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ID Dateup Author Type Category Subject
  126   Fri Nov 20 06:18:51 2009 ranaMiscSUSbandpass filter

seems OK, as long as the current noise doesn't get you. To make sure you have to terminate the input to this filter and then look at the resulting noise in the SR785.

  127   Sun Nov 22 16:02:09 2009 ZachMiscSUSAOSEM noise measurement

On Friday, Rana and I discovered that my transimpedance amp was oscillating like whoa at about 100 kHz. A little research showed this to be due to the input capacitance of the AD743 (~20 pF). To fix this, I put a 20-pF cap in parallel with the 100k feedback resistor, and that seemed to do the trick.


The relevant circuitry is shown in attachment 1. +/- 12 V DC was provided by voltage regulators (7912, 78M12). The voltage across the LED was measured to be V_LED = 1.61 V, and the current through it was I_LED = 31.6 mA, giving Z_LED = 50.9 ohm. The voltage out of the amp with a fully lit PD was V_out = -2.83 V, giving a photocurrent of I_ph = 28.3 uA. 


I was concerned about noise that might be imposed by the bandpass filter, so I compared spectra I took with and without it (that is, AC coupled, no BPF, and DC coupled, with BPF). This comparison is shown in attachment 2. There appears to be no difference apart from the aliasing effects at low frequency.


After this, I took the real measurement, extending the range to 800 Hz, averaging 100x and with a linewidth of 1 Hz (I realize now that I should probably have done this with a smaller linewidth, so that I could see below ~1 Hz. I will repeat the measurement this week with better low-frequency resolution). The result can be seen in attachment 3, calibrated to displacement noise in m/rt(Hz) using the measured 0.05-A/m response of the OSEM in the linear region. The four lines are:


- Bright: noise in the OSEM with a fully lit PD

- Dark: noise in the OSEM with the LED off

- Amp: noise in the transimpedance amp with the input terminated

- V_LED: noise in the LED voltage


The first three spectra were taken at the output of the amplifier and calibrated back to meters using the transimpedance gain and OSEM response in A/m. The last was taken across the LED, and calibrated into meters using the values given in paragraph 2. All measurements were taken with the OSEM under a box and with the lights out.


It appears that we are still limited by our setup. The "Dark" noise is coincident with the amplifier noise, while the "Bright" noise is coincident with the LED noise. That said, it is fairly comforting that all this noise is at the level of around 10^-10 m or less, as we can probably expect the true noise of the OSEM to be lower than this. We will know this for sure once we have a truly quiet setup (starting with ultra-low-noise voltage references).

  128   Tue Nov 24 12:28:52 2009 ZachMiscSUSAOSEM measurement update

  I was able to take some better measurements last night. I took data in two bands: 0-100 Hz, 0-1.6 kHz, each with 800 lines. This gives us a decent idea of what's going on at low and high frequency. Attached are four plots, two from each band. All measurements were taken with a box over both the OSEM and the readout circuit and the lights out.

The first two are low- and high- frequency comparisons of the noise in the full (bright) configuration as measured with no BPF and AC coupling vs with the BPF and DC coupling. There appears to be no difference apart from the expected effect above the pole at 1 kHz.

The next two are plots of the noise in various components and the full scheme calibrated into equivalent displacement noise. Everything is below ~10e-10 m/rt(Hz) with the exception of line peaks, and again it would appear that we are limited by our measurement equipment.

Some notes:

- The "dark" noise seems to be coincident with the "amp" noise with the exception of some extra pickup that increases at high frequency (seems to be line-related).

- The "LED" noise is coincident with the "supply" noise up until its 8-Hz corner frequency, after which it falls off as expected until it hits an apparent floor around 100 Hz.

- The "bright" noise seems to be coincident with the "supply" noise, while the "dark" and "amp" are much lower. This could be because the supply noise only shows up when there is an appreciable voltage at the output of the amp.


