The GSV-SRTS model is a sophisticated theoretical framework designed to address the intricate nature of radiation transfer processes within the layers of a canopy. By expanding upon the traditional radiation transfer equation to include the soil GSV and integrating topographic elements, this model offers a more precise depiction of the inherent variations present in canopy-soil systems. One key feature of the GSV-SRTS model is its utilization of a local slope coordinate system for all internal calculations, ensuring a geometrically accurate representation of how light interacts within a rugged mountainous terrain. To illustrate the application of the GSV-SRTS model, this study presents a 3D discontinuous canopy made up of identical trees situated within the layer defined as 0<z<H in the coordinate system (Fig. 1) The number of trees housed within a pixel is determined randomly using the Poisson distribution (Huang et al., 2008). From a stochastic view, the 3D canopy structure is conceptualized as a spatial stochastic process, symbolized by an indicator function denoted as χ(x,y,z). This function yields a value of 1 if the specific point (x, y, z) falls under vegetation cover, and 0 if it does not.

The adoption of a local slope coordinate system (x′, y′, z′), aligned with the inclined ground surface, is fundamentally necessary to render the radiative transfer problem tractable over slopes. In the conventional horizontal coordinate system, a sloping ground surface presents a moving, tilted boundary condition that vastly complicates the formulation of the extinction coefficient and the boundary conditions for the radiative transfer equation (RTE). By transforming to a coordinate system where the z′ axis is normal to the local slope, the ground surface is redefined as a flat plane (z′=0). This crucial simplification allows the use of a standard volumetric formulation for canopy extinction and enables the application of the stochastic approach for gap probability calculation on this effectively “level” domain. Consequently, all canopy structural statistics and radiative transfer calculations are performed relative to this local frame, and the effects of terrain are encapsulated in the modified solar and viewing direction vectors within this frame, as defined in Eqs. ()–(). For any given global direction Ω, its corresponding local sun zenith angle cos⁡θs and local view zenith angle cos⁡θv are calculated as (Gu and Gillespie, 1998): 1cos⁡θs(v)=cos⁡θgcos⁡θ′s(v)+sin⁡θgsin⁡θ′s(v)cos⁡φsg(vg) where θg and φg represent the slope and aspect, respectively. φsg is the relative azimuth angle between the sun and the sloping background; and φvg is the relative azimuth angle between the viewer and the sloping background. All subsequent RT computations inside the canopy are performed using these local angles, ensuring that all physical interactions are relative to the actual inclined surface.

The Stochastic Radiative Transfer Equations (SRTE) are obtained through horizontal averaging of the 3D RT process, thereby integrating the influence of canopy structure. Solving these equations (Eqs.  and ) necessitates determining the mean radiation intensities for both the entire scene, I‾(z,Ω), and the vegetation-covered area within a pixel, U(z,Ω), at depth z in direction Ω, which are defined as (Vainikko, 1973; Shabanov et al., 2000): 2μ∂I‾(z,Ω)∂z=-σ′(Ω)I‾(z,Ω)+σs4π∫4πI‾(z,Ω′)σ′s(Ω′,Ω)dΩ′+F0(z,Ω)+FH(z,Ω)3μ∂U(z,Ω)∂z=-σ′(Ω)U(z,Ω)+σs4π∫4πU(z,Ω′)σ′s(Ω′,Ω)dΩ′+Q0(z,Ω)+QH(z,Ω) where μ=cos⁡θv, and θ is the local view zenith angle. Ω indicates the direction of incoming direct light, and Ω′ indicates the viewing direction. F0(z,Ω) and FH(z,Ω) in Eq. () represent the diffuse and direct radiation entering through the upper and lower boundaries. Q0(z,Ω) and QH(z,Ω) in Eq. () refer to the mean diffuse and direct radiation that penetrate the upper and lower canopy boundaries and subsequently experience scattering in the heterogeneous medium. σ′(Ω) is the extinction coefficient. The extinction coefficient is geometrically corrected to account for local slope-induced projection effects. This correction reduces effective extinction when illumination or viewing directions are highly oblique relative to the local normal, ensuring the local radiative balance remains consistent with the actual slope geometry, which is shown as follows (Mousivand et al., 2015): 4σ′(Ω)=σ(Ω)max⁡(0,μ′s)max⁡(0,μ′v)

