J. Driggers, S. Dwyer, K. Kawabe, J. Kissel, L. Sun

Executive Summary: 
- The systematic error in the time-independent DELTAL_EXTERNAL is now within 2% between 20 and 1000 Hz (at the time of last measurement). See first image attachment.
- We have good agreement between pyDARM model an today's measurement, but what CAL-CS needs to get to the above good level of uncertainty does not match that model.
- The calculation for the time dependent correction factors is still untrustworthy, but they are not correcting any data stream.
- GDS-CALIB_STRAIN has not been updated with the latest changes, and should not, even though we have a new model parameter set for today.
- Still more work to do, but I expect the bulk of the work will be offline, and we likely won't need to change DELTAL_EXTERNAL anymore.
- We're in the home stretch.


Apologies in advance, today's status story about reducing the systematic error is going to be long.
You can follow along with the second and third attachments.

Going in to today, we had three systematic errors that we want to figure out and reduce:
    (1) The long-standing systematic error in the estimate of the PUM actuator strength, due to residual frequency response in measurements from length-to-angle (L2A) decoupling filters causing parasitic length through angle-to-length (A2L) gains (i.e. our current, far-from-center, ETMX spot position).
    (2) The systematic error in the sensing function, due to the newly resolved, but poorly fit pro-spring SRC detuning.
    (3) The non-integer clock cycle delay from the updated 2018-03-27 model.

In order to explore (3), to see whether it was worth installing a thiran filter -- and to confirm or refute claims that the difference between 7.0 and 7.5 clock-cycle delay contributes substantially to the overall DELTAL EXTERNAL systematic error -- we drove a broad band PCAL to DELTAL EXTERNAL transfer function with the CAL-CS calibration exactly as it was at the end of yesterday, but comparing a 7.0 clock cycle delay against a 8.0 clock cycle delay.
    We found that this had very little effect on the overall DELTAL EXTERNAL systematic error and thus is not a major contributor to the DELTAL EXTERNAL systematic error.

To solve (1), this morning's IFO instability forced our hand to turn off a PUM boost in the L2 LOCK L bank (i.e. change the DARM loop), while simultaneously turning off the L2A filters. Sheila documents her thoughts on this spurious coupling and what motivated the change this very well in LHO aLOG 47982.
After this change, we remeasured the PUM actuation function (i.e. the Length-to-Length) transfer function, and indeed found that the spurious cross-coupling problem is now gone.
As such, we fit this new data and found that the previously installed (2018-03-27) actuation strength was 1.4% too high. 

    2018-03-27 PUM strength gain estimate: 6.118e-10 N/ct

    2018-03-28 PUM strength gain estimate: 6.03e-10 N/ct
    Actuator gain, H_c (N/ct)              | 6.03e-10 (+3.441e-13,-3.433e-13) or (+0.05706%,-0.05694%)
    Residual time delay, tau_A (usec)      | 28.12 (+2.201,-2.191) or (+7.826%,-7.792%)

This fit shown in the first .pdf attachment.

So -- we updated the PUM coefficient in "Npct_O3" module of the H1:CAL-CS_DARM_ANALOG_ETMX_L2 filter bank -- i.e. the PUM path contributing to the front-end actuator model that creates DELTAL CTRL.

However, while this *changed* the DELTAL EXTERNAL systematic error, it did not fix it.

So, we started exploring (2), the systematic error in the sensing function.
(a) We took a new measurement of the sensing function, and for what ever reason, the results of the MCMC fit better matched the detuning spring frequency, leaving much less residual. We'll call this the 2019-03-28 model for the sensing function. 

    Optical gain, H_c (ct/m)                 | 3.293e+06 (+3432,-3462) or (+0.1042%,-0.1052%)
    Optical gain, H_c (mA/pm)                | 4.396 (+0.004582,-0.004623) or (+0.1042%,-0.1052%)
    Cavity pole, f_cc (Hz)                   | 405.9 (+1.127,-1.113) or (+0.2775%,-0.2743%)
    Detuned SRC spring frequency, f_s (Hz)   | 5.62 (+0.05135,-0.05228) or (+0.9136%,-0.9302%)
    Detuned SRC spring quality factor, Q_s   |  5.2 (+136.5,-133.9) or (+3.809%,-3.884%)
    Residual time delay, tau_c (usec)        | -2.071 (+0.7531,-0.742) or (+-36.37%,--35.83%)

This fit is shown in the second .pdf attachment.