Have to think about this for a bit, but the next logical step is to turn the measurement setup into something solid (i.e. soldering, enclosure, etc.).

  129   Tue Nov 24 23:17:02 2009 ZachMiscSUSAD587KN voltage noise

I got a few AD587KN (high-precision 10V reference) samples today from AD. I hooked them up to see how much quieter my DC supply would be. The results are pretty good, with the voltage noise reduced by a factor of 5-10 throughout. The first two attachments below are comparisons of the noise in

1. The +12V regulator (MC78M12) alone

2. The AD785KN reference with V_in = +12 V provided by the regulator

3. The same as in 2, only now with an additional "noise reduction" capacitor (a 1-uF capacitor from pin 8 to ground forms a LPF with an internal 4-k resistor, giving a corner frequency of 40 Hz to reduce high-frequency noise),

plotted with the same frequency ranges and settings as those in the previous post.

The reference comes very close to its noise spec of 100 nV/rt(Hz) @ 100 Hz. The only issue is that it seems to have much more line pickup than the regulator (which seems almost completely insensitive to line noise), and this is worsened by the extra capacitor. Attachment 3 is a close-up of the low-frequency spectrum around 60 Hz. I suspect that this will be alleviated somewhat when I move away from the breadboard phase.

I want to rig this up so that I can stabilize the supply voltage to the transimpedance amp and LED, but in order to do so I will need to build a higher-current source using a power transistor, like either of those shown in attachment 4 (the AD587LN is only able to provide <10mA). 

  130   Thu Nov 26 02:59:35 2009 ranaElectronicsSUSLED Driver circuit

We want to have a simple low noise circuit to drive the LED. Our plan is to use the AD587 followed by a filter/buffer.


from 0.1-10 Hz, produce less RIN in the LED light than shot noise by a factor of 3.

With 35 mA of LED drive, we get ~35 uA of photocurrent (no magnet/flag). The shot noise of 35 uA is ~3.5 pA/rHz.

So the RIN from shot noise is 1e-7. So we shoot for a RIN of 3e-8 from the LED.


The AD587 voltage reference has a relative noise of 1e-7 at 0.1 under very good conditions (perhaps our vacuum system will be so kind). So we have to get a factor of 3 filtration at 0.1 Hz.

The following circuit should it for us: its a 2nd order Butterworth implemented in a Sallen-Key configuration. The noise is reasonable and the cutoff frequency is so low (0.03 Hz) because of the latest in capacitor technology.

We can buy metal poly caps which are as large as 47uF and have a reasonable physical size and tolerance and noise.

On page 2 of the plot you can see that the noise performance of this filter is limited by the input voltage noise of the FET opamp (op1) (AD743 - soon to be obsolete). The noise of the BUF634 (op2) is insignificant in this configuration. What we really need to make this good is a part with just as good of an input current noise spec as the AD743 but 3x less voltage noise at 0.1 Hz. I offer one cookie to whomever can find an opamp that fits those parameters.

These images show the circuit diagram (left) and the proto setup (right):


update: added a 230 Ohm series resistor between the BUF634 output and the LED to step the voltage down to the 1.7V that the LED wants.

  131   Thu Nov 26 15:17:01 2009 KojiElectronicsSUSLED Driver circuit

I found a Quad Opamp OP497 (neither dual nor single!), but this is not enough to expel AD743.
Dis-continuation of OP497 is also close except for one SMD package.

AD743 (reference):
LF Voltage Noise: 0.38 V pp, 0.1 Hz to 10 Hz
Voltage Noise: 2.9 nV/√Hz @ 10 kHz
Current Noise: 6.9 fA/√Hz @ 1 kHz

LF Voltage Noise: 0.3 V pp, 0.1 Hz to 10 Hz
Voltage Noise: 15 nV/√Hz @ 1kHz
Current Noise: 5 fA/√Hz @ 1kHz


What we really need to make this good is a part with just as good of an input current noise spec as the AD743 but 3x less voltage noise at 0.1 Hz. I offer one cookie to whomever can find an opamp that fits those parameters.

