the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 09 Feb 2021
An upward continuation method based on spherical harmonic analysis and its application in the calibration of satellite gravity gradiometry data
Abstract. The ground gravity anomalies can be used to calibrate and validate the satellite gravity gradiometry data. In this study, an upward continuation method of ground gravity data based on spherical harmonic analysis is proposed, which can be applied to the calibration of satellite observations from the European Space Agency's Gravity Field and Steady-State Ocean Circulation Explorer (GOCE). Here, the following process was conducted to apply this method. The accuracy of the upward continuation method based on spherical harmonic analysis was verified using simulated ground gravity anomalies. The DTU13 global gravity anomaly data were used to determine the calibration parameters of the GOCE gravitational gradients based on the spherical harmonic analysis method. The trace and the tensor invariants I2, I3 of the gravitational gradients were used to verify the calibration results. The results revealed that the upward continuation errors based on spherical harmonic analysis were much smaller than the noise level in the measurement bandwidth of the GOCE gravity gradiometer. The scale factors of the Vxx, Vyy, Vzz, and Vyz components were determined at an order of magnitude of approximately 10−2, the Vxz component was approximately 10−3, and the Vxy component was approximately 10−1. The traces of gravitational gradients after calibration were improved when compared with the traces before calibration and were slightly better than the EGG_TRF_2 data released by the European Space Agency (ESA). In addition, the relative errors of the tensor invariants I2, I3 of the gravitational gradients after calibration were significantly better than those before calibration. In conclusion, the upward continuation method based on spherical harmonic analysis could meet the external calibration accuracy requirements of the gradiometer.
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Interactive discussion
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RC1: 'Review of the manuscript se-2020-201: "An upward continuation method based on spherical harmonic analysis and its application in the calibration of satellite gravity gradiometry data"', Anonymous Referee #1, 11 Mar 2021
Dear authors and dear editor,
please find below the review of the manuscript se-2020-201.
General comments
The paper "An upward continuation method based on spherical harmonic analysis and its application in the calibration of satellite gravity gradiometry data"' by Qingliang Qu, Shengwen Yu, Guangbin Zhu, Xiaotao Chang, Miao Zhou and Wei Liu studies the effect of an external calibration of gravity gradients as measured by the satellite mission GOCE. For that purpose, ground gravity data products - i.e. global gridded datasets - are upward continued and converted to gravity gradients in the reference frame of the gradiometer on-board the GOCE satellite. For the change of functional and the upward continuation of the ground gravity reference dataset, the authors propose a conversion to a spherical harmonic series. Within data windows of a fixed length, biases and scale factors are estimated to externally calibrate the measurements with respect to the reference dataset. The trace-free condition as well as rotational invariants I2 and I3 are used to access the quality of the calibration. The authors validate the calibration computing the trace reduction as well as the consistency of the higher order invariants with respect to the derived reference gradients.
In the current version of the manuscript, I see major issues in all main parts, i.e. the Introduction and Motivation, the Methods section, the Results and Discussion part as well as the final Discussion of the results and the derived Conclusions. For that reason I have to suggest to reject the submitted manuscript.
In the first part (i.e. 1. Introduction) I am missing the motivation for the study. It remains open, why there is a need for new, respectively alternative, calibration procedures compared to the existing and published approaches. Although there have been several major updates in GOCE L1B gravity gradient processing, which might result in different conclusions for the external calibration compared to the existing published studies, the authors do not address the recent GOCE L1B updates and do not relate their study to the improved L1B data.
The methods section summerizes the upward continuation via a conversion of global gridded gravity data sets to spherical harmonics, for which well established procedures exist. The benefit of using global gridded data set, compared to directly an existing spherical harmonic model, remains open and is not shown. Furthermore, the presentation of the conversion to spherical harmonics is erroneous, there are some inconsistencies and errors in the presented equations. For the applied model for the external calibration, it is not explicitly stated how this relates to the existing and published approaches. Comparisons are missing.
