Attitude Determination for GRACE-FO: Reprocessing the Level-1A SC and IMU Data
Round 1
Reviewer 1 Report
Dear authors,
thank you for this study and the well presented results. I have two general points and some minor points which I like you to address in a revised version.
Kind regards, reviewer
General points:
What is the benefit of using the Kalman filter approach instead of using a least squares adjustment in this case?
I would have expected the differences between your results and JPL (cf. Figure 8 and discussion) to be smaller. One possible reason could also be the relative weighting of SC and IMU, especially for low frequencies. How are these two observation types weighted against each other, constant for all frequencies or frequency dependent?
Some minor points:
l21 (+l150): range-rate instead of "ranging-rate"
l37: I would propose "imperfect or mis-modelled background models"
l41f: Rather something like "We can not expect better gravity field mapping by current and future gravity missions unless..."
l47f: sensor data processing deficiency
l53: one of the most challenging tasks
Equation (3): Isn't it (H^T * H)^-1 instead of (H^T * H^-1) ?
l243: has to be
l262: lead
l284ff: does this last sentence of this section mean, that you do not combine the SC data as described in equations (8), (9) and (10)? If this is the case, why do you explain the combination here?
l313: delete one "the"
l339: complicated
l339: you often write "we don't repeat again". Why not simply refer to the reference and that's it.
l367: "with a precision of"
Figure 2: how do the spectra (sqrt(PSDs)) of (b), (c) and (d) look like? Is it close to white noise?
Figure 4 caption: spectra instead of spectrums; and power spectral density itsead of power spectrum density
l445: "aiming at analyzing"
Table 2: how often are the biases estimated, are they assumed to be constant over for example one day?
Figure 8 caption: two CPR might correspond to 0.8 hours, not to 3.2 hours. Please check and correct also in the corresponding text parts.
Figure 9: as the differences between JPL and your results are mainly periodic with 1 CPR frequency, a daily mean of this difference makes not much sense in my opinion. You could, for example, compute the daily mean absolute amplitude of this 1 CPR signal in the residual time series. Probably this measure would be more stable.
Figure 9: Probably I don't understand the way you compute the differences in Figure 9 compared to Figure 8. The red lines in Figure 8 (left) show nearly zero mean. Why do the mean values in Figure 9 do not show this?
Figure 10: you should compare the differences with noise specifications of MWI or LRI or ACC in this PSD-plot, or with the spectra of post-fit range-rate residuals after gravity field adjustment.
Author Response
Response to the Reviewer #1
thank you for this study and the well presented results. I have two general points and some minor points which I like you to address in a revised version.
Kind regards, reviewer
General points:
Comment 1: What is the benefit of using the Kalman filter approach instead of using a least squares adjustment in this case?
Response: Thanks very much for the insightful discussion. As this question is quite mathematical and professional, we would try the best to answer it using our limited mathematical knowledge and expect our answer may well resolve the concern of reviewer.
In the sense of Bayesian theory, the Kalman filter can be mathematically equivalent to a traditional least square adjustment given that no-priori information is introduced. However, as priori information is used in our Kalman filter configuration (using IMU to propagate the state equation, see Eq. (14) in the manuscript), this Kalman filter is not equivalent to a least square adjustment anymore. Strictly speaking, our Kalman filter is more comparable to the least square collocation rather than the traditional least square solution. And Kalman filter and the least square collocation are widely used for dealing with the problem of multiple data fusion in the time domain.
For the case of GRACE, we learn that both methods are once used for the attitude determination. For example, Klinger(2018) adopts the least-squares collocation method to achieve the attitude determination, whereas more work as far as the author know prefer the Kalman filter, such as JPL, Goswami et al.(2018), Harvey (2019) and so on. It can be seen from the practice that, Kalman filter is more widely applied, so that it might be more reliable in this respect. More importantly, the least-squares collocation suffers from the problem of a fixed fusion time-length, the illness of the normal equation, and a fixed parameter setting. On the contrary, Kalman filter has no such problems. Moreover, Kalman filter is more flexible, for example, the parameters during the fusion can be even manually adjusted when occasionally some very undesired situations take place.
For those reasons, Kalman filter is chosen by design from the very beginning of this study. But we also get an inspiration from the reviewer’s comment that, it might be also valuable to conduct another attitude fusion using least square collocation to enable a comparison with the current one. Nevertheless, this should be subject to our future work.
