bare Soil	Type: Plowed and leveled farmland Roughness: RMS height = 5 mm; Surface relief = 17 mm Moisture: Volumetric Water Content (VWC) = 23%
Tea Plantation	Height: 0.7 m Crown Diameter: 1 m Row Spacing: 1.2 m LAI: 2.3
Rice	Height: 0.5 m Density: 100 plants/m2 LAI: 3.1
Forest	Type: Single broad-leaved forest Height: 17 m Crown Diameter: 3.4 m Row Spacing: 3 m LAI: 3.8

.

2.2. Calculation of Directional Reflectance Factor

During the data acquisition process, remote sensing sensors are inevitably susceptible to interference from multiple factors, such as sensor dark current, optical vignetting, and signal gain bias. These factors introduce extraneous noise and imaging artifacts into the raw data, obscuring the true signal of the target features. To accurately retrieve the true radiance and directional reflectance of the surface targets, it is essential to perform rigorous radiometric correction to eliminate these distortions and noise from the raw spectral imagery. The specific processing workflow is illustrated in Figure 4a.

UAVs are equipped with high-performance GPS and IMU systems to acquire the approximate position and attitude of the camera simultaneously with multi-view image capture [17]. Commercial software packages, such as Pix4D (v4.3) and Agisoft Metashape Professional (v2.1.0), employ Bundle Block Adjustment (BBA) algorithms based on photogrammetric principles to refine the initial low-accuracy spatial data. This process yields camera projection center coordinates with centimeter-level accuracy and principal optical axis orientations with an accuracy superior to 0.01°. Consequently, high-precision Digital Orthophoto Maps (DOMs) and Digital Surface Models (DSMs) of the target area are generated, as shown in Figure 4c specifically. Based on the precise spatial coordinates of the projection center (Bo, Lo, Ho) and the object point (Ba, La, Ha) within the geodetic coordinate system, the VZA (${\theta }_v$) and VAA (${\varphi }_v$) of the remote sensing sensor can be derived using trigonometric geometric relationships.

$${\theta }_v=arctan\left({\displaystyle \frac{H_o-H_a}{\sqrt{{\left(B_o-B_a\right)}^2+{\left(L_o-L_a\right)}^2}}}\right)$$

(1)

$${\varphi }_v=\left\{\begin{array}{c}arctan\left({\displaystyle \frac{\left|L_o-L_a\right|}{\left|B_o-B_a\right|}}\right) L_o-L_a>0, B_o-B_a>0\\ \pi -arctan\left({\displaystyle \frac{\left|L_o-L_a\right|}{\left|B_o-B_a\right|}}\right) L_o-L_a>0, B_o-B_a\le 0\\ \pi +arctan\left({\displaystyle \frac{\left|L_o-L_a\right|}{\left|B_o-B_a\right|}}\right) { L}_o-L_a\le 0, B_o-B_a\le 0\\ 2\pi -arctan\left({\displaystyle \frac{\left|L_o-L_a\right|}{\left|B_o-B_a\right|}}\right) L_o-L_a\le 0, B_o-B_a>0\end{array}\right.$$

(2)

Driven by the relative motion between the Earth and the Sun, the solar position varies seasonally and diurnally. This position is typically characterized by the Solar Zenith angle (SZA) and Solar Azimuth angle (SAA) within a horizontal coordinate system [18]. Such variations not only induce changes in the radiant intensity reaching the Earth’s surface but also result in distinct illumination geometries. While the solar position can be measured instrumentally, it is commonly calculated using astronomical algorithms; specifically, the SZA (${\theta }_s$) and SAA (${\varphi }_s$) are derived based on the geographic coordinates (Ba, La, Ha) of the observation target and the data acquisition time. Building upon these principles, the Sun–Target–Sensor geometric model is established, as illustrated in Figure 5.

2.3. BRDF Inversion Models and Characteristic Parameters

2.3.1. RPV Model

The RPV model [19] is a nonlinear, semi-empirical BRDF model characterized by robust fitting capabilities, a streamlined mathematical form, minimal parameterization, and computational efficiency. It is primarily employed to simulate the reflectance of non-flat vegetation canopies exhibiting significant backscattering behavior. Furthermore, the model is applicable to anisotropy studies across varying spatial scales, ranging from the centimeter scale in laboratory settings [20,21] to the kilometer scale in spaceborne applications [22]. The RPV model utilizes specific parameters to describe the surface directional reflectance factor, $R\left({\theta }_v,{\varphi }_v{,\theta }_s{,\varphi }_s\right)$, for a given illumination direction ${(\theta }_s{,\varphi }_s)$ and viewing direction (${\theta }_v,{\varphi }_v$), as expressed below:

$$R({\theta }_v,{\varphi }_v{,\theta }_s{,\varphi }_s)={\rho }_0\ast {\displaystyle \frac{{cos}^{k-1}{\theta }_v\ast {cos}^{k-1}{\theta }_s}{{\left(\mathit{cos}{\theta }_v+\mathit{cos}{\theta }_s\right)}^{1-k}}}\ast F(g)\ast R(G)$$

