The Relationship of Time Span and Missing Data on the Noise Model Estimation of GNSS Time Series

Round 1

Reviewer 1 Report

This study focuses on noise model estimation criterion (BIC_tp) derived from the AIC and the BIC by introducing 2π factors. It is more sensitive to abnormal steps (random jumps). Using observation data from 72 GNSS stations from 1992 to 2022 and simulated data, four combined noise models are used to explore the impacts of time series lengths (ranging from2 to 24 years) and data loss (between 2% and 30%) on noise models and velocity estimation. It is worth noting that the paper concludes that for a time series with a minimum time length of 12 years, both the selection of the optimal stochastic noise model and the estimation of the velocity parameters are reliable.

This paper analyzed impact of time span and missing data on the noise model estimation of GNSS time series. It is well structured and well written; the language is also acceptable. The manuscript writes well and can be published after address my following comments:

1. In section 2.2.2 Simulation Time Series：After reference [36],English writing generally does not use "：", you can modify to ".", please correct the whole manuscript.

2. Figure 2 shows the simulated time series, Simulated time series should be modified to simulation time series. Please check all the figures.

3. Formula (2) does not have parameter F, please check for deletion.

4. In the paper, N represents the north direction. It is recommended to modify N to the north direction/ north component in individual figures such as Figures 6, 13, 14, and 15. Please check all the figures.

5. The horizontal axis of Figure 9 is not fully displayed. Please check if Figure 9 replaces the year. Please check all the figures.

6. Please add punctuation after the title in Figure A6 and Figure A7.

7. Check if the journal name of the reference is abbreviated and italicized?

It is acceptable.

Author Response

Response to Reviewer 1 Comments

Manuscript ID: remotesensing-2491080.

Title: Impact of Time Span and Missing Data on the Noise Model Estimation of GNSS time series

We would like to thank Assistant Editor Mr. Milos Miric and the anonymous reviewers for providing an opportunity to revise the manuscript. The comments and suggestions of the reviewers are all valuable and very helpful. We have studied them carefully and have made revisions to improve the manuscript.

Detailed corrections are listed below point by point:

1. In section 2.2.2 Simulation Time Series：After reference [36],English writing generally does not use "：", you can modify to ".", please correct the whole manuscript.

Reply: Corrected. We checked the whole chapter and make related correction. See line 100 in the revised manuscript.

 

2. Figure 2 shows the simulated time series, Simulated time series should be modified to simulation time series. Please check all the figures.

Reply: Corrected. Sorry for the unclear statements.

 

3. Formula (2) does not have parameter F, please check for deletion.

Reply: Corrected. We removed parameter F in the revised manuscript, see line 143 in the revised manuscript.

 

4. In the paper, N represents the north direction. It is recommended to modify N to the north direction/ north component in individual figures such as Figures 6, 13, 14, and 15. Please check all the figures.

Reply: We modify N to the north component in the whole manuscript.

 

5. The horizontal axis of Figure 9 is not fully displayed. Please check if Figure 9 replaces the year. Please check all the figures.

Reply: Corrected. We have re-plotted Figure 9 following your suggestion.

 

6. Please add punctuation after the title in Figure A6 and Figure A7.

Reply: Corrected, see line 542and 545 in the revised manuscript.

 

7. Check if the journal name of the reference is abbreviated and italicized?

Reply: We unify the references and added make some for supplement as follows (highlighted in the revised manuscript):

• Abatzoglou, J. T., McEvoy, D. J., & Redmond, K. T. The west wide drought tracker: drought monitoring at fine spatial scales. B.Am. Meteorol. Soc .2017, 98, 1815-1820.
• Xu C. Reconstruction of gappy GPS coordinate time series using empirical orthogonal functions. J. Geophys. Res. Solid Earth. 2016, 121, 9020-9033.
• Herring, T.A., King, R.W., McClusky, S.C. GAMIT Reference Manual. GPS analysis at MIT. Release 10.4. Massachusetts Institute Technology.2010a.
• Herring T A, King R W, McClusky S C. GLOBK. Global Kalman filter VLBI and GPS analysis program. Version, 2005, 10.
• Lichten, S.M., Bar-Sever, Y.E., Bertiger, E.I., Heflin, M., Hurst, K., Muellerschoen, R.J.,Wu, S.C., Yunck, T.P., Zumberge, J.F. GIPSY-OASIS II: a high precision GPS data processing system and general orbit analysis tool. Technology.2006. 2, 2–4.
• Dong D, Herring T A, King R W. Estimating regional deformation from a combination of space and terrestrial geodetic data. J. Geodesy, 1998, 72, 200-214.
• Bos, M. S., Fernandes, R.M.S., Williams, S.D.P. and Bastos, L. Fast error analysis of continuous GPS observations. J. Geodesy. 2008, 82, 157–166.
• Bozdogan H. Model selection and Akaike's information criterion (AIC): The general theory and its analytical extensions. Psychometrika, 1987, 52, 345-370.
• Wagenmakers, E. J., & Farrell, S. AIC model selection using Akaike weights. Psychon. B. Rev. 2014, 11, 192-196.

