# Advanced methods for subsurface velocity estimation : trans-dimensional inversion and machine learning

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Inversion is a widely adopted tool to estimate the subsurface elastic properties of the Earth from seismic data. However, it faces several obstacles due to lack of adequate data coverage, and various assumptions made in forward modeling and inversion algorithms resulting often in sub-optimal results. One such assumption is the choice of parameterization of the model. In general, it is assumed to be known a priori and kept fixed. This can lead to either over or under parameterization, causing either overfitting or underfitting the data. In the first part of my thesis, I address the problem of model parameterization. Along with searching for models that fit the data, I also solve for the optimum number of model parameters required as dictated by the data. In a deterministic approach, I use the Basis Pursuit Inversion (BPI), which imposes sparsity in the model parameterization by adding a regularization term of L₁ norm of the model vector. The weight of the regularization term plays a dominant role, and I propose an approach for automatic calculation of this weighting factor. Alternately, I also develop a stochastic method, using Bayesian framework to solve my inverse problem in which the model parameters are treated as unknown. Unlike BPI, this method also provides us with estimates of uncertainty. Here, I make use of the Reversible Jump Markov Chain Monte Carlo (RJMCMC) framework, which allows changing the number of model parameters. However, the conventional RJMCMC is generally very slow as it attempts to sample a variable dimensional model space. To address this, I propose a new method called the Reversible Jump Hamiltonian Monte Carlo (RJHMC), which improves the efficiency by combining RJMCMC with a gradient-based Hamiltonian Monte Carlo (HMC). The gradient-based steps ensure quick convergence by allowing the sampling to take large steps guided by the gradient instead of complete random steps. I represent my model space using a layer-based earth model for the 1D problem and using an adaptive ensemble of nuclei along with Voronoi partition for a 2D problem. Subsequently, I use the method to solve the deconvolution problem in 1D, and tomography and Full Waveform Inversion problems in a 2D setting. It also provides estimates of the elastic parameters and marginal distribution of the number of model parameters. I use the 1D RJHMC to estimate density, along with P- and Swave velocities from a pre-stack angle gather. The region contains paleo-residual gas (PRG), which shows same signature as that of normal gas saturation, and can be better differentiated using density. Additionally, I applied trans-dimensional tomography to invert for P-wave velocity structure at an Axial Seamount, which is one of the most volcanically active regions in northeastern Pacific. In addition to BPI and RJMHC, I develop workflows, which take advantage of the hybrid schemes and Machine Learning (ML) algorithms. Solving an elastic FWI problem can be challenging, as it is very computationally expensive in comparison to the more commonly used acoustic formulation. I propose a hybrid scheme, where the initial P-wave velocity result from an acoustic FWI can be used to perform less expensive pre-stack Amplitude vs. Angle (AVA) inversion. This provides us with all three elastic parameters: P-wave velocity, S-wave velocity, and Density. Several inverse problems can be mapped into a neural network architecture, which can be solved using the currently developed deep learning algorithms. The last part of my dissertation describes two machine learning (ML) algorithms that I have developed for seismic inversion. I use a Convolutional Neural Network (CNN) to perform seismic inversion, in which instead of using the traditional way of using input-output pairs to train the network, I use the physics of the forward wave-propagation to guide the training. It circumvents the need for providing the label data during training and makes it unsupervised. In addition to this, I propose to use a Recurrent Neural Network (RNN) to estimate NMO velocities, which is a basic seismic processing technique. Generally, the NMO velocity is hand-picked and requires a lot of human intervention and computation time. Using this workflow with only 10% of data used as training, the network estimates NMO velocities almost instantly for the rest of the dataset