(3)

$$F\left(g\right)={\displaystyle \frac{1-{\theta }^2}{ {[1+{\theta }^2-2\theta cos(\pi -g\left)\right]}^{1.5}}}$$

(4)

Here, the parameter ${\rho }_0$ represents the amplitude of the surface reflectance intensity, ranging from 0 to 1. The parameter $k$ characterizes the overall shape of the BRDF; specifically, values of $k$ > 1 indicate a bell-shaped pattern, while $k$ < 1 corresponds to a bowl-shaped pattern. The parameter $\theta$ governs the asymmetry between forward scattering (0 ≤ $\theta$ ≤ 1) and backscattering (−1 ≤ $\theta$ ≤ 0), describing the angular distribution of solar radiation intensity scattered by the surface targets. In the aforementioned equations, the phase angular $g$ is defined as:

$$\mathit{cosg}=\mathit{cos\theta vcos\theta s}+\mathit{sin\theta vsin\theta scos}(\varphi v-\varphi s)$$

(5)

A simple function is employed to characterize the hotspot effect:

$$R(G)={\displaystyle \frac{1-{\rho }_o}{{\rho }_c+G}}$$

(6)

In the original formulation of the RPV model, the hotspot parameter (${\rho }_c$) is fixed at 1, representing the minimum state where the hotspot effect is effectively neglected. However, when specific observations covering the hotspot region are available, this parameter is employed to explicitly regulate the hotspot effect within the backscattering domain. Subsequent research by Koukal et al. (2014) [23] has confirmed that the 4-parameter version of the RPV model provides a significantly more accurate fit to observational data compared to the corresponding 3-parameter version. In the preceding equation, the geometric factor $G$ is given by:

$$G=\sqrt{tan2\theta v+tan2\theta s-2tan\theta vtan\theta scos(\varphi v-\varphi s)}$$

(7)

It can be deduced from the above equations that when $\theta s$ = $\theta v$ and $\varphi s$ = $\varphi v$—specifically within the hotspot region—the geometric factor $G$ becomes zero. At this point, the function $R\left(G\right)$ attains its maximum value, resulting in a consequent increase in the total reflectance.

2.3.2. BRDF Characteristic Parameters

BRDF characteristics are typically defined by their overall shape (e.g., spindle-shaped, bowl-shaped, or bell-shaped), the hotspot effect, and zenith reflectance. To achieve a quantitative characterization of these features and facilitate the subsequent analysis of spatial scale effects, this study introduces specific quantitative parameters, including the hotspot magnitude and direction, the Hotspot-to-Average Reflectance Ratio ($R_a$), and the Zenith-to-Average Reflectance Ratio ($R_b$).

The $R_a$ is defined as the ratio of the directional reflectance at the hotspot ($R_{hot}$) to the mean reflectance ($R_{mean}$), as expressed in the following equation. This parameter quantifies the influence of hotspot intensity on the overall BRDF shape; specifically, a larger ratio indicates a more concentrated and pronounced hotspot effect.

$$R_a={\displaystyle \frac{R_{hot}}{R_{mean}}}$$

(8)

The $R_b$ is defined as the ratio of the nadir directional reflectance ($R_{top}$) to the $R_{mean}$, as illustrated in the equation below. This parameter reflects the contribution of the vertical (nadir) reflectance intensity to the overall BRDF geometry. Specifically, a larger ratio indicates a more pronounced convex (or bell-shaped) BRDF profile.

$$R_b={\displaystyle \frac{R_{top}}{R_{mean}}}$$

(9)

2.4. Spatial Scale Transformation

Spatial scale transformation refers to the process of transferring data or information from one scale to another [24]. Generally, the transformation from high resolution to low resolution is termed upscaling, while the conversion from low resolution to high resolution is known as downscaling [12]. Research on area-to-area upscaling typically follows one of two strategies: aggregate-then-invert or invert-then-aggregate. This study adopts the invert-then-aggregate approach. In this workflow, high-resolution remote sensing parameters are first retrieved from high-resolution data and subsequently aggregated to generate the corresponding low-resolution information.

