The transfer function when wired as a multiplying circuit is: W = ((X1-X2)*(Y1-Y2) / 10V) + Z2
For this to be true the Z1 pin should be wired to the output W, to provide feedback, which isn't shown explicitly on Tara's general multiplying circuit diagram. Also for testing the chip inputs were wired as differential, not with one leg grounded as shown on the GMC diagram.

The 10 V comes from the default division voltage when the denominator control inputs (U0, U1, U2) are grounded. If you want some added offset to the output you can send it to the Z2 pin.

The input impedance is listed as 50k for all X, Y, and Z pins.

We measured the noise with 0V X/Y inputs, it was around 1 mV/rtHz at 10 Hz, as you can see in Tara's earlier post, slightly improving at higher frequency.
The input noise is listed as 1 uV/rtHz from 100 Hz to 1 MHz. The amplifier gain is listed as 72 dB which is ~ 4000x, and we were at the default denominator of 10V so this corresponds to a noise of 1e-3 * 10 / 4000 = 2.5 uV/rtHz at the input, seems reasonable compared to spec sheet. The signal to be squared in the creak setup (the output of the Michelson) will have to be bandpassed first, probably by an SR560, so gain can be applied there to get in over the multiplier noise floor.

As Tara noted the output does rail for signal amplitudes well below the listed maximum input, so we need a better understanding of how to control the gain.

I was flabbergasted when I saw this. There are many really good seismometers with very simple mechanical design and electronics. This is a nice one with complicated mechanics and electronics.

The low frequency oscillation we mentioned in the previous Log could originate from the creep of the rubber between PZT and the Shim. Because the initial stress caused the creep of the rubber, the Shim relaxed slowly and changed the optical path and caused the low frequency oscillation. This mechanism can explain the phase change between the driving and the signal. Rana recommended to use a spring to replace the rubber. To calculate the spring constant of the spring: Spring constant of the Shim, ks = 3EI/L^3; Amplitude of displacement of PZT ~ A; Amplitude of displacement of the Shim ~ B; the spring constant of the spring ~ k;

k = ks*B/(A-B)

From current dimension, ks ~ 10000 N/m. If we don't want to drive PZT too hard, assume A = 2B; k = ks = 10000 N/m.

We tested AD734 on the diagnostic bread board, the result is good.

We want to square/multiply signals between 10 to 100 Hz, so we use AD734 chip to do the work. The circuit is connected as described here

We try to square the signal. the test signals are sine waves at 10 Hz, 50Hz. The output are nice sine waves, but the gain is high (72dB). The chip rails as the input exceeds 0.5 Vpkpk. We will have to check the signal from the PD in the setup to see if it is higher than 0.5 Vpkpk or not. If so we can change the gain of the chip. Otherwise we can go ahead and use it.

The spectrum of the output, for 10Hz input, there's a peak at 20Hz output. For 50Hz input, there's a peak at 100Hz. The response is flat between this bandwidth.

I tested the mixer, the demodulated signal from input at 10 - 100 Hz might be too small and too distorted to get reliable data.

As we want to square/demodulate signal in 10 - 100 Hz BW. a low frequency mixer might be a good tool. I asked Alastair to buy this mixer for me, and it arrived today.

The lowest acceptable frequency in the design is 500 Hz, but I don't know how well it works at 10 - 100 Hz so I tested it.

==Setup and result==

I used SR785 to generate sine wave, then split it with a T and connected the output to LO and RF of the mixer.

I tested that the mixer works fine at the designed frequency. The plot below shows the result from 1kHz signal input.

Next, I changed the frequency to 10 Hz, 50Hz, and 100Hz.

The demodulated signal is then observed in frequency domain (left column of the plot) and in time domain ( right column of the plot)

I think the peaks at driving frequencies (10Hz, 50Hz,100Hzand their harmonics) appear because of the offset of the sine input signal.

The results for low frequency seem to be too distorted. We will test the AD734 chips tomorrow. I got the package this afternoon.

We measured the tip tilt angle of the blade while the main part of blades was bent flat. REM: ~9 degree; ROM: ~ 7 degree. This angle should be able to cancel by mirror holder.

One block of Al was designed to mount mirror holder with the blades. The SolidWork drawing is attached below.

Two screws (2-56, A2, 3/16) will be used to mount the block onto blades through the two holes in the head of blades.

One screw (8-32) will be used to mount the mirror holder onto the block. The mirror hold is light, the block should be able to hold it firmly.

We finalized the drawing for blade clamping system. The drawings are posted here and in Crackle ATF Wiki. We will submit the drawings to the machine shop tomorrow.

For each blade, the clamping system will consist of: 1)Steel base, 2)Steel pillar, 3) Steel top clamp, 4) Al knife edge top piece,5)Al knife edge bottom piece,and 6) Al end piece.

1) Steel base x1: The steel base is 3"x3"x0.5" . It has 4 counter sunk holes that allow us to mount the steel pillar on it. It has 3" rails on both sides, so we can mount it on the table. Extra clamps can be used to hold the base on the table.

2) Steel pillar x1: It is 5.5" height with 2"x2" square cross section. There are 4 tapped 1/4-20 holes , 1" in depth, on the bottom for mounting it on the base. There are 2 tapped 3/8 , 1" in depth, on top for clamping two clamps along with the blade.

3) Steel top clamping piece x1, This will clamp the blade on the pillar.

4) Aluminum knife edge, top piece x1,

5) Aluminum knife edge, bottom piece x1: (4&5) The two knife edge pieces will be used for loading the mass block on the maraging blade tip. The explanation is written in this entry.

