In the Hapke-HSR model, dry soil is treated as a semi-infinite horizontal surface containing irregularly and randomly distributed absorbing particles (Ding et al., 2022), and the formulas of this model are defined as follows: 1Rd(θs,θv,φ,λ)=ω41cos⁡θs+cos⁡θv{[P(g,g′)(1+B(g))]+H(cos⁡θs)H(cos⁡θv)-1}2P(g,g′)=1+bcos⁡g+c(3cos⁡2g-1)2+b′cos⁡g′+c′(3cos⁡2g′-1)23cos⁡g=cos⁡θscos⁡θv+sin⁡θssin⁡θvcos⁡φ4cos⁡g′=cos⁡θscos⁡θv-sin⁡θssin⁡θvcos⁡φ5B(g)=B01+1htan⁡(g2)6H(x)=1+2x1+2x1-ω where Rd(θs, θv, φ, λ) represents dry soil reflectance, and ω denotes the average SSA. B(g) is the hotspot function, with B0=0.4 and h=0.1 adopted as optimal values. The scattering phase function is represented by P(g,g′), for which the parameters are set as b′=0.4, c=0.4, and c′=0. These parameter values are adopted from previous studies (e.g., Hapke, 2012; Ding et al., 2022) and represent commonly used or empirically optimized values for soil surfaces, providing stable and physically reasonable simulations across a wide range of conditions. The P(g,g′) function is employed in this study because it provides an improved representation of soil anisotropic scattering characteristics. Here, θs, θv, and φ denote the solar zenith angle (SZA), view zenith angle (VZA), and relative azimuth angle (RAA), respectively.

The relationship between the soil SSA and wavelength was established via radiative transfer theory. The association formulas of the SSA and wavelength are as follows: 7ω=1-4πMχsoilλ where M represents the soil particle size and shape-dependent parameter and χsoil is the soil refractive index, which is important for simulating soil optical properties.

In the Hapke-HSR model, the relationship between SSA (ω) and wavelength is further simplified as follows: 8ω1=1-1A0λ+A19ω2=1-1A2λ+A3+Δd where Δd refers to the offset of two soil spectra and where A0, A1, A2 and A3 represent the soil spectral parameters.

Considering the influence of SMC, we assume that the equivalent water thickness represents the transition from dry soil to wet soil (Yang et al., 2011). The formulas are defined as follows: 10Rw(θs,θv,ϕ,λ)=Rd(θs,θv,φ,λ)×e-αwaterf11αwater=4πχwaterλ where Rw(θs, θv, φ, λ) represents wet soil reflectance, χwater is the water refractive index for water, and f is the equivalent water thickness.

The primary challenge in addressing the statistical relationship between the SSA and wavelength of the Hapke-HSR model is to provide a stable parameter χsoil. In this study, we assume that the shape of the dry SSR is consistent with the variation in the SSA with wavelength (Ding et al., 2022). A method is proposed to calculate the parameter χsoil via the dry SSR and improve the Hapke-HSR model. The SSA typically has a significant influence on the SSR, with multiple scattering events often disregarded in the theoretical derivation (Yang, 2022). The relationships between dry SSR and SSA are as follows: 12Rd(θs,θv,φ,λ)=C1×ω13C1=14(cos⁡θs+cos⁡θv){[P(g,g′)(1+B(g))]-1}14ω=1-C2×χsoilλ15C2=4πM16χsoil=λC2×(1-Rd(θs,θv,φ,λ)C1) where C1 and C2 are the shape adjustment parameters of the dry SSR. We use C1=1 and C2=1 as the initial values to simplify the calculation, and we then further calculate the shape adjustment parameters.

