Gradient-Based Efficient Optimization Method for Surface Sensor Arrays in Microseismic Source Mechanism Inversion

The accuracy of microseismic moment tensor inversion critically depends on the spatial configuration of surface sensor arrays, quantified by the condition number 𝜅(𝑮) of the dynamic response matrix. Existing optimization methods suffer from slow convergence and unreliability due to the fundamental lack of gradient information. In this specific context, the analytical gradient of 𝜅(𝑮) with respect to sensor positions via singular value perturbation theory and the chain rule of direction cosines are derived for the first time in this paper. Based on the analytical gradient, three optimization strategies are established: a BFGS quasi-Newton method with quadratic penalty (BP) for smoothly constrained regions; a sequential quadratic programming method (FMC/SQP) achieving the lowest condition numbers across all shapes with approximately 5000 evaluations; and a center-boundary parameterization method (CB) that exploits the structural regularity of optimal configurations to reduce variables from 2𝑁 to 𝑁 − 1, requiring only approximately 300 evaluations (1/16 of FMC). Systematic experiments on five region shapes and two complex constraint scenarios demonstrate that gradient methods achieve a 15–30% reduction in across 4 of 5 shapes and a 5–31-fold improvement in robustness, with FMC reducing the average condition number by 25% at 15.4-fold better robustness than the baseline. Synthetic inversion experiments confirm that reducing from approximately 1.5 × 104 to approximately 50 reduces moment tensor inversion error by approximately 50-fold. The analytical gradient framework is transferable to sensor placement problems across multiple disciplines, and the three-method toolchain offers a graduated accuracy–efficiency trade-off for practical deployment optimization.