6) Aluminum end piece that holds the mirror mount on the blade tip x1: We want to have a steerable mirror for the IFO. So we need a mirror mount. The block will hold the mount and the blade tip together through screws. This piece is uploaded in the above entry.

The assembly (without the blade and the mirror mount) is shown below.

We made a sketch for the weight clamp that will carry the mass block on the end of the blades. This will be done in Solidwork tomorrow.

We plan to load a block of mass under the tip of the blade by using a pair of knife edge pieces so that the rubbing between the mass block and the blade is minimized.

The edge of the blade cannot be too large, or it will be noisy when the blade is driven. On the other hands, if the blade angle is too small (sharper blade), the stress on the blade due to the weight will be too large and cause plastic deformation on the blade, which we don't want. We plan to make it flat ~ 1mm wide, with 120degree open angle.

The yield tensile strength of maraging steel is ~ 1 -2 GPa. With the contact area at the knife edge we can calculate the maximum clamping force.

The width of the edge is ~ 5cm

The thickness of the edge ~ 1mm.

so the maximum force should not exceed ~ 1 GPa x 0.05 m x 0.001 m ~10^4 newton.

We will use spring washers to make sure that we do not tighten the clamps together with too much force and cause plastic deformation on the blade.

We measured the weight needed for pulling the blades down, and measured Q, f0 of the blades. For Rom blade, the weight is 1.279 kg, f0 = 2.27 Hz, Q = 300. For Rem blade, the weight is 2.005kg, f0 = 2.35Hz, Q = 475. The test blades are named Romulus(Rom) and Remus(Rem).

Why do we do this:

The maraging blades are designed to be flat when they are in used, so we need to know how much weight do we need to pull them down to their operating level. The weight will determine the size of the load mass we want in the drawing as well. We plan to mount mirror mount on the load mass, so we can align the mirror for the interferometer's end mirror. Plus, resonance frequencies and Qs of the blades and seismic noise will be used to estimated the noise budget of the setup.

The weight was applied to the blade until the blade horizontally leveled. Then the total weight was recorded. After that, we used shadow sensing technique to determine their resonance frequencies and Q factors.

The results are summarized here:

Blade load mass f0 Q

Rom 1.279 kg 2.27 Hz 300

Rem 2.005 2.35 Hz 475

.

fig1: determining the weight. The blade mounted on the table appears flat with the right weight.

We switched the current metal shim with the thicker Aluminium shim. Now both mirrors are also the same. We tested and showed that the shim is not too hard to be pushed by pzt.

First, the thicker Al shims have bigger bending stiffness and more difficult to bend under the surrounding perturbation. Therefore, the signal we got has less noise from the surrounding perturbation.

By using the PZT we have, we can still drive the shim well. With the driving, we observed intensity oscillates from ~50 mV to ~200 mV.

We also observed a low frequency (~80 mHz) oscillation of the signal. I didn't find the source of this oscillation. The sensitivity of response to driving is lower while the intensity is near the minimum and Maximum and higher while intensity is at the middle.

We made a drawing for a structure hat will hold the maraging blade. The details aren't complete yet. The holes for the clamping will be identified, but the sketch shows the rough idea.

We want to clamp the blade to a structure. The drawing for the clamp will be provided by Ryan (he found it in the dcc.) The structure is consisted of the base and the pillar. Although a monolithic structure is better, it might be to expensive to carve out a big piece of Al block, so Koji suggested that we do it like this. The base will be mounted on the table, and the pillar will be mounted on the base by 4 screws. The height of the pillar is not decided yet. It depends on how big the Al mass block we need to pull down the blade by its weight, and how the mirror for reflecting the beam up will be mounted, but it should be around 6 - 8 inches.

The mass block will be used for mounting the end mirror of the interferometer + a translational stage. This way we can steer the beam with 2 mirrors and adjust the arm length. We will determine the weight, so we can estimate the size of the mass block, assuming we will use Al.

The "general multiplying circuit" box in the diagram shows how each AD734 will be powered/ fed input signal.

For the signal from the PD, we need to bandpass(10-100Hz) it first. We plan to use a SR560. To split the signal to x and y input, we will use a T connector. Then square the signal and band pass it again at 0.1 - 100Hz bandwidth.

For the signal from the function generator which drives the PZT. We will high pass it, by either SR560 or a high pass circuit. We might need a buffer here if the output impedance of the function generator is not high. Split the signal with a T again, and square it.

After both signals are squared, we multiply them together. Send one to X1 input, another signal goes to Y1 input. Then we FFT the output signal from W.

I ordered 5 of AD734 and thinking about how to make a circuit for squaring the signal.

The "chopping" signal readout technique requires that we square the signals. Basically we need to (as rana suggested):

(1) square the signal from PD, (after 10-100Hz bandpass) to convert it to power, and band pass it again.

(2) square the driving signal (might be varied from 0.1- 1Hz.) This is illustrated in the diagram as doubling the frequency ("2 x freq" box.) The driving signal for PZT is offset. So the signal is V drive = A + B xsin (2pi fdrive t) with A > B. This ensures that the voltage on one end of the PZT is always higher than another end. We might need to high pass this signal first, to get a signal with only 2 fdrive frequency after we square it.

(3) multiply signal from (1) and (2) to demodulate the signal.

Basically, 3 multipliers are needed.

The first one is for (1), so the input frequency is ~ 10 -100Hz, and the output is 20-200 Hz.

The second multiplier is for (2), the signal is ~ 0.1 - 1 Hz, but this one might have large DC term after we square it.

The third one is for (3), this one has to multiply 2 low f signals together which is quite similar to (2), so the design can be the same.

I'll consult Frank and/or Koji again before finalize the multiplier circuit.