The Window Averaging Method [23] is a statistical transformation technique applied to the grayscale raster layers of BRDF model parameters. The method operates by centering on a target pixel and progressively expanding the pixel window dimensions from 3 × 3, 5 × 5,…, up to 101 × 101, calculating the mean value of pixels within each window. To mitigate the potential lack of representativeness in local areas, this study partitions the grayscale layer into equal-area blocks, within which the window statistics are performed synchronously. In this framework, the varying window dimensions correspond to distinct spatial scales, and the mean value derived for each block serves as the representative BRDF characteristic parameter for the land surface at that specific scale. The spatial resolution ($P\left(n\right)$) associated with each scale is determined by the following equation:

$$P\left(n\right)=n\times q$$

(10)

where $n$ denotes the side length of the pixel window, and $q$ represents the spatial resolution of the original imagery.

2.5. Quantification of Surface Heterogeneity

The variogram [25] is a fundamental statistic used to characterize the spatial correlation of random fields and stochastic processes. It is defined as the variance of the difference between values at two spatial locations. The calculation principle is expressed as follows:

$$\gamma \left(h\right)={\displaystyle \frac{1}{2N\left(H\right)}}{\sum }_{i=1}^{N\left(h\right)}{\left[Z\left(x_i\right)-Z(x_i-h)\right]}^2$$

(11)

where $\gamma \left(h\right)$ represents the semivariogram of the experimental area; $Z\left(x_i\right)$ denotes the sample value, and $h$ indicates the spatial lag distance between any two samples; $N\left(h\right)$ corresponds to the number of sample pairs separated by distance $h$ used in the calculation. The most commonly employed theoretical models for fitting the experimental semivariogram are bounded models, including the Spherical, Exponential, and Gaussian models. Taking the Spherical model as an example, its mathematical expression is defined as follows:

$$\gamma \left(h\right)\left\{\begin{array}{c} 0 h=0\\ c_0+c_1\left({\displaystyle \frac{3a}{2h}}-{\displaystyle \frac{a^3}{2h^3}}\right) 0<h\le a\\ c_0+c_1 h>a\end{array}\right.$$

(12)

In this equation, $c_0$ represents the nugget, indicating the variability at small scales—specifically, the variation between two samples at a distance approaching zero, which reflects spatial heterogeneity induced by random factors. The term $c_0$ + $c_1$ represents the maximum variability of the target variable. The parameter $a$ denotes the range, defined as the maximum distance over which sample points remain spatially correlated. The geometric interpretation of these parameters is illustrated in Figure 6.

3. Result

3.1. Model Inversion Accuracy

The RPV model employed in this study involves three or four unknown parameters. To estimate these parameters, the Least Squares Method (LSM) was utilized to fit the redundant observational data. Consequently, the set of RPV model parameters yielding the minimum RMSE was identified as the optimal solution. The mathematical expression for RMSE is defined as follows:

$$RMSE=\sqrt{{\displaystyle \frac{\sum _{j=1}^n{\left(R_{obs}\left({\theta }_i,{\theta }_v,\varphi ,\lambda \right)-R_{model}\left({\theta }_i,{\theta }_v,\varphi ,\lambda \right)\right)}^2}{n}}}$$

(13)

In the equation above, $R_{obs}\left({\theta }_i,{\theta }_v,\varphi ,\lambda \right)$ and $R_{model}\left({\theta }_i,{\theta }_v,\varphi ,\lambda \right)$ represent the observed and modeled reflectance values, respectively, and n denotes the number of observations. The RMSE, typically ranging from 0 to 1, quantifies the deviation between the RPV simulated reflectance and the true values. It serves as an indicator of the model fitting quality for multi-angular data, where a lower value implies a superior fit [26].

As illustrated in Figure 7, even in the absence of dedicated observations targeting the hotspot region, the four-parameter RPV model (black line) consistently exhibits a lower RMSE than the three-parameter model (colored lines) across the entire spectral range and all land cover types. This indicates that by incorporating an adjustable parameter, the four-parameter model can more precisely characterize the complex features of surface bidirectional reflectance, thereby enhancing the inversion accuracy. Overall, the results presented in this figure directly corroborate our research hypothesis: the accuracy improvement of the four-parameter RPV model is not strictly contingent upon dedicated hotspot observations. Even without an observation scheme explicitly designed to capture the hotspot, the four-parameter model, leveraging its more flexible parameterization, still yields lower RMSEs across most spectral bands and land covers, demonstrating superior adaptability and robustness.