 

 

Author Response File: Author Response.docx

Reviewer 2 Report

The authors have evaluated the effect of both time span and missing data on the GNSS station coordinate noise model. I consider the following problems as well as list some suggestions for improvement:

 

Major problems:

 

1, the main problem is that the title states the effect of missing series and time span on the noise model. However, almost all of the figures present mainly the effect of time span. All three points in the conclusion stressed the time span of 12 years, but the effect of the missing series is not reflected. Even if there is no effect or little effect, how the authors made sense of it should still be concluded.

 

2, Regarding the 2.2 method: The authors actually used multiple methods and compared them, but only one method is presented here.

 

Other issues:

1, line 21,23: Please write specifically what "significant impact" is.

 

2, line 56: international GNSS service (IGS): Please check whether the English brackets are used correctly in the whole text.

 

3, line 99: (1) data and method?

(2) the author have listed directly from the first level heading to the third level heading. Line 204 and 294 also have the same problem. There should be some concatenation text between them.

 

4, line113: The formatting and line spacing of the mathematical symbols in this paragraph is not normal, you can list the English letters in the text in the form of corner marks instead of inserting formulas. line463 also has the same problem, if you have to insert formulas, please make the formulas the right size and adjust the line spacing.

 

5, the format of figure 2: (mm).

 

6, in the title. What time series of GNSS? Station coordinates, observations or ephemeris?

Although I can understand the author's meaning, I think the full text should be continued to be improved from the language presentation.

Author Response

Response to Reviewer 2 Comments

Manuscript ID: remotesensing-2491080.

Title: Impact of Time Span and Missing Data on the Noise Model Estimation of GNSS time series

We would like to thank Assistant Editor Mr. Milos Miric and the anonymous reviewers for providing an opportunity to revise the manuscript. The comments and suggestions of the reviewers are all valuable and very helpful. We have studied them carefully and have made revisions to improve the manuscript.

Detailed corrections are listed below point by point:

Reviewer 2

1. the main problem is that the title states the effect of missing series and time span on the noise model. However, almost all of the figures present mainly the effect of time span. All three points in the conclusion stressed the time span of 12 years, but the effect of the missing series is not reflected. Even if there is no effect or little effect, how the authors made sense of it should still be concluded.

Reply: Missing data is a persistent issue in the analysis of the GNSS time series. Missing data can disrupt the continuity and the integrity of the time series, which has the potential of biasing the estimated model parameters. It is necessary to investigate the effects of the data gaps on the noise model and velocity estimation as discussed in previous studies.

In Section 3.1.2. “Effect of Missing Data on Noise Model”, to explore the impact of data gaps on the noise model selection and estimated velocity, we simulate 100 time series with various lengths as described in the previous section. Figure3 illustrates probability of detection for different noise models as a function of the missing data (Gap) and length of the simulation time series.

In Section 3.2.2. “Effect of GNSS Missing data on Noise Model”, to further explore the variation in the noise models of real GNSS stations with different missing rates and during time periods. Figure 6, Figure A5 and Figure A6 illustrate evolution of different selected noise models with various missing rates for the component. Figure 9 illustrates the mean velocity change curves with different missing rates, Figure 10 illustrates the velocity uncertainty distribution curves with different missing rates.

 

2. Regarding the 2.2 method: The authors actually used multiple methods and compared them, but only one method is presented here.

Reply: We mainly use BIC_tp method, for AIC and BIC we added related references in the list in the revised version as a supplement, and added related citations as follows：

“29.Akaike H. A new look at the statistical model identification. IEEE transactions on automatic control. 1974, 19,716-723. ”

“30.Schwarz G. Estimating the dimension of a model. The annals of statistics, 1978, 461-464. ”

 

3.line 21,23: Please write specifically what "significant impact" is.