We setup the Michelson interferometer with two identical x and y arms. We drove both mirrors at 2 Hz and observed signal at 10 Hz using a lockin amplifier. We saw no significant difference whether the mirror were dirven or not.

(The pzt for the second mirror is fixed. The wire is soldered back to its electrode.)

We setup the Michelson interferometer, now with similar setups on two arms. The end mirrors on both arms are attached on metal shims. The shims touch the PZTs which are driven by 2Hz, 6Vpkpk sinusoidal signal with 7 V offset.

We use a voltage divider(we planned to make one, but we found a nice one in EE lab lying on the floor, so we borrowed it) to adjust the voltage on one of the PZTs to make sure that both mirrors are driven by the same distance. We adjusted the divider to minimize the signal at 2Hz.

fig 1: With a voltage divider, we can adjust the voltage on the PZT so that both mirrors are pushed by the same distance and the 2Hz common mode is minimized. On the plot, Y axis shows the signal output from the lock in amplifier at 2Hz. The higher value means the stronger signal at 2Hz. X axis is time scale. The setup was 5mV sensitivity range, filter in 300 ms, phase -152.3 degree.

The signal output from the lock in amp has not been calibrated to length yet. We just want to see the qualitative result.

Once we made sure that we minimized the common mode, we tried to measure the possible up converted noise at 10Hz. (We used the internal oscillator in the lockin amplifier for reference signal at 10 Hz.)

First, we did not drive the mirror, so that we could see the signal at 10 Hz due to background. Then, we drove the mirror at 2 Hz, and observed any possible up-converted noise at 10Hz

There is nothing conclusive yet. The 2Hz signal that drives the PZTs are plotted here for comparison. From a quick glance, there is no obvious correlation between the noise and the driving signal.

fig2: Signal from the lock in amp at 10Hz. Setup: sensitivity at 500 uV, in filter 300 ms.

Why are we doing this:

We want to measure any possible up-converted noise when the material under stress is driven at low frequency. For example, the system is driven at 2Hz, there might be broadband noise occurs due to the motion. If there is, we can try driving the system with different amplitude to see if the noise changes or not.

We setup the basic Michelson interferometer with one arm which can driven by a PZT and another one whose position is adjustable.

The laser we got didn't work at the beginning. We found that the power supplier was not functional. Tara borrowed another power supplier for the laser.

The basic Michelson interferometer was setup. One of the mirror attached on copper plate was replaced by a regular mirror with position adjustable. One of the PZT is needed to be fixed.

We observed Dark Fringe by adjusting position of the regular mirror.

We got the signal from a basic Michelson setup with one of the arm being driven by a PZT.

This is the signal from the oscilloscope.

First, we check the signal when there is no voltage applied to the PZT, the signal is plotted in green.

Then, we drove only one of the mirror by PZT. The voltage is 6Vpkpk, with 7V offset.

The signal is plotted in blue when the mirror was driven. We can see strong signal on the scope.

Today we brought a rack from Drever lab to 050 W Bridge. This rack will be used for crackle experiment.

We start setting up the experiment, and we need a rack for electronics equipment, so with Steve's help we got one from Drever lab. We cleaned the rack before brought it in the lab, so there should be no dust.

Next, we will find a lock-in amplifer, maybe a function generator to drive the system.

If the plan is to use feed-forward cancellation instead of noise templates, then the way to optimize the array design is to understand where gravity perturbations are generated. The following plot shows a typical gravity-perturbation field as seen by the test mass. It is a snapshot at a specific moment in time. The gravity-perturbation force is projected onto the line along the arm (Y=0). Green means no gravity perturbation along the arm generated at this point.

The plot shows that the gravity perturbations along the direction of the arm seen by the test mass are generated very close to the test mass (most of it within a radius of 10m), and that it is generated "behind" and "in front of" the mirror. This follows directly from projecting onto the arm direction. As we already know, for feed-forward, we can completely neglect the existence of seismic waves and focus on actual gravity perturbations. In short, for feed-forward, you would place the seismometers inside the blue-red region and don't worry about any locations in the green. The distance between seismometers should be equal to or less than the distance between red and blue extrema. So even though I haven't simulated feed-forward cancellation yet, I already know how to make it work. Obviously, if subtraction goals are more ambitious than what we need for aLIGO, then feed-forward cancellation of NN would completely fail generating more problems than solving problems. Unless someone wants to deploy hundreds to a few thousand seismometers around each test mass.

I wanted to push the limits and see when NN subtraction performance starts to break by changing the number of seismometers and the size of the array. For aLIGO, 10 seismometers in a doubly-wound spiral around the test mass with outer radius 8m is definitely ok. Only if I simulate a seismic field that is stronger by a factor 20 than the 90 percentile curve observed at LHO does it start to get problematic. The subtraction residuals in this case look like

The 20 seismometer spiral is still good, but the 10 seismometer spiral does not work anymore. It gets even worse when you consider arrays with circular shape (and one seismometer at the center near the test mass):

This result is in agreement with previous results that circular arrays have trouble in general to subtract NN from locally generated seismic waves or seismic transients (wavelets).

I should emphasize that the basic assumption is that I know what the minimum seismic wavelength is. Currently I associate the minimum wavelength with a Rayleigh overtone, but scattering could make a difference. It is possible that there are scattered waves with significantly smaller wavelength.

Instead of estimating in the frequency domain, I now have a filter that is defined in frequency domain, but transformed into time domain and then applied to the seismometer data. The filtered seismometer data can then be used for the usual time-domain linear estimators. The results is perfect:

So what's left on the list? Although we don't need this, "historically" I had interest in PCA. Although it is not required anymore, analyzing the eigenvalues of the linear estimators may tell us something about the number of seismometers that we need. And it is simply cool to understand estimation of information in seismic noise fields.

