School Seminars and Colloquia

Bayesian wavelet extraction

Continuum Mechanics

by Dr James Gunning


Institution: CSIRO Petroleum, Clayton
Date: Tue 12th October 2004
Time: 10:00 AM
Location: Room 213, Richard Berry Building

Abstract: We discuss a Bayesian approach for deriving seismic source
waveforms ('wavelets') from processed seismic and well-log data, a problem
known as the 'well-tie' problem in the petroleum industry. The well-tie
problem is most naturally approached from a Bayesian viewpoint, which
naturally integrates prior knowledge about the well tie in the form of
marker constraints, vertical seismic profile (VSP) data, phase prejudices,
and plausible interval velocities. We can perform simultaneous extractions
at multiple (possibly deviated) wells, for multiple offset seismic data
(using a linearised Zoeppritz reflectivity), and can estimate additional
uncertainty parameters such as time-registration errors for stacks or
well-location errors caused by imaging problems. Useful diagnostics
include distribution details for the mis-tie, or noise, amplitude
(critical for inversion studies), and multiple realisations of the
extracted wavelets from the Bayesian posterior, showing the uncertainty in
the wavelet scaling and extent, the time-to-depth map, and the noise
parameters for each stack.

For the statistically minded, the problem can be seen as moderately
computational nonlinear regression problem with a Bayesian model-selection
flavour. Certain perennial issues arise, such as (a) the Lindley paradox
comes into play, so sensible construction of the prior is important, and
(2) evaluation of the marginal probability of the models must be done with
care, and we find the Laplace approximation based on the numerical Hessian
at the modes to be adequate. The posterior distribution is a mixture that
is only mildly non-Gaussian, so sampling can be performed using a Gaussian
mixture without recourse to MCMC methods, although the latter would be
perfectly possible.

For More Information: Dr John Sader: 8344-4042, J.Sader@ms.unimelb.edu.au