Reply: We replace this sentence to“Accurate noise model identification is crucial for obtaining a reliable velocity and its uncertainty estimated from the GNSS time series. Selecting the wrong stochastic noise models can bias the velocity parameters and can possibly mislead the geodynamic and geodesy interpretation (e.g., crustal deformation, seismic source imaging.)”.

4. line 56: international GNSS service (IGS): Please check whether the English brackets are used correctly in the whole text.

Reply: Corrected. Thanks for point it out. See line 56 in the revised manuscript.

 

5. line 99: (1) data and method? (2) the author has listed directly from the first level heading to the third level heading. Line 204 and 294 also have the same problem. There should be some concatenation text between them.

Reply: Corrected. We have corrected it: “2. Materials and Methods/2.1 Simulation GNSS Time Series/2.2 Real GNSS Time Series/2.3. Stochastic Model Selection Criteria”.

 

4. line113: The formatting and line spacing of the mathematical symbols in this paragraph is not normal, you can list the English letters in the text in the form of corner marks instead of inserting formulas. line463 also has the same problem, if you have to insert formulas, please make the formulas the right size and adjust the line spacing.

Reply: Corrected. See line 102 to 109 and 465 to 466 in the revised manuscript.

 

5. the format of figure 2: (mm).

Reply: Corrected. We modified Figure 2, see line 117 to 118 in the revised manuscript.

 

6.in the title. What time series of GNSS? Station coordinates, observations or ephemeris?

Reply: We revised the whole manuscript and uniform it to GNSS time series which is short for “daily coordinate of the GNSS stations”. We have also added related references on the definition of GNSS times series, see as follows:

“The classic trajectory model is used to simulate the GNSS time series [35-37].”

References:

• Bevis, M., Bedford, J., & Caccamise II, D. J. The art and science of trajectory modelling. Geodetic time series analysis in Earth sciences. 2020, 1-27.
• He X, Hua X.H, Lu Tie D.L, Yu K.G, Xuan W. Analysis of the Impact of Time Span on GPS time series Noise Model and Velocity Estimation. Journal of National University of Defense Science and Technology. 2017, 6,12-18.
• Montillet, J.P.; Bos, M. Geodetic time series analysis in earth sciences. Springer. Berlin/Heidelberg, Germany, 2019.

 

 

Author Response File: Author Response.docx

Reviewer 3 Report

 The paper focuses on investigating the impact of noise model selection criteria on identifying stochastic noise properties in GNSS coordinate time series. Specifically, it examines the effectiveness of the noise model estimation criterion BIC_tp, which is designed to be more sensitive to abnormal steps. The study utilizes a combination of observation data from 72 GNSS stations and simulated data, exploring the effects of different time lengths and data loss scenarios on noise models and velocity estimation. In general, this type of research is intriguing, but the authors' presentation of the data lacks standardization, rendering it inappropriate. The paper fails to clearly outline its contributions and novel ideas, making it challenging to assess its originality or uniqueness. Moreover, the absence of a well-defined theory makes it difficult for others to replicate or improve upon the method. Additionally, the noise model, a crucial aspect, is inadequately discussed and lacks precise modeling, such as determining whether it is multiplicative or additive, or analyzing its spectral properties. Properly modeling noise is essential for noise suppression, reduction, and analyzing its adverse effects, which is lacking in this study. Due to the improper noise modeling, nothing can be verified, and the methodology remains unknown. Theoretical and technical aspects should not only be known but also clearly justified, which is lacking in this case. The various types of noise are not adequately addressed, and there are missing sections regarding radio link budget analysis and noise budget analysis. Overall, due to these theoretical and methodological issues, it is challenging to verify the results and provide adequate justifications. Hence, I would reject it and won’t recommend resubmission.

Below are some other major concerns:

1- The English writing style and paper format lack professionalism. For instance, the abstract contains numerous acronyms, which hinder the understanding of the concept by potential readers. Additionally, the headings and subsection titles are improper and fail to effectively convey the intended meaning. The presentation of data, especially graphs and simulation results, is of low quality and lacks labels, indices, units, and line styles. The placement of these simulation results within the text is also inappropriate, with significant gaps between the text and graphs. The formulations used are improper in size and do not adhere to standard formats or use appropriate symbols. Overall, the paper template is unacceptable. Furthermore, the English style requires improvement.

