Differentiable Physical Modeling for Forest Above-Ground Biomass Retrieval by Unifying a Water Cloud Model and Deep Learning.
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| Title: | Differentiable Physical Modeling for Forest Above-Ground Biomass Retrieval by Unifying a Water Cloud Model and Deep Learning. |
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| Authors: | Zhao, Cui1,2 (AUTHOR), Shi, Rui1,2 (AUTHOR), Ji, Yongjie3 (AUTHOR), Zhang, Wei1 (AUTHOR), Zhang, Wangfei1,2 (AUTHOR) zhangwf@swfu.edu.cn, He, Xiahong1,2,3 (AUTHOR), Zhao, Han1,2 (AUTHOR) |
| Source: | Remote Sensing. Mar2026, Vol. 18 Issue 6, p912. 30p. |
| Subjects: | Forest biomass, Biomass estimation, Deep learning, Remote sensing, Synthetic aperture radar, Artificial neural networks |
| Abstract: | Highlights: What are the main findings? The proposed DPM framework, which integrates the WCM with deep neural networks, achieved superior forest AGB retrieval performance in both subtropical and temperate forest regions, with R2 values of 0.60 and 0.48, outperforming traditional physical models and purely data-driven approaches, and demonstrating strong generalization across northern and southern forests. By incorporating automatic differentiation, DPM enables joint optimization of the WCM's physical constraints and neural network parameters, preserving the physical plausibility of the model while flexibly capturing complex nonlinear relationships, thereby significantly reducing overfitting under limited training data. What are the implications of the main findings? DPM offers a novel framework for remote sensing-based forest AGB estimation, fully leveraging neural networks' capacity to model complex nonlinear relationships while maintaining the interpretability of physical models, resulting in improved retrieval accuracy and stability. This study was conducted using C-band SAR data. Due to its relatively short wavelength, C-band has limited penetration ability within dense vegetation canopies, and its backscattering signal is prone to saturation in areas with high biomass, which restricts the model's inversion performance in complex forest environments. Future work could enhance this framework in two ways: first, by integrating longer wavelengths, such as L-band and P-band SAR data; second, by adopting physical models that better describe multiple scattering mechanisms, like the MIMICS model. This study offers a methodological reference for combining deep learning with physical scattering models, and these improvements are expected to further overcome the limitations of C-band in dense vegetation conditions. 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. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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| Abstract: | Highlights: What are the main findings? The proposed DPM framework, which integrates the WCM with deep neural networks, achieved superior forest AGB retrieval performance in both subtropical and temperate forest regions, with R2 values of 0.60 and 0.48, outperforming traditional physical models and purely data-driven approaches, and demonstrating strong generalization across northern and southern forests. By incorporating automatic differentiation, DPM enables joint optimization of the WCM's physical constraints and neural network parameters, preserving the physical plausibility of the model while flexibly capturing complex nonlinear relationships, thereby significantly reducing overfitting under limited training data. What are the implications of the main findings? DPM offers a novel framework for remote sensing-based forest AGB estimation, fully leveraging neural networks' capacity to model complex nonlinear relationships while maintaining the interpretability of physical models, resulting in improved retrieval accuracy and stability. This study was conducted using C-band SAR data. Due to its relatively short wavelength, C-band has limited penetration ability within dense vegetation canopies, and its backscattering signal is prone to saturation in areas with high biomass, which restricts the model's inversion performance in complex forest environments. Future work could enhance this framework in two ways: first, by integrating longer wavelengths, such as L-band and P-band SAR data; second, by adopting physical models that better describe multiple scattering mechanisms, like the MIMICS model. This study offers a methodological reference for combining deep learning with physical scattering models, and these improvements are expected to further overcome the limitations of C-band in dense vegetation conditions. 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. [ABSTRACT FROM AUTHOR] |
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| ISSN: | 20724292 |
| DOI: | 10.3390/rs18060912 |
