Geological Prior Information for Bayesian Tomography
Authors: Bloem Hugo , Tetzlaff Daniel , Curtis Andrew 
Published: March 2021

&lt;p&gt;Seismic tomography is used widely to image the Earth's interior structure and to infer subsurface properties. Tomography is a nonlinear inverse problem, so computationally expensive inversion methods are required to estimate uncertainties in tomographic results. Monte Carlo sampling explores the Bayesian posterior probability distribution function (pdf) which describes the solution to tomographic problems. However, this is expensive, and often intractable for high dimensional model spaces due to the curse of dimensionality &amp;#8211; the exponential increase in computation required to explore parameter space with increasing number of degrees of freedom in the Earth model. Variational methods and some neural network inversion methods use optimisation to estimate the posterior pdf. These methods may be more efficient, but they still suffer from the &amp;#8216;curse&amp;#8217; in high dimensional problems.&lt;/p&gt;&lt;p&gt;The Bayesian solution to tomographic problems combines information available prior to collecting the current data set with information from the geophysical data. To counteract the curse we wish to inject geological prior information that reduces the hypervolume of parameter space to be explored. We use geological process modelling software SedSimple to generate geological training images. This software is designed specifically to produce relatively simple three-dimensional geological structures at reduced computational cost compared to more sophisticated current process models. The output represents the general form of expected geological structures, but not specific detail that might be encountered in any particular volume of the Earth. These geological models are used to train a generative adversarial network (GAN): the GAN then performs high-dimensional regression between the models so that it can generate other, similar models extremely rapidly. The latent space of the trained GAN provides a reduced-dimensionality representation of prior information from SedSimple, which we wish to use as prior information to constrain geophysical imaging problems.&lt;/p&gt;&lt;p&gt;The GAN provides a mapping from latent parameters &lt;strong&gt;a&lt;/strong&gt; to simplistic geological models. Real structure consists of a simplistic model, plus overlying geological complexity parametrised by vector &lt;strong&gt;m&lt;/strong&gt;. We seek the posterior probability of &lt;strong&gt;m&lt;/strong&gt; and &lt;strong&gt;a&lt;/strong&gt; given geophysical data &lt;strong&gt;d&lt;/strong&gt;, written . Assume &lt;strong&gt;d&lt;/strong&gt; is divided into a part &lt;strong&gt;d&lt;/strong&gt;&lt;sub&gt;s&lt;/sub&gt; that is only sensitive to simplistic structures, the remaining data being mainly sensitive to the overlying complexity (e.g. wave travel times versus seismic waveforms). We evaluate the posterior pdf P&amp;#8217;(&lt;strong&gt;a&lt;/strong&gt;&lt;strong&gt;d&lt;/strong&gt;&lt;sub&gt;s&lt;/sub&gt;) of latent parameters &lt;strong&gt;a&lt;/strong&gt; given data &lt;strong&gt;d&lt;/strong&gt;&lt;sub&gt;s&lt;/sub&gt; using the GAN. This pdf is estimated without considering trade-offs between &lt;strong&gt;a&lt;/strong&gt; and &lt;strong&gt;m&lt;/strong&gt; since we limit the data to &lt;strong&gt;d&lt;/strong&gt;&lt;sub&gt;s&lt;/sub&gt; only.&lt;/p&gt;&lt;p&gt;We can expand the posterior pdf using identity P(&lt;strong&gt;m&lt;/strong&gt;,&lt;strong&gt;a&lt;/strong&gt;&lt;strong&gt;d&lt;/strong&gt;) = P(&lt;strong&gt;m&lt;/strong&gt;&lt;strong&gt;a&lt;/strong&gt;,&lt;strong&gt;d&lt;/strong&gt;) P(&lt;strong&gt;a&lt;/strong&gt;&lt;strong&gt;d&lt;/strong&gt;). Assuming the influence of &lt;strong&gt;m&lt;/strong&gt; on &lt;strong&gt;a&lt;/strong&gt; in the posterior is minimal, we can approximate this equation by P(&lt;strong&gt;m&lt;/strong&gt;,&lt;strong&gt;a&lt;/strong&gt;&lt;strong&gt;d&lt;/strong&gt;) = P(&lt;strong&gt;m&lt;/strong&gt;&lt;strong&gt;a&lt;/strong&gt;,&lt;strong&gt;d&lt;/strong&gt;) P&amp;#8217;(&lt;strong&gt;a&lt;/strong&gt;&lt;strong&gt;d&lt;/strong&gt;&lt;sub&gt;s&lt;/sub&gt;). We can estimate P(&lt;strong&gt;m&lt;/strong&gt;&lt;strong&gt;a&lt;/strong&gt;,&lt;strong&gt;d&lt;/strong&gt;) for each value of latent parameters &lt;strong&gt;a&lt;/strong&gt; by using conventional linearised methods if &lt;strong&gt;m&lt;/strong&gt; is very high dimensional, or using fully nonlinear methods if &lt;strong&gt;m&lt;/strong&gt; can be decomposed into lower-dimensional dependencies on the data. This framework allows informative prior information from geological process models to constrain posterior pdf&amp;#8217;s on the full complexity model (&lt;strong&gt;m&lt;/strong&gt;,&lt;strong&gt;a&lt;/strong&gt;) using geophysical data &lt;strong&gt;d&lt;/strong&gt;, as will be illustrated in this talk.&lt;/p&gt;