Regularisation of nonlinear inverse problems: imaging the near-surface weathering layer

The authors give a detailed comparison of damping and difference smoothing as a means of regularising inverse calculations. They show that damping is potentially disastrous in multiparameter inversions since the small singular values may control long-spatial-wavelength features in the solution, whereas difference smoothing avoids this problem entirely by down-weighting the rough singular vectors wherever they happen to lie in the spectrum. Further, they show that regularisation can produce rather different results depending on whether the inversion is done via jumping or creeping. In particular, they find that if the inversion is regularised by difference smoothing, then jumping and creeping will give the same results only if the initial model is smooth. They illustrate these ideas by inverting refracted seismic arrivals to image the Earth's near-surface weathering layer. This 'refraction statics' problem has a fundamental long-wavelength ambiguity, so damping merely introduces undesirable long-wavelength perturbations to the solution.