The terrain-induced azimuthal dependence of scattering is characterized by a topographic modulation function (Sandmeier and Itten, 1997). The modulation function, which depends on the local slope angle and the directions relative to the slope normal, physically approximates the effect of this truncated geometry on the scattering process. It adjusts the standard volumetric scattering phase function to better represent the anisotropic scattering behavior in the vicinity of the sloping ground boundary, where the hemispherical distribution of scatterers is asymmetrically constrained. It is described as follows: 5Ctopo(ϕv)=sin⁡θgcos⁡ϕvgμ′sμ′v which enhances scattering in the downslope direction and weakens it in the upslope direction. The scattering kernel σs(Ω′,Ω) in Eqs. () and () is adjusted by the slope-induced azimuthal modulation function: 6σ′s(Ω′,Ω)=σs(Ω′,Ω)1+Ctopo(ϕv) where σs(Ω′,Ω) is the original differential scattering coefficient (using a Henyey–Greenstein phase function).

The average horizontal intensity across over vegetated area U(z,Ω) or the entire horizontal plane I‾(z,Ω) comprise two components: the direct and the diffuse parts of the incoming irradiance, which can be given as follows (Shabanov et al., 2000): 7U(z,Ω)=Udirδ(Ω′-Ω)+Udif(z,Ω)I‾(z,Ω)=fdirIdirδ(Ω′-Ω)+Idif(z,Ω) where fdir is the fraction of incident direct radiation on the upper boundary; δ(Ω′-Ω) is Dirac's delta function (Dirac, 1981), which approaches infinity when the polar angle in direction Ω matches the direction of direct solar radiation Ω, and zero otherwise; Udir(z) (Idir(z)) and Udif(z,Ω) (Idif(z,Ω)) represent the direct component and the diffuse component over the vegetation-occupied area (or the entire scene area), respectively, which is described in Eq. (). The boundary conditions for the entire scene are simplified as follows (Shabanov et al., 2000; Bird and Hulstrom, 1981). 8I0‾(z,Ω)=Fdir(Ω0)δ(Ω-Ω0)+d0(z,Ω),u<0IH(z,Ω)=R(Ω,Ω′)π∫2π+R(Ω,Ω′)cos⁡θvdΩ+dH(z,Ω),u>09Fdir=S0cos⁡θsT10ρsoil=1π∫2π+R(Ω,Ω′)cos⁡θvdΩ where δ(Ω-Ω0) is the Dirac's delta function, which enables precise estimation of the solar radiation. Fdir(Ω0) represents the intensity of direct radiation incident on the upper boundary. S0 is the solar constant, and T is the atmospheric transmittance. d0(z,Ω) denotes the diffuse radiation intensity from the upper boundary, while dH(z,Ω) represents the intensity entering the canopy through the lower boundary. These two parameters can be calculated using the Second Simulation of the Satellite Signal in the Solar Spectrum (6S) model (Roujean et al., 1992). The 6S model calculates d0(z,Ω) as atmospheric diffuse skylight via RTE accounting for Rayleigh and aerosol scattering and dH(z,Ω) by integrating surface/canopy scattering effects on transmitted radiation. R(Ω,Ω′) is the bidirectional soil reflectance factor, the same as R1×l in Eq. () and ρsoil is the soil hemispherical reflectance. The incoming radiation can be parameterized by two values: the total flux Fdir+dif(Ω0), defined as follows (Shabanov et al., 2000): 11Fdir+dif(Ω0)=∫2π-I0‾(z,Ω)cos⁡θvdΩ=Fdircos⁡θv+∫2π-d0cos⁡θvdΩ

