This thesis presents a number of advances in trans-dimensional Bayesian methods for geophysical inversion problems of varying physics and character.
In Chapter 2, an approach to extracting phase velocity information from seismic am- bient noise observations is presented using a trans-dimensional partition modelling approach. This method is able to extract Love and Rayleigh wave dispersion jointly from three component data as continuous dispersion curves with uncertainties that can be carried forward in subsequent phase velocity map inversions.
In Chapter 3, the development of a new type of trans-dimensional algorithm called trans-dimensional trees is presented. This scheme is targeted towards higher dimen- sional geophysical inversion problems where existing Voronoi cell methods have com- putational deficiencies. This is an abstract trans-dimensional approach with many po-
tential applications. This chapter demonstrates the flexibility of trans-dimensional trees by coupling this with a wavelet parameterisation and compares the results to existing Voronoi cell approaches in simple linearised tomographic problems.
In Chapter 4, the trans-dimensional tree approach is applied to the inversion of Air- borne Electromagnetic data along a section of flight line for a 2D profile of resistivity. This approach uses the trans-dimensional tree with a wavelet parameterisation coupled with a 1D forward model to invert a spatially coherent model, that is, with no lat- eral discontinuities due to the independent 1D inversions. Results show more detail is resolved than an existing inversion using a damped and smoothed least squares optimi- sation approach. This chapter also introduces hierarchical priors and the generation of estimated covariant noise models to further improve and stabilise the inversion. In Chapter 5, the results of the dispersion curves generated in Chapter 2 are inverted for Love and Rayleigh phase velocity maps of Iceland. The Fast Marching Method is used “tightly coupled” to the trans-dimensional tree approach with a wavelet param- eterisation to invert in a fully non-linear fashion. The difference between non-linear and linear (with fixed ray paths) inversions is demonstrated in synthetic examples. This comparison highlights the generally poorer recovery of features and underestimation of anomaly magnitudes of the linear inversions compared to the fully non-linear ap- proach developed in this chapter. Taking this class of inversion further, all periods of interest are inverted jointly in a 3D fully non-linear inversion to take advantage of the correlation of spatial features between neighbouring frequencies.
In Chapter 6, the trans-dimensional tree is applied to a large problem in the spherical domain, through the inversion of global surface wave data. In this problem, the total number of observations is of the order of 5 million ray paths. This preliminary study test the feasibility of inverting these observations in a reasonable length of time. A key benefit of the trans-dimensional tree approach is that the inversion is stabilised through relative Bayesian model choice rather than smoothing or damping constraints, both of which cause the magnitude of fast and slow anomalies to be underestimated and examples are provide of this in the African rift region.
In Chapter 7, a simple general trans-dimensional partition modelling scheme for 1D problems is introduced that is able to adapt to data best explained with discontinuous features or smoothly varying features, and combinations thereof. The parameterisation used in this approach is polynomials expressed as Gauss-Legendre-Lobatto polynomials where the curve(s) are defined by nodal interpolation points. This ensures that the prior is more intuitive compared to polynomial coefficient priors. It is shown that allowing an inversion to consider more complex combinations of polynomials in a partition modelling scheme produces better results in synthetic regression problems. In Chapter 8, a common and difficult problem in seismology is that many problems have non-unique solutions and a classic one is whether observations are best explained by a 1D Earth model with a series of homogeneous layers, or smoothly varying struc- ture. To attempt to address this, some synthetic surface wave dispersion problems are considered. A novel spectral element method is derived for computing surface wave dispersion predictions from arbitrary models expressed as a series of elements with ar- bitrary order. A novelty in this approach is the inclusion of a Laguerre element for representing a half-space that dramatically improves accuracy at longer periods. This method is first validated against known analytic results and existing approaches. Finally in a series of synthetic tests coupling the trans-dimensional approach of Chapter 7 with the spectral element surface wave dispersion forward model, it is examined whether de- cisive posterior information can be obtained in the inversion of simple structures with slowly varying and homogeneous layers in Love wave, Rayleigh wave and joint inver- sions.
Phase velocity determination
2.1 Introduction
This chapter provides a simpler introduction to some of the key concepts relating to Bayesian inversion, Markov chain Monte Carlo (McMC) and trans-dimensional sam- pling that will be used throughout this thesis. Here the problem is one dimensional and involves inverting for dispersion curves from empirical Greens functions, a key component in ambient noise tomography.
Ambient noise tomography is a relatively new technique for surface wave tomography that uses the ambient seismic wave field, excited by ocean swells, storms and wind, to image relatively near surface structure (up to approximately 100km). The reason for the relatively shallow limit is that ambient noise excitation frequencies are generally limited to between 1 and 30 second period, with a dominant spectrum around 7 to 16 seconds [Bensen et al., 2007, Figure 7(a)]. This limit of useful frequencies translates to a similar limit on resolvable depths.
Early work by Aki [1957] on micro-tremors established much of the more recent theo- retical work showing the plausibility of recovering the elastic Green’s function between two recording stations through the cross-correlation of the ambient seismic noise or coda [Lobkis and Weaver, 2001, Derode et al., 2003, Snieder, 2004, Wapenaar, 2004, Larose et al., 2005] (see also Larose et al. [2006] for a review article). The Greens function between stations A and B represents the signal observed at station B of an impulse at A and vice-versa. These virtual seismic events between station pairs enables seismic tomography techniques to be applied in the absence of Earthquakes. Early ap- plications of this approach using observed seismic coda were reported by Campillo and Paul [2003], Paul et al. [2005], and similarly for ambient noise Shapiro and Campillo [2004], Sabra et al. [2005].
There are three preliminary stages for ambient noise tomography, as outlined by Bensen et al. [2007], one of the first papers to make key recommendations for these steps. These steps are, (i) pre-processing the continuously recorded seismograms, (ii) cross correlation of seismograms from two stations to obtain inter-station empirical
Greens functions (EGFs), and (iii) extracting path integrated surface wave dispersion information for two stations using these EGFs.
A key recommendation is that continuously recorded seismograms be pre-processed prior to cross correlation. The purpose of this pre-processing is three fold, firstly to remove instrument response from the seismograms that may amplify/suppress mea- surement of ambient noise, secondly time domain normalisation to remove the effects of local or global seismicity from corrupting latter cross correlation and lastly spec- tral normalisation or whitening to raise the signal to noise ratio of all frequencies of interest.
For focus of this chapter is the estimation of dispersion based upon the Green’s func- tions recovered from cross-correlations of noise, a crucial component of ambient noise tomography. An overview of the pre-processing steps required is also given as they have important consequences for subsequent processing.