Cross-Gradient Constrained Joint Inversion of Seismic Impedance and Resistivity

Individual geophysical inversions reflect only limited physical parameters of the subsurface and often suffer from non-uniqueness. Joint inversion of different geophysical datasets can mitigate this problem by integrating complementary information. In this study, we realize a structurally constrained joint inversion of seismic wave impedance and DC resistivity data using the cross-gradient function. The cross-gradient is discretized via the central difference quotient, and its fundamental properties are analyzed. A two-dimensional joint inversion objective function incorporating cross-gradient constraints is derived and solved using the Lagrange multiplier method. The approach relies solely on structural similarity between the two physical properties and requires no explicit petrophysical relationship. Synthetic model tests are conducted for cases where the two property distributions are structurally consistent and inconsistent, respectively. Results demonstrate that the joint inversion yields more accurate and sharper boundary delineation, geometric shapes closer to the true model, and significantly improved resolution of multiple anomalous bodies, with recovered physical property values approaching the true values and enhanced convergence speed and stability. When the two models are structurally inconsistent, the cross-gradient constraint effectively acts only on regions of common structure without introducing adverse effects or external errors elsewhere. Numerical simulations confirm the validity and reliability of the proposed algorithm in reducing inversion non-uniqueness and improving parameter accuracy.