Considering that the reflectance of vegetation in the visible spectrum is significantly lower than that in the infrared spectrum, the resulting RMSE for visible bands is inherently smaller than that for infrared bands. However, this numerical difference does not necessarily imply superior model fitting performance in the visible region. Consequently, relying solely on RMSE to evaluate the RPV model inversion is insufficiently rigorous, as it fails to strictly describe the overall inversion quality across different magnitudes. To address this limitation, the RRMSE is introduced to quantitatively assess the model inversion performance. The calculation formula is given by:

$$RRMSE={\displaystyle \frac{RMSE}{\overline{y}}}\ast 100\%$$

(14)

where $\overline{y}$ represents the arithmetic mean of all actual observed values. The RRMSE serves to quantify the magnitude of relative deviation between the predicted values and the ground-observed values. Consequently, a lower RRMSE value indicates a higher degree of model inversion accuracy.

Figure 8 displays the RRMSE results across different wavelengths. For all six bands, the RRMSE metrics for the selected land covers did not exceed 25%, with multiple datasets falling below 10%, reaching their minima in the NIR band (840 nm). This intuitively demonstrates the high inversion accuracy of the RPV model across diverse land cover types. A closer examination of the vegetation types (tea plantations, forests, and rice) reveals that their RRMSE values exhibit a decreasing trend as their spectral reflectance intensity increases; that is, inversion accuracy is positively correlated with spectral reflectance intensity. This phenomenon can be attributed to the fact that vegetation with higher reflectance intensity offers a more significant distinction between the target spectral signal and environmental background noise, thereby reducing error interference during the inversion process. In conclusion, validated by both RMSE and RRMSE metrics, the four-parameter model demonstrates robust fitting accuracy and stability for BRDF inversion of different land covers in the study area, effectively characterizing the bidirectional reflectance properties of the target features.

3.2. Scale Effects on BRDF

Currently, investigations into the coupling relationship between BRDF characteristics and spatial scale primarily focus on identifying scale-dependent patterns and scale-sensitive features. This is achieved by analyzing the evolutionary trends of shape factors, BRDF model parameters, and image grayscale statistics across varying image resolutions. Mainstream methodological approaches include the trend analysis method and the upscaling comparison method. Specifically, the trend analysis method utilizes linear or nonlinear fitting techniques to quantify the dynamic variations in BRDF features with respect to spatial scale. By extracting critical abrupt change points—such as inflection points, peaks, and valleys—this method provides a fundamental basis for the quantitative characterization of scale effects.

Leveraging the solar incidence geometry parameters acquired during UAV observations (detailed in Table 3) and the corresponding multispectral imaging data, this study accurately simulated the BRDF characteristics of four typical land covers during the observation window. Given that the model inversion performance was found to be optimal in the infrared spectrum, the NIR band was selected as the representative case for this analysis (as illustrated in Figure 9). The Window Averaging Method was employed to simulate the response characteristics of the RPV model parameters across multiple spatial scales. By doing so, this study aims to reveal the underlying mechanisms governing the spatial scale dependence of BRDF features for these four land cover types. Ultimately, these findings provide empirical support for the subsequent selection of optimal observation scales and the enhancement of multi-scale remote sensing retrieval accuracy.

3.2.1. Scale Evolution of RPV Model Parameters

The RPV model is adopted to interpret the BRDF scale effect, with the core lying in analyzing the nonlinear variation in model parameters across scales, as detailed in Figure 10.

The parameter ${\rho }_0$ characterizes the overall reflectance level of the land surface, with its value determined not only by the spectral reflectance of the specific band but also by the contribution of background reflectance to the directional reflectance. Within the study area, the values for bare soil were relatively low, whereas those for rice were comparatively high. Regarding spatial scale effects, distinct patterns were observed as the pixel scale increased: the value for tea plantations initially exhibited a decreasing trend, attributed to the vegetation reflectance being attenuated by the soil background, and subsequently tended to stabilize due to the uniform distribution of the tea plants; forests, characterized by continuous canopies and high vegetation coverage, showed an initial increase followed by stabilization as the enlargement of the pixel window gradually smoothed out the influence of canopy shadows; bare soil maintained a stable value across all spatial scales due to its surface homogeneity; and finally, although the variation for rice was not statistically significant, it followed a pattern consistent with forests, where the influence of shadows diminished as the pixel scale increased.

The empirical parameter k reflects the extent of surface anisotropy. For forests, tea plantations, and bare soil, k was consistently found to be less than 1, characterizing these surfaces as bowl-shaped. This is primarily caused by substantial heterogeneity within pixels arising from soil background signals, shadowing, and geometric roughness. Conversely, rice exhibited k > 1 (bell-shaped), suggesting a quasi-Lambertian nature due to its dense and uniform canopy structure. With increasing pixel scale, k remained stable for bare soil and rice, whereas it showed a decrease-then-stabilize trend for tea and an increase-then-stabilize trend for forests.