The numerical results presented in the Results section rely on a single reference dataset and a single part of the time series. Consequently, the dependence of the results from the chosen reference dataset remains open, such that it is quite hard to draw conclusions. The numerical test of the upward continuation is not representative for the dataset used in the numerical analysis (in Sect. 3.2), as the real data set includes correlated data and systematic errors. Confirmation of the results with different datasets would be required to interprete the results correctly and reliable. Furthermore, it is not totally clear which version of the actual GOCE data were used.
The discussion and validation of the results is not clear. On the one hand, it is shown, that the trace-free criterion improves for the calibrated gradients. But, the other two chosen validation approaches are - from my point of view - not independent. If I understand the applied procedures correctly, the comparison using the TRF data is very similar to the first trace analysis and no additional data is involved. The higher order invariant analysis is - from my understanding - heavily biased, as consistency to the dataset used for calibration is checked. It is obvious and expected that the calibrated gradients are more consistent to the reference data compared to the un-calibrated gradients.
Finally, in the Conclusion chapter I am missing the actual conclusions. In the current form it is more a summary of the manuscript.
But what follows from the performed study? Are the conclusions and results of the older studies still valid? How will the calibrated data be used? Are there jumps in the calibrated gradient time series?To address this points more directly, please find attached some more text specific comments to emphasize/justify my overall impression.
-
RC2: 'Comment on se-2020-201', Anonymous Referee #2, 07 Apr 2021
“An upward continuation method based on spherical harmonic analysis and its application in the calibration of satellite gravity gradiometry data” by Qingliang Qu, Shengwen Yu, Guangbin Zhu, Xiaotao Chang, Miao Zhou, and Wei Liu
This paper deals with upward continuation of ground gravity data based on spherical harmonic analysis and its application in the calibration of satellite gravity gradient data. By using the simulated ground gravity anomalies, the accuracy of the proposed upward method is verified. Finally, the calibration of GOCE gravity gradient data is executed with the DTU13 global gravity anomaly data. After calibration, the results show improvement and are slightly better than the EGG_TRF_2, which confirm the feasibility of calibration of GOCE gradiometer with the upward continuation of ground gravity data based on spherical harmonic analysis.
The paper is well written for the structure, the introduction provides a fair overview of the existing calibration of GOCE gravity gradient data based on ground gravity data. In addition, the work on the calibration of GOCE gravity gradient data is significant for the GGT’s calibration of future satellite gradiometry mission. Summarizing, I recommend the publication of the manuscript after revision according to the following minor remarks.
The paper needs to be polished by a native speaker.
To make it clear, please use the same color bar in Fig.2 and the same bound for vertical axis (PSD) in Fig.4.
Page 2, line 41, replace “be examined and outlined here” with “be examined and outlined in the following”
Page 2, line 42, replace “Arabelos and Tscherning (Arabelos and Tscherning, 1998)” with “Arabelos and Tscherning (1998)”
Page 2, line 45, replace “was discussed in Bouman et al. and Pail (…)” with “was discussed by Bouman et al. (2003) and Pail (2002)”
Page 2, line 46, replace “Denker (Denker, 2002)” with “Denker (2002)”
Page 2, line 50, replace “compared in Wolf and Denker (Wolf and Denker, 2005)” with “compared in Wolf and Denker (2005)”
Page 2, line 52, replace “on an ellipsoid” with “on an reference ellipsoid”
Page 2, line 54, replace “were applied” with “were used”
Page, line 55, replace “Kern and Haagmans (Kern and Haagmans, 2005)” with “Kern and Haagmans (2005)”
Page 2, line 58-60, change the sentence “To validate …” to “Bouman et al. (2004, 2009, 2011) did much research about the external calibration model based on the regional ground gravity data to validate the SGG data” better?
Page 2, line 66, replace “in Yildiz et al. (Yildiz, 2012;Yildiz et al., 2016)” with “Yildiz (2012) and Yildiz et al. (2016)”
Page 2, line 68, replace “in the high-precision” with “for the high-precision”
Page 2, line 71, replace “Eshagh (Eshagh, 2011). Å prlák et al. (Å prlák et al., 2015)” with “Eshagh (2011). Å prlák et al. (2015)”
Page, line 76, replace “inverse matrix” with “inverse”. It seems that in most cases the “matrix” in “inverse matrix” is omissible.