 2- The title lacks informativeness and fails to indicate the contribution or novelty of the research. The abstract is excessively lengthy and does not clearly state the research's contributions or novelty. An abstract should provide readers with a concise overview of the research scope and be accompanied by relevant metrics. The keywords used are not based on the taxonomy of remote sensing.

3- The literature review is incomplete, and the authors have not effectively presented their solution and methodology in alignment with the existing researches. The paper lacks a comprehensive review of relevant literature, which hinders the understanding of the context and significance of the study. Furthermore, the authors have not clearly outlined their proposed solution or methodology in relation to the research problem. This gap undermines the coherence and effectiveness of the paper, as it becomes difficult to assess the appropriateness and validity of the chosen approach.

4- The materials and methods should be thoroughly analyzed both theoretically and through experiments, and the results should be justified using objective quality assessment techniques. However, in this paper, the material and methods section lacks comprehensive analysis and evaluation. The description of the noise is incomplete, and the model used to represent the noise is insufficient. This inadequate representation of the noise hampers the validity and reliability of the study. Additionally, the simulation results provided are incomplete, making it impossible to verify the findings. Overall, the lack of proper analysis, incomplete description of noise, and insufficient simulation results undermine the credibility and verifiability of the study.

5- The simulation results are presented solely in the form of graphs without proper metric justification based on theory or the characteristics of the noise. The absence of a clear connection between the noise model and the results undermines the ability to provide a scientific justification for the findings. As a result, the simulation results appear more like a report rather than a scientific paper, lacking the necessary scientific rigor and analysis.

 Hence, based on the present state of the paper, I must recommend its rejection. authors can take a look at these papers and can follow suit:

 

F. Wu, H. Luo, H. Jia, F. Zhao, Y. Xiao and X. Gao, "Predicting the Noise Covariance With a Multitask Learning Model for Kalman Filter-Based GNSS/INS Integrated Navigation," in IEEE Transactions on Instrumentation and Measurement, vol. 70, pp. 1-13, 2021, Art no. 8500613, doi: 10.1109/TIM.2020

G. Giangregorio, M. di Bisceglie, P. Addabbo, T. Beltramonte, S. D'Addio and C. Galdi, "Stochastic Modeling and Simulation of Delay–Doppler Maps in GNSS-R Over the Ocean," in IEEE Transactions on Geoscience and Remote Sensing, vol. 54, no. 4, pp. 2056-2069, April 2016

S. Gleason, J. Johnson, C. Ruf and C. Bussy-Virat, "Characterizing Background Signals and Noise in Spaceborne GNSS Reflection Ocean Observations," in IEEE Geoscience and Remote Sensing Letters, vol. 17, no. 4, pp. 587-591, April 2020, 

H. Zhang, Y. Xu, R. Luo and Y. Mao, "Fast GNSS acquisition algorithm based on SFFT with high noise immunity," in China Communications, vol. 20, no. 5, pp. 70-83, May 2023

 

 

 

 

Please refer to the comments above. 

Author Response

Response to Reviewer 3 Comments

Manuscript ID: remotesensing-2491080.

Title: Impact of Time Span and Missing Data on the Noise Model Estimation of GNSS time series

We would like to thank Assistant Editor Mr. Milos Miric and the anonymous reviewers for providing an opportunity to revise the manuscript. The comments and suggestions of the reviewers are all valuable and very helpful. We have studied them carefully and have made revisions to improve the manuscript.

Detailed corrections are listed below point by point:

1. In general, this type of research is intriguing, but the authors' presentation of the data lacks standardization, rendering it inappropriate. The paper fails to clearly outline its contributions and novel ideas, making it challenging to assess its originality or uniqueness.

Reply: We used Hector software (Bos et al., 2013) and the python appendices to simulate the GNSS time series in this work. The simulation time series range from 2 to 30 years (denoted in the following as 2a, 4a, 6a, 8a, 10a, 12a, 14a, 16a, 18a, 20a, 22a, 24a, 26a, 28a, 30a), with 100 stations simulated for each case.

 

Example of 30 year-long simulation time series with different noise models.

In addition. We analyze daily time series from 72 GNSS stations from the Enhanced Solid Earth Science ESDR System.

Spatial Distribution of the selected GNSS stations.