  132   Mon Nov 30 18:53:28 2009 ZachElectronicsSUSLED Driver circuit

Not having much luck. I found the LT1028, which has 10x better low-frequency voltage noise (35 nVpp, 0.1 Hz to 10 Hz), but its current noise is worse by a ridiculous factor of 1000:

Screen_shot_2009-11-30_at_6.51.02_PM.png           Screen_shot_2009-11-30_at_6.51.16_PM.png




I found a Quad Opamp OP497 (neither dual nor single!), but this is not enough to expel AD743.
Dis-continuation of OP497 is also close except for one SMD package.

AD743 (reference):
LF Voltage Noise: 0.38 V pp, 0.1 Hz to 10 Hz
Voltage Noise: 2.9 nV/√Hz @ 10 kHz
Current Noise: 6.9 fA/√Hz @ 1 kHz

LF Voltage Noise: 0.3 V pp, 0.1 Hz to 10 Hz
Voltage Noise: 15 nV/√Hz @ 1kHz
Current Noise: 5 fA/√Hz @ 1kHz


What we really need to make this good is a part with just as good of an input current noise spec as the AD743 but 3x less voltage noise at 0.1 Hz. I offer one cookie to whomever can find an opamp that fits those parameters.


  133   Tue Dec 1 00:49:19 2009 KojiElectronicsSUSLED Driver circuit

The situation is well illustrated in the following application note of Analog Devices:

Low Noise Amplifier Selection Guide for Optimal Noise Performance

Even though the graph is created for 1kHz, it is very clear that AD743
has superb performance for high source impedance purposes:
combination of low current noise and low voltage noise.

If the source impedance of Rana's circuit (400kOhm@DC) are reduced to 20K or so,
OP-27 type OPamp can come into the scope. However this means we need to use 10 times
larger capacitors. This is almost impossible for now, though innovation on the caps
can change the situation.

LT1028 is an AD797 type opamp. This works greatly with the smaller source impedance.

  134   Wed Dec 2 18:29:07 2009 ZachMiscSUSLow-noise LED driver

 Yesterday, I rebuilt Rana's low-noise LED driver in the Bridge elab. It is based on a 2nd-order active lowpass filter (using the Sallen-Key topology). The schematic is shown below. The circuit is essentially the same as the one Rana posted a few days ago, only the R and C values are all around twice what they are in his schematic. This results in the same corner frequency of fc = 0.03 Hz.




I hooked it up and measured Vsk,out = 9.76 V. I then used it to drive a 50-ohm resistor, and measured VLED = 1.71 V, then measured the current to be ILED = 33.5 mA.

After ensuring that it was supplying the correct voltage, I hooked it up to the LED and took a spectrum of the voltage noise across it over the two frequency bands I have been using in previous posts. The following are comparison plots of the noise here and the noise with the simple RC filter used before, calibrated to displacement noise.



RA: although these plots have Displacement in the y-axis, they are NOT measurements of actual displacement noise. They are estimates for the contribution to the displacement noise made by the LED RIN based on measurements of the voltage noise across the LED.



Something is clearly wrong: not only is the new configuration worse at lower frequencies, but the rolloff seems to go as 1/f and not 1/f2. Investigating after dinner...


  135   Thu Dec 3 02:35:39 2009 ranaMiscSUSLow-noise LED driver


 OP27 current noise is too high - use AD743.

  136   Mon Dec 7 02:59:38 2009 ZachMiscSUSLED Driver noise (with AD743)

 I retook the measurement from my last post, this time using an AD743 in place of the OP27 (per Rana's comment). The results are below.



RA: Although it says m/rHz, this is not measured displacement noise, but rather estimated displacement noise due to the LED noise. The previously measured conversion from the LED RIN to apparent displacement is used to convert from the voltage noise of the LED driver to the contribution to the OSEM's displacement readout.



