Differentiable Physical Modeling for Forest Above-Ground Biomass Retrieval by Unifying a Water Cloud Model and Deep Learning

To address the limitations of traditional forest above-ground biomass (AGB) retrieval methods—namely, the restricted accuracy of physical models and the limited generalization ability of purely data-driven models—this study proposes a differentiable physical modeling (DPM) approach for forest AGB estimation. The method adopts the water cloud model (WCM) as a physics-based framework, grounded in radiative transfer theory, and integrates C-band synthetic aperture radar (SAR) data with multispectral imagery. Within the PyTorch tensor computation framework, automatic differentiation (AD) is employed to seamlessly couple the WCM with the deep fully connected neural network (DFCNN), enabling a differentiable implementation of the WCM. Using mean squared error (MSE) as the loss function, the neural network parameters are optimized through backpropagation and gradient descent, thereby constructing an end-to-end trainable DPM model that effectively retrieves forest AGB while preserving physical interpretability and generalization capability. To validate the proposed method, two representative test sites were selected: Simao in Pu’er, Yunnan Province, and Genhe in Inner Mongolia. GF-3 PolSAR and RADARSAT-2 data were used to extract backscattering coefficients and compute the radar vegetation index (RVI), while Landsat 8 OLI imagery was employed to calculate the normalized difference vegetation index (NDVI), difference vegetation index (DVI), and soil-adjusted vegetation index (SAVI). These datasets, together with ASTER GDEM, field-measured biomass, and other relevant datasets, were integrated to construct a multisource dataset combining remote sensing and ground observations. The performance of the DPM model was then compared with the traditional WCM and several data-driven models, including the fully connected neural network (FNN), generalized regression neural network (GRNN), RF, and Adaptive Boosting (AdaBoost). The results indicate that the DPM model achieved R2 = 0.60, RMSE = 24.23 Mg/ha, Bias = 0.4 Mg/ha, and ubRMSE = 22.43 Mg/ha in Simao, and R2 = 0.48, RMSE = 33.29 Mg/ha, Bias = 0.87 Mg/ha, and ubRMSE = 33.28 Mg/ha in Genhe, demonstrating consistently better performance than both the WCM and all tested data-driven models. The DPM model demonstrated consistent performance across ecologically contrasting forest regions. It alleviated the systematic overestimation bias of purely data-driven models and overcame the limitations in predictive accuracy resulting from the simplified structure of the WCM. The differentiability of the WCM enables the loss function errors to be backpropagated through the neural network, thereby allowing the optimization of the physical model parameters. Overall, the DPM framework integrates the advantages of both physical models and data-driven approaches, providing an estimation method with acceptable accuracy for forest AGB retrieval. It also offers theoretical and practical insights for the integration of deep learning and physical knowledge in other research fields.