With this info, and the updated PUM actuator strength, we can then created an updated DARM model parameter file,
    /ligo/svncommon/CalSVN/aligocalibration/trunk/Runs/O3/H1/params
        modelparams_H1_20190328.py
With that parameter file, we could then process the DARM open loop gain and predict the new number of clock cycle delays between actuation and sensing paths. 

These plots and predictions are in the third .pdf attachment.

(b) In parallel, we also began adjusting the overall gain of the existing, installed, 2019-03-27, inverse sensing function because it was obvious from the conclusion of (3) that something was wrong with the sensing function at high frequency. Via trial and error with broadband PCAL2DELTAL transfer functions as our metric, we found that we could get the smallest DELTAL systematic error with the 2019-03-27 inverse sensing function multiplied by 1.04, i.e. a 4% increase in inverse sensing, or 4 decrease in optical gain of the sensing function.
We think we narrowed down the explanation for this with a flaw in the logic of how we normalized yesterday's SRC_D2N filter: 
    - We were confused that pyDARM "normalized the suggested Foton filter at 500 Hz," and instead used Foton to normalize "where the SRC_D2N transfer function 'should be flat,' at 100 Hz."
    - Upon further investigation, the installed 2019-03-27 SRC_D2N filter was never flat, so we accrued a self-inflicted 4% error by normalizing to a frequency that was not representative of the true optical gain.

(c) This all left the high-frequency end of the DELTAL systematic error in excellent shape -- but below 40 Hz, there was still a hefty 4% hump. By this time, also in parallel, Keita had put together a Thiran filter that approximated the 7.5 clock cycled delay -- see his description in LHO aLOG 48008. So I tried switching from the 7.0 integer clock cycle delay to the Thiran filter delay of 7.5, expecting to see very little impact. Confusingly, the thiran filter caused more systematic error than flip flopping between 7.0 and 8.0 integer clock cycles. So, I reverted back to using the integer clock cycles.

Then we went back to (a) -- now that we understood how to properly normalize the SRC_D2N filter, and had a better fit to the pro-spring detuning, we then proceeded to update the inverse sensing function filter.  Note, that at this point, with the updated PUM gain coefficient, and the better spring detuning model, the predicted clock delay between A and C was to be 7.75 cycles -- which we deemed close enough to 8.0 clock cycles.

So We installed the 2019-03-28 SRC_D2N and Gain filters in the H1:CAL-CS_DARM_ERR bank, in FMs 7 and 8, under O3B_D2N and O3BGain, reverted the hand tuned gain to 1.0, and changed the relative path delay to 8.0 cycles, and measured the broad band PCAL2DELTAL transfer function, expecting complete, self-consistent, success.

This didn't work, and we don't yet understand why. 

So, we reverted to the 2019-03-27 inverse sensing filter, with the 4% gain adjustment and 7 clock cycles, with the all stage actuators gains matching the model, but this left a clear 4%, flat across the entire band systematic error, instead of a direct reproduction of what we had earlier. Sheesh!

Tired and exhausted at the inability to install something that was self-consistent, and needing to document our work today, I went with a quick fix of reducing all paths by 4%. 
Thus, we left CALCS DELTAL EXTERNAL with the best systematic error we were able to achieve today:
    - The 2019-03-27 inverse sensing function filter, with a gain of 1.00 (not 1.04)
    - The relative path delay at 7 clock cycles,
    - The PUM coefficient updated with today's measurement, and
    - All actuator gains scaled down by 4% (i.e. multiplied by 0.96).

We'll do out best to understand all this tomorrow.

Now that you know the story, if may be easier to follow along by looking at the second and third image attachments. The story provided an impressive display of how little (at least) I understand how each piece of the puzzle impacts the systematic error for DELTAL. 
But, in the end, RED curves in the broad-band measurements correspond to the state CAL-CS was in for the "final answer" swept sine of the first image attachment.

I'll attach comments with pointers to scripts that produced all this work in due time.