Advanced Numerical Methods for a First-Kind Fredholm Integral Equation in Potential Field Continuation

In this research, surface Au concentration measurements are considered as a spatially correlated geochemical field associated with deep occurrences of disturbing masses using real geological exploration data from the Novo-Khairuzovsky gold deposit in East Kazakhstan. The approach is based on the relationship between potential-field continuation problems and reconstruction of subsurface geological anomalies from surface observations. The considered approaches include Tikhonov and Lavrentiev regularization, SVD, and TSVD. Special attention is given to regularization parameter selection using the L-curve method, Morozov discrepancy principle, and GCV. Comparative computational analysis is performed to evaluate the accuracy, stability, and efficiency of these methods in solving first-kind Fredholm integral equations. Results are assessed using error metrics and spatial visualization of reconstructed fields within a Geographic Information System (ArcGIS), enabling consistent geospatial interpretation. Results show that Lavrentiev regularization with L-curve criterion provides the most stable and reliable reconstruction across all depths, achieving high correlations (R=0.8876 at 100 m and R=0.8049 at 200 m) with low reconstruction errors. Tikhonov regularization performs acceptably at 100 m but becomes less stable at greater depths. Among spectral methods, TSVD improves stability compared with classical SVD, while standard SVD shows weak correlations and larger reconstruction errors due to high noise sensitivity.