I was putting some things together today to program the first lines of a 2D-NN-Wiener-filter simulation. 2D is ok since it is for aLIGO where the focus lies on surface displacement. Wiener filter is ok (instead of adaptive) since I don't want to get slowed down by the usual finding-the-minimum-of-cost-function-and-staying-there problem. We know how to deal with it in principle, one just needs to program a few more things than I need for a Wiener filter.
My first results are based on a simplified version of surface fields. I assume that all waves have the same seismic speed. It is easy to add more complexity to the simulation, but I want to understand filtering simple fields first. I am using the LMS version of Wiener filtering based on estimated correlations.
The seismic field is constructed from a number of plane waves. Again, one day I will see what happens in the case of spherical waves, but let's forget it about it for now. I calculate the seismic field as a function of time, calculate the NN of a test mass, position a few seismometers on the surface, add Gaussian noise to all seismometer data and test mass displacement, and then do the Wiener filtering to estimate NN based on seismic data. A snapshot of the seismic field after one second is shown in the contour plot.
Seismometers are placed randomly around the test mass at (0,0) except for one seismometer that is always located at the origin. This seismometer plays a special role since it is in principle sufficient to use data from this seismometer alone to estimate NN (as explained in P0900113). The plot above shows coherence between seismometer data and test-mass displacement estimated from the simulated time series.
The seismometers measure displacement with SNR~10. This is why the seismometer data looks almost harmonic in the time series (green curve). The fact that any of the seismometer signal is harmonic is a consequence of the seismic waves all having the same speed. An arbitrary sum of these waves produce harmonic displacement at any point of the surface (although with varying amplitude and phase). The figure shows that the Wiener filter is doing a good job. The question is if we can do any better. The answer is 'yes' depending on the insturmental noise of the seismometers.
So what do I mean? Isn't the Wiener filter always the optimal filter? No, it is not. It is the optimal filter only if you have/know nothing else but the seismometer data and the test-mass displacement. The title of the last plot shows two numbers. These are related to coherence via 1/(1/coh^2-1). So the higher the number, the higher the coherence. The first number is calculated fromthe coherence of the estimated NN displacement of the test mass and the true NN displacement. Since there is other stuff shaking the mirror, I can only know in simulations what the true NN is. The second number is calculated from coherence between the seismometer at the origin and true NN. It is higher! This means that the one seismometer at the origin is doing better than the Wiener filter using data from the entire array. How can this be? This can be since the two numbers are not derived from coherence between total test-mass displacement and seismometer data, but only between the NN displacement and seismometer data. Even though this can only be done in simulation, it means that even in reality you should only use the one seismometer at the origin. This strategy is based on our a priori knowledge about how NN is generated by seismic fields. Now I am simulating a rather simple seismic field. So it still needs to be checked if this conclusion is true for more realistic seismic fields.
But even in this simulated case, the Wiener filter performs better if you simulate a shitty seismometer (e.g. SNR~2 instead of 10). I guess this is the case because averaging over many instrumental noise realizations (from many seismometers) gives you more advantage than the ability to produce the NN signal from seismometer data.