Jonathan Kane

Wavelet domain linear inversion with application to well logging

Solving linear inversion problems in geophysics is a major challenge when dealing with non-stationary data. Certain non-stationary data sets can be shown to lie in Besov function spaces and are characterized by their smoothness (dierentiability ) and two other parameters. This information can be input into an inverse problem by posing the problem in the wavelet domain. Contrary to Fourier transforms, wavelets form an unconditional basis for Besov spaces, allowing for a new generation of linear inversion schemes which incorporate smoothness information more precisely. As an example inversion is performed on smoothed and subsampled well log data.

The influence of stacking velocity uncertainties on structural uncertainties

Summary Errors in the velocity eld used to migrate seismic data are a leading cause of errors in the positionining of structural events in the processing of seismic data: uncertainty in the velocity eld leads to structural uncertainty. In this paper, we investigate the broader question of how errors in stacking velocity, time to an event in a stacked section, and the slope of an event in a time section lead to errors in the positioning of structural events for an isotropic medium. We perform a sensitivity analysis, obtaining simple formulas for the errors in structure that are rstorder in the errors in stacking velocity, zero-oset time, and slope. These formulas are geometrically explicit: if we make a small change in stacking velocity (or time or slope), we then know the direction and magnitude of the resulting change to each point on the selected event. Being the result of sensitivity analysis, these formulas are linear. Thus if we had a probability distribution for the errors in velocity (i.e., we knew the uncertainty in velocity), we could use these formulas to obtain a probability distribution for the errors in position for points on the selected event (i.e., the uncertainty in structure). Our analysis focuses on the neighborhood of a single point on an event and assumes a homogeneous velocity eld. Although the analysis is based on a very simple model, numerical experiments show that the relationships are valid approximately for moderate heterogeneities in the velocity eld. In a companion paper (Bube et al., 2004), we use these results to investigate errors in structural location due to uncertainty in weak anisotropy.

Wavelet Domain Geophysical Inversion

We present a non-linear method for solving linear inverse problems by thresholding coefficients in the wavelet domain. Our method is based on the wavelet-vaguelette decomposition of Donoho (1992). Numerical results for a synthetic travel-time inversion problem show that the wavelet based method outperforms traditional least-squares methods of solution.

Joint deconvolution and interpolation of remote sensing data

We present a method for the simultaneous deconvolution and interpolation of remote sensing data in a single joint inverse problem. Joint inversion allows sparsely sampled data to improve deconvolution results and, conversely, allows large-scale blurred data to improve the interpolation of sampled data. Geostatistical interpolation and geostatistically damped deconvolution are special cases such a joint inverse problem. Our method is posed in the Bayesian framework and requires the definition of likelihood functions for each data set involved, as well as a prior model of the parameter field of interest. The solution of such a problem is the posterior probability distribution. We present an algorithm for finding the maximum of this distribution. The particular application we apply our algorithm to is the fusion of digital elevation model and global positioning system data sets. The former data is a larger scale blurred image of topography, while the latter represent point samples of the same field. A synthetic data set is constructed to first show the performance of the method. Real data is then inverted.