Reference:
Goswami S, Klinger B, Weigelt M, et al. 2018. Analysis of attitude errors in GRACE range-rate residuals-a comparison between SCA1B and the reprocessed attitude fused product (SCA1B+ACC1B). IEEE Sensors Letters, 2(2):1-4.
Harvery N, Sakumura C. 2019. Results from a GRACE/GRACE-FO attitude reconstruction Kalman filter. Journal of Geodesy, 93:1881-1896.
Klinger B. 2018. A contribution to GRACE time-variable gravity field recovery: improved Level-1B data pre-processing methodologies. PhD thesis, Verlag der Technischen Universität Graz, Germany.
Comment 2: I would have expected the differences between your results and JPL (cf. Figure 8 and discussion) to be smaller. One possible reason could also be the relative weighting of SC and IMU, especially for low frequencies. How are these two observation types weighted against each other, constant for all frequencies or frequency dependent?
Response: Thanks very much for the advice. SC and IMU mostly compensate each other in the spectrum, specifically, SC is sensitive to the low frequency while IMU is sensitive to the high frequency. In this sense, an optimal combination of SC and IMU will make a better attitude determination, which is also the departure point of this study.
It is worth mentioning that, in the least square solution, SC and IMU are both regarded as the ‘observations’, so that a relative weighting of them is required. However, there is no such a concept of ‘weighting’ in our Kalman filter. This is because IMU is regarded as the ‘dynamical model’ (see Line 293-302 in the revised manuscript), and SC is the only ‘observation’. The mechanism of Kalman filter will automatically tune/adjust itself and make a balance between the ‘model’ and ‘observation’. What will affect the process of ‘self-adjustment’ is the covariance matrix (Q and R, see Eq. (15) and Eq. (18)) rather the traditional ‘weighting’. And apparently, the ‘self-adjustment’ is dynamical, demonstrating that the so called ‘weighting’ is not constant.
We once also suspect that the covariance matrix (Q, R) is responsible for the differences between our product and JPL’s. To this end, many tests by changing (Q, R) are carried out (not shown in the manuscript), but it turns out that the low-frequency (such as 1-CPR) difference always exists. Therefore, we can assert that, (at least) the low frequency difference is not caused by the configuration of the Kalman filter.
Comment 3: Some minor points:
l21 (+l150): range-rate instead of "ranging-rate"
l37: I would propose "imperfect or mis-modelled background models"
l41f: Rather something like "We can not expect better gravity field mapping by current and future gravity missions unless..."
l47f: sensor data processing deficiency
l53: one of the most challenging tasks
Equation (3): Isn't it (H^T * H)^-1 instead of (H^T * H^-1) ?
l243: has to be
l262: lead
l313: delete one "the"
l339: complicated
l339: you often write "we don't repeat again". Why not simply refer to the reference and that's it.
l367: "with a precision of"
Figure 4 caption: spectra instead of spectrums; and power spectral density itsead of power spectrum density
l445: "aiming at analyzing"
Figure 8 caption: two CPR might correspond to 0.8 hours, not to 3.2 hours. Please check and correct also in the corresponding text parts.
Response: Thanks very much for pointing out our incorrect language usage, typo and unclear statement. Your help has largely improved the manuscript. We have made corrections carefully in the revised manuscript point by point as you suggested.
Comment 4: l284ff: does this last sentence of this section mean, that you do not combine the SC data as described in equations (8), (9) and (10)? If this is the case, why do you explain the combination here?
Response: Strictly speaking, we do make the combination, but the combined result is used for identifying outliers of each SC. After removing outliers, every single SC rather than the combined result is sent to the Kalman filter. This statement is also given in the manuscript, see Line 284.
Comment 5: Figure 2: how do the spectra (sqrt(PSDs)) of (b), (c) and (d) look like? Is it close to white noise?
Response: As the reviewer requested, we plot the spectra of (b) as an example, see the below. According to its spectral behaviors, it is very close to white noise.
Comment 6: Table 2: how often are the biases estimated, are they assumed to be constant over for example one day?
Response: The bias (accompanying the attitude quaternion, see Eq. (11) in the manuscript) is estimated for every epoch at a frequency of 8 Hz, which is decided by the mechanism of Kalman filter. Therefore, the bias is not constant.