2.Moreover, the absence of a well-defined theory makes it difficult for others to replicate or improve upon the method. Additionally, the noise model, a crucial aspect, is inadequately discussed and lacks precise modeling, such as determining whether it is multiplicative or additive, or analyzing its spectral properties.

Reply: We have cited relevant references on the methods used in the manuscript.

References as follows:

• Bozdogan, H. (1987). Model selection and Akaike's information criterion (AIC): The general theory and its analytical extensions. Psychometrika, 52(3), 345-370.
• Wagenmakers, E. J., & Farrell, S. (2004). AIC model selection using Akaike weights. Psychonomic bulletin & review, 11, 192-196.
• Akaike H. A new look at the statistical model identification. IEEE transactions on automatic control. 1974, 19,716-723.
• Schwarz G. Estimating the dimension of a model. The annals of statistics, 1978, 461-464.
• Bos, M. S., Fernandes, R. M. S., Williams, S. D. P., & Bastos, L. Fast Error Analysis of Continuous GNSS Obser-vations with Missing Data. J. Geodesy. 2013, 87, 351-360.
• Amiri‐Simkooei A R, Tiberius C C J M, Teunissen P J G. Assessment of noise in GPS coordinate time series: methodology and results. J. Geophys. Res. Solid Earth. 2007, 112(B7).
• Amiri-Simkooei A R. Non-negative least-squares variance component estimation with application to GPS time series. J. Geodesy. 2016, 90, 451-466.
• Bock, Y., Nikolaidis, R.M., de Jonge., P.J. and Bevis, M. Instantaneous geodetic positioning at medium distances with the Global Positioning Systems. J. Geophys. Res. 2000, 105, B12, 28223–28253.
• Bock Y, Moore A W, Argus D, et al. Extended Solid Earth Science ESDR System (ES3): Algorithm Theoretical Basis Document, NASA MEaSUREs project, # NNH17ZDA001N. 2021.
• Bos, M. S., Fernandes, R.M.S., Williams, S.D.P. and Bastos, L. Fast error analysis of continuous GPS observations. J. Geodesy. 2008, 82, 157–166.

 

3.Properly modeling noise is essential for noise suppression, reduction, and analyzing its adverse effects, which is lacking in this study. Due to the improper noise modeling, nothing can be verified, and the methodology remains unknown. Theoretical and technical aspects should not only be known but also clearly justified, which is lacking in this case.

Reply: Properly modeling noise use the optimal model identification, obtaining optimal noise model characteristics through noise reduction in GNSS data. We use the ES3 GNSS time series developed by SOPCA in this work. Their team used the GNSS data processing strategy to model most of the error sources. However, the residual systematic errors are out the scope of this work. We use the BIC_tp in order to select the optimal noise model which was initially proposed in He et al., 2019. (He, X., Bos, M. S., Montillet, J. P., & Fernandes, R. M. S. Investigation of the noise properties at low frequencies in long GNSS time series. J. Geodesy. 2019, 93, 1271-1282.)

 

4. The various types of noise are not adequately addressed, and there are missing sections regarding radio link budget analysis and noise budget analysis. Overall, due to these theoretical and methodological issues, it is challenging to verify the results and provide adequate justifications. Hence, I would reject it and won’t recommend resubmission.

Reply: We have defined the four models used and added relevant references.

Four models:

FNWNRW (Flicker noise plus white noise and random walk noise), FNWN (Flicker noise plus white noise), PLWN (power law noise and white noise), GGMWN (generalized Gauss-Markov and white noise).

References as follows:

• Agnew, D. C. The time‐domain behavior of power‐law noises. Geophys Res Letters.1992, 19, 333-336.
• Mandelbrot, B. B., & Van Ness, J. W. Fractional Brownian motions, fractional noises and applications. SIAM re-view, 1968, 10, 422-437.
• Montillet, J. P., & Bos, M. S. (Eds.). Geodetic time series analysis in earth sciences. Springer.2019.
• He, X., Bos, M. S., Montillet, J. P., & Fernandes, R. M. S. Investigation of the noise properties at low frequencies in long GNSS time series. J. Geodesy. 2019, 93, 1271-1282.
• Bos, M., Fernandes, R., Williams, S. and Bastos, L. The noise properties in GPS time series at European stations revisited. EGU General Assembly, Geophysical Research Abstracts. 2013, EGU2013-12825.

