## A multigrid approach for time domain FWI

### Daniel O. Trad

Full Waveform inversion (FWI) is usually implemented in either time or frequency domains, and in some cases in a hybrid domain. The frequency-domain version (FD) is often used in academia because its formulation is simple to understand in terms of matrices. Another key advantage is that FD FWI permits the use of a multigrid approach consisting of bootstrapping from low to high frequencies where data bandwidth is gradually increased. A limitation of FD FWI, however, is that it becomes very expensive to calculate on 3 dimensions because a large system of equations has to be solved at each frequency. The time-domain (TD) approach, however, seems to be more promising in terms of scalability. In this case, rather than solving one frequency of many shots at a time, all frequencies for each shot are solved at a time. However, this statement is an oversimplification, since parallelization and cycle skipping turn the algorithm into a very different dataflow. In practice, the time domain algorithm permits to solve many shots simultaneously as well, by using parallel clusters with Graphics Processing Units (GPUs), leading to a very efficient implementation with coarse-grained parallelism across nodes and fine-grained parallelism across GPUs. On the other hand, additional complications arise because bandpass filtering the data is not sufficient to replicate the frequency-domain multigrid approach. In this report, we will discuss a time-domain implementation of Full Waveform Inversion to understand how these factors can be taken into account.