To incorporate the impact of multiple scattering within the soil, we first calculate the χsoil parameter via Eq. (16). This parameter is then utilized in the improved Hapke-HSR model to determine the dry SSR. We adjust the dry SSR estimated with the improved Hapke-HSR model via the measured SSR and then calculate the adjusted dry SSR. The relationship between the dry SSR simulated via the improved Hapke-HSR model and the measured dry SSR can be expressed via the following formula: 17Rd′(θs,θv,φ,λ)=C3×Rd(θs,θv,φ,λ)+C4 where C3 and C4 are spectral shape adjustment parameters for dry soil reflectance, obtained through linear regression between simulated and measured reflectance, and used as empirical correction terms to compensate for discrepancies arising from the simplified treatment of multiple scattering in the Hapke-HSR model. Here, Rd(θs, θv, φ, λ) denotes the reflectance simulated by the improved Hapke-HSR model, and Rd′(θs, θv, φ, λ) represents the measured dry soil reflectance.

Figure 1a shows the variation in the parameter χsoil and soil SSA calculated with the dry SSR (i.e., dup20_009) considering the wavelength. The parameter χsoil increases with wavelength, the slope clearly increases at wavelengths of 2.0–2.5 µm, and there are obvious peak values in the absorption band of water (centred at 1.47, 1.90 and 2.21 µm). The soil SSA is highly similar to the dry SSR. With increasing wavelength, the SSA increases significantly in the spectral range of 0.4–1.36 µm. In the spectral range of 1.36–2.5 µm, obvious valleys occur in the absorption band of water, which is similar to that of dry SSR. In general, the SSA and dry SSR curves display very high similarity in terms of shape, but the SSA curve is flatter. Figure 1b shows the measured dry SSR and the SSR estimated with the improved Hapke-HSR model. Clearly, the SSR modelled by the improved Hapke-HSR model matches well with the measured dry SSR and can characterize the dry SSR characteristics well. The accuracy of this improved model in fitting these typical data is shown in Table 4 (R2=1.00, RMSE = 0.001). These analyses suggest that calculating the parameter χsoil via dry SSR data is feasible and can solve the problems associated with the statistical relationship between the SSA and wavelength of the Hapke-HSR model.

In this section, eight different databases provided by Dupiau et al. (2022) are used to verify the Hapke-HSR + MARMIT-2 model (Table 2). These eight datasets were acquired primarily via the Analytical Spectra Devices (ASD) FieldSpec spectroradiometer. The soil types in the datasets are diverse, consisting primarily of clay, silt, and sand as the main components. The Bab16, Dup20, and Liu02 databases provide the soil components of each sample, such as organic matter, iron, nitrogen and organic carbon. These eight databases provide dry and wet SSR data for 219 soil samples, spanning 1984 spectra in the 0.4–2.5 µm range. The data quality of these databases is high, but there are some uncertainties in the 2.4–2.5 µm range. Therefore, these databases offer crucial data for validating the ability of the Hapke-HSR + MARMIT-2 model to describe SSR features.

In this section, we first analyse the effects of the main input parameters of the Hapke-HSR + MARMIT-2 (HM) model on the soil reflectance properties. Table 3 shows the main input variables of the HM model. The Hapke-HSR model has many input variables, and the variables of this model were optimized in previous studies (Ding et al., 2022; Dupiau et al., 2022; Hapke, 2012; Verhoef et al., 2018); five main variables were used to describe the soil spectral and angular features. The MARMIT-2 model includes three main parameters that describe the influence of the SMC on SSR. We analyse the effects of different parameters in the HM model on the simulated SSR.

