Interpretation of Self-Potential Anomalies over 2D Inclined Thin Sheet structure using Particle Swarm Optimization with Non-linear Polynomial Modeling of Background Effects

Interpretation of self-potential (SP) anomalies is challenging due to the presence of spatially coherent background noise that can obscure or distort the source signal. These systematic background effects, analogous to regional components in gravity or magnetic methods, arise from measurement errors, heterogeneous subsurface conditions, or interactions among multiple anomalous sources. They often exhibit non-linear behavior that cannot be adequately addressed using a constant or linear slope. Earlier approaches attempted to remove such coherent patterns through baseline corrections or linear de-trending.This study presents an incremental algorithmic development for the interpretation of SP anomalies associated with a 2D inclined thin-sheet structure, explicitly accounting for non-linear background contributions while jointly estimating source geometry. Using the metaheuristic technique of Particle Swarm Optimization (PSO), the background field is parameterized as a second-order polynomial, with coefficients representing a constant offset, linear gradient, and quadratic curvature. These coefficients are simultaneously optimized with the source parameters using an L2-norm type misfit. The method is particularly stable with respect to depth and half-width of causative body; however, dip, location, and electric dipole moment can become ambiguous in the presence of noise. Therefore, separation of background trends from the signal is crucial for recovering the actual source parameters accurately. To assess solution stability, the spread of solution ensemble obtained from multiple independent PSO runs under identical conditions is analyzed. To further evaluate parameter sensitivity and interdependence, a correlation matrix is computed and crossplots are plotted. Among the background components, quadratic curvature exhibits the strongest coupling with the recovered source parameters, whereas the constant offset shows minimal influence compared with source-only optimization.When extended to multiple-source SP anomaly data, quadratic background modeling proved inadequate. Using a single quadratic polynomial failed to capture complex regional–local interactions, while assigning separate quadratic backgrounds to individual sources unnecessarily increased the dimensionality of the model space. To address this problem, a residual-based higher-order background modeling approach is implemented. In this approach, source parameters are optimized first, and the resulting residual field is iteratively approximated using a polynomial of the lowest order necessary to capture systematic background effects within the optimization framework, thereby avoiding the enforcement of a fixed polynomial degree.The proposed method is evaluated using synthetic SP anomaly data under both noise-free and noisy conditions and is subsequently validated using field datasets, including a single-source anomaly from the Surda region, India, and a multiple-source anomaly from KTB region, Germany.

Overall, the proposed approach offers benefit in improved recovery of source parameters by effectively decoupling source and background responses. The polynomial background represents long-wavelength, spatially coherent variations of physical origin superimposed on target signal. However, this approach increases model dimensionality, may lead to overfitting if search bounds are not appropriately constrained, and may result in non-unique polynomial representations in multi-source cases.

Keywords: Self-Potential (SP) method, 2D inclined thin-sheet, Non-linear background, Particle Swarm Optimization, Residual modeling.