In our last post I showed our first ever depth migration of synthetic data over a simple syncline. Although the main reflector was correctly positioned the artefacts from both the line ends and the tails of the “bow tie” were distracting from the result.
The swing artefact coming in from the line ends is quite obviously due to the truncation of the event and the resulting unbalanced migration operator. The common solution to this problem is to apply some sort of a spatial taper to the input data so that the closer you get to the end of the line the less energy there is to swing up into the shallow portion. What should the length of the taper be? The answer I think is subject to a whole host of factors including the geology at the line ends. It would be good to revisit this question from a theoretical point of view later but for the time being I’ve simply added a “line_end_taper” parameter. In any case it’s certainly an effective technique:
Another method I tried was to restrict the aperture approaching the line ends in such a way that the operator was always symmetric. And, like a lot of things I try … it didn’t work. Another one to revisit later.
The more problematic artefact in the first example was that coming from the tails of the bow tie. I was pretty sure that this was due to some mismatch between the forward (modelling) and reverse (migration) operators. In fact this is all that it could be. Given that the migration seems to be working cleanly in the more simple parts of the model this seems to point the finger at the modelling side of things. A former colleague and migration expert now (academic) always said to me that Kirchhoff migration does not like truncations. OK, I get that but what if the truncation is legitimate? After some time pondering this I realised that such truncations just aren’t physical.
By performing ray tracing I was working in the high frequency limit. To create the synthetic I was convolving the ray tracing “spikes” with a desired wavelet (a 20 Hz Ricker wavelet). This is how the unphysicality (I just checked and that is a word) was introduced. Taking a look at the ray paths that lead to a triplication towards the fringes of the bow tie:
The normal incidence ray to the left compared to those on the edge of the syncline has a greater chance of experiencing a full Fresnel zone reflection for a given frequency (say 20 Hz). So the answer then is to use a bit more physics in the model. Fortunately, I have some finite difference (FD) modelling algorithms in my toolkit. So I dusted off my acoustic FD solver and used that to acquire a normal incidence section:
With both the line_end_taper option and the amplitudes correctly modelled by using the FD modelling the final result is now almost free from those unsightly artefacts.
So that’s another nice little milestone on this journey to developing our depth migration capability. Stay tuned for some further algorithmic developments as well what we’re doing on the hardware side of things.