Rasmus Bødker Madsen

A thesis for the degree of Doctor of Philosophy defended August, 2018.

The PhD School of Science, Faculty of Science, Climate and Computational Geophysics, Niels Bohr Institute, University of Copenhagen

Supervisor:
Thomas Mejer Hansen

Probabilistic Linear Inversion of Reflection Seismic Data

Seismicacquisition is the major provider of geophysical data for large-scale hydrocarbon exploration purposes. Large quantities of data-rich seismic surveys hampers a proper integration of geophysical data and geological prior knowledge, due to computational demands. To remedy this, simplified and approximate models, and mathematically convenient priors are applied to allow computationally faster solutions. It is in particular difficult to obtain reliable uncertainty estimates of the physical parameters in the subsurface using linear-least squares solutions. These models present a trade-off between obtaining a realistic depiction of the subsurface and having an analytic (computationally efficient) solution to the inverse problem. This thesis addresses this issue and presents new methodologies and work-flows, which could potentially enable a better description of the subsurface. The work is manifested in three main directions.

Firstly, errors from using a simplified linear forward are considered. The work is split between linear assumptions of the non-linear Zoeppritz equations and errors arising during processing of seismic data. Errors from both sources are deemed significant in seismic inversion, and in both cases, the error is estimated and described with a Gaussian distribution. This enables more trustworthy posterior solutions.

Secondly, non-stationarity in the Gaussian prior distribution is considered. Using localized marginal maximum likelihood estimators, a non-stationary global measure of the variance is obtained. This estimate is used as a plug-in variance in the prior distribution. Improved posterior results are obtained using the non-stationary estimate compared to a traditional stationary measure.

Finally, the strategy of inferring the noise model as part of the inversion result is investigated. It is demonstrated that the shape of the noise is decisive for the reliability of the obtained posterior solutions. In all three presented work areas, the results indicate that a proper description of the noise on the data is vital in bridging the gap between computationally feasible solutions and realistic posterior distributions. A better description and accounting for modeling errors can potentially significantly improve currently available linear probabilistic inversion methods.