## Computationally Efficient methods for Uncertainty Quantification in Seismic Inversion

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Full waveform inversion is an iterative optimization technique used to estimate subsurface physical parameters in the earth. A seismic energy source is generated in a borehole or on the surface of the earth which causes a seismic wave to propagate into the underground material. The transmitted wave then reflects off of material interfaces (rocks and fluids) and the returning wave is recorded at geophones. The inverse problem involves estimating parameters that describe this wave propagation (such as velocity) to minimize the misfit between the measured data and data we simulate from our mathematical model. The seismic velocity inversion problem is difficult because it contains sources of uncertainty, due to the instruments used to record the data and our mathematical model for seismic wave propagation. Using uncertainty quantification (UQ), we construct distributions of earth velocity models. Distributions give information about how probable an Earth model is, given the recorded seismic data. This rich information impacts real-world decision making, such as where to drill a well to produce oil and gas. UQ methods based on repeated sampling to construct estimates of the distribution, such as Markov chain Monte Carlo (MCMC), are desirable because they do not impose restrictions on the shape of the distribution. How ever, MCMC methods are computationally expensive because they require solving the wave equation repeatedly to generate simulated seismic wave data. This dissertation focuses on techniques to reduce the computational expense of MCMC methods for the seismic velocity inversion problem. Two-stage MCMC uses an inexpensive filter to cheaply reject unacceptable velocity models. The operator upscaling method, an inexpensive surrogate for the wave equation, is one such filter. We find that two-stage MCMC with the operator upscaling filter is effective at producing the same uncertainty information as traditional one-stage MCMC, but reduces the computational cost by between 20% and 45%. A neural network, in conjunction with operator upscaling, is another choice of filter. We find that the neural network filter reduces the computational cost of MCMC by 65% for our experiment, which includes the time needed to generate the training set and the neural network. The size of the problem we can solve using two-stage MCMC is limited by the random walk sampler. Hamiltonian Monte Carlo (HMC) and the No-U-Turn sampler (NUTS) use gradient information and Hamiltonian dynamics to steer the sampler, thereby eliminating the inefficient random walk behavior. Discretizing Hamiltonian dynamics requires two user specified parameters: trajectory length and step size. The NUTS algorithm avoids setting the trajectory length in advance by constructing variable-length paths. We find that the NUTS algorithm for seismic inversion results in superior decrease in the residual over traditional HMC while removing the need for costly tuning runs. However, constructing the gradient for the seismic inverse problem is computationally expensive. In two-stage, neural network-enhanced HMC we replace the costly gradient computation with a neural network. Additionally, we use the neural network to reject unacceptable samples as in two-stage MCMC. We find that the two-stage neural network HMC scheme reduces the computational cost by over 80% when compared to traditional HMC for a 100-unknown layered problem.