The parameter $\theta$ characterizes the asymmetry of the BRDF, specifically representing the relative strength of forward versus backward scattering. With the exception of bare soil, the $\theta$ values for the three vegetation types are positive (indicating a predominance of backscattering), which is consistent with the hotspot effect characteristic of vegetation canopies; in contrast, bare soil exhibits a negative value, indicative of forward scattering properties. As the spatial scale increases, the $\theta$ values for tea plantations and forests display an initial decrease followed by stabilization, suggesting that the canopy structure tends toward homogeneity at larger scales, thereby weakening the asymmetry; meanwhile, bare soil and rice show minimal variation, demonstrating a distinct scale insensitivity.

3.2.2. Scale Evolution of Near-Principal Plane Reflectance

For any land surface, BRDF effects are most pronounced in the principal plane [27], with the hotspot and dark spot representing the most information-rich regions within the bidirectional distribution [28]. Given that the intensity and nature of the hotspot effect vary with illumination geometry, investigating its characteristics across spatial scales is critical for land cover classification and vegetation canopy structure retrieval.

Figure 11 illustrates the response relationship between the VZA, ranging from −60° to 60°, and reflectance in the near-principal plane for four typical land cover types across varying spatial scales (0.5 m, 1 m, 1.5 m, 2 m, and 3 m). The distribution curves reveal that all four land cover types exhibit a typical hotspot effect, characterized by a distinct reflectance peak at the observation angle coinciding with the solar incidence angle, which aligns with the expected bidirectional reflectance behavior of targets in the near-principal plane. Furthermore, the response of reflectance to increasing spatial scale varies significantly by land cover. Specifically, tea plantations show an overall decreasing trend in near-principal plane reflectance as the scale increases from 0.5 m to 3 m, with values diminishing across all VZAs. Conversely, forests exhibit an overall upward trend, with reflectance values rising as the scale increases. The impact on bare soil is localized; reflectance increases with scale primarily at the hotspot position, while remaining stable at other angles. Rice, however, shows no significant overall variation, with high consistency among curves at different scales. Notably, compared to the other three categories, rice exhibits significantly enhanced specular reflection features in the near-principal plane. This phenomenon is attributed to the rice canopy displaying near-Lambertian surface characteristics at the current pixel scales, resulting in a more prominent specular reflection effect.

3.2.3. Scale Evolution of BRDF Characteristic Parameters

Synthesizing the results from Figure 10, Figure 11 and Figure 12, distinct evolutionary patterns emerge as the pixel scale increases: for tea plantations, the overall reflectance level decreases, yet the magnitude of the decrease at the hotspot is less than that at the zenith angular positions, which enhances the hotspot concentration capability and causes the BRDF shape to become more concave, evolving into a more pronounced bowl-shaped structure; conversely, forests show an increase in overall reflectance, but the increase at the hotspot is smaller than at the zenith angular positions, thereby weakening the hotspot concentration capability and resulting in a progressively convex BRDF shape that manifests as a shallow bowl-shaped structure; bare soil exhibits increased reflectance at the hotspot while remaining relatively stable at other positions, a dynamic that enhances hotspot concentration capability and drives a decrease in the shape parameter $R_b$, leading to a morphological transition from a bell-shaped to a bowl-shaped structure; and finally, the surface of rice exhibits increasingly prominent Lambertian characteristics, which attenuates the hotspot concentration capability and causes the $R_b$ to gradually rise.

3.3. The Physical Mechanism for Selecting the Optimal Observation Scale

3.3.1. Analysis of the Variogram

Owing to the high spatial resolution characterizing UAV remote sensing, the acquired imagery captures intricate details of land surface features, resulting in pronounced inter-pixel variability. Consequently, the variogram method was employed to quantitatively analyze the spatial heterogeneity of different land covers. The data employed in this study consist of DOM for various land cover types. Their spatial resolution is consistent with the spatial scale analysis presented above. The results are presented in Figure 13, leading to the following conclusions:

