Adjoint state method and numerical algorithms for full waveform inversion of seismic data

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Full waveform inversion (FWI) is a model-based data fitting procedure that is widely used in exploration geophysics to obtain high-resolution subsurface images of the Earth. On one hand, the FWI method is a promising technique with great potentials. For example, it is able to image the subsurface with high resolution up to half of the propagated wavelength. Moreover, it requires minimal preprocessing of the recorded seismic data and takes into account both direct and reflected waves in the inversion procedure. On the other hand, the FWI method has some limitations such as the requirement of an accurate initial earth model; the high computational cost, especially for 3D inversion and non-uniqueness of solutions since the FWI is an underdetermined inverse problem. **This proposal will address these difficulties through the development of efficient and accurate numerical methods for forward seismic modeling, the utilization of the adjoint state method for efficient gradient calculation and building an accurate initial model using a hybrid time tomography method. Together with the recent developments on gradient-based optimization algorithm, an efficient computational framework will be developed. Within this program we mainly focus on acoustic and elastic wave equations, however extension to more complicated models is possible. Mathematically, we formulate the FWI as a partial differential equation (PDE)-constrained nonlinear optimization problem, where the misfit function measuring the difference between observational and synthetic data is iteratively minimized by a gradient-based optimization algorithm. **Due to the absence of low frequency components in the seismic data, a key limitation of FWI is that the starting model must be accurate and contain spatial wavenumbers from 0 up to the minimum wave number that can be reconstructed by the seismic data. A reliable initial model is critical in avoiding spurious local minima and in compensating for the absence of low frequency components in the data. We will address this difficulty through the development of a hybrid time tomography method to obtain an accurate initial model for FWI.**Another limitation of the FWI method is its high computational cost. We will address this difficulty from two aspects: (1) develop efficient numerical methods for the forward problem so the computational cost in a single FWI iteration can be reduced; (2) apply gradient-based optimization algorithm to reduce the number of iterations. Utilization of gradient-based algorithm necessitates an accurate derivative of the misfit function, which is efficiently calculated by adjoint state method. We will derive the adjoint equation using both perturbation theory and Lagrange multiplier method, and then develop efficient numerical methods to solve the adjoint equation for the adjoint variable. **FWI is a severely underdetermined inverse problem with many solutions. This problem is related to the large number of model parameters and the absence of low frequency components in data. We will address this issue through the development of regularity strategies, such as the incorporation of well-log data in the misfit function.**In summary, this research program will study and make contributions in the following areas: efficient and accurate numerical methods for seismic equations; effective boundary reflection absorption; hybrid travel-time tomography methods; mathematical analysis of inverse problem regularization; and, efficient and accurate adjoint state method. The results of this research program will find applications in reservoir exploitation and prospect evaluation, and provide ample interdisciplinary training opportunities in applied mathematics and geophysics for highly qualified personnel.