We consider efficient methods for computing solutions to and estimating uncertainties in dynamic inverse problems, where the parameters of interest may change during the measurement procedure. Compared to static inverse problems, incorporating prior information in both space and time in a Bayesian framework can become computationally intensive, in part, due to the large number of unknown parameters. In these problems, explicit computation of the square root and/or inverse of the prior covariance matrix is not possible. In this work, we develop efficient, iterative, matrix-free methods based on the generalized Golub-Kahan bidiagonalization that allow automatic regularization parameter and variance estimation. We demonstrate that these methods can be more flexible than standard methods and develop efficient implementations that can exploit structure in the prior, as well as possible structure in the forward model. Numerical examples from photoacoustic tomography, deblurring, and passive seismic tomography demonstrate the range of applicability and effectiveness of the described approaches. Specifically, in passive seismic tomography, we demonstrate our approach on both synthetic and real data. To demonstrate the scalability of our algorithm, we solve a dynamic inverse problem with approximately 43, 000 measurements and 7.8 million unknowns in under 40 seconds on a standard desktop.
Progression of an immature block cave in proximity to a mature sublevel cave was monitored over an 18-month period using double-difference tomography. This method utilizes existing microseismic data to generate time-lapse images of seismic wave velocity changes within the rock mass. The velocity changes are caused by damage zones forming or local high-stress concentrations. In this study, images were generated on monthly intervals. An average of 2000 microseismic events recorded by 25 stations were used to generate each image. Results showed a development of the initial undercut followed by the growing cave and interaction with the adjacent sublevel cave. The goal of the research is to provide a tool that can augment both numerical modeling results and underground geotechnical measurements to allow the mine operator to produce in the safest and most efficient manner possible.
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