Electromagnetic integral equation approach based on contraction operator and solution optimization in Krylov subspace

SUMMARY The paper presents a new code for modelling electromagnetic fields in complicated 3-D environments and provides examples of the code application. The code is based on an integral equation (IE) for the scattered electromagnetic field, presented in the form used by the Modified Iterative Dissipative Method (MIDM). This IE possesses contraction properties that allow it to be solved iteratively. As a result, for an arbitrary earth model and any source of the electromagnetic field, the sequence of approximations converges to the solution at any frequency. The system of linear equations that represents a finite-dimensional counterpart of the continuous IE is derived using a projection definition of the system matrix. According to this definition, the matrix is calculated by integrating the Green's function over the ‘source’ and ‘receiver’ cells of the numerical grid. Such a system preserves contraction properties of the continuous equation and can be solved using the same iterative technique. The condition number of the system matrix and, therefore, the convergence rate depends only on the physical properties of the model under consideration. In particular, these parameters remain independent of the numerical grid used for numerical simulation. Applied to the system of linear equations, the iterative perturbation approach generates a sequence of approximations, converging to the solution. The number of iterations is significantly reduced by finding the best possible approximant inside the Krylov subspace, which spans either all accumulated iterates or, if it is necessary to save the memory, only a limited number of the latest iterates. Optimization significantly reduces the number of iterates and weakens its dependence on the lateral contrast of the model. Unlike more traditional conjugate gradient approaches, the iterations are terminated when the approximate solution reaches the requested relative accuracy. The number of the required iterates, which for simple iterations, is approximately proportional to the square root of the lateral contrast of conductivity, in case of optimization, becomes approximately proportional to the logarithm of the lateral contrast. The effect of optimization increases with the complexity of the model under consideration. The new code is applied to a number of problems of subsurface, borehole and marine geophysics. Some of the results are compared with independent computations using different approaches.