Much better than I expected, but also not good enough. All spectra in this plot (except for the constant noise model) are averages over 10 simulation runs. The NN is the average NN, and the two "res." curves show the residual after subtraction. It seems that the frequency-dependent linear estimator is working since subtraction performance is consistent with the (frequency-dependent) SNR. Remember that the total integrated SNR=NN/noise is much smaller than 1 due to the low-frequency noise, and therefore you don't achieve any subtraction using the simple time-domain linear estimators. Now the final step is to improve the subtraction performance a little more. I don't have clever ideas how to do this, but there will be a way.

So far, the test mass noise was white noise such that SNR = NN/noise was about 10. Now the simulation generates more realistic TM noise with the following spectrum:

The time series look like:

So the TM displacement is completely dominated by the low-frequency noise (which I cut off below 3Hz to avoid divergent noise). None of the TM noise is correlated with NN. Now this should be true for aLIGO since it is suspension-thermal and radiation-pressure noise limited at lowest frequencies, but who knows. If it was really limited by seismic noise, then we would also deal with the problem that NN and TM noise are correlated.

Anyway, changing to this more realistic TM noise means that nothing works anymore. The linear estimator tries to subtract the dominant low-frequency noise instead of NN. You cannot solve this problem simply by high-pass filtering the data. The NN subtraction problem becomes genuinely frequency-dependent. So what I will start to do now is to program a frequency-dependent linear estimator. I am really curious how well this is going to work. I also need to change my figures of merit. A simple plot of standard-deviation subtraction residuals will always look bad. This is because you cannot subtract any of the NN at lowest frequencies (since TM noise is so strong there). So I need to plot spectra of subtraction noise and make sure that the residuals lie below or at least close to the TM noise spectrum.

The simulation is not a good representation of a real detector. The first step to make it a little more realistic is to simulate variables that are actually measured. So for example, instead of using TM acceleration in my simulation, I need to simulate TM displacement. This is not a big change in terms of simulating the problem, but it forces me to program filters that correct the seismometer data for any transfer functions between seismometers and GWD data before the linear estimation is calculated. This has been programmed now. Just to mention, the last more important step to make the simulation more realistic is to simulate seismic and thermal noise as additional TM displacement. Currently, I am only adding white noise to the TM displacement. If the TM displacement noise is not white, then you would have to modify the optimal linear estimator in the usual way (correlations substituted by integrals in frequency domain using freqeuncy-dependent noise weights).

I am now also applying 5Hz high-pass filters here and there to reduce numerical errors accumulating in time-series integrations. The next three plots are just a check that the results still make sense after all these changes. The first plot is shows the subtraction residuals without correcting for any frequency dependence in the transfer functions between TM displacement and seismometer data:

The dashed line indicates the expected minimum of NN subtraction residuals, which is determined by the TM-displacement noise (which in reality would be seismic noise, thermal noise and GW). The next plot is shows the residuals if one applies filters to take the conversion from TM acceleration into displacement into account:

This is already sufficient for the spiral array to perform more or less optimally. In all simulations, I am injecting a merry mix of wavelets and spherical waves at different frequencies. So the displacement field is as complex as it can get. Last but not least, I modified the filters such that they also take the frequency-dependent exponential suppression of NN into account (because of TM being suspended some distance above ground):

The spiral array was already close to optimal, but the performance of the circular array did improve quite a bit (although 10 simulation runs may not be enough to compare this convincingly with the previous case).

I wanted to write down what I learned from our filter discussion yesterday. There seem to be two different approaches, but the subject is sufficiently complex to be wrong about details. Anyway, I currently believe that one can distinguish between real filters that operate during run time, and estimation algorithms that cannot be implemented in this way since they are acausal. For simplicity, let's focus on FIR filter and linear estimation to represent the two cases.

A) FIR filters

A FIR filter has M tap coefficients per channel. If the data is sampled, then you would take the past M samples (including sample at present time t) of each channel, run them through the FIR and subtract the FIR output from the test-mass sample at time t. This can also be implemented in a feed-forward system so that the test-mass data is not sampled. Test-mass data is only used initially to calclulate the FIR coefficients, unless the FIR is part of an adaptive algorithm. For adaptive filters, you would factor out anything from the FIR that you know already (e.g. your best estimates of transfer functions) and only let it do the optimization around this starting value.

The FIR filter can only work if transfer functions do not change much over time. This is not the case though for Newtonian noise. Imagine the following case:

(S1)-----(TM)----------(S2)

where you have two seismometers around a test mass along a line, one of them can be closer to the test mass than the other. We need to monitor the vertical displacement to estimate NN parallel to the line (at least when surface fields are dominant). If a plane wave propagates upwards, perpendicular to the line, then there will be no NN parallel to this line (because of symmetry). The seismic signals at S1 and S2 are identical. Now a plane wave propagating parallel to the line will produce NN. If the distance between the seismometers happens to be the length of the plane wave, then again, the seismometers will show identical seismic signals, but this time there is NN. An FIR filter would give the same NN prediction in these two cases, but NN is actually different (being absent in the first case). So it is pretty obvious that FIR alone cannot handle this situation.

What is the purpose of the FIR anyway? In the case of noise subtraction, it is a clever time-domain representation of transfer functions. Clever means optimal if the FIR is a Wiener filter. So it contains information of the channels between sensors and test mass, but it does not care at all about information content in the sensor data. This information is (intentionally if you want) averaged out when you calculate the FIR filter coefficients.