Comment 7: Figure 9: as the differences between JPL and your results are mainly periodic with 1 CPR frequency, a daily mean of this difference makes not much sense in my opinion. You could, for example, compute the daily mean absolute amplitude of this 1 CPR signal in the residual time series. Probably this measure would be more stable.
Figure 9: Probably I don't understand the way you compute the differences in Figure 9 compared to Figure 8. The red lines in Figure 8 (left) show nearly zero mean. Why do the mean values in Figure 9 do not show this?
Response: Thanks for the questions. We have to clarify that, Fig. 8 is plotted by removing their time mean, which has already been demonstrated in the caption, see the manuscript. On the contrary, Fig. 9 plots only the time (daily) mean. In this sense, Fig. 8 and Fig. 9 are basically two things.
Comment 8: Figure 10: you should compare the differences with noise specifications of MWI or LRI or ACC in this PSD-plot, or with the spectra of post-fit range-rate residuals after gravity field adjustment.
Response: Thanks very much for the advice. We have added the post-fit residuals of MWI into Fig. 10 in the revised manuscript, according to your suggestion.
Author Response File: 
Author Response.doc
Reviewer 2 Report
Dear Editor,
Please find below my comments related to the manuscript titled: “Attitude Determination for GRACE-FO: Reprocessing the Level-1A SC and IMU Data”. The paper deals with the improvement of GRACE-FO attitude product. The paper is well written and presents the methodology and results in detail. My main issue is that according to Figure 11 the new product HUGG-01, performs almost as well as CSR-RL06 output. How do we know that HUGG-01 is a real improvement? Since we do not have an application of the product in a real-world case, e.g. in some basins, we cannot actually know if an improvement is achieved in the final output. I think authors should comment and clarify this issue.
Some minor comments:
Line 8: IMU I believe stands for Inertial Measurement Unit. It should be good to put the full name and then provide the abbreviation.
Line 481: Kalman filter on the IMU
Line 603: contain
Figure 11: Please provide the time period those images refer to.
Author Response
Response to the Reviewer #2
Comment 1: Please find below my comments related to the manuscript titled: “Attitude Determination for GRACE-FO: Reprocessing the Level-1A SC and IMU Data”. The paper deals with the improvement of GRACE-FO attitude product. The paper is well written and presents the methodology and results in detail. My main issue is that according to Figure 11 the new product HUGG-01, performs almost as well as CSR-RL06 output. How do we know that HUGG-01 is a real improvement? Since we do not have an application of the product in a real-world case, e.g. in some basins, we cannot actually know if an improvement is achieved in the final output. I think authors should comment and clarify this issue.
Response: Thanks very much for the comments. We clarify that, the accuracy of the recovered gravity field model is related to many factors, such as the background model and payload data processing. Apparently, one can hardly expect to improve the current gravity recovery precision by solely improve one factor like the attitude information in this study, since the satellite itself is an integrated and comprehensive system. This has been also supported by our initial results shown by Fig. 9 and Fig. 10 in the manuscript.
Nevertheless, we still think our product might be slightly better than the official one, solely from the perspective of the payload performance, see Fig. 8, where our product contains less high-frequency noise. Although this improvement can not be sensed by the current gravity mission, it does not mean the future gravity mission can not do it. But meanwhile, we also admit that, the assessments shown in this study is only an initial one. Just as the reviewer suggested, a better assessment should be computing a time-series of gravity fields (probably at least 5 years) and investigating what kind of improvement can be achieved in the real world. Such a work makes a great sense, but also requires lots of time to compute the temporal gravity fields considering the limited revision time the editor gave us. And meanwhile, it might be out of the current scope of this study, so we would try to finish it in the next step. Thanks very much for the constructive suggestion.
Comment 2: Some minor comments:
Line 8: IMU I believe stands for Inertial Measurement Unit. It should be good to put the full name and then provide the abbreviation.
Line 481: Kalman filter on the IMU
Line 603: contain
Figure 11: Please provide the time period those images refer to.
Response: Thanks very much for the careful review, and we have already made corrections according to your suggestions in the revised manuscript.
Reviewer 3 Report
Review of the paper “Attitude Determination for GRACE-FO: Reprocessing the Level-1A SC and IMU Data” submitted to Remote Sensing.