The horizontal average intensity U(z,Ω) over the vegetation-covered area can be decomposed into the direct (Udir(z)) and diffuse (Udif(z)) components of the incoming solar radiation, defined as follows (Shabanov et al., 2000): 12Udir(z)+σ(Ω)cos⁡θv∫0zK(z,z′,Ω)Udir(z′,Ω)dz′=1Udif(z,Ω)+σ(Ω)cos⁡θv∫0zK(z,z′,Ω)Udif(z′,Ω)dz′=1cos⁡θv∫0zK(z,z′,Ω)Sdif(z′,Ω)dz′+F0(z,Ω),μ<0Udif(z,Ω)+σ(Ω)cos⁡θv∫zHK(z,z′,Ω)Udif(z′,Ω)dz′=1cos⁡θv∫zHK(z,z′,Ω)Sdif(z′,Ω)dz′+FH(z,Ω),μ>013Sdif(z′,Ω)=∫4πσs(Ω→Ω′)Udif(z′,Ω′)dΩ′ where Sdif(z′,Ω) denotes the spherical integration of scattering. K(z,z′,Ω) is the Conditional Probability Correlation Function (PCF). This core function quantifies the probability of spatial correlation between vegetation elements at depths z and z′ along direction Ω, thereby statistically capturing the effect of canopy 3D structure and heterogeneity on the radiation field. The stochastic canopy structure is embedded in the SRTE solution via the PCF, which encodes spatial correlations of vegetation presence derived from χ(x,y,z). This function directly modulates the integral terms in Eq. (), ensuring canopy heterogeneity and terrain-slope effects are propagated through radiation intensities. Its mathematical form is shown as follows (Shabanov et al., 2007): 14q(z,z′,Ω)=1S∬Sχ(x,y,z)χ(x′,y′,z′)⋅f(θg,Δh)dxdy15f(θg,Δh)=exp⁡-ΔhH⋅tan⁡θg16p(z)=1S∬Sχ(x,y,z)dxdy17K(z,z′,Ω)=q(z,z′,Ω)p(z)=∫Sχ(x,y,z)χ(x′,y′,z′)dxdy∬Sχ(x,y,z)dxdy where p(z) is the canopy density at canopy depth z, χ(x,y,z) is the indicator function (1 for vegetation, 0 otherwise). The point (x,y,z) moves travels a distance L in the direction Ω to reach (x′, y′, z′). The bracket 〈⋅〉 denotes the ensemble average over all realizations of χ(x,y,z) within a finite pixel. Δh is the vertical elevation difference between two points, H is the total vertical height of the vegetation canopy.

Soil reflectance properties are important elements that should be carefully formulated in canopy radiative transport process modelling (Knyazikhin et al., 1998b; Verhoef and Bach, 2007). However, modelling soil reflectance is challenging due to the complex optical properties of soil in shortwave bands. The GSV soil reflectance model addresses this by providing an effective method for approximating the reflectance spectra of both dry and wet soils (Jiang and Fang, 2019). Coupling GSV with SRT is critical because: (1) soil drives reflectance at low LAI (<2), requiring precise spectral representation; (2) SRT resolves canopy heterogeneity unaddressed by homogeneous models; (3) slope effects jointly modulate soil exposure and canopy gap probability. GSV-SRTS integrates these via local slope coordinates and coupled soil-canopy radiation.

The GSV refers to the soil spectral vector, which can be obtained by integrating dry and wet soil data. The global soil reflectance matrix Rm×l can be expressed as follows (Jiang and Fang, 2019): 18Rm×l≈Um×m∑m×lVl×l where m represents the number of dry soil samples; l represents the number of bands; and Rm×l represents the dry soil spectral matrix. Um×m, Σm×l, and Vl×l can be calculated by the Python callable function svd (singlular value decomposition).

Principal component analysis is used to obtain Σm×n and Vn×l, where Vn×l takes the first few vectors. Um×mΣm×ncan be written as Cm×n. For our GSV model, instead of m row vectors, C1×(n+1) is needed, which can be expressed as follows (Jiang and Fang, 2019): 19C1×(n+1)=R1×dV(n+1)×d-1 where R1×d represents the soil spectral data from the region of interest. The first several rows of V(n+1)×d can account for the first n rows of V(n+1)×d because we take the first several vectors through principal component analysis. The n+1 row of V(n+1)×d represents the vector of the n rows of wet soil and one soil spectral data of the study area. Finally, the reflectance spectrum of the soil background can be calculated as follows (Jiang and Fang, 2019): 20R1×l=C1×(n+1)V(n+1)×l where R1×1 is the final soil spectral vector.