B) Linear estimation

So how to deal with information content in sensor data from multiple input channels? We will assume that an FIR can be applied to factor out the transfer functions from this problem. In the surface NN case, this would be the 1/f^2 from NN acceleration to test-mass displacement, and the exp(-2*pi*f*h/c) - h being the height of the test mass above ground - which accounts for the frequency-dependent exponential suppression of NN. Since the information content of the seismic field changes continuously, we cannot train a filter that would be able to represent this information for all times. So it is obvious, that this information needs to be updated continuously.

The problem is very similar to GW data analysis. What we are going to do is to construct a NN template that depends on a few template parameters. We estimate these parameters (maximum likelihood) and then we subtract our best-estimate of the NN signal from the data. This cannot be implemented as feed forward and relies on chopping the data into stretches of M samples (not necessarily the same value for M as in the FIR case). Now what are the template parameters? These are the coefficients used to combine the data stretches of the N sensors. This is great since the templates depend linearly on these parameters. And it is trivial to calculate the maximum-liklihood estimates of the template parameters. The formula is in fact analogous to calculating the Wiener-filter coefficients (optimal linear estimates). If we only use one parameter per channel (as discussed yesterday) or if one should rather chop the sensor data into even smaller stretches and introduce additional template coefficients will depend on the sensor data and how nature links them to the test mass. Results of my current simulation suggest that only one parameter per channel is required.

When I realized that the NN subtraction is a linear estimation problem with templates etc, I immediately realized that one could do higher-order noise subtraction so that we will never be limited by other contributions to the test mass displacement (and here I essentially mean GWs since you don't need to subtract NN below other GWD noise, but maybe below the GW spectrum if other instrumental noise is also weaker). Something to look at in the future (if this scenario is likely or not, i.e. NN > GW > other noise).

Here is the hour of truth (I think). I ran simulations of wavelets. These are not anymore characterized by a specific frequency, but by a corner frequency. The spectra of these wavelets almost look like a pendulum transfer function, where the resonance frequency now has the meaning of a corner frequency. The width of the peak at the corner frequency depends on the width of the wavelets. These wavelets propagate (without dispersion) from somewhere at some time into and out of the grid. There are always 12 wavelets at four different corner frequencies (same as for the other waves in my previous posts). The NN now has the following time series:

You can see that from time to time a stronger wavelet would pass by and lead to a pulse like excitation of the NN. Now, the first news is that the achieved subtraction factor drops significant compared to the stationary cases (plane waves and spherical waves):

And the 4*pi, 10 seismometer spiral dropped below an average factor of 0.88. But I promised to introduce an absolute figure to quantify subtraction performance. What I am now doing is to subtract the filtered array NN estimation from the real NN and take its standard deviation. The standard deviation of the residual NN should not be larger than the standard deviation of the other noise that is part of the TM displacement. In addition to NN, I add a 1e-16 stddev noise to the TM motion. Here is the absolute filter performance:

As you can see, subtraction still works sufficiently well! I am now pretty much puzzled since I did not expect this at all. Ok, subtraction factors decreased a lot, but they are still good enough. REMINDER: I am using a SINGLE-TAP (multi input channel) Wiener filter to do the subtraction. It is amazing. Ideas to make the problem even more complex and to challenge the filter even more are welcome.

I had to rebuild some of the guts of my simulation to prepare it for the big changes that are to come later this week. So I only have two results to report today. The code can now take arbitrary waveforms. I tested it with spherical waves. I injected 12 spherical waves into the field, all originating 50m away from the test mass with arbitrary azimuths. The 12 waves are distributed over 4 frequencies, {10,14,18,22}Hz with equal spectral density (so 3 waves per frequency). The displacement field is far more complex than the plane-wave fields and looks more like a rough endoplasmic reticulum:

The spatial spectra are not so much different from the plane-wave spectra:

The white dots now indicate the back-azimuth of the injected waves, not their propagation direction. And we can finally compare subtraction performance for plane-wave and spherical-wave fields:

Here the plane-wave simulation is done with 12 plane waves at the same 4 frequencies as the spherical waves, and in both cases I chose a 20 seismometer 4*pi spiral array. Note that the subtraction performance is pretty much identical since the NN was generally stronger in the spherical-wave simulation (dots 5 and 20 in the right figure lie somewhere in between the upper right group of dots in the left figure). This makes me wonder if I shouldn't switch to some absolute measure for the subtraction performance, so that the absolute value of NN does not matter anymore. In the end, we don't want to achieve a subtraction factor, but a subtraction level (i.e. the target sensitivity of the GW detectors).

Anyway, the result is very interesting. I always thought that spherical waves (i.e. local sources) would make everything far more complicated. In fact, it does not. And also the fact that the field consists of waves at 4 different frequencies does not do much harm. (subtration performance decreased a little). Remember that I am currently using a single-tap FIR filter if you want. I thought that you need more taps once you have more frequencies. I was wrong. The next step is the wavelet simulation. This will eventually lead to a final verdict about single-tap v. mutli-tap filtering.

Considering equal areas covered by seismic arrays, the number of seismometers relates to the density of seismometers and therefore to the amount of aliasing when measuring spatial spectra. In the following, I considered four cases:

1) 10 seismometers randomly placed (as usual, one of them always at the origin)
2) 10 seismometers in a spiral winding one time around the origin
3) The same number winding two times around the origin (in which case the array does not really look like a spiral anymore):

4) And since isotropy issues start to get important, the forth case is a circular array with one of the 10 seismometers at the origin, the others evenly spaced on the circle.

Just as a reminder, there was not much of a difference in NN subtraction performance when comparing spiral v. random array in case of 20 seismometers. Now we can check if this is still the case for a smaller number of seismometers, and what the difference is between 10 seismometers and 20 seismometers. Initially we were flirting with the idea to use a single seismometer for NN subtraction, which does not work (for horiz NN from surface fields), but maybe we can do it with a few seismometers around the test mass instead of 20 covering a large area. Let's check.

