基于正则化约束项的球坐标系大地电磁快速正演

随着跨大陆尺度的大地电磁（Magnetotellurics,简称MT）勘探的广泛开展,为了克服地球曲率带来的误差,有必要开展基于球坐标系下的3D MT正反演研究.该类正、反演问题的一个重要特点是所采用的频率往往比较低,在电磁场满足的偏微分方程中跟电导率有关的项几乎可以忽略.当采用数值方法进行该类电磁正演时,由于数值离散误差,正演算法无法模拟电性变化带来的电荷积累.因此在采用迭代求解器求解该类正演问题时,即使采用了传统的迭代电流散度校正技术,迭代求解器的收敛依然很慢.针对以上问题,本文显式地将散度校正项添加到原始控制方程中来对控制方程进行约束（为方便称之为正则化约束项）,以保证每次迭代电流的散度为零.此方法避免了额外求解散度方程,以期显著提高球坐标系下3D MT正演效率.在正演中,采用球谐函数高阶项P₁⁰来近似MT的场源,在球坐标系下对加入了正则化约束项的正演方程进行有限差分离散.本文首先设计了一个一维层状结构模型,对本文所提算法的数值解与解析解进行了对比.然后设计了一个简单低阻模型和一个基于实测数据反演结果的Cascadia模型,测试了本文算法的数值表现.通过结果对比,验证了本文算法的正确性.数值表现测试结果显示相比于传统算法,本文算法在计算时间和迭代次数上都显著减少,而且不会随周期变化而发生显著变化.

Efficient magnetotelluric forward modeling in the spherical coordinate system based on a regularization constraint

With the availability of Magnetotelluric (MT) data on transcontinental scale, it is necessary to carry out 3D MT forward modeling and inversion in the spherical coordinate system to avoid the errors caused by the curvature of the Earth. The conductivity term in the curl-curl equation governing the electromagnetic (EM) fields can be neglected, due to the low frequencies used for this kind of forward modeling and inversion problems. And it is difficult to model the charge accumulation caused by the conductivity discontinuity when the numerical methods are used. The convergence of iterative solver would be slowly or even diverged for numerical solution, even preconditioned properly and with the application of divergence correction. To address this problem, the scaled correction term is explicitly added to the curl-curl equation (called the regularization constraint term for convenience) to ensure the divergence of current of each iteration is zero. The method avoids additional solving of the divergence equation and significantly improves the efficiency of 3D MT forward modeling in spherical coordinate. The regularized curl-curl equation is discretized with finite difference methods in the spherical coordinate and a high order spherical harmonic function P10 is used to approximate the MT source for forward modeling. First, a layered model with the analytical solution is designed. Then a simple low resistivity model and a Cascadia model based on an inversion model from the measured data are designed to test the numerical performance of the algorithm. The comparison with analytical solution verifies the correctness of the algorithm. The numerical performance tests show that the algorithm significantly reduces the computational time and the number of iterations compared to the conventional algorithm based on iterative divergence correction. More importantly, the algorithm is stable over all the considered periods.