Page 3, line 79, replace “the possibilities of spherical harmonic analysis for” with “the feasibility of applying spherical harmonic analysis to”
Page 9, line 220, replace “Veicherts et al. (Veicherts et al., 2011)” with “Veicherts et al. (2011)”
Page 14, line 306, replace “than before calibration” with “than that before calibration” ?
Citation: https://doi.org/10.5194/se-2020-201-RC2 -
AC1: 'Reply on RC2', Qingliang Qu, 11 May 2021
The comment was uploaded in the form of a supplement: https://se.copernicus.org/preprints/se-2020-201/se-2020-201-AC1-supplement.pdf
-
AC1: 'Reply on RC2', Qingliang Qu, 11 May 2021
Interactive discussion
Status: closed
-
RC1: 'Review of the manuscript se-2020-201: "An upward continuation method based on spherical harmonic analysis and its application in the calibration of satellite gravity gradiometry data"', Anonymous Referee #1, 11 Mar 2021
Dear authors and dear editor,
please find below the review of the manuscript se-2020-201.
General comments
The paper "An upward continuation method based on spherical harmonic analysis and its application in the calibration of satellite gravity gradiometry data"' by Qingliang Qu, Shengwen Yu, Guangbin Zhu, Xiaotao Chang, Miao Zhou and Wei Liu studies the effect of an external calibration of gravity gradients as measured by the satellite mission GOCE. For that purpose, ground gravity data products - i.e. global gridded datasets - are upward continued and converted to gravity gradients in the reference frame of the gradiometer on-board the GOCE satellite. For the change of functional and the upward continuation of the ground gravity reference dataset, the authors propose a conversion to a spherical harmonic series. Within data windows of a fixed length, biases and scale factors are estimated to externally calibrate the measurements with respect to the reference dataset. The trace-free condition as well as rotational invariants I2 and I3 are used to access the quality of the calibration. The authors validate the calibration computing the trace reduction as well as the consistency of the higher order invariants with respect to the derived reference gradients.
In the current version of the manuscript, I see major issues in all main parts, i.e. the Introduction and Motivation, the Methods section, the Results and Discussion part as well as the final Discussion of the results and the derived Conclusions. For that reason I have to suggest to reject the submitted manuscript.
In the first part (i.e. 1. Introduction) I am missing the motivation for the study. It remains open, why there is a need for new, respectively alternative, calibration procedures compared to the existing and published approaches. Although there have been several major updates in GOCE L1B gravity gradient processing, which might result in different conclusions for the external calibration compared to the existing published studies, the authors do not address the recent GOCE L1B updates and do not relate their study to the improved L1B data.
The methods section summerizes the upward continuation via a conversion of global gridded gravity data sets to spherical harmonics, for which well established procedures exist. The benefit of using global gridded data set, compared to directly an existing spherical harmonic model, remains open and is not shown. Furthermore, the presentation of the conversion to spherical harmonics is erroneous, there are some inconsistencies and errors in the presented equations. For the applied model for the external calibration, it is not explicitly stated how this relates to the existing and published approaches. Comparisons are missing.
The numerical results presented in the Results section rely on a single reference dataset and a single part of the time series. Consequently, the dependence of the results from the chosen reference dataset remains open, such that it is quite hard to draw conclusions. The numerical test of the upward continuation is not representative for the dataset used in the numerical analysis (in Sect. 3.2), as the real data set includes correlated data and systematic errors. Confirmation of the results with different datasets would be required to interprete the results correctly and reliable. Furthermore, it is not totally clear which version of the actual GOCE data were used.
The discussion and validation of the results is not clear. On the one hand, it is shown, that the trace-free criterion improves for the calibrated gradients. But, the other two chosen validation approaches are - from my point of view - not independent. If I understand the applied procedures correctly, the comparison using the TRF data is very similar to the first trace analysis and no additional data is involved. The higher order invariant analysis is - from my understanding - heavily biased, as consistency to the dataset used for calibration is checked. It is obvious and expected that the calibrated gradients are more consistent to the reference data compared to the un-calibrated gradients.
