Magnetotelluric inversion based on minimum entropy and evaluation of solutions

The Magnetotelluric (MT) sounding is widely used in geophysical research and plays an important role in the exploration of oil and gas resources and the basic research of deep geological structures. Due to the use of diffusion fields in the MT method, its inversion exhibits significant non-uniqueness and lacks effective solution evaluation methods, making it difficult to achieve checkboard test similar to that in seismic tomography research. Currently, the most widely used inversion methods include the following: Occam’s inversion (Constable et al., 1987), which yields a model with the smallest roughness for a specified misfit, providing a stable and rapidly convergent solution; reduced basis Occam’s inversion (REBOCC), which transforms the linearized inverse problem from the model space to the data space (Siripunvaraporn & Egbert, 2000); and nonlinear conjugate gradients (NLCG), which avoid excessive evaluations of the full Jacobian matrix and the complete solution of a linearized inverse problem at each step of iteration (Rodi & Mackie, 2001). These inversion methods usually cast the mathematical constraint of the model with the minimum model, or the smoothest model into the objective function to relieve the non-uniqueness of the inverse problem and provide stable and smooth inversion results. However, these methods make it difficult to image the shape and sharp boundary of geological structures clearly. Although MT inversion based on Bayesian theory can effectively obtain sharp boundary information and estimate model uncertainties, its significant computing cost limits practical application in large-scale data.

Entropy can measure the average level of "uncertainty" in random variable systems and evaluate the stability of the model system. Zhdanov (2002) proposed introducing minimum entropy constraints into geophysical inversion as a means of regularization. This method aims to generate more focused and clear inversion results by limiting the entropy of the model. In geophysical diffusion field inversion, this method has been widely used and has shown remarkable results. Based on the constraint of minimum entropy, we further introduce statistical methods to calculate the uncertainty of the model. Assuming that each parameter of the model follows a Gaussian normal distribution, the probability density distribution of the model can be regarded as the superposition of the probability distributions of all parameters. On this basis, we introduce the variance of the model parameters as the inversion parameter into the objective function based on the minimum entropy constraint, thereby effectively quantifying the uncertainty of the inversion model. To verify the effectiveness of this method, we designed a checkboard model for synthetic test and applied it to the Gonghe geothermal basin on the northeastern edge of the Tibet Plateau. The results show that the algorithm can effectively characterize the spatial distribution characteristics of partial melt in geothermal basins. Our research not only obtained a focused resistivity inversion model, but also quantitatively evaluated the reliability of the results, providing a new strategy for accurate imaging of complex geological structures and quantitative analysis of the uncertainty of inversion results.