Three-Dimensional Spectral Induced Polarization (SIP) Forward Modelling Based on Piecewise Linear Continuous Geoelectric Model Using Finite Elements and Recursive Inversion

Petrophysical parameters of rocks and ores, influenced by composition, porosity, temperature, and pressure, are generally distributed uniformly or continuously in space—relatively homogeneous within individual geological units and varying smoothly across stratigraphic transition zones and contact boundaries. Based on this geological characteristic, this paper establishes a three-dimensional (3-D) piecewise linear continuous spectral parameter model to compute forward responses of apparent spectral parameters under low-frequency current excitation. The calculation follows a two-step workflow: finite-element forward simulation of multi-frequency apparent complex resistivity, followed by recursive inversion to obtain apparent spectral parameters. The subsurface medium is discretized with hexahedral meshes, with four Cole–Cole parameters (zero-frequency resistivity, chargeability, time constant, and frequency exponent) assigned to each mesh node. Linear interpolation is adopted for complex resistivity and potential within each element, ensuring piecewise linear continuity of both physical properties and simulated fields. To improve accuracy, the total complex potential is decomposed into a primary field from the source current and a secondary field from complex conductivity variations, and the corresponding boundary value problem and variational form are derived. On this basis, we implement the finite-element algorithm for 3-D piecewise linear continuous media and the recursive inversion algorithm for spectral parameters, and develop an interactive 3-D SIP forward modeling program. Comparison with analytical solutions for a continuous layered model shows good agreement, with relative errors below 1.5% for the real part and 3.8% for the imaginary part of apparent complex resistivity. Two numerical cases—a cubic anomaly in homogeneous half-space and a sandbox model—further verify the performance of the proposed method.