Thomas Bayes versus the wedge model: An example inference using a geostatistical prior function

Jason M. McCrank, Gary F. Margrave, and Don C. Lawton

ABSTRACT

The Bayesian inference is used to estimate model parameters in a synthetic example. The model is a thin bed that follows a wedge shaped trend. However, rather than a uniform trend, the model's thickness at each lateral position across the wedge is randomized. Additionally, the velocity at each model location is normally random with a specified mean and variance. Synthetic seismic data are generated with a zero-offset convolutional model and are then used to invert for bed-thickness. The Bayesian likelihood function is defined using the known relationship between bed-thickness, wedge velocity, wavelet tuning frequency, and seismic amplitude. Population of the amplitude/bed-thickness joint probabilistic density likelihood function is achieved using Monte Carlo methods, and the Bayesian prior is weighted by the known geostatistical trend of the wedge. The inversion result is a probabilistic density function describing the probabilities of wedge thicknesses. Results show that the maximum a posteriori parameter estimate is more accurate than the estimate found with the raw statistical trend or with a deterministic method of data inversion. Additionally, the posterior probabilistic density function can be used to perform statistical analysis, opening a pathway for quantified analysis of parameter uncertainty.

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