Discovering Acoustic Impedance Inversion Equation

Classical acoustic impedance inversion (AII) methods rely on mathematical and physical models to estimate subsurface acoustic impedance distribution. However, these methods face difficulties in accurately fitting complex impedance data. While deep learning (DL) methods have achieved higher accuracy and efficiency in impedance inversion, they remain closed box models, lacking interpretability. Hence, it is difficult to analyze the reason why they are (or are not) effective. To address the limitations of both traditional and DL methods, this article proposes a novel approach, acoustic impedance inversion with symbolic regression (AII-SR), which discovers partial differential equations (PDEs) from impedance data to model AII. To discover these PDEs, AII-SR adopts a dual-learning framework. It employs a forward model based on the Robinson convolution principle to ensure the physical consistency and reliability of predictions. Subsequently, AII-SR creates an inversion model that combines a symbolic regression-based PDE generator with a physics-informed solving neural network (PSNN) to identify PDEs that accurately fit the impedance data. Experiments demonstrate that AII-SR outperforms DL methods, such as SSEI, temporal convolutional network (TCN), and Se-UNet, in terms of accuracy and interpretability. AII-SR generates concise, interpretable mathematical expressions in the form of PDEs, offering profound insights into the physical relationships between seismic and impedance data.