Figure 3 illustrates the impact of the main variables of the Hapke-HSR + MARMIT-2 (HM) model on the SSR. For the influence of angle variation, the SSR calculated via the HM model increases with increasing SZA parameters. The increase in SSR is obvious in the range of 0.4–1.36 µm; however, the rate of increase in SSR subsequently decreases. The impact of VZA parameters on SSR is consistent with the influence of the SZA parameters since the Hapke-HSR model is reciprocal. Furthermore, the influence of the RAA parameters on SSR is basically the same as that of the SZA parameters with increasing RAA parameters; however, the changes in soil reflectance are slightly different. With increasing parameter b, the SSR simulated by the HM model decreases, with a more pronounced reduction in the range of 0.4–1.36 µm than in 1.36–2.5 µm. Both parameters b and c influence the BRDF shape through the phase function. However, b primarily controls the overall anisotropy of scattering, whereas c governs the forward–backward asymmetry. Under the observational configurations considered in this study, b shows a stronger and more stable influence on reflectance, while the sensitivity of c is relatively weaker. Therefore, the discussion focuses on parameter b (Ding et al., 2022). The SSR simulated with the HM model also decreases with increasing M; this finding corroborates the experimental observations regarding spectral variations due to soil particle size reported by Sun et al. (2023). However, the variation in the parameter M of the SSR is basically the same at different wavelengths. A possible reason is that the influence of the parameter M in the range of 0.4–2.5 µm on the SSR is consistent, which may be related to the structure of Eq. (7). For the variables related to the SMC, with increasing parameter δ, the SSR decreases, whereas in the strong absorption band of water, this effect is smaller. A possible reason for this result is that the absorption of water weakens the impact of the parameter δ on the SSR. As the parameter L increases, the SSR decreases in the range of 1.0–2.5 µm, whereas the variation in the parameter L has no effect on the soil reflectance in the range of 0.4–1.0 µm. With increasing parameter ε, the SSR decreases. In the strong absorption band of water, the simulated SSR quickly decreases. In summary, the main parameters of the Hapke-HSR model are related to the influence of dry SSR and angular variation characteristics, and the variables of the MARMIT-2 model mainly account for the influence of SMC on the SSR. Therefore, the HM model can characterize the spectral and angular reflectance attributes of dry and wet soils by coupling the Hapke-HSR and MARMIT-2 models.

In previous studies, we assessed the Hapke-HSR model to describe soil BRDF features (Ding et al., 2022). Therefore, we focus mainly on evaluating the soil spectral characteristics in this paper. In addition, these eight soil databases do not provide angle-related information. Therefore, we use SZA = 45°, VZA = 0° and RAA = 0° as fixed values because SSR is usually measured in the nadir direction. Figure 4 shows that the Hapke-HSR, MARMIT-2 and Hapke-HSR + MARMIT-2 (HM) models effectively fit the influence of the typical fitted SSR (dup20_009-001) at SMCs = 0 %, 14.45 %, 27.34 %, 31.6 %, 36.2 %, 40.34 %, 45.07 %, 49.25 %, and 57.06 %, respectively. The SSR decreases significantly with increasing SMC, and the main absorption bands (centred at 1.47 and 1.90 µm) of water become wider. The outcomes simulated using the Hapke-HSR, MARMIT-2 and HM models can be used to determine the change in SSR with increasing SMC and are highly in line with the measured SSR values. On the basis of a comparison of the results, the HM model fits the measured SSR better than the Hapke-HSR and MARMIT-2 models do, especially at SMC ≥ 30 %. Compared with the MARMIT-2 model, the HM model yields slightly better results. The main reason is that the MARMIT-2 model yields very high accuracy in characterizing SSR characteristics. The ability of the Hapke-HSR model to accurately fit the measured SSR decreases with increasing SMC. The correlation analysis results indicate that the fitting ability of these two models meets the relevant requirements. However, at SMC ≥ 30 %, the fitting capability of the MARMIT-2 model is relatively low due to slight underestimations in the range of 0.4–1.36 µm, and there is an overestimation in the strong water absorption band; moreover, the Hapke-HSR model has difficulty capturing the SSR characteristics in the absorption band of water, which leads to significant underestimation of the fitted SSR at a wavelength of approximately 1.90 µm. However, the HM model effectively considers the influence of these factors, resulting in high accuracy for characterizing SSR attributes. To show the differences between the SSR values simulated with these three models and the measured spectral reflectance values, we calculate the bias between them, as shown in Appendix Fig. A1. In addition, the Hapke-HSR and HM models are applied to simulate the dry SSR, and the SSR simulated via the HM model is more consistent with the measured results than the SSR obtained with the Hapke-HSR model is.