 

5. The English writing style and paper format lack professionalism. For instance, the abstract contains numerous acronyms, which hinder the understanding of the concept by potential readers. Additionally, the headings and subsection titles are improper and fail to effectively convey the intended meaning. The presentation of data, especially graphs and simulation results, is of low quality and lacks labels, indices, units, and line styles. The placement of these simulation results within the text is also inappropriate, with significant gaps between the text and graphs. The formulations used are improper in size and do not adhere to standard formats or use appropriate symbols. Overall, the paper template is unacceptable. Furthermore, the English style requires improvement.

Reply: We polish the English expression again with MDPI English service.

 

6. The title lacks informativeness and fails to indicate the contribution or novelty of the research. The abstract is excessively lengthy and does not clearly state the research's contributions or novelty. An abstract should provide readers with a concise overview of the research scope and be accompanied by relevant metrics. The keywords used are not based on the taxonomy of remote sensing.

Reply: We revised the abstract and the revised summary is as follows:

Accurate noise model identification for GNSS time series is crucial for obtaining a reliable GNSS velocity field and its uncertainty, which can support reasonable geodynamic and Geodesy interpretation. In this study, by thoroughly considering time span and missing data effect on the noise model of GNSS time series, we used four combined noise models to analyze the duration of the time series (ranging from 2 to 24 years) and the data gap (between 2% and 30%) effect on noise model selection and velocity estimation at 72 GNSS stations spanning from 1992 to 2022 in global region together with simulation data. Our results showed that the selected noise model have better convergence when GNSS time series is getting longer. With longer time series, the GNSS velocity uncertainty estimation with different data gaps is more homogenous to a certain order of magnitude. When the GNSS time series length less than 8 years, it showed that the flicker noise and random walk noise and white noise (FNRWWN), flicker noise and white noise (FNWN), and power law noise and white noise (PLWN) models are wrongly estimated as a Gauss–Markov and white noise (GGMWN) model, which can affect the accuracy of GNSS velocity estimated from GNSS time series. When the GNSS time series length more than 12 years, the RW noise components are most likely to be detected. As the duration increases, the impact of RW on velocity uncertainty decreases. Finally, our finding that the selection of the stochastic noise model and velocity estimation are reliable for a time series with a minimum du-ration of 12 years.

For “The keywords used are not based on the taxonomy of remote sensing.”, our work is submitted to Special Issues “International GNSS Service Validation, Application and Calibration”, and our work in related to the topic of “Precise non-linear motion modelling of GNSS”, “GNSS signal processing”.

 

7. The literature review is incomplete, and the authors have not effectively presented their solution and methodology in alignment with the existing researches. The paper lacks a comprehensive review of relevant literature, which hinders the understanding of the context and significance of the study. Furthermore, the authors have not clearly outlined their proposed solution or methodology in relation to the research problem. This gap undermines the coherence and effectiveness of the paper, as it becomes difficult to assess the appropriateness and validity of the chosen approach.

Reply: We conducted detailed reference research in the introduction section of the article and cited 34 relevant references (about impact of time span and missing data on the noise model estimation of GNSS time series).

 

8. However, in this paper, the material and methods section lack comprehensive analysis and evaluation. The description of the noise is incomplete, and the model used to represent the noise is insufficient. This inadequate representation of the noise hampers the validity and reliability of the study. Additionally, the simulation results provided are incomplete, making it impossible to verify the findings. Overall, the lack of proper analysis, incomplete description of noise, and insufficient simulation results undermine the credibility and verifiability of the study.

Reply: Bos (2008) propose an estimation method that combined the Akaike information criteria (AIC) and Bayesian information criteria (BIC) noise model to overcome the bias of MLE (for details see he et al. 2019), where the larger the MLE value, the greater the bias. AIC and BIC are used to estimate the noise model. The smaller the AIC and BIC values, the closer the corresponding model is to the true model (He X,2017). Studies (Bozdogan H,1987) have shown that the model selected with the AIC could not converge to the true model. Other works (Wagenmakers et al,2014) have shown that BIC could not accurately identify the GGM models due to a divergence in the formula. Based on these results, we follow the conclusions of He et al (2019) recommending the use of an alternative information criterion, the BIC_tp:

                    

where L is the likelihood function, n is the length of the time series, and k is the number of variables in the model, v is the number of model parameters. More information can be obtained in Montillet, J.P. (2019). More information can be obtained in [27,37].