Here are the four performance graphs for the four cases (in the order given above):

All in all, the subtraction still works very well. We only need to subtract say 90% of the NN, but we still see average subtractions of more than 99%. That's great, but I expect these numbers to drop quite a bit once we add spherical waves and wavelets to the field. Although all arrays perform pretty well, the twice-winding spiral seems to be the best choice. Intuitively this makes a lot of sense. NN drops with 1/r^3 as a function of distance r to the TM, and so you want to gather information more accurately from regions very close to the TM, which leads you to the idea to increase seismometer density close to the TM. I am not sure though if this explanation is the correct one.

A spiral shape is a very good choice for array configurations to measure spatial spectra. It produces small aliasing. How important is array configuration for NN subtraction? Again: plane waves, wave speeds {100,200,600}m/s, 2D, SNR~10. The array response looks like Stonehenge:

A spiral array is doing a fairly good job to measure spatial spectra:

The injected waves are now represented by dots with radii proportional to the wave amplitudes (there is always a total of 12 waves, so some dots are not large enough to be seen). The spatial spectra are calculated from covariance matrices, so theory goes that spatial spectra get better using matched-filtering methods (another thing to look at next week...).

Now the comparison between NN subtraction using 20 seismometers, 19 of which randomly placed, one at the origin, and NN subtraction using 20 seismometers in a spiral:

A little surprising to me is that the NN subtraction performance is not substantially better using a spiral configuration of seismometers. The subtraction results show less variation, but this could simply be because the random configuration is changing between simulation runs. So the result is that we don't need to worry much about array configuration. At least when all waves have the same frequency. We need to look at this again when we start injecting wavelets with more complicated spectra. Then it is more challenging to ensure that we obtain information at all wavelengths. The next question is how much NN subtracion depends on the number of seismometers.

I just want to catch up on my conclusion that a single seismometer cannot be used to do the filtering of horizontal NN at the surface. The reason is that there is 90° phase delay of NN compared to ground displacement at the test mass. The first reaction to this shoulb be, "Why the hack phase delay? Wouldn't gravity perturbations become important before the seismic field reaches the TM?". The answer is surprising, but it is "No". The way NN builds up from plane waves does not show anything like phase advance. Then you may say that whatever is true for plane waves must be true for any other field since you can always expand your field into plane waves. This however is not true for reasons I am going to explain in a later post. All right, but to say that seismic dispalcement is 90° ahead of NN really depends on which directoin of NN you look at. The interferometer arm has a direcion e_x. Now the plane seismic wave is propagating along e_k. Now depending on e_k, you may get an additional "-" sign between seismic dispalcement and NN in the direction of e_x. This is the real show killer. If there was a universal 90° between seismic displacement and NN, then we could use a single seismometer to subtract NN. We would just take its data from 90° into the past. But now the problem is that we would need to look either 90° into the past or future depending on propagation direction of the seismic wave. And here two plots of a single-wave simulation. The first plots with -pi/2<angle(e_x,e_k)<pi/2, the second with pi/2<angle(e_x,e_k)<3*pi/2:

All right. The next problem I wanted to look at was if the ability of the seismic array to produce spatial spectra is somehow correlated with its NN subtraction performance. Now whatever the result is, its implications are very important. Array design is usually done to maximize its accuracy to produce spatial spectra. So the general question is what our guidelines are going to be? Information theory or good spatial spectra? I was always advertizing the information theory approach, but it is scary if you think about it, because the array is theoretically not good for anything useful to seismology, but it may still somehow provide the information that we need for our purposes.

Ok, who wins? Again, the current state of the simulation is to produce plane waves all at the same frequency, but with varying speeds. The challenge is really the mode variation (i.e. varying speeds) and not so much varying frequencies. You can always switch to fft methods as soon as you inject waves at a variety of frequencies. Also, I am simulating arrays of 20 seismometers that are randomly located (within a 30m*30m area) including one seismometer that is always at the origin. One of my next steps will be to study the importance of array design. So here is how well these arrays can do in terms of measuring spatial spectra:

The circles indicate seismic speeds of {100,250,500,1000}m/s and the white dots the injected waves (representing two modes, one at 200m/s, the other at 600m/s). The results are not good at all (as bad as the maps from the geophone array that we had at LLO). It is not really surprising that the results are bad, since seismometer locations are random, but I did not expect that they would be so pathetic. Now, what about NN subtraction performance?

The numbers indicate the count of simulation runs. The two spatial spectra above have indices 3 (left figure) and 4 (right figure). So you see that everything is fine with NN subtraction, and that spatial spectra can still be really bad. This is great news since we are now deep in information theory. We should not get to excited at this point since we still need to make the simulation more realistic, but I think that we have produced a first powerful clue that the strategy to monitor seismic sources instead of the seismic field may actually work.

It turns out that I had to do some clean-up of my NN code:

1) The SNRs were wrong. The problem is that after summing all kinds of seismic waves and modes, the total field should have a certain spectral density, which is specified by the user. Now the code works and the seismic field has the correct spectral density no matter how you construct it.

2) I started with a pretty unrealistic scenario. The noise on the test mass, and by this I mean everything but the NN, was too strong. Since this is a simulation of NN subtraction, we should rather assume that NN is much stronger than anything else.

3) I filtered out the wrong kind of NN. I am now projecting NN onto the direction of the arm, and then I let the filter try to subtract it. It turns out, and it is fairly easy to prove this with paper and pencil, that a single seismometer CANNOT never ever be used to subtract NN. This is because of a phase-offset between the seismic displacement at the origin and NN at the origin. It is easy to show that the single-seismometer method only works for the vertical NN or underground for body waves.