Seems better than before, but not quite what expected. I observed that the transfer function of the S-K filter was what it should be up until a decade or two above the corner frequency, after which it appeared to spring zeroes out of nowhere and level off at high frequency. I tried to see what would happen if I changed the resistor values, and the following plot is what I got.


This plot seems familiar from my electronics courses, but I haven't put a finger on what is causing this behavior yet. I'm sure that the answer is somewhere in H&H (or in the brain of a kind soul who happens to be reading this--wink wink).

  137   Mon Dec 7 12:01:14 2009 ranaMiscSUSAOSEM LED Driver noise (with AD743)

The low frequency noise looks pretty good now. The funny shape is most likely a thermal transient due to having not enough insulation. You need to droop some Kleenex over the circuit to stop the thermal air currents and then put a second box over the first box. Then its probably best to sit outside of the room when taking the measurement to reduce the human noise.

  138   Tue Dec 8 10:07:39 2009 ZachElectronicsSUSSallen-Key filter attenuation limit

 Last night, I was going to retake the noise measurement with the added elements that Rana suggested, but instead I spent a ridiculous amount of time trying to figure out what is going on with my Sallen-Key filter. I now know a lot more about their limitations, but am still at a loss as to what is happening in this case. The problem is that no one seems to be using a corner frequency anywhere near this low (at least not anyone trying to explain these filters). The following is a comparison of an ideal 2nd-order Butterworth filter and a real one using an LT1464.


The filter behaves as expected a ways past the corner frequency, rolling off at 40 dB/decade. As the signal increases in frequency, the capacitors' impedances decrease, until (at point 'b'), they fall below the output impedance of the amp, causing the response to climb at 40 dB/decade. This happens for a short while, until the unity-gain bandwidth frequency (~ 1-10 MHz, 4 MHz for the AD743) of the op amp is reached, and the filter can attenuate no more ('c'), so the response flattens out to 0 dB/decade.

Different component values affect the high-frequency behavior of the filter, as shown below.



This makes sense, since with smaller capacitors it takes a higher frequency to fall below the output impedance of the amp. In any case, though, the final flatline always happens near the UGB frequency. The following is a plot of the transfer functions I measured (also in last post). I did not change the C values--only the R's--so the corner frequency is different in each case. What I observe is some R-dependent maximum attenuation, which sets on well before the 4-MHz UGB frequency of the AD743. The strangest part is that for small enough R this maximum "attenuation" is actually a positive gain.


I suppose it is not a huge deal, as I can increase the stopband attenuation as R while only increasing the Johnson noise as 2 * sqrt(R), but it would be nice to get some insight into what 's going on.

NOTE: While it appears that the high-frequency flatline in the other plots occurs for a different reason, it still seems to be R-dependent. I could not find any explanation for what determines this asymptotic behavior.

  139   Tue Dec 15 01:48:22 2009 ZachMiscSUSAOSEM noise measurement

 I tracked down some more AD743s at Wilson House 2.0 today (thanks to Rich). I was then able to simultaneously use 743s to both drive the LED and amplify the readout. Below is an 800-line DC - 25 Hz noise spectrum of

- The voltage across the LED (DC level: VLED = 1.59 V) -- AC coupled

- The output of the amp with a fully lit PD (DC level: Vout,full = -2.95 V) -- DC coupled, through bandpass filter

- The output of the amp with the LED out -- DC coupled, through bandpass filter

- The output of the amp with an open input -- DC coupled, through bandpass filter

all calibrated to equivalent displacement noise. For the LED plot, this was done by using the measured current ratio between the LED and the PD when fully lit along with the measured OSEM response of 0.05 A/m. The other three were converted using this response along with the transimpedance gain of the amp (100,000 V/A). For all measurements, the OSEM was covered by a box, and the circuit was draped with a cloth and put under a box within another box (to reduce air currents).


