Constructing accurate earth models from seismic data is a challenging task. Traditional methods rely on ray based approximations of the wave equation and reach their limit in geologically complex areas. Full waveform inversion (FWI) on the other side seeks to minimize the misﬁt between modeled and observed data without such approximation.
While superior in accuracy, FWI uses a gradient based iterative scheme that makes it also very computationally expensive. In this thesis we analyse and test an Alternating Direction Implicit (ADI) scheme in order to reduce the costs of the two dimensional time domain algorithm for solving the acoustic wave equation. The ADI scheme can be seen as an intermediate between explicit and implicit ﬁnite diﬀerence modeling schemes. Compared to full implicit schemes the ADI scheme only requires the solution of much smaller matrices and is thus less computationally demanding. Using ADI we can handle coarser discretization compared to an explicit method. Although order of convergence and CFL conditions for the examined explicit method and ADI scheme are comparable, we observe that the ADI scheme is less prone to dispersion. Furhter, our algorithm is eﬃciently parallelized with vectorization and threading techniques. In a numerical comparison, we can demonstrate a runtime advantage of the ADI scheme over an explicit method of the same accuracy.
With the modeling in place, we test and compare several inverse schemes in the second part of the thesis. With the goal of avoiding local minima and improving speed of convergence, we use diﬀerent minimization functions and hierarchical approaches. In several tests, we demonstrate superior results of the L1 norm compared to the L2 norm – especially in the presence of noise. Furthermore we show positive eﬀects for applying three diﬀerent multiscale approaches to the inverse problem. These methods focus on low frequency, early recording, or far oﬀset during early iterations of the minimization and then proceed iteratively towards the full problem. We achieve best results with the frequency based multiscale scheme, for which we also provide a heuristical method of choosing iteratively increasing frequency bands.
Finally, we demonstrate the eﬀectiveness of the diﬀerent methods ﬁrst on the Marmousi model and then on an extract of the 2004 BP model, where we are able to recover both high contrast top salt structures and lower contrast inclusions accurately.