Traveltime tomography of crosshole ground-penetrating radar based on an arctangent functional with compactness constraints

Regularization has been an effective technique to provide unique and stable solutions for crosshole ground-penetrating radar (GPR) traveltime tomography. The traditional form of this method, in which a low-order differential operator was used, commonly yields a smooth solution that may not be appropriate when anomalies occur in block patterns, such as voids or irregular objects. The minimum support (MS) functional can be used to improve the resolution of blocky structures; however, in crosshole GPR traveltime tomography, the MS functional is unable to resolve residual artifacts, whose departure from an a priori model are smaller than the focusing parameter selected from a trade-off curve. In addition, it would result in severe instability and yield a trade-off curve with poorly defined corners when the focusing parameter nears the precision of the apparatus. We have developed a new stabilizing functional based on the arctangent (AT) function that effectively removes the artificially small values in the crosshole GPR traveltime tomography, and ultimately is more efficient because it does not require the user to select a focusing parameter. We inverted three 2D synthetic data sets based on the reweighted regularized conjugate gradient algorithm. Compared with the low-order differential and MS functional, the user will be able to clearly distinguish the anomaly boundary using this method, which will yield results that are closer to the actual structure. We also discussed the impact of some influencing factors caused by the noise contained in the data, the central frequency of the antenna, the anomalous trends, and the ray coverage angle. We further inverted an experimental data set to test the effectiveness and robustness of the method.