We did the simulation of the stacks by defining a transfer function for one stack (green plot) and another similar transfer function for the other stack.
We simulated the ground motion by filtering a white noise with a low pass filter with a cutoff frequency at 10Hz. (blue plot) (the ground motion for the 2 stacks are completely uncorrelated)
We simulated the electronic white noise for the seismic measurements. (black plot)
We filtered the ground motion (without the measurements electronic noise) with the stack's transfer function and subtracted them to find the mirror response (red plot), which is the target signal for the wiener filter.
We computed the static wiener filter with the target signal (distance between the mirrors) and the input data (seismic measurements = ground motion + electronic noise).
We filtered the input and plotted the output (light blue plot).
We subtracted the target and the output to find the residual (magenta plot).
We didn't figure out why the residual is above the electronic noise only under ~6hz. We tried to increase and decrease the electronic noise and the residual follows the noise still only under ~6Hz.
It also shows that the residues are above the target at frequencies over 20Hz. This means that we are injecting noise here.
We tried to whiten the target and the input (using an high pass filter) to make the wiener filter to care even of higher frequencies.
The residues are more omogeneously following the target.
We also plotted the Wiener filter transfer function without making whitening and with making whitening. It shows that if we do whitening we inject no noise at high frequency. But we loose efficency at low frequencies.
We shouldn't care about high frequency, because the seismometers response is not good over 50Hz. So, instead of whitening, we should simply apply a low pass filter to the filter output to do not inject noise and keep a good reduction at low frequencies.