Near-source response of a resistive layer to a vertical or horizontal electric dipole excitation

Frequency and time-domain analytical expressions are derived for the electromagnetic field response of a resistive layer inserted in an otherwise homogeneous whole-space as observed in the vicinity of an electric dipole source. The analysis of closed-form solutions demonstrated that at near-source-receiver offsets, the spectral and spatial distribution of the fields is better described by a superposition of images, associated with the dipolar character of the charges distributed in the boundaries, rather than the guided mode behavior dominating the response in the far offset regime. Approximate solutions of the fields in the frequency domain were derived using the saddle point method of integration. The formulas describing the fields were in good agreement with semianalytical calculations. However, a lower frequency bound was found, below which the expressions are inaccurate, and thereby they cannot be used to obtain time-domain solutions. A kernel modulation scheme was used instead, which yields an infinite series representation for the fields. The expressions thus derived produce accurate fields to very low frequencies, and thereby they were also used to obtain time-domain formulas. The analysis indicated that for a vertical electric dipole (VED) excitation, the late time response of the image field associated with the charge density induced on the upper boundary appears to cancel the direct field, thus providing the response of the layer. For a horizontal electric dipole (HED) source, the superimposed contributions of the transverse electric (TE) and transverse magnetic (TM) modes appeared to oppose the image field, resulting in the direct field dominating over the response of the layer, and thereby masking any sensitivity to the properties of the layer in this configuration.