This chapter has presented an approach to robustly extract surface wave dispersion information jointly for Love and Rayleigh ambient noise data. The Bayesian approach allows minimal pre-processing of the data and robust prior information to effect a stable and precise estimate of dispersion with uncertainty estimates that can be used for subsequent tomographic inversion.
In this study the use of focusing parameters in the Dirichlet priors has not been ad-
dressed so this could be further explored either with α in the Dirichlet priors with
fixed values greater than one or the use of a hierarchical prior on this parameters. The focus of this study has been on phase velocity alone however the model param- eterisation permits easy calculation of the group velocity which could be either used as additional constraint in the inversion by incorporating traditional Frequency-Time Analysis (FTAN) [Dziewonski and Hales, 1972] in the likelihood. Alternatively the group velocities can be estimated from the posterior phase velocity curves as in Figure 2.26. While the group velocity estimate obtained from the phase velocity curves is in good agreement with results obtained with other methods, there is room for improve- ment of the group velocity results. These could be improved by raising the order of the polynomial further while preserving monotonicity [Dougherty et al., 1989] to enable smoother group velocity estimates.
A simple independent Gaussian noise model with a base level of noise estimated from a quiescent part of the real spectrum. An obvious extension would be to use estimates of covariance errors in the spectrum which could be done per station pair or across the entire array.
Lastly the optimal value of the Bessel function scaling to fit the observed real spectrum could be inverted for as part of the inversion. However, a more interesting approach would be to attempt to recover the amplitude envelope of the spectrum similar to the recovery of the source time function in Dettmer et al. [2015]. This could provide additional information, such as frequency dependent attenuation, that could be used in
Trans-dimensional Trees
3.1 Introduction
In the previous chapter a trans-dimensional inversion was developed for a one dimen- sional geophysical problem. Extending trans-dimensional inversion to higher dimen- sions has typically involved the use of Voronoi cells [Okabe et al., 1992, Samet, 2006] and at first glance this would appear well suited to trans-dimensional geophysical inver- sions as these cells have a long history in large scale geophysical inversion problems, for example Sambridge and Gudmundsson [1998].
Using Voronoi cells, by specifying the location of the nuclei as well as the value (or values) of Earth properties within each cell, a mobile Voronoi model can be used to represent Earth properties spatially in 2D [Bodin et al., 2012a]. In the first 3D applica- tion, Piana Agostinetti et al. [2015] have recently extended the Voronoi cell approach to local earthquake tomography. These Voronoi cell parameterisations are grid free and locally adapt to regions of increased heterogeneity tempered by the resolving power of the data. Although the application of the trans-dimensional Voronoi cell method is now well established for seismic imaging, there are a number of short comings that hinder its application as the number of data and complexity of the Earth model in- creases.
In ray based seismic tomography, numerical integration along ray paths requires the evaluation of the model at hundreds to thousands of spatial points per observation. For each point along the ray the Voronoi cell parameterisation of the Earth properties is needed, for example, seismic wave speed, and this involves determining in which cell the point resides. A naive algorithm would simply find the nearest Voronoi nuclei
by computing the distance to every nuclei of the model and this results in an O(n)
operation, wherenis the number of Voronoi cells. In 2D problems, a Delaunay trian-
gulation can be used to speed up the cell look up operation to anO(logn)operation
[Sambridge and Gudmundsson, 1998]. Even with fast algorithms for incrementally maintaining the Delaunay triangulation [Lawson, 1977], the accounting cost of main- taining the triangulation can be prohibitive as the number of cells increases.
A second feature of the Voronoi cell approach is that they do not lend themselves well to representing a continuous field. In a Voronoi cell parameterisation, the Earth properties within each cell are often represented with constant values, although in principle, any order polynomial is possible. This means that each Earth model consists of an irregular polygonal mesh with discontinuities, both in the function and in its derivatives, at the interfaces between cells. Typically, any single Earth model in the ensemble is rather crude and implausible and it is only by averaging over many such crude representations that it is possible to generate a continuous field. This means that the Voronoi cell approach must utilise multiple independent Markov chains or very large numbers of samples in a single chain in order to produce a continuous field through spatial averaging.
Use of Voronoi cells in 3D imaging has two additional complications. The first is that there is no analogue of fast 2D incremental Delaunay calculation algorithms [Sam- bridge et al., 1995] and so Voronoi cells must be determined from “scratch” each time the mesh is updated, further adding to the computational burden. The second is that the shape of Voronoi cells in 3D is particularly sensitive to the choice of spatial scaling between lateral and radial directions. For example, Voronoi cells built around nuclei at depth can easily protrude to the surface.
In this chapter a new class of parameterisation for trans-dimensional imaging problems is introduced which overcomes the limitations of Voronoi cells while providing a gen- eral efficient framework for dealing with 1D, 2D and 3D problems in Cartesian or spherical geometries. The new framework allows a great deal of flexibility in terms of the choice of basis functions, including multi-scale parameterisations such as wavelets and sub-division surfaces. Due to these new efficiencies and flexibility, with the new algorithm, trans-dimensional inversion of larger scale 3D tomographic problems are more tractable.