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

Abstract

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.

Flexible Coupling in Joint Inversions: A Bayesian Structure Decoupling Algorithm

“…The application of the SD algorithm to field measurements confirms the performance of the algorithm in distinguishing structures between two parameters, here DTR and EAL, at the depth levels of the 1-D profile where the two observables show clearly different oscillations. Our application of the SD algorithm to the reconstruction of borehole measurements confirms the feasibility of the RjMcMC approach for characterizing the lithological interfaces in terms of changepoint locations, as done in Reading and Gallagher (2013) using a standard RjMcMC algorithm As for many Monte Carlo algorithms, the main shortcoming of our approach is the computational time. Our model parameterization, which comprises three different set of interfaces, and our recipe, which encompasses three different blocks of moves for perturbing the current model, increase the computation time due to the enlarged model space and the need of testing all the different moves.…”

“…In particular, a sharp increase of the value of EAL in the sandstone formation (about 300-302 m) is not reflected in any kind of variation in the DTR value in the In this case study, the algorithm works as a simple "changepoint" regression (as in the first synthetic test), where we search for the family of most probable layered models that predict DTR and EAL data, considering constant DTR and EAL within each layer. This assumption, and the use of a RjMcMC algorithm, recalls the methodology used in Reading and Gallagher (2013), with the most interesting difference being, in our implementation, the potential for exploiting decoupled structures. The model is composed of a variable number of interfaces, belonging to the three different qualities, Q i , i = 1, 2, 3, where Q 1 interfaces contain information about both DTR and EAL, while Q 2 and Q 3 interfaces represent DTR and EAL only, respectively.…”

“…Trans‐D algorithms have been applied to changepoint problems (Hopcroft et al, ; Gallagher, ) obtaining relevant results in estimating both the existence and the number of changepoints in observed time series. It is worth noticing that a similar algorithm can be used for locating changes in borehole measurements, that is, not only for time series but also for 1‐D profiles (e.g., Reading & Gallagher, ). Here we jointly invert the two time series for retrieving the time period where changes in the observations are decoupled.…”

Seismic inversion and uncertainty analysis using a transdimensional Markov chain Monte Carlo method

“…By studying the abrupt changes of borehole geophysical logs, Reading and Gallagher (2013) found that interfaces are sharply defined if there exists a large lithology contrast. By a tomography study using surface wave, Galetti et al (2015) found out that high uncertainty was observed along a discontinuity in the velocity field and the uncertainty map exhibits some spatial detail of velocity anomaly.…”

“…The uncertainty in geophysical inverse problems was estimated by various stochastic inversion algorithms, such as MCMC (Liu and Stock, 1993;Malinverno and Briggs, 2004;Chen and Dickens, 2009;Gunning et al, 2010;Kwon and Snieder, 2011), SA (Dosso, 2002;Dosso and Nielsen, 2002;Bhattacharya et al, 2003;Roy et al, 2005;Varela et al, 2006), PSO (Fernández-Martínez et al, 2012;Rumpf and Tronicke, 2015), and rjMCMC Reading and Gallagher, 2013;Dadi, 2014;Galetti et al, 2015;Dadi et al, 2015).…”