Cost effective strategies for problems in computational geophysics : seismic modeling and imaging

The first part of my thesis focuses on seismic modeling in fractured media. Several recent developments in finite elements such as usage of high degree polynomials to approximate the wavefield, diagonalization of mass matrices to be inverted through mass lumping techniques and usage of high order time-stepping schemes, have made these methods (along with their classical advantages) more attractive when compared to the finite difference methods (FDM). Discontinuous Galerkin finite element method (DGM) and spectral element method (SEM) have particularly attracted researchers in the field of numerical wave propagation. SEM uses continuous basis functions, which do not allow for discontinuities in the displacement field. Hence it can be used to simulate wave propagation only in non-fractured media. On the other hand, DGM allows for discontinuities in the displacement field to simulate fractures or faults but with a significant increase in computation cost and memory requirement. Here, I formulate and analyze two new, improved finite element techniques (FEM) for the numerical solution of elastic wave propagation in fractured and non-fractured media. Enriched Galerkin (EGM) and hybrid Galerkin (HGM) formulations are proposed for solving elastic wave propagation that has advantages similar to those of DGM but with a computational cost comparable to that of SEM. EGM uses the same bilinear form as DGM, and discontinuous piecewise constants or bilinear functions enrich the continuous Galerkin finite element spaces. EGM satisfies local equilibrium while reducing the degrees of freedom in DGM formulations. HGM employs DGM in areas containing fractures and SEM in regions without fractures. The coupling between the domains at the interfaces is satisfied through interface conditions. The degree of reduction in computation time depends primarily on the density of fractures in the medium. I apply these methods to model wave propagation in 2D/3D fractured media and validate their efficiency with numerical examples. Fractured reservoirs are more complicated due to the presence of fractures and pores. Biot’s fundamental theory on wave propagation in fluid-saturated porous media is still well accepted and forms the basis of this work. To examine the effects of fluid-filled cracks and fractures, I next propose to combine poroelasticity with the linear slip theory for simulating wave propagation in fractured porous media. This study provides an equivalent anisotropic medium model for the description of porous rock with fractures in the seismic frequency band. I solve Biot’s poroelastic wave equations using a velocity-stress staggered grid finite difference algorithm. Through numerical examples, I show that fractures and pores strongly influence wave propagation, induce anisotropy, and poroelastic behavior in wavefields. I also validate the presence of two compressional waves as predicted by Biot’s theory along with the converted waves due to faults. Compared to elastic methods, this approach provides a considerably concise and more accurate model for fractured reservoirs. The second part of my thesis centers on developing cost-effective solutions for seismic migrations and anisotropic moveout corrections. Least-squares migration (LSM) is a linearized inversion problem that iteratively minimizes a misfit functional as a function of the model perturbation. The success of the inversion largely depends on our ability to handle large systems of equations given the massive computation costs. I propose a suite of unsupervised machine learning (ML) approaches that leverage the existing physics-based models and machine learning optimizers to achieve more accurate and cheaper solutions. First, I use a special kind of unsupervised recurrent neural network and its variant, Hopfield neural networks, and the Boltzmann machine, to solve the problems of Kirchhoff and post-stack reverse time migrations. Physics-based forward models can be used to derive the weights and biases of the neural network. The optimal configuration of the neural network after training corresponds to the minimum energy of the network and thus gives the reflectivity solution of the migration problem. I next implement a fast image-domain target-oriented least-squares reverse time migration (LSRTM) workflow using a conjugate Hopfield network. The method computes a low-cost target-oriented Hessian matrix using plane-wave Green’s functions. I recover a more accurate image in the presence of a truncated Hessian matrix. I further implement pre-stack LSRTM in a deep learning framework and adopt various machine learning optimizers to achieve more accurate and cheaper solutions than conventional methods. Using a time-domain formulation, I show that mini-batch gradients can reduce the computation cost by using a subset of total shots for each iteration. Mini-batches not only reduce source cross-talk but are also less memory intensive. Combining mini-batch gradients with Adam optimizer and Huber loss function can improve the efficiency of pre-stack LSRTM. I demonstrate high accuracy predictions on complex synthetic models that can generate noisy data. Finally, I develop a Hough transform neural network based technique for normal moveout correction in vertically transverse isotropic (VTI) media. This technique offers advantages when compared to the time and computational costs required in a conventional anisotropic normal moveout correction. Using a Hough transform based neural network, I simultaneously fit all the non-hyperbolic reflection moveout curves using intermediate to long offsets. I apply the network to synthetic VTI datasets and demonstrate the practical feasibility of anisotropic moveout correction that is independent of travel-time picking and velocity analysis