Finally, in the Conclusion chapter I am missing the actual conclusions. In the current form it is more a summary of the manuscript.
But what follows from the performed study? Are the conclusions and results of the older studies still valid? How will the calibrated data be used? Are there jumps in the calibrated gradient time series?To address this points more directly, please find attached some more text specific comments to emphasize/justify my overall impression.
-
RC2: 'Comment on se-2020-201', Anonymous Referee #2, 07 Apr 2021
“An upward continuation method based on spherical harmonic analysis and its application in the calibration of satellite gravity gradiometry data” by Qingliang Qu, Shengwen Yu, Guangbin Zhu, Xiaotao Chang, Miao Zhou, and Wei Liu
This paper deals with upward continuation of ground gravity data based on spherical harmonic analysis and its application in the calibration of satellite gravity gradient data. By using the simulated ground gravity anomalies, the accuracy of the proposed upward method is verified. Finally, the calibration of GOCE gravity gradient data is executed with the DTU13 global gravity anomaly data. After calibration, the results show improvement and are slightly better than the EGG_TRF_2, which confirm the feasibility of calibration of GOCE gradiometer with the upward continuation of ground gravity data based on spherical harmonic analysis.
The paper is well written for the structure, the introduction provides a fair overview of the existing calibration of GOCE gravity gradient data based on ground gravity data. In addition, the work on the calibration of GOCE gravity gradient data is significant for the GGT’s calibration of future satellite gradiometry mission. Summarizing, I recommend the publication of the manuscript after revision according to the following minor remarks.
The paper needs to be polished by a native speaker.
To make it clear, please use the same color bar in Fig.2 and the same bound for vertical axis (PSD) in Fig.4.
Page 2, line 41, replace “be examined and outlined here” with “be examined and outlined in the following”
Page 2, line 42, replace “Arabelos and Tscherning (Arabelos and Tscherning, 1998)” with “Arabelos and Tscherning (1998)”
Page 2, line 45, replace “was discussed in Bouman et al. and Pail (…)” with “was discussed by Bouman et al. (2003) and Pail (2002)”
Page 2, line 46, replace “Denker (Denker, 2002)” with “Denker (2002)”
Page 2, line 50, replace “compared in Wolf and Denker (Wolf and Denker, 2005)” with “compared in Wolf and Denker (2005)”
Page 2, line 52, replace “on an ellipsoid” with “on an reference ellipsoid”
Page 2, line 54, replace “were applied” with “were used”
Page, line 55, replace “Kern and Haagmans (Kern and Haagmans, 2005)” with “Kern and Haagmans (2005)”
Page 2, line 58-60, change the sentence “To validate …” to “Bouman et al. (2004, 2009, 2011) did much research about the external calibration model based on the regional ground gravity data to validate the SGG data” better?
Page 2, line 66, replace “in Yildiz et al. (Yildiz, 2012;Yildiz et al., 2016)” with “Yildiz (2012) and Yildiz et al. (2016)”
Page 2, line 68, replace “in the high-precision” with “for the high-precision”
Page 2, line 71, replace “Eshagh (Eshagh, 2011). Å prlák et al. (Å prlák et al., 2015)” with “Eshagh (2011). Å prlák et al. (2015)”
Page, line 76, replace “inverse matrix” with “inverse”. It seems that in most cases the “matrix” in “inverse matrix” is omissible.
Page 3, line 79, replace “the possibilities of spherical harmonic analysis for” with “the feasibility of applying spherical harmonic analysis to”
Page 9, line 220, replace “Veicherts et al. (Veicherts et al., 2011)” with “Veicherts et al. (2011)”
Page 14, line 306, replace “than before calibration” with “than that before calibration” ?
Citation: https://doi.org/10.5194/se-2020-201-RC2 -
AC1: 'Reply on RC2', Qingliang Qu, 11 May 2021
The comment was uploaded in the form of a supplement: https://se.copernicus.org/preprints/se-2020-201/se-2020-201-AC1-supplement.pdf
-
AC1: 'Reply on RC2', Qingliang Qu, 11 May 2021
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