This plot is just the prove for the phase-offset between horizontal NN and gnd displacement at origin. The offset is depends on the wave content of the seismic field:

The S0 points in the following plot are now obsolete. As you can see, the Wiener filter performs excellently now because of the high NN/rest ratio of TM dispalcement. The numbers in the titel now tell you how much NN power is subtracted. So a '1' is pretty nice...

One question is why the filter performance varies from simulation to simulation. Can't we guarantee that the filter always works? Yes we can. One just needs to understand that the plot shows the subtraction efficiency. Now it can happen that a seismic field does not produce much NN, and then we don't need to subtract much. Let's check if the filter performance somehow correlates with NN amplitude:

As you can see, it seems like most of the performance variation can be explained by a changing amplitude of the NN itself. The filter cannot subtract much only in cases when you don't really need to subtract. And it subtracts nicely when NN is strong.

So I was curious about comparing the performance of the array-based NN Wiener filter versus the single seismometer filter (the seismometer that sits at the test mass). I considered two different instrumental scenarios (seismometers have SNR 10 or SNR 1), and two different seismic scenarios (seismic field does or does not contain high-wavenumber waves, i.e. speed = 100m/s). Remember that this is a 2D simulation, so you can only distinguish between the various modes by their speeds. The simulated field always contains Rayleigh waves (i.e. waves with c=200m/s), and body waves (c=600m/s and higher).

There are 4 combinations of instrumental and seismic scenarios. I already found yesterday that the array Wiener filter is better when seismometers are bad. Here are two plots, left figure without high-k waves, right figure with high-k waves, for the SNR 1 case:

'gamma' is the coherence between the NN and either the Wiener-filtered data or data from seismometer 0. There is not much of a difference between the two figures, so mode content does not play a very important role here. Now the same figures for seismometers with SNR 10:

Here, the single seismometer filter is much better. A value of 10 in the plots mean that the filter gets about 95% of NN power correctly. A value of 100 means that it gets about 99.5% correctly. For the high SNR case, the single seismometer filter is not so much better as the Wiener filter when the seismic field contains high-k waves. I am not sure why this is the case.

The next steps are
A) Simulate spherical waves
B) Simulate wavelets with plane wavefronts (requires implementation of FFT and multi-component FIR filter)
C) Simulate wavelets with spherical wavefronts

Other goals of this simulation are
A) Test PCA
B) Compare filter performance with quality of spatial spectra (i.e. we want to know if the array needs to be able to measure good spatial spectra in order to do good NN filtering)

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.

This morning I installed the polarizing optics for the REFL isolation. I thought I would need another QWP to linearize the output beam of the HeNe, but the JDSU supposedly has a 500:1 linear polarization ratio out of the box, so all I had to do was turn the head to the right orientation. Below is a raytrace of the setup.

I hooked up the PDs to the scope to see if things were working correctly, and though the DC levels are off (i.e. the contrast is not great due to the rather hodgepodge setup), the AC response looks correct. Here is a screenshot. Here, both of the mirrors were being driven common-mode at 2-Hz with the 3:5 ratio I figured out yesterday. You can still see some 2-Hz harmonics here (particularly ~8 Hz), but the majority of the signal is just the ambient noise.

EDIT: I just realized that, interestingly enough, the dominant low-frequency signal here is probably not exactly 8 Hz, but slightly above; if you look at the previous entry, when both mirrors are driven at 2 Hz, the strongest peak is at a bit above 8 Hz, where there is no peak in the single-mirror case (though there is one at 8 Hz existing as a 2-Hz overtone there). I am not sure what this is from.

After yesterday's success with the PZT we scavenged from the Drever cabinets, I decided to repair the second one and make a duplicate mirror assembly. It isn't quite as pretty as the last one since I had to solder it (so I won't show a closeup), but all in all it came out pretty well. I used the other shim that matched the one we used for the first mirror, and glued the fully-silvered mirror that bore the PZTs to the top of it. (I later found out that Rana put the new shim stock on my desk, but we will use the stuff that's in place now until we get some kinks worked out. In any case, we have yet to use any nice 5101 mirrors.) I also installed a beam dump to block the second ghost beam from hitting the PD. Here is a shot of the second mirror assembly and another of the full setup:

I brought some SR560s in to set up the control loop, but it was difficult to bias the PZTs enough (so that the polarity never reversed) and still have enough range on the SR560 output; we need a voltage amp to really get going with these guys. In the meantime, I thought it would be somewhat useful to characterize the relative strengths of the PZTs by driving them both and maximizing the CMRR. I drove them at 2 Hz with the Tektronix FG, using an offset of +5V, and I determined that the best amplitude ratio was about 3:5. Below is the voltage spectrum of the PD output while driving one and both mirrors, respectively, showing a CMRR of about 36. I am certain that we can do better, but it is difficult to tweak it in view of the excess noise at the moment with the loop open.

