Chapter V: Noise Subtraction with Neural Networks
5.2 Mock Data
seismic motion∆xLF(t)into the gravitational wave band via terms likex2LF(t)xHF(t) coming from terms likex3in thesin(x)expansion.
In both situations, monitoring auxiliary channels can allow us to subtract the noise they cause. We often refer to these auxiliary channels as witness channels and the noise they cause as the target channel. For example in the bilinear noise mechanism, the spot displacement ∆y and the angular motion of the mirror∆θ are witnesses channels and the length change∆Lis the target channel.
Our approach is to remove the subtractable noise offline by measuring auxiliary channels, processing them to calculate the subractable noise, and then removing it.
Linear regression has been tried with some success [4], but since the couplings are nonlinear in nature, non-linear regression may lead to better performance. Neural networks are a form of non-linear regression that has had success in everything from image recognition [5] to natural language processing [6] to predicting financial markets [7] (which is actually a somewhat similar problem to reducing the noise in DARM)
their witnesses, the acs control signals, by a 1/f4transfer function for the mirror suspension.
It is important that our network can learn these filters both for simplicity and because we can not account for all the filtering in the interferometer. Our simplest mock data ensures that we can handle these filters and is
y(t)= H[x(t)], (5.4)
where yis the target, xis the witness, andH is a bandstop or bandpass buttersworth filter.
Bilinear Coupling
Motivated, by the angle-to-length mechanism, we consider mock data with a N pairs of bilinearly coupled, perfect witnesses
y(t)= ÕN
i=1
xi(t)xN+i(t) (5.5)
where yis the target and xi are the witnesses.
We start withxithat have white spectra and then make the mock data progressively more challenging giving them colored spectra representative of the beam spot motion and angular position of the mirrors, adding white sensing noise to each witness and adding background noise (sampled from the aLIGO noise curve) to the target. We will refer to this mock data set as the colored bilinear mock data.
Finally in our most realistic bilinear mock dataset. we try to model the angle-to-length noise as it occurs in aLIGO. We will refer to this mock data set as the bilinear ifo mock data.
There are 2 mirrors in each arm of the interferometer. The spot position on each mirror has two degrees of freedom (i.e pit and yaw on each mirror) and the angular motion of each mirror has 2 degrees of freedom (i.e pit and yaw on each mirror), so there are 16 perfect witnesses.
To determine the coupling to DARM, we decompose the mirror spot positions into a common/differential and hard/soft mode basis (see [8–10] for a description of hard/soft modes). Similarly, we decompose the angular motion into the common/differential, hard/soft basis. For geometric reasons, the soft modes can essentially be filtered out by control loops and we consider only the hard modes to couple to DARM y(t); we
take the coupling to be hard*hard and c*d
y(t)= xc,pspotxd,pangle(t)+xd,pspotxc,pangle(t)+xc,yspotxd,yangle(t)+xd,yspotxc,yangle(t), (5.6) where the hard modesxare labeled with c/d for common/differential, p/y for pitch/yaw, and angle/spot to distinguish between the angular and spot position true motion.
We will not have access to the perfect witnesses to perform the regression.
The true angular motion and true spot motion can (in theory) be reconstructed from the Alignment Sensing and Control (ASC) system measurements and Internal Seismic Isolation (ISI) system measurements. We use our knowledge of the these systems to model the witnesses in our most realistic bilinear mock dataset.
The ASC system is the control system responsible for maintaining the mirror alignment at high frequencies. The angle witnesses are essentially ASC error signals measuring the hard and soft modes of the mirror rotation, which differ from the true angular motion by a 1/f4transfer function for the suspension. In our mock data, we model the ASC error signals by giving them a few known resonances and most of their power at high frequencies. To get the witnesses for the ASC channels, we also multiply by a fixed, unknown mixing matrix to account for unknown cross coupling during the measurement of the error signals and add a small amount of white measurement noise.
The ISI system is the control system responsible for removing low frequency seismic motion from the test masses. Movement of the objects holding the test mass, such as the suspension, couples into angular motion of the mirrors, which also produces beam spot motion. In our mock data, we model ISI channels by giving them a few known resonances and most of their power at low frequencies. We model the true spot position as a linear combination of the eight ASC error signals (again multiplied by a different mixing matrix to model cross couplings and supplemented with white measurement noise) and eight ISI channels (supplemented with measurement noise) plus and unknown DC offset. It is not immediately obvious which ISI channels will be necessary for reconstructing the true spot position so we also include eight irrelevant ISI channels that our network will have to learn to discard.
Hence, the coupling in this model is essentially between low frequency ISI signals and high frequency ASC signals, the coupling between two high frequency ASC signals will be not effect the target band (10-20 Hz) in DARM that we are trying to clean. This is also why we didn’t also give the true angular motion a low frequency
ISI component; the coupling between two low frequency ISI signals is not in the 10-20 Hz band.
5.3 Neural Networks