The discrete ordinates method, a standard approach for solving radiative transfer problems, was used to numerically solve the coupled SRTS system with its boundary conditions. The incoming radiation is parameterized by the total flux Fdir+dif(Ω0) (Shabanov et al., 2000). The horizontal average intensities I‾(z,Ω) and U(z,Ω) are divided into direct and diffuse components for computational solution. The final simulated BRF is calculated from the resulting upwelling radiation intensity at the canopy top (z=0) for the viewing direction Ωv: 21BRF(Ωv,Ωs)=π⋅I‾(0,Ωv)Fdir(Ωs)+Ediff where Ediff=∫2πd0(Ω)|cos⁡θv|dΩ is the total diffuse irradiance on the horizontal plane.

To comprehensively evaluate the proposed GSV-SRTS model, the simulations were conducted as follows: First, the overall accuracy of GSV-SRTS was evaluated against DART using scatterplots to validate the model feasibility. Next, GSV-SRTS was compared with the typical mountain CR models to validate its advantage in various terrain and canopy conditions after incorporating the SRT theory and soil spectral vector. The angular distribution of GSV-SRTS was analyzed to reveal the response mechanism of terrain and canopy conditions to multi-angle observations. Finally, multi-resolution satellite imagery was used to evaluate model suitability in real mountainous areas.

The DART model is a highly regarded analysis tool utilized for modeling the transmission of radiation signals within soil-leaf-canopy systems (Gastellu-Etchegorry et al., 2012). It has the capability to accurately simulate the canopy reflectance in various types of terrain and canopy structures. During the validation of the model, DART simulations were used as the standard to assess the reliability of the GSV-SRTS model as well as other comparable models. In the virtual scenarios created by DART, specific parameters such as the solar zenith angle of 30° and solar azimuth angle of 0° were set. Additionally, to evaluate the performance of the model in multi-angle observations, the view zenith angle was varied from 0 to 60°, while the view azimuth angle ranged from 0 to 360°, encompassing the entire observation hemisphere. To investigate the accuracy of the model across different sloping surfaces, the slope angle was adjusted from 0 to 50° in increments of 10°, covering gentle, moderate, and steep slopes. The aspect was fixed at 0, 90, 180 and 270°, corresponding to surfaces facing north, east south and west respectively. Various values of Leaf Area Index (LAI) ranging from 0.5 to 4, canopy height from 2 to 30 m, and canopy density from 0.1 to 0.8 were set to represent sparse, moderate, and dense vegetation coverage scenarios. The virtual vegetation scenes were standardized at 100m×100 m with diverse canopy structures, terrains, and viewing conditions. The specifics of the input parameters for generating scenes in DART are outlined in Table 1. In comparison to traditional mountain canopy reflectance models, the SLCT and GOST2 models were utilized, which are based on radiative transfer and geometrical optical theories for sloping terrains. All results from the compared models were juxtaposed with the DART simulations in the same virtual scenes as the benchmark for accuracy assessment.

To discover the influence of canopy structure, multi-angle CR simulations were conducted. The virtual vegetation scenes were set with stable terrain and view conditions. The sun zenith angle was set to 30° and the sun azimuth angle was set to 0°. The view zenith angle was set to range from 0 to 60°, and the view azimuth angle ranged from 0 to 360° to cover the hemispheric viewing space. And the canopy densities were set to 0.2, 0.5 and 0.8 to demonstrate sparse, medium and dense vegetation coverage. On multi-angle CR simulations, the terrain and canopy condition effects were assessed explicitly.

To assess the effectiveness of the proposed model, it is imperative to simulate canopy reflectance in a real mountainous setting. The chosen location for this study is the Wanglang Field Observation Station, situated in the northeastern region of the Tibetan Plateau at coordinates approximately 32.94° N and 104.07° E This area boasts a diverse landscape characterized by towering mountain peaks and deep gorges, with elevations ranging from 2000 to 4200 m and slopes that can vary from gentle inclines to steep gradients of up to 76°. The various vegetation types that thrive in this unique environment include deciduous forests, coniferous forests, mixed forests combining both coniferous and deciduous species, as well as scrublands. Field measurements were stratified by elevation, slope, and aspect to cover dominant vegetation types. Canopy parameters (LAI, height) were measured via hemispherical photography and UAV LiDAR, with soil spectra sampled from exposed sites aligned with target pixels.