• The magnitude of spatial heterogeneity varies distinctively across different land cover types. Bare soil, characterized by a homogeneous cover composition and a smooth surface, exhibits relatively low spatial heterogeneity; in contrast, the heterogeneity of vegetation is generally higher than that of bare soil, primarily governed by factors such as plant density and leaf size. Furthermore, significant variations exist among specific vegetation types: the tea plants in this study were in their juvenile stage with small canopy volumes, making their heterogeneity analysis susceptible to the influence of the underlying soil background; forests exhibited high spatial heterogeneity, attributed to the diversity of tree species, variations in growth status, and extensive shadowing caused by leaf occlusion; and conversely, rice paddies demonstrated low spatial heterogeneity, as their dense growth minimizes the influence of the soil background, while their uniform growth status contributes to a spatially homogeneous texture.
• Spatial heterogeneity exhibits a trend of initial increase followed by stabilization as the lag distance increases. At short lag distances, pixel pairs demonstrate strong homogeneity (or high spatial autocorrelation); however, heterogeneity progressively intensifies as the separation distance grows. When the heterogeneity reaches its stable plateau, the spatial correlation between points diminishes to its minimum, while the variability reaches its maximum. The lag distance corresponding to this point is defined as the Range, representing the maximum extent of spatial correlation.
• The magnitude of variation in spatial heterogeneity differs distinctively across land cover types. The bare soil within the study area corresponds to agricultural land, where the soil within a single plot exhibits uniform properties, such as moisture content and elemental composition; consequently, the variability between any two spatial points remains within the same order of magnitude. In contrast, variations in vegetation heterogeneity are driven not only by external factors like shadowing and background effects but also by intrinsic physicochemical parameters, including chlorophyll content and LAI. These factors collectively contribute to significantly larger differences between spatial points.

3.3.2. Relationship Between BRDF Scale Characteristics and Spatial Heterogeneity

As illustrated in Figure 10 and Figure 12, the BRDF characteristics of the terrestrial targets exhibit significant variations as the spatial scale increases, yet they eventually converge to a stable state at a specific scale. This evolution is highly consistent with the changing trend of their spatial heterogeneity (Figure 13). To quantitatively reveal the intrinsic linkage between the BRDF scale characteristics and spatial heterogeneity, this study introduces Kendall’s Rank Correlation Coefficient to conduct a correlation analysis between the BRDF feature parameters (${\rho }_0$, $k$, $\theta$, $R_a$, $R_b$) and the spatial heterogeneity indicator ($\gamma \left(h\right)$). Kendall’s rank correlation coefficient is a non-parametric statistical metric utilized to measure the strength of the ordinal association between two variables. Unlike the Pearson correlation coefficient, it does not assume a normal distribution for the variables, making it particularly well-suited for analyzing relationships that are non-linear but exhibit a monotonic trend. Its values range from [−1, 1]; an absolute value closer to 1 indicates a stronger concordance (or discordance) in the evolutionary trends of the two variables, whereas a value of 0 signifies no correlation. In this study, this coefficient is employed to evaluate whether the internal structural complexity (heterogeneity) and the external radiometric characteristics (BRDF parameters) of the targets demonstrate a synergistic evolutionary pattern across varying spatial scales. The corresponding analytical results are presented in Figure 14.

Based on Figure 14, the following detailed observations can be derived:

(1)

Significant correlations. Whether for vegetation or bare soil, statistically significant correlations exist between their spatial heterogeneity and the BRDF model parameters. This is visually evidenced by the deeply saturated, flattened ellipses within the correlogram.

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Discrepancies across land cover types. For the forest, spatial heterogeneity demonstrates a strong positive correlation with the RPV model parameters (highlighted by the red regions). Specifically, the correlation coefficient between $\gamma \left(h\right)$ and $\theta$ reaches up to 0.73. This suggests that the increasing structural heterogeneity of the forest canopy at larger observation scales directly drives the intensification of surface anisotropy. Conversely, for the tea plantation, the spatial heterogeneity exhibits a strong negative correlation with the RPV parameters (indicated by the blue regions), with the coefficient between $\gamma \left(h\right)$ and $\theta$ dropping to −0.84. This discrepancy is primarily attributed to the juvenile stage of the tea plants, where the bare soil background between the crop rows exerts a distinct interference mechanism on the illumination and shadowing patterns compared to tall woodlands. For the rice paddy, while the correlation remains significant, the overall coefficients are comparatively lower (e.g., 0.42 between $\gamma \left(h\right)$ and $k$), which is intrinsically linked to its naturally lower spatial heterogeneity.

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Responses of the shape factors. The shape factors $R_a$ and $R_b$ also exhibit acute sensitivity to scale variations. Taking the tea plantation as an example, $\gamma \left(h\right)$ is negatively correlated with $R_b$ and positively correlated with $R_a$. This implies that modifications in the spatial structure simultaneously stretch and compress the geometric shape of the BRDF.

