Generalized Tikhonov's algorithm for accurate calculation of one-dimensional transient responses directly in time domain

In the last two decades, forward modelling for the time domain (transient) electromagnetic method has concentrated almost entirely on multi-dimensional models and algorithms. At the same time, the interpretation of real field data is still mainly one dimensional. This is caused by the lack of an efficient multi-dimensional acquisition procedure supported by sufficiently fast and reliable inversion software, on the one hand, and by the great efficiency of one-dimensional field set up and interpretation of the data on the other hand. The latter is particularly true for the short offset transient electromagnetic method, which is much less sensitive to multi-dimensional effects, compared to long offset methods. The most commonly used one-dimensional forward modelling algorithms are based on the spectral method, which requires calculating rapidly oscillating Fourier–Bessel (Hankel) integrals. Due to the very fast decay of short offset responses, the integrals become computationally unstable at late times of the transient process. Although this problem has been successfully solved for practically feasible measurement times of conventional short offset systems using transverse electric and mixed transverse electric and transverse magnetic fields, it turned out crucial for novel methods based on the use of unimodal transverse magnetic fields. These methods are much more sensitive to geoelectric parameters of the Earth in general and those of resistive targets, in particular, but they generate responses, which drop at late times significantly faster than those of conventional methods. Such behaviour of transverse magnetic fields represents severe computational problem for the spectral method, but is successfully solved by direct time domain algorithms. This article describes a generalization of the well-known Tikhonov's solution to a boundary value problem directly in time domain, which is applied to an arbitrary one-dimensional earth model excited by an arbitrary source. Contrary to existing spectral algorithms, the described method allows accurate calculations of both transverse electric and transverse magnetic transient responses at arbitrarily late times. On the other hand, it is more time efficient than finite-difference/finite element direct time domain algorithms and provides analytical late-stage asymptotic solutions.