Satellite products from Landsat 9-OLI, Sentinel-2, and Jilin-1 with spatial resolutions of 30, 10, and 0.5 m, respectively, are presented in Fig. 2. Cloud-free remote sensing images from these sensors, covering the experimental area, served as reference data for model performance assessment. The Landsat 9-OLI image was acquired on 30 July 2024, during the vegetation growing season, with solar zenith and azimuth angles of 25.5 and 119.6°, respectively. It was obtained from the Earth Resources Observation and Science (EROS) Center Science Processing Architecture (ESPA) on-demand interface (https://espa.cr.usgs.gov/, last access: 27 April 2026), which provides surface reflectance products corrected using the Second Simulation of the Satellite Signal in the Solar Spectrum Vector (6SV) model (Vermote et al., 2016). The Sentinel-2 image was captured on 8 July 2022, with corresponding solar zenith and azimuth angles of 18.73 and 119.2°, and was downloaded from the Copernicus Open Access Hub (https://dataspace.copernicus.eu/, last access: 21 April 2026). The Jilin-1 remote sensing images were acquired and mosaicked from the Jilin-1 Satellite Network (https://www.jl1mall.com/, last access: 27 April 2026), with data collected between June and August from 2022 to 2024. These three remote sensing images are accompanied by corresponding digital elevation models (DEMs) for topographic analysis. Topographic factors for the study area were derived from the following sources: the ASTER Global Digital Elevation Model (GDEM) version 2 (spatial resolution of 1 arc-second, approximately 30 m) was used for the Landsat 9-OLI image; the Copernicus DEM GLO-10 product (10 m resolution) was acquired from the Copernicus Open Access Hub to match the Sentinel-2 image; and high-resolution DEM data were obtained via unmanned aerial vehicle (UAV) laser scanning and resampled to a 0.5 m resolution to align with the Jilin-1 imagery.

The input parameters for driving the SRTS model are summarized in Table 2. These parameters have been carefully defined using a combination of field measurements, remote sensing products, and relevant literature. For example, the leaf linear characteristic dimension (SL) has been designated as 0.05–02 m based on typical values for vegetation (Jacquemoud et al., 2009). Canopy structure parameters included in the model consist of a leaf area index (LAI) ranging from 2–4 m2m-2, canopy height (CH) varying from 5–30 m to encompass the range of measurements, and an average leaf angle (ALA) of 30° which follows an ellipsoidal distribution assumption (Campbell, 1990). In order to accurately capture the topographic variability of the study area, parameters such as slopes (θg) ranging from 0–60° and aspects (φg) from 0–360° have been incorporated into the model. These parameters ensure that the full terrain characteristics are taken into consideration during the simulations. Furthermore, atmospheric parameters for the study area have been derived using the MODTRAN atmospheric radiative transfer model (Berk et al., 2004). This ensures that the atmospheric conditions are accurately represented in the model simulations. The illumination geometry has been fixed based on remote sensing observations, providing a consistent and reliable input for the model. By incorporating all of these parameters into the SRTS model, we are able to accurately simulate and analyze the surface reflectance properties of the study area.

The input parameters for the model are outlined in Table 2, with a summary provided for reference. A dataset based on these parameters was used to generate 10 000 combinations of input values. Each combination was then used to compare the remote sensing reflectance with the simulated reflectance stored in a lookup table (LUT) on a pixel-by-pixel basis. In order to mitigate the challenges posed by the ill-posed nature of the model inversions (Vermote et al., 1997) the final outcome was determined by averaging the results from the 100 best-matching simulations. The degree of agreement between the observed and simulated reflectance was evaluated using the following cost function: 22Cost=Simr-ObsrObsr2+Simn-ObsnObsn2 where Simr and Simn are the simulated reflectances in the red and NIR bands, respectively, and Obsr and Obsn are the observed reflectances in the red and NIR bands, respectively.