I rigged up a way to use the small ThorLabs PZTs we took from the 40m yesterday. After an hour or so of going back and forth from the ATF to the SUS lab with random optical hardware to find something suitable, I finally found a solution using one of the fancy translation stages we have for the eventual gyro modematching. Here's a shot of the whole assembly:

That was just to find a way to mount to the magnetic base we are using; I still needed a way to actually hold the PZT and connect it to the mirror on the shim "blade". We knew we wanted to have something give-y like rubber between the PZT and the blade itself to suppress high-frequency noise in the actuator, so I found a piece of rubber grommet to do the trick. The grommet had a hole in it, of course, so I wrapped it in a piece of shrinkwrap so that I could glue it along the flatter surface to the PZT. On the other end, I needed something firm attached to the PZT with which to hold it (gripping the PZT itself might damage it and in any case would reduce the range of motion). I chose to use a polyester film capacitor---with the leads trimmed---and glued it to the other side. Here is a closeup:

This thing is supposed to put out 4.5 um with 150 V applied, so I figured I could get a decent signal using a drive on the order of 5-10 V (since we are using 633-nm light, this is on the order of a fringe). I installed a PDA100A at the AS port of the interferometer and realigned the beams from both arms to overlap. The manufacturer warns never to reverse the polarity of the PZT leads, so I applied a ~6 Vpp drive with an offset of +5V. I could clearly see an output coherent with the drive on the scope over a wide range of frequencies. I decided to plug it into the Agilent and look at the spectrum. Here is an example of one with a 3-Hz drive signal. There is a lot of upconversion because the mirror is swinging through a couple of fringes. I was able to change the overtone structure by adjusting the drive amplitude and offset (so that it stayed roughly linear).

For the heck of it, I thought I might try and measure a transfer function from the PZT to the PD signal. It can be seen below. Even with maximum integration, the ambient noise is very high at the moment, and turning up the drive doesn't help since the thing quickly loses linearity, but to the naked eye the TF looks roughly like what one would expect from a driven pendulum with a resonance somewhere around 100-200 Hz. Rana and I noticed that the simple system with the shim clamped to the base and the mirror glued to its top had a fairly high Q, but the thing is now damped by the rubber contact, so the resonance is not very evident in the TF.

From these very simple trials, I would guess that these PZTs will work quite nicely once we can close the loop and operate at the dark fringe. I have unfastened the second unit from the mirror on which we found them, and I will try and put a new wire on the ground lead tomorrow so that we can test it.

We grabbed the old quasi-EUCLID HeNe setup from the 40m's SP table and brought some of the parts over to the SUS lab (where Alastair's fibers and the Cryo people are).

We have a single blade spring set up in the Y arm of the michelson. We have aligned it for maximum contrast by eye. We also made sure to keep the REFL beam from going back into the laser.

We experimented with a couple of glues to see what worked. In the end we have attached a junky, mostly-reflective, silvered mirror using super glue to the tip of the blade.

Tomorrow morning Zach is going to use the cast-off PZT stacks that we got from Vass and see if they can be used (word is that they're (AE0203D04F - Piezoelectric Actuator, Max Displacement 4.6 µm, 3.5 x 4.5 x 5 mm).The spec sheet says that it takes a maximum of 150 V to give a 4 micron displacement (enough for us). They also say that the PZT will fail quickly if reverse biased at all or put in a high humidity environment.

For our first trick, we're going to just drive one arm of the interferometer and measure the signal with the old analog lockin amp. Next, we are thinking to use the purple box as the DAQ to do more sophisticated things.

Nanometrics has a couple of seismometers which are cheaper than the T240 which may be of interest to us: better than the Guralp CMG-40T, but cheaper and easier to use than the STS-2.

I've turned the main turbo back on and aligned the readout now. The vacuum is already down to 1e-6torr. It seems that the the offset pin on the bottom of the mass is causing the fiber to move a lot as the isolation mass rotates. The movement does not appear much to the eye, but is taking up most of our readout range at the moment.

We can wait to see if this motion dies down. If not we may be forced to replace this intermediate mass with one where the pin is in the center.

Explanation of the legend: lvdt1_fine_cal, lvdt2_fine_cal are left and right sensors they are right on top of each other and general tilt spectrum.
The red curve shows common mode. It has some noise of its own mostly due to imperfect cancellation between left and right sensors, but mostly it shows what electronics is definitely capable of.
Bracket refers to standalone LVDT mounted on a bracket and shows what a single LVDT can do - it is calibrated the same was as the other two. The pale pink curve on the bottom is hard limit from
amplifier and ADC sensitivity.

This did not use any feedback.

The large peak in the middle is the tiltmeter proper frequency. We tuned it higher so it is easier to compare performance between open loop and close loop cases.

In an effort to (1) train Jan and Sanjit to use the elog and (2) actually write down some useful info, I'm going to put some highly useful info into the elog. We'll see what happens after that....

The deal: we have a Trillium, an STS-2, a GS-13 and the Ranger Seismometers, and we want to make nifty breakout boxes for each of them. These aren't meant to be sophisticated, they'll just be converter boxes from the many-pin milspec connectors that each of the seismometers has to several BNCs so that we can read out the signals. These will also give us the potential to add active control for things like the mass positioning at some later time. For now however, the basics only.

I suggest buying several boxes which are like Pomona Boxes, but cheaper. Digi-Key has them. I don't know how to link to my search results, so I'll list off the filters I applied / searched for in the Digi-Key catalog:

Hammond Manufacturing, Boxes, Series=1550 (we don't have to go for this series of boxes, but it seems sensible and middle-of-the-line), unpainted, watertight.

Then we have a handy-dandy list of possible sizes of nice little boxes.

The final criteria, which Sanjit is working on, is how big the boxes need to be. Sanjit is taking a look at the pinouts for each seismometer and determining how many BNC connectors we could possibly need for each breakout box. Jan's guess is 8, plus power. So we need a box big enough to comfortably fit that many connectors.

From the attached plot you can see the usual 1/f noise at low frequency, which is most likely caused by current setting resistor which is cooled by air currents.
Voltage references and the output driver are the secondary suspects.

The 1/f rise is much smaller than what we see from the tiltmeter, so that slope must be due to mechanical noise as we expected from correlation plots. Once the magnets arrive we can reduce the number of wires to test the size of the effect.

Also, the standalone LVDT is a fairly good proximity detector - putting a hand close to it produces a very large change in the reading. It would be interesting to see how differential LVDT coils perform in this regard.