Groups, algebras, and the non-linearity of geophysical inverse problems

Mathematical methods from the theory of continuous groups are used to determine whether a non-linear inverse problem, in the form of a functional, can be transformed into a linear inverse problem. If such transformations exist they can be constructed from the solutions of a linear system of differential equations. An illustration of the methodology is given by the linearization of the functional relating basement topography to observed surface gravity. The linearized inversion of gravity data for basement topography is applied to observations from Yucca Mountain, Nevada. A 2.0 km step in the basement to the west of Yucca Mountain, corresponding to the Bare Mountain fault, matches the Bouguer gravity anomaly. The resolution and uncertainty associated with the estimates of basement topography indicate that the structure directly beneath the gravity line is well constrained.     

A transformational approach to geophysical inverse problems

A set of coordinate transformations is used to linearize a general geophysical inverse problem. Statistical and analytic techniques are employed to estimate the parameters of such linearization transformations. In the transformed space, techniques from linear inverse theory may be utilized. Consequently, important concepts, such as model parameter covariance, model parameter resolution and averaging kernels, may be carried over to non-linear inverse problems. I apply the approach to a set of seismic cross-borehole traveltimes gathered at the Conoco Borehole Test Facility. the seismic survey was conducted within the Fort Riley formation, a limestone with thin interbedded shales. Between the boreholes, the velocity structure of the Fort Riley formation consists of a high-velocity region overlying a section of lower velocity. It is found that model parameter resolution is poorest and spatial averaging lengths are greatest in the underlying low-velocity region.     

Non-linear altimetric geoid inversion for lithospheric elastic thickness and crustal density

The inversion of high-resolution geoid anomaly maps derived from satellite altimetry should allow one to retrieve the lithospheric elastic thickness, T _e , and crustal density, _c . Indeed, the bending of a lithospheric plate under the load of a seamount depends on both parameters, and the associated geoid anomaly is correspondingly dependent on the two parameters. The difference between the observed and modelled geoid signatures is estimated by a cost function, J , of the two variables, T _e and _c . We show that this cost function forms a valley structure along which many local minima appear, the global minimum of J corresponding to the true values of the lithospheric parameters. Classical gradient methods fail to find this global minimum because they converge to the first local minimum of J encountered, so that the final parameter estimate strongly depends on the starting pair of values ( T _e ,   _c ). We here implement a non-linear optimization algorithm to recover these two parameters from altimetry data. We demonstrate from the inversion of synthetic data that this approach ensures robust estimates of T _e and _c by activating two search phases alternately: a gradient phase to find a local minimum of J , and a tunnelling phase through high values of the cost function. The accuracy of the solution can be improved by a search in an iteratively restricted parameter subspace. Applying our non-linear inversion to the Great Meteor Seamount geoid data, we further show that the inverse problem is intrinsically ill-posed. As a consequence, minute geoid (or gravity) data errors can induce large changes in any recovery of lithospheric elastic thickness and crustal density.     

A probabilistic approach to the inversion of data from a seismic array and its application to volcanic signals

Array techniques are particularly well‐suited for detecting and quantifying the complex seismic wavefields associated with volcanic activity such as volcanic tremor and long‐period events. The methods based on the analysis of the signal in the frequency domain, or spectral methods, have the main advantages of both resolving closely spaced sources and reducing the necessary computer time, but may severely fail in the analysis of monochromatic, non‐stationary signals. Conversely, the time‐domain methods, based on the maximization of a multichannel coherence estimate, can be applied even for short‐duration pulses. However, for both the time and the frequency domain approaches, an exhaustive definition of the errors associated with the slowness vector estimate is not yet available. Such a definition become crucial once the slowness vector estimates are used to infer source location and extent. In this work we develop a method based on a probabilistic formalism, which allows for a complete definition of the uncertainties associated with the estimate of frequency–slowness power spectra from measurement of the zero‐lag cross‐correlation. The method is based on the estimate of the theoretical frequency–slowness power spectrum, which is expressed as the convolution of the true signal slowness with the array response pattern. Using a Bayesian formalism, the a posteriori probability density function for signal slowness is expressed as the difference, in the least‐squares sense, between the model spectrum and that derived from application of the zero‐lag cross‐correlation technique. The method is tested using synthetic waveforms resembling the quasi‐monochromatic signals often associated with the volcanic activity. Examples of application to data from Stromboli volcano, Italy, allow for the estimate of source location and extent of the explosive activity.     

