Transdimensional change-point modeling as a tool to investigate uncertainty in applied geophysical inference: An example using borehole geophysical logs

Recently developed methods for inferring abrupt changes in data series enable such change points in time or space to be identified, and also allow us to estimate noise levels of the observed data. The inferred probability distributions of these parameters provide insights into the capacity of the observed data to constrain the geophysical analysis and hence the magnitudes, and likely sources, of uncertainty. We carry out a change-point analysis of sections of four borehole geophysical logs (density, neutron absorption, sonic interval time, and electrical resistivity) using transdimensional Bayesian Markov chain Monte Carlo to sample a model parameter space. The output is an ensemble of values which approximate the posterior distribution of model parameters. We compare the modeled change points, borehole log parameters, and the variance of the noise distribution of each log with the observed lithology classes down the borehole to make an appraisal of the uncertainty characteristics inherent in the data. Our two examples, one with well-defined lithology changes and one with more subtle contrasts, show quantitatively the nature of the lithology contrasts for which the geophysical borehole log data will produce a detectable response in terms of inferred change points. We highlight the different components of variation in the observed data: due to the geologic process (dominant lithology changes) that we hope to be able to infer, geologic noise due to variability within each lithology, and analytical noise due to the measurement process. This inference process will be a practical addition to the analytical tool box for borehole and other geophysical data series. It reveals the level of uncertainties in the relationships between the data and the observed lithologies and would be of great use in planning and interpreting the results of subsequent routine processing. INTRODUCTION The inference of geophysical properties from observed data may be represented by two end-member schools of thought: (1) “Real world” techniques, aimed at the routine processing of geophysical data to infer earth structure using commercially produced software and (2) “Demonstration algorithms,” aimed at exploring what can be gained through the inference process using a wide variety of techniques, which typically are implemented in an academic environment. Although this is a sweeping simplification, it has sufficient validity to illustrate a key difference in the way that geophysical uncertainty is currently approached in practice. In most routinely applied geophysical modeling, the output is a single model that best fits the data and uncertainty is viewed as an addendum to this result (Loke and Barker, 1996; Li and Oldenburg, 1998, 2000; Zelt and Barton, 1998; Loke et al., 2010). In contrast, much of the research into demonstration algorithms addresses the nonuniqueness of models constrained by the available data as the dominant theme (Stoffa and Sen, 1991; Sen and Stoffa, 1992; Yamanaka and Ishida, 1996; Sambridge, 1999). Some algorithms also enable the investigation of uncertainty as part of the exploration of the parameter space (Dosso and Dettmer, 2011; Guo et al., 2011; Bodin et al., 2012b). In an industry context, we recognize the practical need for a quick result in geophysical modeling. This need often precludes the routine use of some of the more intricate demonstration algorithms. However, we advocate that any geophysical practitioner should seek information regarding the uncertainties in the data from which Manuscript received by the Editor 15 September 2012; revised manuscript received 2 February 2013; published online 24 May 2013. University of Tasmania, School of Earth Sciences and CODES Centre of Excellence, Hobart, Australia. E-mail: [email protected]. Université de Rennes 1, Géosciences, Rennes, France. E-mail: [email protected]. © 2013 Society of Exploration Geophysicists. All rights reserved. WB89 GEOPHYSICS, VOL. 78, NO. 3 (MAY-JUNE 2013); P. WB89–WB99, 7 FIGS., 2 TABLES. 10.1190/GEO2012-0384.1