Asymptotic ray theory for transient diffusive electromagnetic fields

We develop an asymptotic ray theory for transient diffusive electromagnetic fields in isotropic media. The formulation is first derived in the time Laplace transform domain by introducing an ansatz procedure, whereby appropriate expansions for the electric and the magnetic field strengths are substituted in the original field equations. We arrive at consistent recurrence formulas for the sequences of field amplitude vectors that multiply appropriately chosen asymptotic sequences of algebraic powers of the Laplace transform parameter. These representations differ for the two types of fields (electric and magnetic fields) and for the two types of sources (electric current source and magnetic current source). The exponential part of the field expressions contains the diffusive equivalent of the eikonal function in the asymptotic ray theory of wave propagation. This function satisfies the diffusive equivalent of the eikonal equation. Next, we derive the transport equations for the vectorial electric and magnetic field amplitudes of the successive orders. Transient field representations within the asymptotic ray approximation are then obtained by carrying out the inverse Laplace transformation to the time domain by inspection. The ray approximation thus obtained is asymptotic for ''early times''. We consider as an example the case of the electric and the magnetic dipole radiation in a homogeneous medium. Here an exact solution exists, which we show to exhibit the structure of the original ansatz but with a finite number of terms. The asymptotic ray theory for transient diffusive electromagnetic fields is expected to lend itself to important applications in surface, surface-to-borehole, and crosswell transient electromagnetic prospecting.