Convolution-free electromagnetic system response in the time domain

ABSTRACT In transient electromagnetic (TEM) measurements, the resolution of shallow subsurface resistivity structures is largely determined by the early time responses. However, the early times are disproportionately affected by system properties, such as the transmitter waveform, system filters, and signal gating. In the time domain, modeling these effects requires convolution operations, which can be computationally intensive. In this paper, convolution operations are cast as matrix products using natural splines and leveraging the fact that timing (i.e., waveform timing, gate times, model time discretization) is fixed. Matrices representing different effects, such as the waveform and gating, can then be combined via matrix multiplication. As such, once the matrix representation of the convolution(s) is constructed, all subsequent forward responses can be carried out convolution free. Validation tests show that the matrix approach produces results that are largely within 2% of standard routines. The accuracy and flexibility of the approach make it a practical solution for TEM modeling, be it 1D, 2D, or 3D applications.