The spatial resolution of remote sensing observations fundamentally determines the accuracy and application value of derived products. When spatial resolution is too coarse, the mixed pixel effect intensifies and fine-scale details are obscured, making it impossible to accurately extract key parameters such as radiometric characteristics and geometric morphology; this, in turn, compromises downstream applications like land cover classification and quantitative retrieval. Conversely, while excessively high spatial resolution captures fine structural details, it significantly amplifies the influence of geometric structural differences and spatial heterogeneity. Therefore, selecting an optimal observation scale tailored to specific land cover types is a prerequisite for fully characterizing BRDF features and ensuring the accuracy of remote sensing inversions. Based on the previous research, the following analysis can be obtained:

• In the initial stage of observation (at small spatial scales), land surface BRDF characteristics exhibit significant variations as the spatial scale increases; the magnitude and trend of these variations are primarily determined by the degree of spatial heterogeneity intrinsic to the land cover. For instance, land covers with high spatial heterogeneity (e.g., forests) display more drastic fluctuations in their BRDF characteristics with scale, whereas those with lower heterogeneity (e.g., rice, homogeneous bare soil) show relatively gradual scale-dependent changes. As the spatial scale continues to increase and reaches a specific threshold, the BRDF characteristics tend to stabilize, ceasing to show significant variations with further scale enlargement. The mechanism underlying this stabilization threshold is essentially the averaging out of internal spatial heterogeneity; at this point, the proportion of components within a single pixel tends to become fixed, the mixed pixel effect reaches a dynamic equilibrium, and the overall radiometric reflection patterns of the land surface are stably presented.
• The spatial scale at which BRDF characteristics stabilize (i.e., the optimal observation scale) exhibits a strict positive correlation with the degree of spatial heterogeneity. For land covers with high spatial heterogeneity, such as forests and tea plantations, the internal composition is diverse and highly variable; consequently, a larger observation scale is required to achieve the effective averaging out of these internal components. This results in a higher threshold for BRDF stabilization and, correspondingly, a larger optimal observation scale. Conversely, for land covers with low spatial heterogeneity, such as homogeneous bare soil or regularly planted monocultures (e.g., rice), the internal components are fewer and uniformly distributed. In these cases, a smaller observation scale suffices for the proportion of internal components to become fixed, allowing BRDF characteristics to stabilize earlier and resulting in a relatively smaller optimal observation scale.

4. Discussion

4.1. The LESS Model Simulation

The LESS (Large-scale remote sensing data and image simulation framework) model accurately characterizes the interactions between solar radiation and heterogeneous land surfaces, making it a pivotal tool for quantitative remote sensing modeling and retrieval in recent years [29]. To validate the accuracy of the RPV model at optimal observational scales, this study employed the LESS model to simulate the BRDF of the four investigated land covers. High-fidelity 3D scenes were constructed to accurately replicate the canopy and surface structures of these targets. Driven by solar geometry parameters acquired from UAV observations, multi-angular reflectance datasets were generated across varying VZAs within the principal plane. These LESS simulations served as a benchmark for comparing and evaluating the fitting performance of the RPV model-derived BRDF.

As evident from the comparison in Figure 15, the simulated reflectance curves from the LESS and RPV models exhibit a high degree of agreement across varying VZA for all four land cover types. Specifically, for forest canopies, reflectance simulated by both models consistently varies with the observation zenith angle, both demonstrating typical BRDF bowl-shaped characteristics. For bare soil and rice, the distribution trends and magnitudes of the scatter points (blue vs. red) demonstrate excellent consistency across both negative and positive VZA ranges. Although minor numerical fluctuations are observed in specific angular intervals, primarily attributed to subtle differences in surface heterogeneity, these do not compromise the overall fitting performance. Collectively, these results confirm that the RPV model is capable of accurately characterizing the true reflectance anisotropy of diverse land covers under the current observational conditions.

4.2. Selection of the Optimal Observation Scale

Based on the findings in Section 3.3.2, the optimal observation scales for different land covers in the NIR band were preliminarily identified as follows: approximately 1.3 m for bare soil, 1.5 m for tea plantations, 1 m for rice, and 2 m for forests. Considering the practical operational mode of UAV missions, which typically involve simultaneous sampling across multiple bands, it is necessary to further investigate whether the optimal observation scales for each land cover type exhibit consistency across different spectral bands. Based on the experimental variogram results presented in Figure 16, significant disparities exist in the spatial heterogeneity across different spectral bands. The detailed analysis is as follows:

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Discrepancies in Sill Variance ($c_0+c_1$): Distinct spectral bands exhibit substantial differences in the intensity of characterized surface heterogeneity once the variograms reach a stable state. Specifically, the sill value of the NIR band is significantly higher than that of the visible bands. The core mechanism lies in the high-reflectance characteristics of vegetation foliage in the NIR region, which creates a pronounced contrast with shadowed areas, thereby amplifying the captured surface heterogeneity information. In contrast, visible bands are modulated by the pigment absorption of leaves, resulting in relatively dampened reflectance gradients and consequently lower sill levels. Although minor fluctuations in sill values occur within the visible spectrum (blue, green, and red bands) due to varying absorption intensities, these internal variations are far less prominent than the discrepancy between the NIR and visible domains.

