Emulation: A Bayesian tool for joint inversion

Where several different kinds of geophysical datasets have been acquired from a particular region, each of these can contain valuable information about the Earth, which may not be present in the other datasets. Jointly determining a common model, therefore, often gives a more thorough and more constrained description of the Earth structure than considering each dataset individually. For example, a seismic velocity inversion is only weakly constrained by first arrival seismic refraction data, but considering it alongside Magneto-Telluric (MT) and gravity data can greatly assist in the constraint (Jegen-Kulcsar et al., 2009). Strategies for joint inversion are therefore an active area of research. To date, most schemes for accomplishing this have been deterministic in nature. Using a deterministic technique often means that it is conceptually difficult to include prior beliefs about the system under determination, uncertainties both in measurement and the relationship between the different physical quantities (velocity, resistivity, density), and the discrepancy between the model and the real Earth. Statistical strategies such as MCMC (Markov Chain Monte Carlo) model searches exist for assessing this kind of problem, but the number of potentially computationally expensive forward model runs required to effectively sample the whole model space and thus achieve a meaningful result is normally prohibitively high (> 10⁵), even for simple 1D models, so such schemes are not generally implemented. However, a technique known as emulation is used in various scientific fields eg. cosmology (Vernon and Goldstein, 2009), whereby computationally expensive forward modelling code (a simulator) is approximated by an uncertainty-calibrated computationally cheap function. Here we apply emulation to the problem of stochastic joint model determination. We show that emulation can be used to quickly exclude large areas of implausible model space, allowing fast updating of beliefs about an Earth structure. It thus provides a means by which the input model space for a deterministic inversion or MCMC scheme can be greatly reduced. We also show how an emulator can, by itself, effectively constrain a region of the Earth. We demonstrate the concept using a 1D model.