The funny low-frequency junk from the previous driver spectrum is gone--thanks to the isolation from air currents--but the line seems a bit higher overall (trying to figure out why). There also still seems to be the funny effect of noise added by the LED above the level of the voltage noise across it, and I think it's somewhat strange that the "Dark" noise is lower than the amp noise with no input. We can probably still do better..

  140   Tue Dec 15 09:28:39 2009 ranaMiscSUSPD front end noise

This is the LISO estimate of the PD front end noise. It could be improved somewhat by using a higher value resistor, but there's no point.

The shot noise level for the 20-40 uA of current we have is more than 1 pA/rHz so we should be OK above 30 mHz.

So even the level of the dark noise below seems too high and also the 10,000 V/A statement. The feedback resistor we had used to be 100k...

  141   Tue Dec 15 10:58:56 2009 ZachMiscSUSPD front end noise

10k was a typo--fixed.


This is the LISO estimate of the PD front end noise. It could be improved somewhat by using a higher value resistor, but there's no point.

The shot noise level for the 20-40 uA of current we have is more than 1 pA/rHz so we should be OK above 30 mHz.

So even the level of the dark noise below seems too high and also the 10,000 V/A statement. The feedback resistor we had used to be 100k...


  142   Thu Dec 17 10:50:16 2009 ZachMiscSUSAOSEM noise measurement

 I retook the measurement from the previous post, since the noise in the amp appeared much higher than it should. It looks much better now, but still not great. Above around 3 Hz, the amp noise is at the LISO-predicted level of ~8e-12 m/rHz equivalent displacement noise. Below this, it seems to show some 1/f-ish noise back to DC. LISO predicts some low-frequency noise as well, due to the increase in the AD743's current and voltage noise there, but it doesn't begin until below 1 Hz, and it doesn't seem quite as steep. I'm trying to figure out what is causing this, but the best solution might be to move to a more solid, soldered arrangement.
LED: Voltage across the LED (DC value 1.61 V), AC coupled
Bright: Noise at output of amp, LED on and PD connected, DC coupled with bandpass
Dark: Noise at output of amp, LED off and PD connected, DC coupled with bandpass
Amp: Noise at output of amp, PD disconnected, AC coupled (I checked this DC coupled and through the bandpass, as well, but there is no significant difference)
Amp (theoretical): LISO-predicted noise at the output of the amp with no input


  143   Fri Dec 18 11:56:12 2009 ZachMiscSUSNew AOSEM LED problem

 I picked up the other AOSEM from the 40m today, so that I could compare it with the one I've measured in an attempt to get to the bottom of the noisy LED problem. It began uneventfully: I measured the impedance of the LED by connecting an ammeter in series and slowly increasing the voltage. I got ILED = 36 mA at about VLED = 1.7, giving ZLED ~ 47 ohms.

Then, I powered up the LED driver, and tested it with a 50-ohm resistor (as usual), measuring V ~ 1.7 across it. Having confirmed that everything was working properly, I hooked the LED up, and measured NO current. I hooked directly back up to the DC supply and found the same result. The thing appears to be blown, but I have no idea how. I went through every precaution I have been taking with the other OSEM, which worked fine when I switched it back in. Crap.

One thing I noticed before I hooked anything up was that the small white pieces attached to the LED and PD on either side of the OSEM opening were very loose when compared to the other OSEM. When I first measured no current, I tried applying some pressure to the LED side, and some current flowed across, but only about 1/10 of what it should have been.

  144   Tue Jan 5 21:14:46 2010 ZachElectronicsGeneralWaitDone Error ??

 The SR785 I am using for the AOSEM noise measurements (the one that was in the TCS lab) doesn't seem to want to boot up all the way. After sitting at the "Backup OK" screen for ~30 secs, a dialog box pops up reading "WaitDone Error", after which the machine reboots. This continues forever.

I remember this having happened when I first liberated it from the TCS lab a few weeks ago. I turned the thing off for a while and it eventually worked just fine. I tried the same thing now and it didn't work. I am going to give it another go in the morning.

Has anyone experienced this error before?

