A new time-domain approach for the electromagnetic induction problem in a three-dimensional heterogeneous earth

Abstract

We present a new time-domain approach to the forward modelling of 3-D electromagnetic induction in a heterogeneous conducting sphere excited by external and internal sources. This method utilizes the standard decomposition of the magnetic field into toroidal and poloidal parts, and spherical harmonic expansions of both the magnetic fields and the conductivity heterogeneity. Resulting induction equations for the spherical harmonics are solved simultaneously in the time domain. Coupling terms between the electromagnetic fields and the conductivity structure are re-expanded in spherical harmonics, so that the terms can be calculated by matrix multiplications at each time step of the computation. A finite difference approximation was used to solve the set of diffusion equations for the spherical harmonics up to degree 20. This method can be efficiently used to analyse transient geomagnetic variations to estimate the 3-D conductivity structure of the Earth. In order to validate the present approach, we solved an induction problem in simple four-layer mantle models, which consist of the surface layer (r = 6371 - 6351 km, σ = 1 S m-1), the upper mantle (r = 6351 - 5971 km, σ = 0.01 Sm-1), the transition layer (r = 5971 - 5671 km, σ = 0.01-1 S m-1), and the lower mantle (r = 5671 - 3481 km, σ = 1 S m-1). Conductivity heterogeneities are considered in the surface layer or the transition layer. For these models, temporal variations of the Gauss coefficients in response to a sudden application of P10-type external field were calculated, and the impulse response function of each harmonic component was obtained by differentiating the calculated variations with time. The response functions of the primary induced components, g10, have large initial values and monotonously decay with time. Changes of the decay rate reflect the radial distribution of the electrical conductivity. For the surface heterogeneous models, temporal variations of other secondary induced components have two peaks, where the first peak (observed at 0.001-0.01 hr after the application of the external field) corresponds to the surface-layer induced phase and the second peak (1-50 hr after the onset) reflects the deeper structure. The difference of the response functions between the models with the conductivity jump at 400 km and 700 km depths becomes apparent after about 1000 s elapsed. The differences can be used to estimate the electrical conductivity structure around the transition layer. Considering that all the induced components except g10 are generated by the surface heterogeneous layer, the surface layer should be included even for calculating the long period response functions for periods much longer than the characteristic time of the surface layer. For the model in which the transition layer is heterogeneous, the signal starts at about 1000 s after the onset and lasts more than about 100 hr. Fourier transform of the time-domain response functions gives the response function in the frequency domain, which can be compared with the previous solutions. Real and imaginary parts of the spatial distribution of the induced magnetic field in frequency domain were calculated from the present results, and compared with those calculated by the staggered-grid finite difference method. This comparison indicates that the surface induced phases are equally detected in the both approaches even for periods as long as 10 days.