基于松弛变量的不等式约束三维电阻率并行反演方法研究

 三维电阻率反演存在多解性问题，且求解耗时较长，制约着电阻率法探测的实际应用。针对该问题，提出在反演方程中施加不等式约束、设计并行算法的解决思路。在传统的常规光滑约束反演基础上，引入松弛变量，将携带有地下介质电阻率变化范围上下限值信息的不等式约束施加到反演方程。利用原始对偶内点法原理，构造新的目标函数，可将模型电阻率值限定在不等式约束范围内并直接在约束规定的可行域内进行寻优，从理论上可压制反演的多解性。同时，设计偏导数矩阵求解并行算法和Cholesky分解并行算法，可使反演速度提高50%以上。综上形成了基于松弛变量的不等式约束三维电阻率并行反演方法，并开展相应的数值反演试验与工程应用验证。结果表明上述反演方法充分利用不等式约束的先验信息，在去除反演成像的假异常、压制反演的多解性、改善反演准确性和计算效率等方面有明显效果。

Parallel 3D electrical resistivity inversion method with#br# inequality constraint based on slack variables

Multiple solutions may be resulted from the inversion of 3D electrical resistivity and the calculation is time-consuming，which restricts the application of resistivity detection. An idea is presented to solve this problem，with the inequality constraints to the inversion equations and the parallel algorithms being applied. Based on the traditional smooth constraints，the relaxed variables were introduced to apply the inequality constraints to the inverse equations，which carry the upper and lower limits information of subsurface media. A new objective function was obtained using the primal dual interior point method. The resistivity values in the function were defined within the inequality constrains range，and the parameters were optimized within the feasible region defined by the constraints. The method theoretically suppressed the multiplicity of inverse solution. The parallel calculation algorithm for partial derivative matrix and the parallel algorithm for Cholesky decomposition to the overall coefficient matrix were designed，which speeds the inversion more than fifty percent. Based on the above research，the parallel 3D electrical resistivity inversion method with inequality constraint based on relax variables was developed，and numerical tests and engineering application were carried out. Results showed that the above method made full use of the inequality constrains and removed the false anomalies，suppressed the multiplicity and improved the accuracy and calculation efficiency.