# Using Bayesian Inference to Quantify How Measurements Affect the Uncertainty of Inversion Results

Monday, April 22, 2002 - 11:00am - 12:00pm

Keller 3-180

Alberto Malinverno (Schlumberger-Doll Research)

In many practical applications, parameters in an Earth model are estimated by inverting geophysical measurements. The inversion process is generally nonunique: many values of the Earth model parameters may fit the measurements equally well. In other words, the model parameters estimated by inversion have an inherent uncertainty. Clearly, the more informative and accurate the measurements are, the less the uncertainty in the inversion.

Bayesian inference is well suited to quantify how much measurements reduce inversion uncertainty. The fundamental object of Bayesian inference is a posterior distribution of the Earth model parameters; this posterior distribution is proportional to the product of a prior distribution (which contains information on the overall variability of and correlations among the model parameters) and of a likelihood function (which quantifies how well an Earth model fits a set of measurements). More information in the prior distribution or in the measurements will result in a corresponding reduction in the spread of the posterior distribution.

I illustrate a Bayesian inference approach by showing how to invert seismic data collected in a vertical seismic profile, where seismic sources are at the surface and receivers are in a well. The object is to predict elastic properties (compressional and shear wave velocity and density) in a one-dimensional layered Earth model beneath the deepest receiver. To quantify posterior uncertainties, I use a Monte Carlo Markov chain method that samples the posterior distribution of layered models. These sampled layered models agree with prior information and fit the seismic data, and their overall variability defines the uncertainty in the predicted elastic properties. It is then easy to show how much the uncertainty in the predicted elastic properties is reduced by increasing the seismic data coverage (in this case, increasing the interval spanned by the seismic source positions at the surface).

References:

(1) On parsimony in Bayesian inference: Malinverno, A., 2000, A Bayesian criterion for simplicity in inverse problem parametrization, Geophys. J. Int., v. 140, 267�285.

(2) On Markov chain Monte Carlo sampling: Malinverno, A. & Leaney, S., 2000, A Monte Carlo method to quantify uncertainty in the inversion of zero-offset VSP data, in SEG 70th Annual Meeting, Calgary, Alberta, The Society of Exploration Geophysicists, Tulsa, Oklahoma. Available at

http://seg.org/meetings/past/seg2000/techprog/pdf/papr0039.pdf

Bayesian inference is well suited to quantify how much measurements reduce inversion uncertainty. The fundamental object of Bayesian inference is a posterior distribution of the Earth model parameters; this posterior distribution is proportional to the product of a prior distribution (which contains information on the overall variability of and correlations among the model parameters) and of a likelihood function (which quantifies how well an Earth model fits a set of measurements). More information in the prior distribution or in the measurements will result in a corresponding reduction in the spread of the posterior distribution.

I illustrate a Bayesian inference approach by showing how to invert seismic data collected in a vertical seismic profile, where seismic sources are at the surface and receivers are in a well. The object is to predict elastic properties (compressional and shear wave velocity and density) in a one-dimensional layered Earth model beneath the deepest receiver. To quantify posterior uncertainties, I use a Monte Carlo Markov chain method that samples the posterior distribution of layered models. These sampled layered models agree with prior information and fit the seismic data, and their overall variability defines the uncertainty in the predicted elastic properties. It is then easy to show how much the uncertainty in the predicted elastic properties is reduced by increasing the seismic data coverage (in this case, increasing the interval spanned by the seismic source positions at the surface).

References:

(1) On parsimony in Bayesian inference: Malinverno, A., 2000, A Bayesian criterion for simplicity in inverse problem parametrization, Geophys. J. Int., v. 140, 267�285.

(2) On Markov chain Monte Carlo sampling: Malinverno, A. & Leaney, S., 2000, A Monte Carlo method to quantify uncertainty in the inversion of zero-offset VSP data, in SEG 70th Annual Meeting, Calgary, Alberta, The Society of Exploration Geophysicists, Tulsa, Oklahoma. Available at

http://seg.org/meetings/past/seg2000/techprog/pdf/papr0039.pdf