Scale Factor Determination for the GRACE Follow-On Laser Ranging Interferometer Including Thermal Coupling

Round 1

Reviewer 1 Report

This paper proposes a new calibration method for the LRI data of GRACE FO. Different from the calibration method adopted by the official institutions, this paper introduces the thermal coupled noise into the calibration model of scale factor and clock error. A linear function model is established to describe the relationship between laser frequency and temperature. The empirical parameters in the function model are determined by the measured data and the least square fitting in the ground test. I have only few concerns as follows.

 

1.     In this paper, a physical model of laser frequency variation with temperature is proposed. In order to reduce the dependence of KBR data during LRI calibration, in-orbit estimation of empirical parameters in the model must be realized. How to achieve this?

 

2.     The manuscript directly present the equation 34. How can we obtain this equation? By the way, what is the purpose of introducing a high-pass filter here?

Author Response

The authors thank the reviewer for her/his positive feedback.

1. We are not sure if this question refers to future missions or to the current GRACE-FO. In the current GRACE-FO, we derived the in-orbit empirical parameters from the difference between KBR and LRI ranging data. Four different types of empirical parameters can be distinguished and were estimated in this paper : A) a drift of the RLU frequency (for fixed RLU temperature setpoint) or equivalently a RLU setpoint drift for a fixed RLU laser frequency B) step-like changes in the RLU frequency C) an exponential decay of the cavity frequency likely from a change in the spacer D) temperature-related coupling factors that describe actually tone errors and not the laser frequency.
    For GRACE-FO, in-orbit  estimation of empirical parameters can only be done with the KBR as reference. In future missions, with laser ranging only, one might calibrate the empirical parameters (in particular of type A and C)  thoroughly on ground.
    Although we provide a model (telemetry-based laser frequency) that uses temperature and laser setpoints to estimate the laser frequency, it is not our recommended way for computing the laser frequency - because the laser unit has not the highest thermal stability and the model is falsified by temperature variations on the satellite. We recommend to use a model for the cavity frequency (exponential cavity decay model), because the cavity is well-shielded.
    
2. Equation 34 follows from eq. 13 in section 3, where we showed that an error in the scale (or frequency) couples with the absolute inter-satellite distance L. The second term of eq. 34 is a linearization of the non-linear time dependency. The high-pass filter is to remove long-term drifts. We modified the paragraph to make the derivation/definition more clear.

Reviewer 2 Report

This is an excellent paper that addresses a very important aspect of processing Earth gravity field data from the GRACE-FO mission as well as the instrument design and calibration of the laser interferometer and its dependence on the microwave data. The authors seem to have expert-level knowledge and insight.  The writing is of impressive quality and clarity that is rare to see in engineering and scientific papers.  The modeling and analysis seem very sound and thorough and, despite its complexity, can be followed by a reader.

Although it is "average" on the Novelty (since it addresses an existing problem) and Interest to readers (since it speaks to a narrow community of detailed data analysts), is very high in its significance, quality, and overall merit and needs to be documented and available to the community.

Congratulations on an impressive paper that reflects the depth of your work.

(no additional files attached)

 

Author Response

Thank you very much for your very positive feedback!

Reviewer 3 Report

In the context of GRACE-FO, the article presents the characterization of the scale factor of the Laser ranging interferometer instrument (LRI) as well as the description of the thermal influence on this parameter. The authors give a very good overview of the measurement principle of the LRI and derive the influence of the conversion factor between optical observable, e.g. the differential phase measurement of the heterodyne Mach-Zender type interferometer, and the range observable as well as the influence of the unmodeled internal delays of the LRI that provide a so-called timeshift. The conversion factor is the so-called Scale Factor. A very exhaustive and comprehensive characterization of the scale factor and timshift is then derived from different means of characterization and it is proposed in conclusion different ways of characterizing the scale factor without the needs of the MWI, present in GRACE-FO but which is not foreseen for future geodesy mission.

The reviewer strongly recommends the publication of the article, after minor comments presented in the following:

- Introduction: Please provide a brief description of the tone error (or relevant reference) and define 1/rev.

- Figure 2, and related paragraph: The tone requirement for the MWI needs to be better explain. Why this requirement, related to the MWI, should be higher than the effective errors arising from the scale factor error and the timeshift? Please justify the reason why the ENBW is 21.038e-6 Hz. On the figure the “equivalent” ASD of the MWI tone error requirement should not be an horizontal line, but a peak centered at 2.f_orb with a bandwidth of the ENBW, please update the figure.

- Part 10: The reviewer suggests inserting a table summarizing the different performances on scale factor estimation with the different models used to calculate the in-flight laser frequency in the paper, and the different techniques presented in this part (e.g. iodine spectroscopy, cavity resonance frequency measurement,…).

Author Response

The authors thank the reviewer for her/his positive feedback.

1. We added the following sentences to the introduction:
    These variations [of the thermal environment] have a strong sinusoidal component at the orbital frequency of f_{orb}\approx 0.18 mHz, often called 1/rev or 1 CPR, and at higher integer multiples of the orbital frequency (e.g. 2/rev = 2 CPR). The resulting errors from the sinusoidal variations are commonly called tone errors and may arise not only directly from temperature but also from local geomagnetic field, gravitational potential or thermo-elastic deformations, all being strongly modulated at 1 and 2 CPR.
2. We modified that section in the manuscript to better motivate the tone requirement. The KBR tone requirement is lower since we show LRI errors. For the LRI, the noise and tone requirements are more relaxed and apply only to higher frequencies. We updated the figure and corresponding text to make this clear.
    Regarding the ENBW: There is nothing specific about the 21 µHz, however, one has to know the ENBW if one wants to convert a a spectrum quantity with units of meter to m/sqrt(Hz), i.e. a spectral density, or vice versa. The ENBW is a property of the discrete fourier transform and depends on the window function, on the time-series length and on the number-of-averages. General information about  the ENBW can be found in the new reference [17].
3. We added a summarizing table, please find it in the updated manuscript.