Temporal–Spatial Partial Differential Equation Modeling of Land Cover Dynamics via Satellite Image Time Series and Sparse Regression
Land cover dynamics play a critical role in understanding environmental changes, but accurately modeling these dynamics remains a challenge due to the complex interactions between temporal and spatial factors. In this study, we propose a novel temporal–spatial partial differential equation (TS-PDE)...
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MDPI AG
2025-03-01
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| Series: | Remote Sensing |
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| Online Access: | https://www.mdpi.com/2072-4292/17/7/1211 |
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| author | Ming Kang Zheng Zhang Zhitao Zhao Keli Shi Junfang Zhao Ping Tang |
| author_facet | Ming Kang Zheng Zhang Zhitao Zhao Keli Shi Junfang Zhao Ping Tang |
| author_sort | Ming Kang |
| collection | DOAJ |
| description | Land cover dynamics play a critical role in understanding environmental changes, but accurately modeling these dynamics remains a challenge due to the complex interactions between temporal and spatial factors. In this study, we propose a novel temporal–spatial partial differential equation (TS-PDE) modeling method combining sparse regression to uncover the governing equations behind long-term satellite image time series. By integrating temporal and spatial differential terms, the TS-PDE framework captures the intricate interactivity of these factors, overcoming the limitations of traditional pixel-wise prediction methods. Our approach leverages <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>×</mo><mn>1</mn></mrow></semantics></math></inline-formula> convolutional kernels within a convolutional neural network (CNN) solver to approximate derivatives, enabling the discovery of interpretable equations that generalize across temporal–spatial domains. Using MODIS and Planet satellite data, we demonstrate the effectiveness of the TS-PDE method in predicting the value of the normalized difference vegetation index (NDVI) and interpreting the physical significance of the derived equations. The numerical results show that the model achieves good performance, with mean structural similarity index (SSIM) values exceeding 0.82, mean peak signal-to-noise ratio (PSNR) values ranging from 28.5 to 32.8, and mean mean squared error (MSE) values approximating <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>9</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow></semantics></math></inline-formula> for low-resolution MODIS images. For high-resolution Planet images, this study emphasizes the efficacy of TS-PDE in terms of PSNR, SSIM, and MSE metrics, with all datasets exhibiting an average SSIM value of over 0.81, an average PSNR maximum of 30.9, and an average MSE of less than 0.0042. The experimental findings demonstrate the capability of TS-PDE in deriving governing equations and providing effective predictions for the regional-scale dynamics of these time series images. The findings of this study provide potential insights into the mathematical modeling of land cover dynamics. |
| format | Article |
| id | doaj-art-9ef467cd0c4349d9b7fdfea4d8dfbe9f |
| institution | OA Journals |
| issn | 2072-4292 |
| language | English |
| publishDate | 2025-03-01 |
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| series | Remote Sensing |
| spelling | doaj-art-9ef467cd0c4349d9b7fdfea4d8dfbe9f2025-08-20T02:15:54ZengMDPI AGRemote Sensing2072-42922025-03-01177121110.3390/rs17071211Temporal–Spatial Partial Differential Equation Modeling of Land Cover Dynamics via Satellite Image Time Series and Sparse RegressionMing Kang0Zheng Zhang1Zhitao Zhao2Keli Shi3Junfang Zhao4Ping Tang5School of Science, China University of Geosciences (Beijing), Beijing 100083, ChinaAerospace Information Research Institute (AIR), Chinese Academy of Sciences (CAS), Beijing 100094, ChinaAerospace Information Research Institute (AIR), Chinese Academy of Sciences (CAS), Beijing 100094, ChinaAerospace Information Research Institute (AIR), Chinese Academy of Sciences (CAS), Beijing 100094, ChinaSchool of Science, China University of Geosciences (Beijing), Beijing 100083, ChinaAerospace Information Research Institute (AIR), Chinese Academy of Sciences (CAS), Beijing 100094, ChinaLand cover dynamics play a critical role in understanding environmental changes, but accurately modeling these dynamics remains a challenge due to the complex interactions between temporal and spatial factors. In this study, we propose a novel temporal–spatial partial differential equation (TS-PDE) modeling method combining sparse regression to uncover the governing equations behind long-term satellite image time series. By integrating temporal and spatial differential terms, the TS-PDE framework captures the intricate interactivity of these factors, overcoming the limitations of traditional pixel-wise prediction methods. Our approach leverages <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>×</mo><mn>1</mn></mrow></semantics></math></inline-formula> convolutional kernels within a convolutional neural network (CNN) solver to approximate derivatives, enabling the discovery of interpretable equations that generalize across temporal–spatial domains. Using MODIS and Planet satellite data, we demonstrate the effectiveness of the TS-PDE method in predicting the value of the normalized difference vegetation index (NDVI) and interpreting the physical significance of the derived equations. The numerical results show that the model achieves good performance, with mean structural similarity index (SSIM) values exceeding 0.82, mean peak signal-to-noise ratio (PSNR) values ranging from 28.5 to 32.8, and mean mean squared error (MSE) values approximating <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>9</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow></semantics></math></inline-formula> for low-resolution MODIS images. For high-resolution Planet images, this study emphasizes the efficacy of TS-PDE in terms of PSNR, SSIM, and MSE metrics, with all datasets exhibiting an average SSIM value of over 0.81, an average PSNR maximum of 30.9, and an average MSE of less than 0.0042. The experimental findings demonstrate the capability of TS-PDE in deriving governing equations and providing effective predictions for the regional-scale dynamics of these time series images. The findings of this study provide potential insights into the mathematical modeling of land cover dynamics.https://www.mdpi.com/2072-4292/17/7/1211partial differential equation modelingland cover dynamicssparse regressiontemporal–spatial modelingsatellite image time series |
| spellingShingle | Ming Kang Zheng Zhang Zhitao Zhao Keli Shi Junfang Zhao Ping Tang Temporal–Spatial Partial Differential Equation Modeling of Land Cover Dynamics via Satellite Image Time Series and Sparse Regression Remote Sensing partial differential equation modeling land cover dynamics sparse regression temporal–spatial modeling satellite image time series |
| title | Temporal–Spatial Partial Differential Equation Modeling of Land Cover Dynamics via Satellite Image Time Series and Sparse Regression |
| title_full | Temporal–Spatial Partial Differential Equation Modeling of Land Cover Dynamics via Satellite Image Time Series and Sparse Regression |
| title_fullStr | Temporal–Spatial Partial Differential Equation Modeling of Land Cover Dynamics via Satellite Image Time Series and Sparse Regression |
| title_full_unstemmed | Temporal–Spatial Partial Differential Equation Modeling of Land Cover Dynamics via Satellite Image Time Series and Sparse Regression |
| title_short | Temporal–Spatial Partial Differential Equation Modeling of Land Cover Dynamics via Satellite Image Time Series and Sparse Regression |
| title_sort | temporal spatial partial differential equation modeling of land cover dynamics via satellite image time series and sparse regression |
| topic | partial differential equation modeling land cover dynamics sparse regression temporal–spatial modeling satellite image time series |
| url | https://www.mdpi.com/2072-4292/17/7/1211 |
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