A linear approach to simulate a constant-Q model in time-domain ground-penetrating radar modeling using multiple Debye poles: Theory and application in karst carbonate rocks

Abstract

Dispersion is an important effect contributing to attenuating ground-penetrating radar (GPR) signals and must be incorporated in wave propagation. The constant-[Formula: see text] model (quality factor [Formula: see text]) is valid in rocks for typical GPR frequency ranges. In time-domain wave modeling, to simulate a constant-[Formula: see text] medium, using multiple Debye poles, the number of poles and their parameters must be previously estimated to honor target values for [Formula: see text] and velocity in the demanded frequency band. For electromagnetic waves, this problem is usually formulated as a highly nonlinear optimization problem presenting several solutions, which is solved using global optimization methods. An efficient linear approach to estimate multiple Debye poles to simulate constant-[Formula: see text] media for GPR data is presented. The resulting inverse problem is easy to solve because the number of poles is relatively small. The approach allows for a novel rescaled solution that depends only on the frequency band and number of poles. Solutions satisfying specific target values for velocity and [Formula: see text] are obtained from the rescaled solution by multiplying this solution by a proportionality factor that depends only on the target values. Choosing the number of poles is done so that velocity [Formula: see text] and [Formula: see text] curves, as a function of frequency [Formula: see text], are approximately invariant in the specified frequency band. Using the criterion that [Formula: see text] and [Formula: see text] curves must present maximum percentage deviations from the respective target values smaller than 5%, we find that, for the 200 MHz antenna, whose useful frequency band is 50–800 MHz, only three poles are sufficient. The theory is validated by modeling dispersion effects using synthetic models. The importance of introducing dispersion effects is illustrated by presenting a trial-and-error modeling approach to approximately reproduce a 200 MHz radargram acquired in fractured and karstified carbonate rocks.