The paper discusses one of the most important error sources in the GRACE-FO mission, related to attitude determination. The paper is very well prepared and provides substantially novel aspects to merit a publication in a peer-reviewed paper. Moreover, the authors publish the combined attitude files, which is a great asset of the paper. I recommend the publication of the paper after addressing a minor comment.
Some authors, such as Loomis et al (https://doi.org/10.1029/2019GL085488) recommend that C20 and C30 series should be replaced in GRACE-FO solutions by the solutions provided from satellite laser ranging (SLR). Typically, the problem with inhomogeneous heating of GRACE spacecraft is assigned to be the main reason for the erroneous values of GRACE-based C20 and C30. Some other authors indicate that the aliasing with the S2 tide can be the reason, or the wrong orientation of the accelerometers by even several degrees.
Do the improved GRACE attitude corrections solve the issue with C20/C30 series? Could you show how the C20 and C30 series change when improving the attitude? Are they more similar to the SLR-based series?
Author Response
Response to the Reviewer #3
Comment 1:
Review of the paper “Attitude Determination for GRACE-FO: Reprocessing the Level-1A SC and IMU Data” submitted to Remote Sensing.
The paper discusses one of the most important error sources in the GRACE-FO mission, related to attitude determination. The paper is very well prepared and provides substantially novel aspects to merit a publication in a peer-reviewed paper. Moreover, the authors publish the combined attitude files, which is a great asset of the paper. I recommend the publication of the paper after addressing a minor comment.
Some authors, such as Loomis et al (https://doi.org/10.1029/2019GL085488) recommend that C20 and C30 series should be replaced in GRACE-FO solutions by the solutions provided from satellite laser ranging (SLR). Typically, the problem with inhomogeneous heating of GRACE spacecraft is assigned to be the main reason for the erroneous values of GRACE-based C20 and C30. Some other authors indicate that the aliasing with the S2 tide can be the reason, or the wrong orientation of the accelerometers by even several degrees.
Do the improved GRACE attitude corrections solve the issue with C20/C30 series? Could you show how the C20 and C30 series change when improving the attitude? Are they more similar to the SLR-based series?
Response: Thank you very much for the interesting discussion. Just as the reviewer mentioned, the issue of C20/C30 is quite complex, and there are possibly several different arguments. We also try to explain it with our expertise.
In theory, the core of gravity field inversion is the establishment of the observation equation by solving the Hill equation. It can be seen from the general solution of Hill equation that, the low-frequency terms of [t, t2, tcoswt, tsinwt] are included, which are also called as the resonance terms. In another word, all signals (no matter it is signal or noise) will leak into these resonance terms. Whenever there is an anomaly/event taking place in the satellite platform, the error caused by the event will leak into the resonance terms. Thus, it is difficult to distinguish the signal/noise in those terms, and such a low-frequency error will be completely absorbed by the recovered gravity field coefficients. In particular, C20 and C30 are the maximum terms for potential coefficients, so they are also most affected by the low-frequency error. For GRACE-FO, the failure of the GRACE-D accelerometer will further exacerbate this situation. Therefore, it is reasonable to replace C20 and C30 by using the SLR results for the science applications.
However, the main improvement of the HUGG-01 attitude data is the high-frequency part as shown in Fig 8, which we believe has only a limited influence on the C20 or C30 terms according to the aforementioned theoretical analysis. The reviewer also suggests us to compute the C20/C30 time-series, we feel that the work is interesting and might make sense for verifying the aforementioned assumption. Nevertheless, we would like to clarify that a time-series of temporal gravity fields needs to be computed (the computation is huge) before plotting the C20/C30 time-series. Because of the limited revision time the editor gave us, we can only leverage the results (gravity fields of three months) in hand to make such a study. Please see the figure below.
Fig 1 The series of C20
Fig 2 The series of C30
Figures 1 and 2 show the C20 and C30 results of June 2018, January 2019, and March 2019. It can be seen from the figure that there is a difference between the results of the GRACE-FO and SLR. In addition, GRACE/GRACE-FO is a circular orbit through the polar regions, thus it is not sensitive to C20 and C30. In summary, it is reasonable to use SLR results for replacement.
Author Response File: Author Response.doc
