STS1 is located at the vertex. x-axis along the x arm.
GUR1 is located at the IMC MC2 mirror. Same orientation.
=> 1. Only the x-direction has good coherence (to be expected)
2. Only good coherence at 1.5-4Hz (huh?)
So probably other noise sources are dominating. Let's look into noise projections. Remember IMC autoalignment is off.
A quick adaptive filter run with only the GUR1 and STS1 witnesses applied only to MCL didn't really do anything. Some more thought needs to be invested into the AA and shaping filters.
The possible explanation to this effect is the following:
Seismic noise mainly consists of the Love and Rayleigh surface waves. In the simulations we've taken 2 perpendicular Love waves and 2 perpendicular Rayleigh waves that interfere under the test mirrors. This interference produces both translational and tilt movements. Then we can see the coherence between translational motion and cavity length.
1. The coherence at big frequencies is small due to the passive isolation.
2. The coherence at 1 Hz is 0 due to the wire resonance.
3. The coherence between 1 and 10 Hz is reasonable. At the real 40m's measurements we can see only good coherence for gur1_x and sts1_x but this is the matter of adjusting seismic waves amplitude and direction. In the simulation we've assumed that all waves are of the same amplitude. The really interesting thing is that
4. The coherence below 0.8 Hz began to grow. We don't see this in real measurements.
But let's simulate the seismometer measurements. It measures not only translational motion but also tilt and with amplitude proportional to g / omega^2. On the Figure below the spectrum of translation motion, tilt and tilt as seen by seismometer is presented. We can see that at low frequencies tilt begins to dominate over the translational motion. We assumed the speed of waves in the region 30 - 60 m/sec.
Let's now plot the coherence between the cavity length and seismometer signal.
We can see that the coherence between seismic signal from measured by seismometer and cavity length is gone below 1 Hz where tilt becomes important.
Now let's try to filter out the seismic noise from the cavity length using both static Wiener filtering and adaptive Mfxlms algorithm. For both filters we've used AA filter before the filters and also AI filter after adaptive filter. The downsampling ratio was 4, the sample frequency 256. We can see that nothing is really subtracted due to the pollution of the seismometer signal due to tilt motion.
Assume we do the same computational experiment but with the seismometers that measure only ground translational motion and tilt do not affect on them. Then we have a reasonable subraction of seismic noise at low frequencies even with the filters of the length 100 as shown on the figure below.
Let's look through an order of magnitude analysis. Assume ground motion consists of only one wave with amplitude A and only vertical movement: z(t) = A*sin(2 pi 0.1 t). So the frequency of the wave is 0.1 Hz. If A = 10-6 m => the amplitude of the suspended mirror motion is also approximately 10-6 m, as we have no isolation at low frequencies. The tilt angle has the amplitude alpha = 2*pi*A/lambda, where lambda - wavelength of the ground wave, lambda = v/f = 40/0.1 = 400 m, v - speed of the wave, f - frequency. Then alpha = 10-8 rad. If the distance between ground and mirror suspension point is 1 m, then mirror motion amplitude due to tilt is B = 10-8 m << A.
It turns out that tilt does not effect much on the cavity length compared to the ground translational motion, but it affects a lot on the seismometer signals, that are used as witness signals in the filtering. For that reason we need tiltmeters to filter seismometer signals in order to obtain pure translational ground motion.