  145   Wed Jan 6 14:43:38 2010 ZachElectronicsGeneralWaitDone Error ??

Same thing happened when I went in this morning. I left it for a while (on this time) and it seems to be working. We might want to have this machine looked at.


 The SR785 I am using for the AOSEM noise measurements (the one that was in the TCS lab) doesn't seem to want to boot up all the way. After sitting at the "Backup OK" screen for ~30 secs, a dialog box pops up reading "WaitDone Error", after which the machine reboots. This continues forever.

I remember this having happened when I first liberated it from the TCS lab a few weeks ago. I turned the thing off for a while and it eventually worked just fine. I tried the same thing now and it didn't work. I am going to give it another go in the morning.

Has anyone experienced this error before?


  146   Wed Jan 13 14:13:03 2010 ZachMiscSUSASOSEM comparison

EDIT: I have calibrated the y axis of the plot to meters

Last night, I got around to testing some of the other AOSEM samples, to see how the noise varied between them. What I found was rather strange: the noise in all the new ones (#s 2-6) was about the same, but they were all quite a bit noisier than the previous one I have been testing (#1). The only difference between them, as far as I can tell, is that the first specimen has a coil wound around it already, while the others just have a rubber band. Also, the newer ones all have an impedance of ~ 44-45 ohms, while I measured 47 ohms for the first (though, among the new ones, the slight variation in Z seems to have no correlation with the small differences in noise level). For those wondering, YES, I did remeasure the noise in the 1st one; I am not using old data.

Either my meddling with the old one has somehow made it quieter or something is amiss.


  147   Tue Jan 26 22:22:17 2010 ZachMiscSUSAOSEM LED, PD current comparison

Norna and Rich: I am sorry for taking so long to get you this measurement. I plan to do the noise measurements on the standalone LEDs this week.

The following table gives the current through each OSEM's LED (measured using the voltage drop across the 238-ohm resistor in series), as well as the measured photocurrent (the DC output of the amplifier divided by its transimpedance gain of 100,000 V/A), and the ratio of the two. The plot from the previous post is reproduced below for analysis--I realized that I did still have this plot saved in units of V/rHz. In some cases (e.g. #3, #4), the noise level seems to be correlated to the photocurrent, but not all of them follow this pattern. The issue of #1 being significantly quieter than the other set remains, as well.

AOSEM LED and PD Current Comparison
1 35.1 31.4 1120
2 35.5 38.5 923
3 35.5 57.6 616
4 35.4 30.5 1160
5 35.4 42.9 826
6 35.5 48.2 737
7 35.5 39.2 906
8 35.4 39.8


9 35.2 34.0 1040
10 35.5 43.8 810




  148   Fri Feb 12 11:33:04 2010 MottMiscSUSState of the Shaker

The Shaker project is coming along nicely.  I am currently looking into using the built-in ability to download a waveform to the front end to do the sweeps, but we are running into memory problems, and I get the sense from Tony that it was not really designed to do this.  Currently we are able to download a waveform to the frontend, run the generator according to it, and make a measurement over a full run of the waveform.  If we can crack the limited time constraint and figure out the averaging, this is going to be the most straightforward solution.

I am working, in parallel, with Gert (who is out of the office at the moment) on using pure script to do this, although I am worried about starting and stopping the generator so frequently.  Apart from anything else, there is a slight hang in the frontend when the generator start method is called; it is not noticeable when the button is pushed in the app, but I think it adds quite a bit of latency to the program.  I am still waiting to hear from Gert about how to acquire a time series; hopefully we can figure that out by early next week, since it is critical.  I am also not entirely sure how we force the program to do all the analysis on the time series after it is acquired.  Ideally we would want the analysis to run in parallel and update the frontend continuously, but I am not sure this is possible with VBA (I don't think you can do multithreaded programming) and I am not sure I would know how to do so even if it is!



  149   Fri Feb 12 11:35:19 2010 MottMiscSUSPiezos

Engineering was very helpful showing me how to make the leads we need for the piezos; I will go crimp some more at the beginning of next week. 

