Statistical inverse inference for solving the first kind of nonlinear Fredholm integral equation

Inverse problems, widely applied in geology, oceanography, and geophysics, are often ill-posed, presenting significant challenges. This study addresses the first kind of nonlinear Fredholm integral equation derived from gravity measurement and magnetic relief problems, which are highly sensitive to data perturbations. Conventional methods, such as Newton iteration with regularization, provide stable solutions but fail to account for data randomness, yielding only single approximations. We propose a Bayesian statistical inversion framework to overcome these limitations, transforming the problem into posterior distribution characterization. We estimate solutions and quantify their uncertainty using Markov chain Monte Carlo (MCMC) methods, specifically the preconditioned Crank–Nicolson (pCN) algorithm. Numerical simulations demonstrate the method's effectiveness in gravity and magnetic problems, offering a robust tool for uncertainty analysis in stochastic inverse problems.