Table 4 shows that the Hapke-HSR, MARMIT-2 and HM models fit the parameters and statistical results of the SSR. With increasing SMC, the parameters L and ε in the MARMIT-2 model increase significantly, whereas the parameter δshows little variation. The parameter f increases significantly. Moreover, the SSR fitting accuracy of the Hapke-HSR and MARMIT-2 models decreases with increasing SMC, especially at SMC ≥ 30 %. This finding may be because the MARMIT-2 model ignores the variations in soil scattering characteristics, particle size and shape with increasing SMC. In the Hapke-HSR model, a dry soil surface is overlaid with a water layer to reflect the influence of the SMC on the SSR. This simple assumption limits the ability of the Hapke-HSR model to fit the variable characteristics of the SMC. The overall R2 values for the Hapke-HSR model in SSR fitting vary from 0.952 to 0.971, with RMSE values varying from 0.016 to 0.019, and the MARMIT-2 model achieves R2 values between 0.957 and 0.995 in SSR fitting, with RMSE values ranging from 0.007 to 0.021 and negligible bias. These results indicate that these two models can effectively characterize the variation in SSR with SMC and yield high fitting accuracy. However, the HM model is accurate (RMSE = 0.008), presenting a high R2 (R2=0.991) and a small bias in relation to the measured SSR. This is because the HM model considers the variations in the soil scattering characteristics, particle size and particle shape with increasing SMC. Therefore, the HM model can effectively characterize the attributes of SSR and exhibits greater accuracy than the Hapke-HSR and MARMIT-2 models do, especially at SMC ≥ 30 %.

Figure 5 shows that the Hapke-HSR, MARMIT-2 and HM models fit the typical measured SSR (bab16_014-008) at SMC = 0 %, 5 %, 10.7 %, 16 %, 21.1 %, 30.8 %, and 45.5 %, respectively. This set of typical data is suspected to have a specular reflection effect when SMC = 30.8 % and 45.5 %. Therefore, we further validate the capacity of the HM model for describing the relevant SSR attributes. The outcomes of the Hapke-HSR, MARMIT-2 and HM models match the typical measured SSR when the SMC < 30 %. The Hapke-HSR and MARMIT-2 models cannot effectively consider the specular reflectance characteristics at high SMCs (SMC ≥ 30 %), and the results of the HM model display greater consistency with the measured SSR values. When the SMC is high and there is a specular reflectance effect, the fitting capability of the Hapke-HSR model is significantly underestimated in the range of 0.5–1.2 µm and two strong water absorption bands; moreover, there is a slight overestimation in the range of 1.5–1.9 µm. The outcomes of the MARMIT-2 model are marginally underestimated across the 0.4–1.36 µm interval, and there is a slight overestimation across the spectral interval of 1.36–2.5 µm, particularly in the strong water absorption region. The HM model can match the measured SSR well, especially at SMC = 30.8 % and 45.5 %, possibly because this model accounts for the specular scattering characteristics of high SMCs (SMC ≥ 30 %) on the basis of the coefficient (b) and soil particle shape-dependent parameter (M). To better show the differences between the SSR values simulated with these three models and the measured SSR values, we calculate the bias between them, as shown in Appendix Fig. A2. The results indicate that the HM model can describe SSR features effectively at SMC ≥ 30 %, and the simulated values exhibit very high consistency with the measured SSR values.

Table 5 shows that the Hapke-HSR, MARMIT-2 and HM models fit to the SSR parameters and statistical results. The overall precision of the Hapke-HSR and MARMIT-2 models in terms of fitting measured SSR was high (R2=0.943–0.946, RMSE = 0.006–0.022), especially at SMC < 30 %; however, these two models were not suitable at SMC ≥ 30 %, which needs to be improved by accounting for specular reflectance. The HM model shows greater accuracy in fitting the variation in the SMC than the Hapke-HSR and MARMIT-2 models; the overall R2 is 0.995, the RMSE is 0.009, and the bias is negligible. When SMC = 30.8 % and 45.5 %, the measured SSR are suspected to have a specular reflection effect, and the HM model maintains a higher fitting precision (R2=0.990–0.993, RMSE = 0.009) than the Hapke-HSR and MARMIT-2 models (R2=0.846–0.973, RMSE = 0.019–0.053). These results indicate that these two models can be combined by coupling the Hapke-HSR and MARMIT-2 models (i.e., HM model), which can effectively determine the variation in SSR with increasing SMC, particularly in the presence of specular reflectance.