The new structures should be coming in soon, so we will have a dedicated structure for the piezo damping, at which point we can really get cracking.

  150   Thu Feb 18 17:43:15 2010 MottMiscSUSPiezos


I finished crimping all the connectors we will need for the piezos.  We are now just waiting for the new structures to arrive so we can start gluing the piezos on.

  151   Sat Jun 26 13:59:57 2010 Vladimir DergachevMiscSUSovernight tiltmeter plots
And here are the plots from overnight run:



* I picked a nice drift segment out of the whole run which showed some
junks in the beginning and a few near the end, possibly caused by external

* The drift with oscillations is still there. It is likely they are


* The best spectrum comes from lvdt2. It is likely that LVDT1 receives
extra noise from clamping zeners in the fine channel of its preamplifier.


* The full spectrum reaches the limit of the coarse channel only at
high frequencies:


The fine channel of LVDT2 dips a little bit lower, which is easier to
see on linear X scale:


The electronics operates around 6.6 kHz and the conversion to DC is
done digitally. Thus we should see flat noise floor from it, except for
the effect of voltage references which are used both in ADCs and in the
triangle driver and any other noise source that affects the amplitude
(such as a current setting resistor in the triangle driver).

Riccardo - I think it would help to isolate the effects of mechanics
noise from driving electronics if we had a test fixture for LVDTs.
Something like a U bracket for the excitation part of the LVDT and a
screw-on cover for the pickup coil. It would be nice to have a choice
between a plain cover, a cover with a slot and a PEEK cover.

This essentially follows the suggestion Eric made at the last meeting,
except I would avoid usage of all-analog readout as I am not confident I
can debug it easily. We can still do it as a confirmation once we know
what the baseline curves are from our current system.
  152   Wed Jul 7 00:14:55 2010 Vladimir DergachevNoise HuntingSUS2e-9 rad/sqrt(Hz)
Here is a fresh plot:


Changes: reduced noise in preamps, reduced gain in fine channels so that
they just reach 1e-10 rad/sqrt(Hz) - this produces large overlap without
having to trim LVDTs.

We are now under 2e-9 rad/sqrt(Hz), but there is still work to be done as
can be clearly seen from the correllation plot:


Or from overlay of coarse and fine channels:


The negative correllation values are clearly electronic (likely excitation
coil driver).

Some of it subtracts:


but there is a lot of room until we reach the pink line (which is the
limit of readout electronics for fine channels).
  153   Thu Jul 8 16:41:29 2010 JanMiscSeismometryRanger Pics

This Ranger is now lying in pieces in West Bridge:



First we removed the lid. You can see some isolation cylinder around the central metal 
part. This cylinder can be taken out of the dark metal enclosure together with the interior 
of the Ranger.


Magnets are fastened all around the isolation cylinder. One of the magnets was missing 
(purposefully?). The magnets are oriented such that a repelling force is formed between 
these magnets and the suspended mass. The purpose of the magnets is to decrease the 
resonance frequency of the suspended mass.


Next, we opened the bottom of the cylinder. You can now see the suspended mass. 
On some of the following pictures you can also find a copper ring (flexure) that was 
partially screwed to the mass and partially to the cylinder. Another flexure ring is 
screwed to the top of the test mass. I guess that the rings are used to fix the horizontal 
position of the mass without producing a significant force in vertical direction. The 
bottom part also has the calibration coil.


Desoldering the wires from the calibration coil, we could completely remove the mass 
from the isolation cylinder. We then found how the mass is suspended, the readout 
coil, etc.:

DSC02509.JPG DSC02513.JPG

  154   Sat Jul 17 13:41:45 2010 JanMiscSeismometryRanger

Just wanted to mention that the Ranger is reassembled. It was straight-forward except for the fact that the Ranger did not work when we put the pieces together the first time. The last (important) screws that you turn fasten the copper rings to the mass (at bottom and top). We observed a nice oscillation of the mass around equilibrium, but only before the very last screw was fixed. Since the copper rings are for horizontal alignment of the mass, I guess what happens is that the mass drifts a little towards the walls of the Ranger while turning the screws. Eventually the mass touches the walls. You can fix this problem since the two copper rings are not perfectly identical in shape, and/or they are not perfectly circular. So I just changed the orientation of one copper ring and the mass kept oscillating nicely when all screws were fastened.

