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

Abstract

We give a detailed comparison of damping and difference smoothing as means of regularising inverse calculations. We show that damping is potentially disastrous in multiparameter inversions since the small singular values may control long-spatiai-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, we show that regularisation can produce rather different results depending on whether the inversion is done via jumping or creeping. In particular, we 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. We 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.

Three‐dimensional seismic refraction tomography: A comparison of two methods applied to data from the Faeroe Basin

“…Regularization is an approach by which constraints, in addition to the data, are applied to an inverse problem to treat the underdetermined part of the solution [e.g., Scales et al, 1990;Phillips and Fehler, 1991]. Usually the constraints result in the final model satisfying some property in addition to fitting the data.…”

“…Usually the constraints result in the final model satisfying some property in addition to fitting the data. This property is often chosen so that the model has "minimum structure" since we seek models that include only structure that is required to fit the data according to its noise level [Scales et al, 1990]. Minimum struc- A total of 53,479 first arrivals were picked (Figure 3)• Pick uncertainties were estimated using an automated scheme based on an empirical relationship between pick uncertainty and signal-to-noise ratio (SNR) [Zelt and Forsyth, 1994]o The average SNR is 7.2, and the average uncertainty is 56 ms, ranging between 40 and 100 ms.…”

VSP traveltime inversion: Near‐surface issues

“…We used a linearized Occam's inversion scheme, which can be thought of as a first-order Taylor series expansion of the traveltime vector in slowness space (Menke, 1984). We used the "jumping" method (e.g., Scales et al, 1990) to solve the inverse problem. The jumping method allows us to apply constraints directly to the slowness model:…”

Refraction tomography using a waveform-inversion back-propagation technique

“…Ray-based refraction tomography sometimes encounters an illposed inverse problem, which can be resolved by introducing regularization ͑see Scales et al, 1990;Zhang and Toksöz, 1998͒. Raybased traveltime-tomography methods are only valid for smooth media ͑Zelt and Barton, 1998͒; moreover, the minimization of an additional damping term, in the case of regularization being applied, penalizes the model roughness.…”