Linearised inversion with GPUs

Dynamic boundaries

As in Figures 2 and 3, we see inversion improve the images considerably, particularly when dealing with low frequency artifacts. However the remaining random boundary artifacts still seem to stack out fairly slowly, at a rate of about $\sqrt N$ , where N is the iteration number. One option we have is to change these boundaries as a function of iteration, now we would expect to see a quicker reduction in boundary artifacts between iterations. This can be done by seeding our random boundary by iteration number, as well as shot position.

Our algorithm becomes Algorithm 2, giving us an additional propagation per iteration (recalculating the random source wavefield slices). For a typical model size this will increase computation time by around 14%. When doing phase encoded linearised inversion we can get this for free, since recalculation of the initial residual is needed during each iteration (Krebs et al., 2009). This system can be referred to as dynamic random boundaries, as opposed to static random boundaries. Furthermore our operator is now non-linear since we have altered our velocity function and hence our operator. This means theoretically we now have to use a non-linear solver. However since the operator difference is only manifested in the image noise there are some cases where a conjugate direction solver gives acceptable convergence characteristics. We see this system become useful in areas of poor shot sampling, where boundary artifact stacking-out can be slow, however in many cases the extra computation does not seem to outweigh improved convergence. As a function of iteration number we see slightly better residual performance (when using steepest descent for both), but when we look at this as a function of cost we see very little difference. Then, by comparing static boundaries with a conjugate direction solver to dynamic boundaries with a steepest descent solver, we get worse performance with dynamic boundaries as a function of cost.

It should be noted that comparing the $l_2$ norm of the residuals in this case is slightly misleading, since the high frequency noise we have slightly reduced is not well represented by this measure. When looking at the images more differences are notable than implied by this scalar fitting methodology. Additionally if we used a better non-linear solver we would fully expect to see more comparable performance.

Linearised inversion with GPUs