  156   Sat Jul 17 22:35:16 2010 Vladimir DergachevNoise HuntingSUSStandalone LVDT measurements
I have finally assembled the bracket to hold a standalone LVDT.
All plots are at:


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.
  155   Sat Jul 17 22:35:16 2010 Vladimir DergachevNoise HuntingSUSStandalone LVDT measurements
  157   Tue Jul 27 00:22:04 2010 JanMiscSeismometryRanger

The Ranger is in West Bridge again. This time we will keep it as long as it takes to add capacitive sensors to it.

  158   Mon Aug 23 22:07:39 2010 JenneThings to BuySeismometryBoxes for Seismometer Breakout Boxes

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. 

  159   Wed Sep 1 22:49:36 2010 Vladimir DergachevMiscSUSLatest tiltmeter spectrum
Latest tiltmeter sensitivity plot.

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.
  160   Thu Sep 2 13:27:43 2010 Vladimir DergachevMiscSUShysteresis measurements

There is no evidence of hysteresis in the latest measurements. The plot below zooms in to the few cycles in the middle of the upper plot.

The thick black areas are created by the proper mode of the tiltmeter and air currents.

  161   Tue Oct 12 10:51:36 2010 AlastairMiscBladesUseful document

 We're going to use this elog to store some of our lab work on blades that will be going on in the SUS lab.

As a first entry here is a useful document on the ALIGO blade design on the DCC:  LIGO T030107  by M.V.Plissi

  162   Tue Nov 2 15:23:33 2010 AlastairLab InfrastructureGeneralMain turbo back on

 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.

  163   Tue Dec 21 06:58:09 2010 ranaThings to BuySeismometryTrillium Noise Plot

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.


  164   Fri Dec 31 15:23:50 2010 ranaMiscCreakNew Creak doc

I have updated the Creak doc (T0900167) in the DCC.

To get started, I think I can just disassemble the long range, polarization using Michelson project from the SURF project.

For the first version with blades, I'm going to just use some shim stock from McMaster: I'll try aluminum and carbon steel since they should creak.

  165   Tue Jan 4 23:29:22 2011 ranaMiscCreakStarted Creak-MICH

Zach, Rana

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.

  166   Thu Jan 6 01:03:24 2011 ZachDailyProgressCreakPZT installed/tested

  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.
  167   Thu Jan 6 21:58:12 2011 ZachDailyProgressCreakSecond mirror/PZT installed, rough CMRR measured

 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:

2011-01-06_18.52.51.jpg 2011-01-06_18.52.33.jpg

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 +5Vand 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.




  168   Fri Jan 7 20:14:39 2011 ZachDailyProgressCreakREFL installed

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.


  169   Wed Jan 19 17:13:11 2011 JanComputingSeismometryFirst steps towards full NN Wiener-filter simulator

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.

  170   Thu Jan 20 16:05:14 2011 JanComputingSeismometryWiener v. seismometer 0

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)

  171   Fri Jan 21 12:43:17 2011 JanComputingSeismometrycleaned up code and new results

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.

  172   Sat Jan 22 20:19:51 2011 JanComputingSeismometrySpatial spectra

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.

  173   Sun Jan 23 09:03:43 2011 JanComputingSeismometryphase offset NN<->xi

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:



  174   Sun Jan 23 10:27:07 2011 JanComputingSeismometryspiral v. random

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.

  175   Sun Jan 23 12:59:18 2011 JanComputingSeismometryLess seismometers, less subtraction?

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.


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