Figure 6 presents a comparison of the SSR results obtained from the Hapke-HSR, MARMIT-2 and HM models and the measured SSR values from eight databases. These three models are generally highly accurate in terms of capturing SSR features. However, the HM and MARMIT-2 models (R2=∼0.993) fit the measured SSR data with slightly greater correlation accuracy than did the Hapke-HSR model (R2=0.957), and the RMSE values of the HM (RMSE = 0.010) and MARMIT-2 (RMSE = 0.012) models were significantly lower than the RMSE of the Hapke-HSR (RMSE = 0.027) model. Additionally, the MRE values of the HM and MARMIT-2 models are approximately 5.74 % and 6.43 % lower than that of the Hapke-HSR model, respectively. These findings indicate that the HM model yields the highest level of accuracy in fitting the measured SSR, followed by the MARMIT-2 model, whereas the Hapke-HSR model has the worst fitting effect on the basis of the measured SSR. The main reason is that the Hapke-HSR model includes a simple assumption regarding the effect of the SMC on SSR. In addition, the SSR simulated by the Hapke-HSR and MARMIT-2 models is considerably uncertain at high SMCs (e.g., Fig. 5f–g), whereas the HM model results display greater consistency with the fitted SSR value. In general, the HM and MARMIT-2 models excellently characterize the SSR attributes of soil and yield greater accuracy than the Hapke-HSR model for the eight soil databases does, and the SSR estimates produced by the HM model are marginally more accurate than those of the MARMIT-2 model are.

The MARMIT-2 and HM models achieve excellent fitting accuracy at SMC levels ranging from 0 %-30 %, whereas the Hapke-HSR and MARMIT-2 models exhibit moderate fitting capability at SMC ≥ 30 % (e.g., Figs. 3 and 4). Therefore, focus is placed on comparing the fitting results of the above three models under the condition of an SMC ≥ 30 %. Figure 7 shows the comprehensive results obtained with the Hapke-HSR, MARMIT-2 and HM models at SMC ≥ 30 %, and these three models exhibit strong agreement with the measured SSR (R2>0.90), with RMSE values ranging from 0.007–0.028. However, the accuracy of the HM model (R2=0.993, RMSE = 0.007) for fitting SSR data is slightly better than that of the MARMIT-2 model (R2=0.983, RMSE = 0.012) and significantly greater than that of the Hapke-HSR model (R2=0.909, RMSE = 0.028). Compared with those of the MARMIT-2 and Hapke-HSR models, the RMSE values of the HM model are 41.7 % and 66.7 % lower, and the MRE values are 2.158 % and 9.702 % lower, respectively. Moreover, the HM model has the ability to improved the inadequate fitting outcomes of the Hapke-HSR model. These findings show that the HM model can describe SSR attributes more effectively than the other models can, especially at SMC ≥ 30 %. The key factor is that the HM model combines the strengths of both the Hapke-HSR and MARMIT-2 models to better describe the changes in SSR with increasing SMC. The MARMIT-2 model also exhibits higher accuracy at SMC ≥ 30 % since it fully considers the effect of the SMC on SSR.

