An objective rationale for the choice of regularisation parameter with application to global multiple-frequency <i>S</i>-wave tomography

Abstract

In a linear ill-posed inverse problem, the regularisation parameter (damping) controls the balance between minimising both the residual data misfit and the model norm. Poor knowledge of data uncertainties often makes the selection of damping rather arbitrary. To go beyond that subjectivity, an objective rationale for the choice of damping is presented, which is based on the coherency of delay-time estimates in different frequency bands. Our method is tailored to the problem of global multiple-frequency tomography (MFT), using a data set of 287 078 S-wave delay times measured in five frequency bands (10, 15, 22, 34, and 51 s central periods). Whereas for each ray path the delay-time estimates should vary coherently from one period to the other, the noise most likely is not coherent. Thus, the lack of coherency of the information in different frequency bands is exploited, using an analogy with the cross-validation method, to identify models dominated by noise. In addition, a sharp change of behaviour of the model ℓ∞-norm, as the damping becomes lower than a threshold value, is interpreted as the signature of data noise starting to significantly pollute at least one model component. Models with damping larger than this threshold are diagnosed as being constructed with poor data exploitation. Finally, a preferred model is selected from the remaining range of permitted model solutions. This choice is quasi-objective in terms of model interpretation, as the selected model shows a high degree of similarity with almost all other permitted models (correlation superior to 98% up to spherical harmonic degree 80). The obtained tomographic model is displayed in the mid lower-mantle (660–1910 km depth), and is shown to be compatible with three other recent global shear-velocity models. A wider application of the presented rationale should permit us to converge towards more objective seismic imaging of Earth's mantle.