Markov Chain Monte Carlo in Practice
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Green's expression for the acceptance probability, on his p. The method only requires the ability to evaluate the prior pdf and the likelihood function for any value of m , and there is no need to compute the marginal likelihood integral of eq. These jumps are done by occasionally proposing to add a layer interface and split a layer in two, or to delete a layer interface and merge two layers. In the following, I give a schematic outline of how the method of Green can be implemented for a nonlinear geophysical inverse problem.
The details are given in the appendixes: Appendix A describes the parameters that need to be specified a priori and the criteria used to set them; Appendix B describes the method used to choose candidate models, paying particular attention to making the MCMC sampling efficient; and Appendix C gives an explicit expression for the acceptance probability of eq.
To start, the Markov chain needs an arbitrary initial earth model m e. The sampling then follows this loop:. C5 Appendix C. A simple way to diagnose the end of the burn-in period in the problem discussed here is to monitor how well the data predicted by the current model g m match the measured data d. As the chain is started from an arbitrary point, at the beginning the fit to the data will be very poor, and it will improve as the iterations progress. Once the burn-in period is over, the parameter vectors output by the chain should be distributed as in the posterior pdf.
The next question is obviously when to stop, i. I will use a simple practical criterion, and continue the MCMC iterations until the characteristics of the posterior pdf stop changing significantly.
Concrete Examples of Monte Carlo Sampling
Similar MCMC algorithms have been described and applied to geophysical inverse problems by other authors. The two main differences with the approach described here are that these authors did not consider the issue of model simplicity by evaluating the posterior probability of having different numbers of layers, and that in their implementation the candidate models in the chain are chosen from the prior pdf.
As noted in Appendix B , different ways to choose a candidate can significantly affect the efficiency of Metropolis—Hastings MCMC sampling, and setting the proposal pdf on the basis of an approximation to the posterior pdf as done here is more efficient than picking candidates from the prior.
The purpose of this exercise is to check that the inferences one makes a posteriori from the Markov chain Monte Carlo sampling results are consistent with a true, known earth model. The synthetic measurements were computed with the method described earlier in the forward model section, adding to the log-apparent resistivities random Gaussian noise with zero mean and a standard deviation equal to 10 per cent of the apparent resistivity value. The prior pdf used in the inversion was set assuming that little was known a priori. The prior probability of the number of layers is uniform from one up to layers, and any configuration of layer interfaces is equally probable a priori in a depth interval of 0.
The MCMC sampling algorithm was started from a two-layer model with an interface in the middle and layer resistivities equal to the prior mean. The sampling was then run for iterations, which took about 3 h on a Sun Ultra 60 workstation. After these many iterations, the results of the sampling did not appear to change appreciably. The progress of the algorithm is illustrated in Fig.
The starting model fit the data very poorly the fractional misfit being about 2. Progress of the algorithm in terms of how well each sampled layered model fits the measured data a and of how many layers each model contains b. The horizontal separation between layered media corresponds to a factor of in resistivity. Figs 3 and 4 also show that the chain mostly samples models that contain relatively few layers, and only occasionally returns models with more than 10 layers. A histogram of the number of layers sampled by the chain is in Fig.
No models with fewer than three layers are ever sampled, because three layers are the absolute minimum needed to adequately fit the data; 91 per cent of the models sampled have 15 or fewer layers. This histogram approximates the posterior pdf of the number of layers; note that the prior pdf is uniform. While the most probable number of layers in Fig.
A three-layer structure is also evident in a posterior histogram of the depths of the layer interfaces Fig. The many sampled layered media that have more than three layers contain minor departures from this basic three-layer structure. Image obtained by superimposing the values of resistivity in the layered media sampled by the MCMC algorithm for the synthetic data in Fig.
This image is an estimated display of the posterior marginal pdf of resistivity at different depths.
The dotted white line shows the range of resistivity and thickness of the middle layer that are consistent with the data of Fig. Histogram of the depths to interfaces in the layered media sampled by the MCMC algorithm for the synthetic data in Fig. This histogram approximates the posterior pdf of the depths to layer interfaces.
