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Message ID: 11407     Entry time: Tue Jul 14 10:23:27 2015
 Author: Ignacio Type: Update Category: General Subject: Optimal detector array placement thoughts

Over the past few days, I've been thinking about how to workout the details conerning Rana's request about a 'map' of the vicinity of the 40m interferometer. This map will take the positions of N randomly placed seismic sensors as well as the signals measured by each one of them and the calculated cross correlations between the sensors and between the sensors and the test mass of interest to give out a displacement vector with new sensor positions that are close to optimum for better seismic (and Newtonian) noise cancellation.

Now, I believe that much of the mathematical details have been already work out by Jenne in her thesis. She explains that the quantity of interest that we wish to minimize in order to find an optimal array is the following,

$R = \sqrt{1-\frac{\vec{C}_{SN}^T C_{SS}^{-1}\vec{C}_{SN} }{C_{NN}}}$

where  $\vec{C}_{SN}$ is the cross-correlation vector between the seismic detectors and the seismic (or Newtonian) noise, $C_{SS}$ is the cross-correlation matrix between the sensors and $C_{NN}$ is the seismic (or Newtonian) noise variance.

I looked at the paper that Jenne cited from which she obtained the above quantity and noted that it is a bit different as it contains an extra term inside the square root, it is given by

$R' = \sqrt{1-\frac{\vec{C}_{SN}^T (C_{SS}^{-1}+C_{\Sigma\Sigma})\vec{C}_{SN} }{C_{NN}}}$

where the new term, $C_{\Sigma\Sigma}$ is the matrix describing the self noise of the sensors. I think Jenne set this term to zero since we can always perform a huddle test on our detectors and know the self noise, thus effectively subtracting it from the signals of interest that we use to calculate the other cross correlation quantities.

Anyways, the quantity $R$ above is a function of the positions of the sensors. In order to apply it to our situation, I'm planning on:

1) Performing the huddle tests on our sensors, redoing it for the accelerometers and then the seismometers (once the data aquisition system is working... )

2) Randomly (well not randomly, there are some assumptions we can make as to what might work best in terms of sensor placement) place the sensors around the interferometer. I'm planning on using all six Wilcoxon 731A accelerometers, the two Guralps and the STS seismometer (any more?).

3) Measure the ground signals and use wiener filtering in order to cancel out their self noises.

4) From the measured signals and their present positions we should be able to figure out where to move the sensors in order to optimize subtraction.

i have also been messing around with Jenne's code on seismic field simulations with the hopes of simulating a version of the seismic field around the 40m in order to understand the NN of the site a little better... maybe. While the data aquisition gets back to a working state, I'm planning on using my simulated NN curve as a way to play around with sensor optimization before its done experimentally.

i have as well been thinking and learning a little bit about source characterization through machine learning methods, specially using neural networks as Masha did back in her SURF project on 2012. I have also been looking at Support vector machines. The reasons why I have been looking at machine learning algorithms is because of the nature of the everchanging seismic field around the interferometer. Suppose we find a pretty good sensor array that we like. How do we make sure that this array is any good at some time t after it has been found? If the array mostly deals with the usual seismic background (quiet) of the site of interest, we could incorporate machine learning techniques in order to mitigate any of the more random disturbances that happen around the sites, like delivery trucks, earthquakes, etc.

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