References as follows:

27. He, X., Bos, M. S., Montillet, J. P., & Fernandes, R. M. S. Investigation of the noise properties at low frequencies in long GNSS time series. J. Geodesy. 2019, 93, 1271-1282.
28. Montillet, J.P.; Bos, M. Geodetic time series analysis in earth sciences. Springer. Berlin/Heidelberg, Germany, 2019.

44.Bos, M. S., Fernandes, R.M.S., Williams, S.D.P. and Bastos, L. Fast error analysis of continuous GPS observations. J. Geodesy. 2008, 82, 157–166.

We have defined the four models FNRWWN (Flicker noise plus white noise and random walk noise), FNWN (Flicker noise plus white noise), GGMWN (generalized Gauss-Markov and white noise) and PLWN (power law noise and white noise) in the manuscript.

References as follows:

1. He, X., Bos, M. S., Montillet, J. P., Fernandes, R., Melbourne, T., Jiang, W., & Li, W. Spatial variations of stochas-tic noise properties in GPS time series. Remote Sens.2021, 13,4534.
2. Agnew, D. C. The time‐domain behavior of power‐law noises. Geophys Res Letters.1992, 19, 333-336.
3. Mandelbrot, B. B., & Van Ness, J. W. Fractional Brownian motions, fractional noises and applications. SIAM re-view, 1968, 10, 422-437.
4. Montillet, J. P., & Bos, M. S. Geodetic time series analysis in earth sciences. Springer.2019.

 

9. The simulation results are presented solely in the form of graphs without proper metric justification based on theory or the characteristics of the noise. The absence of a clear connection between the noise model and the results undermines the ability to provide a scientific justification for the findings. As a result, the simulation results appear more like a report rather than a scientific paper, lacking the necessary scientific rigor and analysis. Hence, based on the present state of the paper, I must recommend its rejection. authors can take a look at these papers and can follow suit:

Reply: We have added relevant content according to your kind comments in the appendix (e.g., GNSS daily time series, site Information of the analyzed 72 GNSS Station and GNSS time series simulation parameters).

 

10. Hence, based on the present state of the paper, I must recommend its rejection. authors can take a look at these papers and can follow suit:
11. Wu, H. Luo, H. Jia, F. Zhao, Y. Xiao and X. Gao, "Predicting the Noise Covariance with a Multitask Learning Model for Kalman Filter-Based GNSS/INS Integrated Navigation," in IEEE Transactions on Instrumentation and Measurement, vol. 70, pp. 1-13, 2021, Art no. 8500613, doi: 10.1109/TIM.2020
12. Giangregorio, M. di Bisceglie, P. Addabbo, T. Beltramonte, S. D'Addio and C. Galdi, “Stochastic Modeling and Simulation of Delay–Doppler Maps in GNSS-R Over the Ocean” in IEEE Transactions on Geoscience and Remote Sensing, vol. 54, no. 4, pp. 2056-2069, April 2016.
13. Gleason, J. Johnson, C. Ruf and C. Bussy-Virat, "Characterizing Background Signals and Noise in Spaceborne GNSS Reflection Ocean Observations," in IEEE Geoscience and Remote Sensing Letters, vol. 17, no. 4, pp. 587-591, April 2020,
14. Zhang, Y. Xu, R. Luo and Y. Mao, "Fast GNSS acquisition algorithm based on SFFT with high noise immunity," in China Communications, vol. 20, no. 5, pp. 70-83, May 2023.

Reply: Thanks for your comments, we checked it and added some citations in the revised manuscript.

Related References:

• Abatzoglou, J. T., McEvoy, D. J., & Redmond, K. T. The west wide drought tracker: drought monitoring at fine spatial scales. B.Am. Meteorol. Soc .2017, 98, 1815-1820.
• Xu C. Reconstruction of gappy GPS coordinate time series using empirical orthogonal functions. J. Geophys. Res. Solid Earth. 2016, 121, 9020-9033.
• Herring, T.A., King, R.W., McClusky, S.C. GAMIT Reference Manual. GPS analysis at MIT. Release 10.4. Massachusetts Institute Technology.2010a.
• Herring T A, King R W, McClusky S C. GLOBK. Global Kalman filter VLBI and GPS analysis program. Version, 2005, 10.
• Lichten, S.M., Bar-Sever, Y.E., Bertiger, E.I., Heflin, M., Hurst, K., Muellerschoen, R.J.,Wu, S.C., Yunck, T.P., Zumberge, J.F. GIPSY-OASIS II: a high precision GPS data processing system and general orbit analysis tool. Technology.2006. 2, 2–4.
• Dong D, Herring T A, King R W. Estimating regional deformation from a combination of space and terrestrial geodetic data. J. Geodesy, 1998, 72, 200-214.
• Bos, M. S., Fernandes, R.M.S., Williams, S.D.P. and Bastos, L. Fast error analysis of continuous GPS observations. J. Geodesy. 2008, 82, 157–166.
• Bozdogan H. Model selection and Akaike's information criterion (AIC): The general theory and its analytical extensions. Psychometrika, 1987, 52, 345-370.
• Wagenmakers, E. J., & Farrell, S. AIC model selection using Akaike weights. Psychon. B. Rev. 2014, 11, 192-196.