Next, we further evaluate the ability of the Hapke-HSR, MARMIT-2 and HM models to fit the measured SSR for different bands (0.4–2.5 µm) at SMC ≥ 30 %. Figure 8 shows the comparison results between the simulated SSR values of the three models and the measured SSR values. The results of these three models are highly consistent with the measured values. The R2 values of these three models are generally very high, in the range of 0.4–2.5 µm. The R2 value of the HM model is the largest, followed by those of the MARMIT-2 model and, finally, the Hapke-HSR model. However, the consistency between the outcomes of these three models and the measured values in the strong absorption band of water (centred at 1.90 µm) is significantly lower than that for the other bands. The RMSE results from the HM model were the smallest, followed by those of the MARMIT-2 model. The maximum RMSE value of the Hapke-HSR model was approximately 0.05 because the Hapke-HSR model uses a simplistic assumption to reflect the effect of the SMC. However, the HM and MARMIT-2 models significantly outperform the Hapke-HSR model, specifically at two major absorption bands of water (centred at 1.47 and 1.90 µm), since the HM and MARMIT-2 models fully consider the changes in SSR characteristics with variations in the SMC. The MRE trends of the three models are basically similar to the RMSE trends, and the HM model demonstrates the highest level of accuracy, with the MARMIT-2 model following closely behind. The MRE value of the Hapke-HSR model is approximately 60 % at the major absorption band of water (centred at 1.90 µm), which is significantly greater than that of the MARMIT-2 model (28 %) and the HM model (15 %). The bias values of the HM model approach 0.4–2.5 µm, whereas those of the Hapke-HSR model exhibit a large range of variation. These studies indicate that the HM model results are more in line with the fitted SSR, whereas the Hapke-HSR model results are more different from the measured SSR values. In general, the variations between the outputs of these three models and the measured values are in the wavelength ranges of 0.4–0.6 and 2.4–2.5 µm and represent the two major absorption bands of water. The soil reflectance is low over the spectral region from 0.40.6 µm, and the soil reflectance variation remains insignificant in this wavelength range as the SMC increases. The soil measurements have great uncertainty in the range of 2.4–2.5 µm, resulting in poor correlations between the fitting results of these three models and the measured values. The SSR changes rapidly in the strong absorption band of water, which leads to great uncertainty in the fitting results of these models. Compared with the observed SSR model, the Hapke-HSR model has the lowest accuracy, followed by the MARMIT-2 model. The HM and MARMIT-2 models are better than the Hapke-HSR model at SMC ≥ 30 % because these two models fully consider the variation in SSR characteristics with the variation in SMC.

Finally, we analysed the fitting performance of the Hapke-HSR, MARMIT-2, and HM models across eight soil spectral databases, as shown in Fig. 9. All three models achieve high accuracy (NRMSE < 9 %), with the HM model generally outperforming the individual models. The performance varies across datasets, reflecting differences in soil properties, reflectance levels, and sensitivity to soil moisture. The HM model shows clear improvements for most datasets (e.g., Bab16, Dup20, Hum15, Liu02, Mar12, and Phil14), indicating the effectiveness of coupling directional scattering and moisture-related processes. However, the improvement is less pronounced for certain datasets. For example, the MARMIT-2 model already achieves high accuracy for the Les08 dataset, resulting in limited additional improvement. For the Lob02 dataset, the relatively low reflectance leads to larger NRMSE values, which reduces the apparent gain. Overall, these results demonstrate the robustness of the proposed framework across diverse datasets, while also highlighting its dependence on dataset characteristics.

In this section, we analyse the imaginary component of the soil index (χsoil parameter) calculated from the dry soil reflectance for eight different databases with the wavelengths shown in Fig. 10. For different soil databases, the change trend of the χsoil parameter is basically the same. The χsoil parameter increases with wavelength, and there is an obvious peak at the two strong absorption bands of water (i.e., centred at 1.47 and 1.90 µm). However, there are still some differences between different soil databases. The values of the parameter χsoil are greater in the strong absorption band of water for the Bab16 and Liu02 databases, whereas the parameter values χsoil are lower for the Hum15, Lob02 and Phil14 databases in the strong absorption band of water; moreover, the influence of the SMC on these three databases is small. For the same soil database, the change in the parameter χsoil is small, and only the difference between the Bab16 database results is large. In addition, the parameter χsoil is obviously greater in the two water absorption bands, and the overall change range also decreases. This decrease may further affect the accuracy of fitting the measured SSR. Therefore, the method of determining the parameter χsoil in each soil database should be theoretically feasible. The parameter χsoil is derived based on the assumption that the spectral shape of dry soil reflectance is similar to that of the single scattering albedo. Therefore, this approach depends on the availability of dry soil reflectance, which is often difficult to obtain from field measurements or satellite observations. Providing a representative dry soil reflectance remains an important challenge for future work. In addition, the use of an averaged χsoil does not explicitly account for variability in soil properties within the same soil type, such as differences in organic carbon content or texture, which may influence spectral absorption. This simplification may contribute to the observed decrease in model accuracy. In future work, the definition of χsoil could be refined by incorporating soil-specific properties or by grouping spectrally similar soils, which is expected to provide a more accurate representation of absorption characteristics and further improve model performance.