To validate the overall simulation accuracy of the GSV-SRTS model, the GSV-SRTS simulations were compared against the DART simulations. As shown in Figs. 3 and 4, the scatter plots were generated using comprehensive input parameter combinations, including varying slopes, aspects, canopy densities, and observation directions. The results showed that GSV-SRTS achieved the highest R2 value of 0.9136 (0.9052) and the lowest RMSE of 0.0146 (0.0106) in the red (NIR) band, outperforming other compared models.

As shown in Fig. 4, the model comparison across different slopes shown that the GSV-SRTS and DART simulations have good consistency over different sloping terrains. In comparisons between DART and other models, the performance of GOST2 and SLCT deteriorates as slope increases. The SLCT model, which assumes trees grow perpendicular to the ground, exhibits distortions in simulating crown gap fraction on steeper slopes. This in turn affects the accurate characterization of radiation contributions over sloping terrain. GOST2 assumes that trees follow the Neyman type-A distribution pattern (Fan et al., 2015). Although the GOST2 model can accurately describe the spatial distribution of the trees, the scenes generated by DART where trees are randomly distributed is more suitable for SRTS. Specifically, in the red (NIR) band, GSV-SRTS precision increases from 0.8702 (0.8635) at 0° to 0.9318 (0.9234) at slopes ≥30°. This comparison demonstrates enhanced capability of GSV-SRTS in complex terrain conditions.

To specifically compare the reflectance under different view zenith angles, the different slopes were set from 0 to 50° with an interval of 10°, as illustrated in Fig. 5. As the terrain slope increases, the GSV-SRTS model achieves the best match with DART simulations in the red and NIR bands, despite SLCT performing better on flat terrain. As the slope steepens, the fundamental differences in physical approach of each model become increasingly apparent. The SLCT model relies on the simplified plane-parallel RT theory, assuming homogeneous leaf distribution and ignoring slope-induced shadowing effects. This leads to systematic underestimation of multiple scattering contributions in steeper terrain. The GOST2 assumes that the trees follow the Neyman type-A distribution, so when view zenith angle is near nadir, the probability of observing the background is larger than that of the other compared models. The GSV-SRTS model accounts for terrain conditions by coupling a voxel-based canopy representation with a Monte Carlo photon-tracking approach, enabling it to resolve anisotropic scattering patterns caused by tilted surfaces. This ability to adjust and remain accurate even in more complex conditions makes GSV-SRTS especially well-suited for sloping terrains.

To reveal the effects of terrain factors and viewing conditions to canopy reflectance, the angular distributions of reflectance under various scenarios were simulated by the GSV-SRTS model. The results are presented in the form of polar coordinate maps. In these maps, the radial distance corresponds to the zenith angle, while the angular position represents the viewing azimuth angle. The polar contour maps for the red and NIR bands are displayed in Figs. 6 and 7, respectively.

As shown in Fig. 6, reflectance in the red band ranges primarily between 0.07 and 0.19. The peak of reflectance occurs at the hotspot, where the view zenith angle is 30° and the azimuth angle is 0°. At the same azimuth angle, reflectance decreases as the zenith angle increases. This angular dependence occurs due to geometric shadowing, path length effects and volume scattering reduction. Photons penetrating the canopy at off-nadir angles traverse longer effective path lengths through the vegetation medium, enhancing absorption by the canopy.

When the aspect is 0° (facing the illumination direction), increasing the slope expands the high-reflectance zone and intensifies maximum reflectance at the azimuth angle of 0°. This amplification occurs because steeper slopes facing the light source create optimal conditions for enhanced canopy reflectance. The tilted surface exposes a larger area to direct solar radiation, enhancing the sunlight received by slope-facing vegetation by a factor of 1/cos⁡θ (where θ is slope angle) (Dubayah and Rich, 1995). On the slope of 20°, solar radiation increases by 64 % compared to that in flat terrain.

