Forward Modeling of Frequency-domain Helicopter-borne Electromagnetic Data using an Improved Finite Difference Method

In this work, we simulate the frequency-domain helicopter-borne electromagnetic (HEM) data over the two-dimensional (2D) and three-dimensional (3D) earth models. In order to achieve this aim, the vector Helmholtz equation is used to avoid the convergence problems in Maxwell’s equations, and the corresponding fields are divided into primary and secondary components. We use the finite difference method on a staggered grid to discretize the equations, which can be performed in two ways including the conventional and improved finite difference methods. The former is very complex in terms of programming, which causes errors. Furthermore, it requires different programming loops over each point of the grid, which increases the program’s running time. The latter is the improved finite difference method (IFDM), in which pre-made derivative matrices can be used. These pre-made derivative matrices can be incorporated into the derivative equations and convert them directly from the derivative form to the matrix form. After having the matrix form system of linear equations, Ax = b is solved by the quasi-minimal residual (QMR). IFDM does not have the complexities of the conventional method, and requires much less execution time to form a stiffness or coefficient matrix. Moreover, its programing process is simple. Our code uses parallel computing, which gives us the ability to calculate the fields for all transmitter positions at the same time, and because we use sparse matrices thorough the code memory space, requires to store the files is less than 100 MB compared with normal matrices that require more than 15 GB space in the same grid size. We implement IFDM to simulate the earth’s responses. In order to validate, we compare our results with various models including the 3D and 2D models, and anisotropic conductivity. The results show a good fit in comparison with the FDM solution of Newman and the appropriate fit integral equations solution of Avdeev that is because of the different solution methods.