We used the average parameter χsoil (i.e., Fig. 10a) to validate the HM model to characterize the SSR attributes; this model is called the HM_mean model in the following section. Figure 11 shows that the HM and HM_mean models fit the influence of the typical measured SSR value (i.e., bab16_056-051) at SMC = 0 %, 9.1 %, 17 %, 29.9 %, 39.3 %, and 52.4 %, respectively. This set of typical data is thought to have a specular reflection effect when SMC = 52.4 %. The HM and HM_mean models match well with the typical measured SSR values. However, the HM model shows greater consistency with the fitted SSR value than does the HM_mean model, especially at SMC < 10 %. The HM_mean model results in significant underestimation and overestimation at SMC = 0 % and 9.1 %, respectively, because the average parameter χsoil is obviously greater in the water absorption band (Fig. 10a), which further affects the accuracy of the HM_mean model. The HM and HM_mean model fitting results can capture the change in SSR with increasing SMC and are highly in line with the measured SSR values at high SMCs, which may be caused by the obvious SSR broadening in the strong absorption band of water with increasing SMC. In general, the HM model, which uses the average parameter χsoil, can still effectively describe the SSR characteristics, especially at high SMCs. However, the average parameter χsoil leads to a significant broadening of the strong absorption band of water at low SMCs, which further leads to obvious overestimation or underestimation of the SSR fitted by the HM_mean model in the strong absorption band of water.

Table 6 shows that the HM and HM_mean models fit the SSR parameters and statistical outputs. The overall accuracy of the HM and HM_mean models in terms of fitting the measured SSR is high (R2=0.991–0.993, RMSE = 0.005–0.011). According to the simulation results, the HM model is more accurate with respect to the measured SSR than the HM_mean model is, especially at SMC < 10 %, because the average parameter χsoil is obviously greater in the water absorption band. The HM and HM_mean models can effectively describe SSR features, especially high SMCs (R2 > 0.98 and RMSE < 0.01). When SMC = 52.4 %, the measured soil spectral data are suspected to have a specular reflection effect, and the HM model maintains a higher fitting accuracy (R2=0.991, RMSE = 0.005) than does the Hapke_mean model (R2=0.988, RMSE = 0.006). These results indicate that the method of assuming an average parameter χsoil in each soil database should be theoretically feasible. However, all the soil types may have large differences in the parameter χsoil. How to select the parameter χsoil of each soil type will be particularly important in our subsequent study.

