Uncertainty quantification in Bayesian inverse problems with model and data dimension reduction

Abstract

The prediction of rock properties in the subsurface from geophysical data generally requires the solution of a mathematical inverse problem. Because of the large size of geophysical (seismic) data sets and subsurface models, it is common to reduce the dimension of the problem by applying dimension reduction methods and considering a reparameterization of the model and/or the data. Especially for high-dimensional nonlinear inverse problems, in which the analytical solution of the problem is not available in a closed form and iterative sampling or optimization methods must be applied to approximate the solution, model and/or data reduction reduce the computational cost of the inversion. However, part of the information in the data or in the model can be lost by working in the reduced model and/or data space. We have focused on the uncertainty quantification in the solution of the inverse problem with data and/or model order reduction. We operate in a Bayesian setting for the inversion and uncertainty quantification and validate the proposed approach in the linear case, in which the posterior distribution of the model variables can be analytically written and the uncertainty of the model predictions can be exactly assessed. To quantify the changes in the uncertainty in the inverse problem in the reduced space, we compare the uncertainty in the solution with and without data and/or model reduction. We then extend the approach to nonlinear inverse problems in which the solution is computed using an ensemble-based method. Examples of applications to linearized acoustic and nonlinear elastic inversion allow quantifying the impact of the application of reduction methods to model and data vectors on the uncertainty of inverse problem solutions. Examples of applications to linearized acoustic and nonlinear elastic inversion are shown.

Combining discrete cosine transform and convolutional neural networks to speed up the Hamiltonian Monte Carlo inversion of pre‐stack seismic data

“…As a side note, we point out that the parameterization of an AVA inversion must always constitute a compromise between model resolution and model uncertainty. This means that both the model uncertainty and the model resolution decrease as the number of considered DCT coefficients decreases (Grana et al ., 2019). In other words, the loss of information due to the parameter space reduction could lead to underfitting the observed data, underestimation of the actual uncertainty in the elastic space and in a decrease of the model resolution.…”

“…We refer the reader to Grana et al . (2019) and Aleardi (2020) for a more in‐depth discussion of this topic in the context of post‐ and pre‐stack seismic inversion.…”

“…A detailed discussion on how the model and data compressions affect the AVA inversion results is beyond the scope of this work, and for this reason, we refer the reader to Grana et al . (2019) for more theoretical insights. Figure 2 shows the explained variability of the generated Vp , Vs and density models as the number of retained DCT coefficients increases.…”

A gradient‐based Markov chain Monte Carlo algorithm for elastic pre‐stack inversion with data and model space reduction

“…In this case, the full state space is projected onto a limited number of basis functions and the algorithm generates samples in this reduced domain. This technique must be applied taking in mind that part of the information in the original (unreduced) parameter space could be lost in the reduced space, and, for this reason, the model parameterization must always constitute a compromise between model resolution and model uncertainty (Dejtrakulwong et al ., 2012; Lochbühler et al ., 2014; Aleardi, 2019; Grana et al ., 2019; Aleardi, 2020b).…”

Samples, Symmetries and Extensions: Exploring Parameter Space in Nonlinear Problems

“…If we are to believe conclusions reached using sampling methods, it is therefore important to reduce the computational power required to explore relevant parts of . This has been attempted previously by reducing the dimensionality of the problem to be solved (Douma et al, 1996;Grana et al, 2019), tightening a priori constraints on parameter values (Curtis & Wood 2004;Walker & Curtis, 2014;Nawaz & Curtis 2019;Linde et al, 2015), developing more efficient forward computations (Rawlinson & Sambridge 2005;Nissen-Meyer et al, 2014;van Driel et al, 2015;Krischer et al, 2017), approximating the forward function with an emulator that can be explored more rapidly (Das et al, 2018;Moseley et al, 2020), or improving predictions of where samples might usefully be located in unexplored parts of (e.g., Fichtner et al, 2019;Khoshkholgh et al, 2020). However, in all such studies the same principle holds: where the parameter space has not been sampled, the value of is unknown.…”