The results of Fig. The flat parts of the measured apparent resistivity curve in Fig. The main non-uniqueness in the results of Fig. In the case of three layers with a resistive middle layer, the shape of the hump in the measured apparent resistivity curve depends primarily on the product resistivity-thickness of the middle layer. To determine the range of resistivity and thickness consistent with the measurements of Fig. The residual misfit is below the expected value of 0. This interval of resistivities and thicknesses is marked by a dotted line in Fig. Fractional misfit to the data of Fig.
The data are fitted below the expected misfit level of 0. This interval of resistivities and thicknesses is marked by a dotted white line in the posterior image of Fig. The main difference is that these authors considered only models with three layers and carried out a search on a finite interval for each model parameter three layer resistivities and two layer thicknesses. Letting the number of layers vary and not restricting the resistivity value to a finite interval results in a greater range of plausible models.
Schott et al. All these smooth solutions show a broad transition both at the top and at the bottom of the middle resistive layer. Regularization smears these transitions uniformly throughout the depth interval being investigated, and thus does not allow for distinguishing layer interfaces that are well resolved from those that are not.
In addition, the posterior uncertainties of resistivity in Fig. In contrast, the results of Fig. As in the synthetic data example, I assume that very little is known a priori : the prior probability of the number of layers is uniform from one up to layers, and any configuration of layer interfaces is equally probable a priori in a depth interval of 0. The measurement errors were set to the values given by Constable et al.
The MCMC sampling started from an initial featureless two-layer model and continued for iterations. As in the synthetic data example, after these many iterations the results of the sampling did not appear to change significantly. Field measurements from the Wauchope DC resistivity sounding, central Australia, after Constable et al.
At least four layers are required by the data, and most layered models sampled by the chain have less than 15 layers. The posterior image of resistivity Fig.
Bayesian inference using Markov chain Monte Carlo: Part 1: Foundations and Implementation
While the resistivities of the layers and the depth to the interfaces are relatively well constrained a posteriori down to about m depth, there are large posterior uncertainties in the resistivities and the interface depths of the deeper layers. Image obtained by superimposing the values of resistivity in the layered media sampled by the MCMC algorithm for the field data in Fig. The continuous curves are the median value of resistivity thick line and the 5 per cent and 95 per cent bounds thin lines. Histogram of the depths to interfaces in the layered media sampled by the MCMC algorithm for the field data in Fig.
The inversion results presented here are generally in agreement with those of Constable et al. These authors analyse the correlation of the inverted resistivities and thicknesses and note that only the product resistivity-thickness is well determined for the resistive fifth layer. The MCMC sampling carried out here shows this fundamental non-uniqueness for the resistive crustal layer with a top at about m and a bottom between and 10 m Fig.
While Constable et al.
Markov chain Monte Carlo in practice: a roundtable discussion
Finally, the median posterior resistivity in Fig. The main difference is that the change in resistivity around m depth is more abrupt in the results obtained here, because there is no smoothing enforced a priori. In this paper, I presented an extension of the often used Bayesian parameter estimation to include the number of free parameters in the earth model being inverted specifically, the number of layers. As the posterior probability of earth models that fit the data becomes lower as the number of layers increases, this procedure ensures that parsimonious model descriptions are preferred without imposing an additional simplicity requirement.
This formulation has been implemented using a Markov chain Monte Carlo algorithm that returns a sample of layered models distributed as in the posterior pdf. This sample spans the space of earth models that fit the data within a specified measurement error Fig. The images in the background are histograms of the apparent resistivity curves predicted by the layered media sampled by the MCMC algorithm in the synthetic a and field data example b.
The formulation presented here addresses two basic problems in solving the non-uniqueness half of inverse problems: the difficulty of setting a prior distribution when little is known a priori and the dependence of the posterior uncertainty on a particular earth model parametrization. While Bayesian inference has many desirable qualities for geophysical data inversion, setting the prior pdf on the basis of available knowledge is often problematic e.
Backus ; Ulrych et al. If little is known a priori , the prior hypothesis should allow for a variety of possible earth models and for a broad range for the parameters in these models.
But this does not seem generally feasible: to solve the problem in practice, it seems that one must choose a particular parametrization e. This paper shows how it is possible to set a simple prior pdf with earth model parameters that are effectively unknown a priori such as the number of layers being left as such.
The a posteriori span of all the unknown parameters is then effectively determined by the data.