Finally, we calculate the overall average parameter χsoil to determine the dependence of the HM model on the dry SSR. First, we normalize all χsoil parameters to the same order of magnitude and then average these indices in each band. Finally, we use all the SSR data to verify the accuracy of this method. Table 7 shows that the HM_mean model fit the statistical results for all the SSR data. Compared with the measured SSR, the HM_mean model has high fitting accuracy. The R2 value of the HM_mean model is 0.988, and the RMSE is 0.014, indicating negligible bias. However, the fitting accuracy of the overall SSR data of the HM_mean model is lower than that of the MARMIT-2 and HM models (i.e., Fig. 6b–c). The main reason is that there are notable discrepancies in the parameter χsoil among various soil types (Fig. 10). The model operation can be simplified by averaging all the χsoil parameters of the soil, but this approach also results in an accuracy decline for the HM_mean model. For SMC values ≥ 30 %, the HM_mean model results in a lower RMSE than that for SMC values < 30 %, which is consistent with the findings illustrated in Fig. 11 and Table 6. However, the NRMSE and MRE values of the HM_mean model at SMC ≥ 30 % were lower than those at SMC < 30 %, possibly because the dry SSR is greater than that of wet soil. In conclusion, the HM_mean model demonstrates proficiency in describing SSR attributes, and the overall average parameter χsoil of the soil can be used to determine the dependence of the HM model on the SSR. In this study, we based our analysis solely on the soil databases, which has limitations. In future research, we will consider soil properties or spectrally similar soils and account for more factors affecting the χsoil parameter to improve the accuracy of the Hapke-HSR + MARMIT-2 model.

Finally, we analyse the impact of the soil parameters on the BRDF shape obtained with the Hapke-HSR + MARMIT-2 model. Considering that we previously used various BRDF data sources to examine the role of the Hapke-HSR model in modelling soil BRDF features, we analyse how the model parameters affect the shape of the BRDF curve (Ding et al., 2022). Figure 12 shows the effects of parameters b, M, δ, L and ε in the Hapke-HSR + MARMIT-2 (HM) model on the soil BRDF shape in the principal plane (PP). With increasing parameter b, the soil reflectance gradually decreases in the forwards direction but increases in the backwards reflection direction. The parameter b has a relatively large contribution to the anisotropy characteristics of the soil. When the parameter M increases, the soil reflectance continuously decreases. The soil anisotropy is strongest when the parameter M is relatively small. With the increase in the parameter δ, the impact of the parameter δ on the soil reflectance is relatively large in the range of 0–0.01, and the anisotropy of the soil is strong; however, the influence of the parameter δ on the SSR is relatively low, and the soil anisotropy is significantly weak. As the parameter L increases, L does not impact the soil reflectance, which corresponds with the results in Fig. 3g. With increasing value of the parameter ε, the equal interval of the soil reflectance decreases since the influence of the surface coverage fraction of water is proportionally related to this factor. In summary, the variation in the parameters b and M in the Hapke-HSR model has a notable effect on the soil BRDF shape, whereas the parameters of the MARMIT-2 model have a relatively minimal effect on the soil BRDF shape. Therefore, the ability of the HM model to describe the features of the soil BRDF is basically consistent with that of the Hapke-HSR model. This occurs because the MARMIT-2 model does not include additional BRDF-related information, whereas the Hapke-HSR model includes input parameters for angle-related information. In future studies, we will comprehensively assess the ability of the HM model to represent soil BRDF features, especially in the forwards direction for wet soil.

Compared with existing models, the proposed Hapke-HSR + MARMIT-2 model provides a physically consistent integration of directional scattering and moisture-related processes. Semi-empirical models such as the BSM and general spectral vectors (GSV) capture spectral variability but lack explicit representation of angular effects (Ding et al., 2022; Jiang and Fang, 2019), which is critical for multi-angular observations and physically based parameter inversion. Moreover, soil reflectance acts as a key background component in canopy reflectance and influences vegetation parameter retrieval. By improving its spectral and directional representation under varying moisture conditions, the proposed framework can provide more reliable inputs for coupled soil–vegetation models (e.g., PROSAIL), thereby reducing uncertainties in vegetation parameter inversion. This study has certain limitations. To simplify the analysis, the effects of surface roughness and porosity are not explicitly considered (Ding et al., 2023; Ding, 2026). Despite this, the model achieves high accuracy, indicating its effectiveness in representing soil reflectance. The evaluation is conducted under a fixed observation geometry due to dataset limitations, which may constrain the characterization of angular effects. Nevertheless, the framework inherently accounts for viewing geometry through the Hapke formulation and can be extended to varying observation conditions. Future work will incorporate multi-angular datasets to further evaluate model performance.