Seismic imaging methods allow to reconstruct the earth¡¯s subsurface parameters based on partial mea- surements of elastic waves at, or near, the surface. Full waveform inversion (FWI) is a numerical optimization problem that uses the whole waveform information of all arrivals to update the subsur- face parameters that govern seismic wave propagation. FWI has shown to provide high quality images and now constitutes a production imaging tool in reservoir exploration. However, FWI still faces many challenges, specially in the topics of the image resolution, computational cost and non-linear optimiza- tion. This thesis concerns some aspects to reduce the computational cost and the use of regularization techniques in the optimization problem.
Currently, the main limitation to perform 3D elastic full waveform inversion on a production level is the computational cost it represents. With this in mind, we provide two contributions. First, we develop a self adjoint formulation of the isotropic first order velocity-stress elastic equations that allow to implement only one forward modeling operator in the gradient computation. Second, solving the forward problem is the most computationally expensive part of FWI, and its cost is proportional to the number of sources (> 10 3 in 3D). To gain efficiency, instead of solving the wave equation for each source, it is possible to solve the wave equation for a linear combination of the sources. This is known as source encoding, and has been widely used when combined with steepest descent or conjugate gradient algorithms in the optimization process. With the purpose of reducing even more the computational cost, we combine Newton and quasi-Newton optimization methods with source encoding techniques. We implement this in a 2D frequency domain acoustic modeling engine based on frequency, but the results are easily extendible to the 3D frequency scenario. Our synthetic numerical tests were carried out in the BP-2004 salt model which is a realistic configuration inspired by the Gulf of Mexico, and the real data we used is an ocean bottom cable dataset recorded from the Valhall field in the North Sea. We find that the lowest computational cost needed to attain a predefined misfit value is provided by l-BFGS, with periodic restarts. However, with noisy data, the most accurate and robust direction of descent is provided by Newton methods, with and without source encoding.
The optimization process requires regularization constraints because the model is not entirely con- strained by the data, and more than one model may fit the observed data equally well. We see that the total variation of the model as a regularization term provides and adequate description of earth models. To improve the quality of the images of the earth parameters, we propose a local total vari- ation denoising algorithm based on the incorporation of the information provided by a migrated image.