However, when the slope reaches 20° and the aspect shifts to 90° (panel g) or 270° (panel i), high reflectance concentrates in specific sections of the upper hemisphere. Reflectance is higher between azimuth angles of 0–90° in panel (g) and 270–360° in panel (i) because the probability of sunlight reaching the ground becomes larger. For slopes with an aspect of 90°, maximum reflectance occurs when the solar azimuth angle is approximately 90°. The PP (principal plane) of incidence is the same with the normal vector of the slope, minimizing the shadowing effects. This geometry enhances the backscatter direction observed by the sensor. It also creates minimal cast shadows and maximum visible illuminated areas, increasing the apparent reflectance. Similarly, slopes with a 270° aspect exhibit peak reflectance near a solar azimuth angle of 270°, following the same geometric principle. Notably, polar contour maps of a 20° slope with aspects of 90 and 270° exhibit symmetry about the vertical axis. This symmetry stems from the fact that the relative azimuth angle between the illumination and viewing directions is the same in both instances. When the sun-sensor geometry is fixed, the reflectance remains consistent, showing mirror-like symmetry around the azimuth of 0°. This confirms that light interacts with slopes in the same way, regardless of their facing direction.

In the NIR band, reflectance displays similar trends yet maintains consistently higher values relative to the red band. As shown in Fig. 7, reflectance ranges primarily from 0.31 to 0.56, with higher values compared to the red band. This difference primarily stems from reduced chlorophyll absorption in the NIR spectrum, enabling more effective light penetration through multiple canopy layers. These distinct interactions lead to characteristically higher NIR reflectance and lower red reflectance in vegetation canopies.

When the slope is 20° with the aspect of 90 or 270°, canopy reflectance increases in the hemispherical distribution at the small zenith angles. This demonstrates that steeper slopes facing the sunward direction receive more solar illumination. The consistent slope and illumination direction, combined with symmetrical aspect, result in a symmetrical distribution of gap fractions, which in turn leads to a symmetrical hemispherical reflectance distribution. For a terrain characterized by the slope of 20° facing the aspect of 90°, lower reflectance is concentrated along the 90–180° view azimuth angles. In contrast, when the aspect shifts to 270°, reflectance decreases along the 180–270° view azimuth directions. When the slope faces 90°, sensors viewing along the azimuths of 90–180° observe the sunlit slope but detect reduced reflectance. Because when the solar azimuth equals aspect, the illumination angle on the inclined surface is minimized, reducing casting shadows and increasing the projected sunlit area. This geometry enhances the backscattering direction observed by sensors. The same physical principles apply to the aspect of 270°, where reflectance maxima occur near solar azimuth of 270°, demonstrating symmetrical behaviour across the PP.

While the hotspot fundamentally results from aligned sun-view geometry reducing mutual shadowing, sloping terrain distorts its signature: local solar/view angles (Eq. 1) shift peak reflectance toward the illuminated slope face, amplified by topographic modulation (Eq. 5). GSV-SRTS resolves this by coupling slope-corrected geometry with stochastic gap probability, capturing asymmetric hotspot broadening on inclined surfaces.

These results demonstrated the importance of incorporating topographic factors into GSV-SRTS in mountainous conditions, where terrain-induced factors, solar and view geometries significantly affect canopy reflectance patterns. The ability of GSV-SRTS to account for terrain-related variations ensures more accurate RT process simulations.

The gap probability, a critical factor influencing the radiative properties of vegetation and soil in remote sensing applications, was utilized in a study to investigate how canopy structure impacts surface reflectance simulations. Using the GSV-SRTS model, the hemispherical distributions of surface reflectance in the red and NIR bands were simulated under different canopy densities. Figure 8 illustrates that canopy reflectance peaks in the hotspot direction, regardless of changes in slope. This directional peak is a result of decreased canopy densities and shadowing from the ground when the sensor's viewing angle aligns with the direction of solar illumination. The GSV-SRTS model accurately captures the reduced probability of observing shaded areas by integrating terrain-modified gap probability and geometries into its simulations. Increasing canopy density leads to a noticeable increase in NIR reflectance (panels b, d, and f) particularly at the nadir, due to enhanced photon interactions within the canopy and changes in gap probability affecting radiation penetration. In contrast, red band reflectance (panels a, c, and e) remains low across all canopy densities, reflecting the strong absorption of red light by chlorophyll. Therefore, both spectral regions demonstrate significant bidirectional reflectance distribution function (BRDF) effects, with NIR reflectance being more sensitive to changes in canopy density.

