Our response to the referees/CQG:
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Dear Ms. Tapp,

We are pleased to know that you have made an initial decision on this submission. We have made some changes to the paper to address the referee comments and we think they have improved the clarity of the paper for the reader.

Below, please find detailed responses to referee comments as well as a complete list of the changes we have made.

Best regards,
Darkhan Tuyenbayev

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Responses to Referee 1:

The manuscript on "Improving LIGO calibration accuracy..." described the calibration model adopted for LIGO detectors and deals with an important issue of temporal dependence. The authors show that, if it is not taken into account, the uncertainty in the calibration could be rather significant. The error in the calibration translates directly into systematic bias in the parameter estimation of the GW signals.
This work is very important and it should be published. In general I have found that the article is well written and quite clear, however I  still have few questions on the presentation and on the results themselves. It would be nice to see the answers before the manuscript is published (as it may improve the clarity).

1. End of the page 3, beginning of the page 4. The parameters kc, fc are introduced and explained twice.

-- The introduction of kappa_c and f_c, before equation (3), was removed.

2. Last paragraph on the page 4. How the "estimated systematic errors" presented in the figures 2,3 were obtained. What did you compare it against?

-- The second sentence in the paragraph that starts with “Not compensating for variations in the DARM control loop parameters can…” was expanded to clarify how the systematic errors are calculated.

3. Equation 8. Two questions: (i) it would be nice to see explicitly the time dependence on the r.h.s.  (ii) which fpcal was used there 1 or 2 or both? (I presume fpcal1)

-- (i) A sentence was added to let the reader know that all of the values that have a subscript “0” are calculated at the reference time. Time-dependences were explicitly added to the sensing and actuation function definitions at the beginning of section 3 and propagated to the equations that follow. The subscript “t” was added to d_err to indicate that the frequency domain value for this signal is calculated at time “t”.

-- (ii) That is correct, for this parameter we use fpcal1. The number “1” was added to the subscript in the equation.

4. Equation 10, again it would be nice to see explicitly the time dependence on the r.h.s.

-- Similarly to item 3 (ii), explicit time dependences were added to the sensing and actuation function definitions and to the Fourier transforms of the signals (subscript “t”).

5. End of the page 7. "The frequency band around 35 Hz..." was it optimized? meaning was there an investigation on the optimal central frequency and spacing between frequencies of the injected signals?

-- The frequency band was chosen from rough estimates based on transfer function measurements of the actuation stages and the overall DARM loop. Currently, we are working on quantifying the choice of frequencies. We hope to use the results of this ongoing study to adjust the line frequencies in future LIGO observing runs.

6. General question: I presume that the calibration lines (signals) are not injected continuously. If not, what was the cadence and duration of each injection?

-- The calibration lines are injected continuously, and the calculated time-dependent parameter values are used in reconstruction of Delta L_ext. In the analyses looking for gravitational wave signals these lines were notched out.

7. General question: would it help to inject more than 4 lines to have redundancy and to check consistency of your results (or to have more calibration points in the overall frequency band).

-- Additional lines at higher frequencies (1 kHz and 3 kHz) were injected using Photon Calibrator systems; they have been used for consistency checks and for assessing high-frequency calibration errors.

8. Figure 7. The negative %, do they mean that the model underestimates the magnitude and phase?

-- Negative % means that the measured transfer function has smaller magnitude compared with the model at the reference time. In this case the reference time model overestimates the magnitude. To improve the clarity, the second to last paragraph of the “Results” section was modified.

9. General question: Nothing is said about the magnitude of the injected signals. It would be nice to have few lines which discuss that.

-- The magnitudes of the excitations were chosen to give a signal-to-noise ratio of 100 with 10-second FFTs. A sentence was added to the 2nd paragraph of section 4, “Results,” to clarify this point.


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Responses to Referee 2:

COMMENTS TO THE AUTHOR(S)
Overall: This kind of detector calibration described here has certainly been done more or less in the same way for prototype interferometers and first generation detectors, although the procedure is more complex because of the quadruple pendulum.  Still, this paper is informative since the detector calibration and its error are of great interests to general physicists and astrophysicists. Interferometer experts would be more interested in the effect of noise in each system (noise propagation), which is mostly neglected in the argument.

Details:
1.       Eq. (2): Delta is missing. Also authors should mention and clarify that the final product is time-domain signal. Make the argument consistent with Eq. (14) --- readers would think this equation is NOT used to reconstruct the final signal and get confused.

-- The missing delta was added to equation (2).

2.       Page 4, 2nd sentence: What is the main cause of the alignment drift? Thermally driven mechanical drift and/or electrical drift? General readers (especially astronomers) would wonder why the detector drifts in spite of the many control loops, requiring continuous calibration. Also explain the mechanism that changes the cavity pole frequency.

-- An explanation was added after equation (3) concerning thermal effects and external lab conditions contributing to changes in the coupled cavity pole frequency.

3.       Page 4, near the last sentence: give the order of fc (~360Hz) here for easier understanding.

-- The approximate coupled-cavity pole frequency was added to the text, as suggested.

4.       Figure 2,3,4: I understand these calculations do not include noise --- for G>>1, the feedback signal should give higher accuracy signal and the figure should look different??

-- Correct, figures 2-4 show only systematic errors due to changes in calibration model parameters and doesn't not include statistical uncertainties arising from noise. It is true that changes in the optical gain and coupled-cavity response affect the systematic errors to a lesser degree below the unity gain frequency. This can be seen in figures 2 and 3. At the same time, if the actuation function has changed and the change is not accounted for, then the errors will mostly appear at lower frequencies, as seen in figure 4. We hope that the text surrounding figures 2-4 explains this clearly enough.
    Note that the final calibration uncertainty (discussed in citation [5]) is reported as quadrature sum of the above systematic error and any statistical uncertainties estimated.

5.       Eq.(8): Explain the deviation of this equation. Define C0 and G0 clearly.

-- A sentence was added above equation (8) explaining how it was derived. Definitions for C_0 and G_0 were added and a sentence was inserted earlier in the paper noting that the subscript “0” denotes a quantity calculated at the reference time, t_0.

6.       Eq. (14): I do not understand why d_ctrl has to appear here. It would be smarter to use D and to make things consistent with the earlier discussions (where only d_err appears).

-- Equation (14) was modified to be consistent with equations (1) and (2), i.e. d_ctrl was replaced in favor of d_err. The text surrounding equation (14) was modified and a paragraph was added to explain how compensating for only scalar factors can be achieved without updating time-domain filters, but compensating for all of the time-varying parameters requires continuously updating the P_i(t) time-domain filter.

7.       Figure 7 and corresponding main text: Better explanations are needed. The “models are generated from” “measurements”. And “the errors” “are estimated by calculating the ratios between measurements and the models”. I did not understand what is being done here.

-- The purpose of this paper is to study how the model, that is parametrized in terms of time-varying factors, changes when one or more of the factors deviate from their reference-time values. Thus, the plots show ratios between the reference-time model and the model including changes in the factors.
    To make this more clear to the reader we have modified the figure caption and the second to last paragraph on page 9 (in the revised submission), noting that the models are based on the measurements taken at the reference time.

8.       Figure 8 and corresponding main text: Use consistent terminology --- magnitude “variation” or “error” (relative?)?

-- The figure caption was updated for consistency and the text referencing figure (8) was modified for clarity.