Sequential integrated inversion of refraction and wide-angle reflection traveltimes and gravity data for two-dimensional velocity structures

A new algorithm is presented for the integrated 2-D inversion of seismic traveltime and gravity data. The algorithm adopts the 'maximum likelihood' regularization scheme. We construct a 'probability density function' which includes three kinds of information: information derived from gravity measurements; information derived from the seismic traveltime inversion procedure applied to the model; and information on the physical correlation among the density and the velocity parameters. We assume a linear relation between density and velocity, which can be node-dependent; that is, we can choose different relationships for different parts of the velocity–density grid. In addition, our procedure allows us to consider a covariance matrix related to the error propagation in linking density to velocity. We use seismic data to estimate starting velocity values and the position of boundary nodes. Subsequently, the sequential integrated inversion (SII) optimizes the layer velocities and densities for our models. The procedure is applicable, as an additional step, to any type of seismic tomographic inversion.
We illustrate the method by comparing the velocity models recovered from a standard seismic traveltime inversion with those retrieved using our algorithm. The inversion of synthetic data calculated for a 2-D isotropic, laterally inhomogeneous model shows the stability and accuracy of this procedure, demonstrates the improvements to the recovery of true velocity anomalies, and proves that this technique can efficiently overcome some of the limitations of both gravity and seismic traveltime inversions, when they are used independently.
An interpretation of field data from the 1994 Vesuvius test experiment is also presented. At depths down to 4.5 km, the model retrieved after a SII shows a more detailed structure than the model obtained from an interpretation of seismic traveltime only, and yields additional information for a further study of the area.     

Automatic differentiation in geophysical inverse problems

Automatic differentiation (AD) is the technique whereby output variables of a computer code evaluating any complicated function (e.g. the solution to a differential equation) can be differentiated with respect to the input variables. Often AD tools take the form of source to source translators and produce computer code without the need for deriving and hand coding of explicit mathematical formulae by the user. The power of AD lies in the fact that it combines the generality of finite difference techniques and the accuracy and efficiency of analytical derivatives, while at the same time eliminating 'human' coding errors. It also provides the possibility of accurate, efficient derivative calculation from complex 'forward' codes where no analytical derivatives are possible and finite difference techniques are too cumbersome. AD is already having a major impact in areas such as optimization, meteorology and oceanography. Similarly it has considerable potential for use in non-linear inverse problems in geophysics where linearization is desirable, or for sensitivity analysis of large numerical simulation codes, for example, wave propagation and geodynamic modelling. At present, however, AD tools appear to be little used in the geosciences. Here we report on experiments using a state of the art AD tool to perform source to source code translation in a range of geoscience problems. These include calculating derivatives for Gibbs free energy minimization, seismic receiver function inversion, and seismic ray tracing. Issues of accuracy and efficiency are discussed.     

Joint inversion of receiver function and surface wave dispersion observations

We implement a method to invert jointly teleseismic P wave receiver functions and surface wave group and phase velocities for a mutually consistent estimate of earth structure. Receiver functions are primarily sensitive to shear wave velocity contrasts and vertical traveltimes, and surface wave dispersion measurements are sensitive to vertical shear wave velocity averages. Their combination may bridge resolution gaps associated with each individual data set. We formulate a linearized shear velocity inversion that is solved using a damped leastsquares scheme that incorporates a priori smoothness constraints for velocities in adjacent layers. The data sets are equalized for the number of data points and physical units in the inversion process. The combination of information produces a relatively simple model with a minimal number of sharp velocity contrasts. We illustrate the approach using noisefree and realistic noise simulations and conclude with an inversion of observations from the Saudi Arabian Shield. Inversion results for station SODA, located in the Arabian Shield, include a crust with a sharp gradient near the surface (shear velocity changing from 1.8 to 3.5 km s¹ in 3 km) underlain by a 5kmthick layer with a shear velocity of 3.5 km s¹ and a 27kmthick layer with a shear velocity of 3.8 km s¹, and an upper mantle with an average shear velocity of 4.7 km s¹. The crustmantle transition has a significant gradient, with velocity values varying from 3.8 to 4.7 km s¹ between 35 and 40 km depth. Our results are compatible with independent inversions for crustal structure using refraction data.     

Smooth inversion for ground surface temperature histories: estimating the optimum regularization parameter by generalized cross-validation

The inversion of recent borehole temperatures has proved to be a successful tool to determine ancient ground surface temperature histories. To take into account heterogeneity of thermal properties and their non-linear dependence on temperature itself, a versatile 1-D inversion technique based on a finite-difference approach has been developed. Regularization of the generally ill-posed problem is obtained by an appropriate version of Tikhonov regularization of variable order. In this approach, a regularization parameter has to be determined, representing a trade-off between data fit and model smoothness. We propose to select this parameter by generalized cross-validation. The resulting technique is employed in case studies from the Kola ultradeep drilling site, and another borehole from northeastern Poland. Comparing the results from both sites corroborates the hypothesis that subglacial ground surface temperatures as met in Kola often are much higher than the ones in areas exposed to atmospheric conditions (Poland).