Moreover, reflectance in the backward-scattering direction consistently exceeds that in the forward-scattering direction. This indicates sunlit crowns become more visible at this direction and mutual shading effects are minimized in this viewing geometry, where the sunlit portions dominate the field of view of sensors. Such directional reflectance features are especially pronounced under dense canopies, where shadowing and multiple scattering further enhance the contrast between forward and backward observations. These reflectance characteristics provide critical understanding of the 3D structure and complexity of forest canopies.

To assess the effectiveness of the GSV-SRTS model in a practical mountainous setting and explore how well the model works at different levels of detail, we utilized a combination of satellite imagery sources with varying resolutions. Specifically, the Landsat 9-OLI imagery with a resolution of 30 m, the Sentinel-2 imagery with a resolution of 10 m, and the Jilin-1 imagery with a resolution of 0.5 m were employed to establish the reference reflectance values for our model evaluation. Additional information regarding the specific geographic area of focus for our study and the technical details of the images can be found in Sect. 3.4.

As illustrated in Fig. 9, the GSV-SRTS-simulated reflectance shows lower R2 values of 0.7129 (0.7287) and higher RMSE values of 0.0291 (0.0376) in the red (NIR) band when compared with the Landsat9-OLI image reflectance. For the Sentinel-2 image, the SRTS-simulated reflectance achieves R2 values of 0.8067 (0.8097) and RMSE values of 0.0271 (0.0292) in the red (NIR) band. The SRTS-simulated reflectance shows the highest consistency with the reflectance of the JiLin-1 image, with R2 values of 0.9078 (0.9143) and RMSE values of 0.0201 (0.0212) in the red (NIR) band.

Even for medium-low resolution pixels, GSV-SRTS maintains advantages by statistically aggregating sub-pixel heterogeneity into radiative parameters, avoiding homogenization biases. Its terrain-aware structure ensures accurate BRDF shapes on slopes, where homogeneous models exhibit systematic errors. However, the reflectance simulation accuracy of the GSV-SRTS model enhances with higher spatial resolution of remote sensing images, owing to the ability of the underlying SRT theory and GSV model to resolve finer details in microscale settings. This compatibility stems from the core mechanics of the GSV-SRTS model, which simulates photon interactions within the canopy in detail. The model incorporates key processes, including multiple scattering between leaves, leaf orientation, shadowing effects, and canopy structure variation. And the soil background is fully considered and accurately modelled. Consequently, the GSV-SRTS model is highly compatible with high-resolution imagery. In contrast, coarser resolution images tend to smooth out small-scale features, limiting the ability of the model to leverage its detailed photon-tracking capabilities. This limitation is clearly evidenced by the lower R2 values obtained with Sentinel-2 and Landsat 9-OLI data, which are primarily attributed to the homogenization of intra-pixel spatial details – terrain variation, canopy structure, and soil background – during the pixel integration process. Our model simulates these sub-pixel heterogeneities explicitly, but the validation data from medium-resolution sensors represents an aggregated average. This does not diminish the model's utility for coarser resolutions; rather, it highlights that the GSV-SRTS model is particularly advantageous for applications where understanding or quantifying the impact of sub-pixel heterogeneity is essential, such as in scaling studies, high-fidelity scene simulation, or informing the parameterization of larger-scale biogeophysical models. As a result, the GSV-SRTS model achieves optimal accuracy with high-resolution images, producing more realistic reflectance estimates. Therefore, the integration of high-resolution data with the photon-sensitive framework of the GSV-SRTS model enables precise capture of subtle canopy reflectance variations, optimizing model performance and achieving highly precise results.

As a result, the GSV-SRTS model achieves optimal accuracy with high-resolution images, producing more realistic reflectance estimates. Therefore, the integration of high-resolution data with the photon-sensitive framework of the GSV-SRTS model enables precise capture of subtle canopy reflectance variations, optimizing model performance and achieving highly precise results.