Finite-difference modeling of magnetotelluric fields: Error estimates for uniform and nonuniform grids

In the finite-difference (FD) method, one solves a set of discrete approximations to continuous differential equations; thus, the solutions only approximate the true values. For the magnetotelluric (MT) method, errors in the electric and magnetic fields computed by the staggered FD method are precisely quantifiable for a model with uniform conductivity. In this case, the errors in the electric and magnetic fields are equal in magnitude but increase with rising node separation. In this paper, I show that errors in MT responses, which rely on ratios of the field values, depend strongly on the method used to interpolate electric field values to the surface where the magnetic field is sampled. Analytic expressions for the FD estimates of the MT responses for a half-space are derived and compared for three different methods of electric field interpolation. The best results are achieved when the electric field values just above and below the surface are interpolated exponentially. For a half-space, the FD estimates of the MT responses are independent of node separation and are precisely equal to the analytic values when the electric field is interpolated exponentially. For models with sharp conductivity contrasts, the errors in the responses derived using this interpolation method increase with rising node spacing but still perform better than other examined interpolation methods. Varying the vertical node separation within a half-space model degrades the solution accuracy. The magnitude of the error depends primarily on the magnitude of the change in vertical node spacing. Lateral variations in the grid spacing do not necessarily yield errors in the FD solutions to the MT equations.