(2)

Discrepancies in Correlation Range ($a$): As indicated by the colored vertical bars in Figure 16, the inflection points where the variogram curves stabilize (i.e., the range) vary significantly across bands, suggesting that the optimal observation scale is band-dependent. This finding underscores that the optimal observation scale is not solely governed by the physical geometry of the targets (such as crown diameter, tree spacing, and row spacing) but also possesses distinct spectral-scale characteristics.

Based on the preceding analysis, the colored bands annotated in Figure 16 represent the characteristic structural scales of each land cover type across different spectral bands. However, given the simultaneous sampling of the sensor’s multiple cameras, it is imperative to ensure that all spectral bands are observed at or above their optimal spatial scales. Therefore, this study initially selected the maximum variogram range among the six bands (or larger) as the optimal observation scale range for a specific land cover, denoted by the gray bands in Figure 16. Nevertheless, as demonstrated earlier, the optimal observation scale for each land cover should be a definitive threshold—rather than a broad range—that maintains the stability of BRDF characteristics while fully preserving detailed target information. Consequently, this paper proposes a “Maximum Threshold Observation Scale” strategy, which designates the maximum variogram range across the variograms of all bands as the optimal observation scale. The core logic underpinning this strategy is that the observation scale corresponding to the maximum range encompasses the optimal scale requirements of all individual bands. This ensures that the observational results across all bands can adequately reflect the key spatial structural features of surface heterogeneity. Although the “Maximum Threshold Observation Scale” strategy may result in the loss of surface-target details information in certain bands, the UAV remote sensing data adopted in this study features a relatively high spatial resolution. As can be observed in Figure 16, this strategy does not significantly affect the optimal observation scales across different bands. For instance, in the forest plot, the variogram ranges across different bands, spanning from 1.5 to 2 m. By selecting 2 m as the unified observation scale, we can guarantee that observations for all bands are captured at or above their respective optimal scales, thereby averting biases in heterogeneity characterization caused by scale mismatches.

In summary, through joint multi-band determination, the optimal observation scales for various land covers under the current observation conditions can be quantitatively identified: approximately 6 m for bare soil, 2 m for forest, 1.5 m for tea plantation, and 1 m for rice. The findings of this study underscore that neglecting spectral effects and scale mismatch issues in multi-band remote sensing data will significantly compromise the accuracy of BRDF inversion, land surface classification, and ecological parameter estimation. Adopting the “Maximum Threshold Observation Scale” strategy provides a crucial methodological framework for the scale correction and heterogeneity analysis of multi-band remote sensing data, fundamentally contributing to the enhanced reliability and applicability of quantitative remote sensing in studies of complex surface heterogeneity.

5. Conclusions

To investigate the spatial scale effects of surface BRDF, this study selected four representative land cover types. By employing spatial upscaling transformation and integrating qualitative with quantitative analyses, we systematically characterized the variation in BRDF across spatial scales and subsequently determined the optimal observation scales for each land cover type. The results indicate that surface BRDF characteristics exhibit significant variations across spatial scales, yet these variations follow a distinct and systematic pattern. Furthermore, the study elucidates the intrinsic correlation between surface spatial heterogeneity and BRDF scale effects, demonstrating that they share consistent fitting trends. At coarser observational scales, surface feature information tends to become blurred. Addressing this, our study explicitly determines the optimal observation scales for various land covers. These optimal scales are defined as the critical range that maintains the stability of BRDF characteristics while preserving the integrity of surface details, thereby providing a scientific reference for high-precision remote sensing applications.

By applying a unified technical framework to analyze diverse land covers, this study demonstrates the robust applicability of the proposed method. Across critical metrics—including BRDF model fitting accuracy, parameter variation trends, and spatial heterogeneity analysis—the results show strong consistency with both remote sensing theory and in situ measurements. These findings effectively validate the reliability of this approach for determining optimal observation scales. However, this study is subject to several limitations. First, the analysis relied on a single spatial upscaling method, lacking comparative validation against alternative approaches. Second, the study objects were restricted to homogeneous land cover types, leaving complex scenarios such as mixed-species forests unexplored. Third, the absence of dedicated observations in the hotspot direction resulted in an incomplete characterization of reflectance features in this critical angular domain. Fourthly, this validation approach entails significant limitations, and there is a lack of cross-validation with existing achievements. The current cross-validation between LESS simulation outputs and RPV model results essentially operates within two idealized simulation frameworks, thereby failing to test the models against authentic field environments. Future research will prioritize field-based empirical observations, employing multi-angle goniometers to acquire measured multi-angular reflectance data for typical land covers. By benchmarking model inversions against these empirical datasets, we aim to further evaluate the RPV model’s retrieval accuracy, quantify systematic errors, and refine model parameters.