 

 

Author Response File: Author Response.docx

Reviewer 4 Report

General comments

The authors analyzed 72 stations from the Enhanced Solid Earth Science ESDR System. The time series data or velocity data for each site are not presented as a plot or table. As a result, it is impossible to check the simulation results. Most of the results shown in this manuscript cannot be checked with the current presentation.

The obtained results are based on completely ignoring the residual systematic errors in the time series. Will the obtained conclusions be adequate in this case?

The authors do not provide a detailed account of the limitations to their research, it has led the reviewer to be questioning the validity of the research and the appropriateness of the research methods used. Maybe I'm wrong.

Various terms are used without clear definitions: AIC and BIC, FNRWWN, FNWN, PLWN, GGMWN

Detailed comments

L125 “The simulated station time series range from 2 to 30 years (denoted in the following as 2a, 4a, 6a, 8a, 10a, 12a, 14a, 16a, 18a, 20a, 22a, 24a, 26a, 28a, 30a), with 100 stations simulated for each case”. How the simulation was practically carried out: description or outer sources.

 

L130 “Figure 2. The 30 year-long simulated time series with different noise models”. The situation with the PLWN model is incomprehensible.

 

Author Response

Response to Reviewer 4 Comments

Manuscript ID: remotesensing-2491080.

Title: Impact of Time Span and Missing Data on the Noise Model Estimation of GNSS time series

We would like to thank Assistant Editor Mr. Milos Miric and the anonymous reviewers for providing an opportunity to revise the manuscript. The comments and suggestions of the reviewers are all valuable and very helpful. We have studied them carefully and have made revisions to improve the manuscript.

Detailed corrections are listed below point by point:

1.The authors analyzed 72 stations from the Enhanced Solid Earth Science ESDR System. The time series data or velocity data for each site are not presented as a plot or table. As a result, it is impossible to check the simulation results. Most of the results shown in this manuscript cannot be checked with the current presentation.

Reply: Corrected. We have added i) time series plots in the appendix (see Figure A1. GNSS time series of GODE station.), ii) Table A1 gives the site information of the analyzed 72 GNSS station, iii) and Table A2 of GNSS time series simulation parameters.

 

2.The obtained results are based on completely ignoring the residual systematic errors in the time series. Will the obtained conclusions be adequate in this case?

Reply: This is a good question. Our previous work based on He et al 2017 (Review of current GPS methodologies for producing accurate time series and their error sources) explore the error sources on GNSS time series, and systematic errors is hard to eliminate completely. We use the ES3 GNSS time series developed by SOPCA in this work. Their team used the GNSS data processing strategy to model most of the error sources. However, the residual systematic errors are out the scope of this work. We will investigate it in the future work. Thanks.

 

3.The authors do not provide a detailed account of the limitations to their research; it has led the reviewer to be questioning the validity of the research and the appropriateness of the research methods used. Maybe I'm wrong.

Reply: The main work of this paper is the impact of time span and missing data on the noise model estimation of GNSS time series. The limitations to our research：One limitation is clearly to model the residual systematic errors as explained in the previous question. Another limitation is the presence of common mode error or the various loading effects which are also not the focus of this work. Here, the core of this work focuses only on the stochastic noise model. We will investigate the limitations of this article in the future studies.

 

4.Various terms are used without clear definitions: AIC () and BIC, FNRWWN, FNWN, PLWN, GGMWN.

Reply: We give the full names of the corresponding terms and relevant references.

AIC (Akaike information criteria), BIC (Bayesian information criteria), FNWNRW (Flicker noise plus white noise and random walk noise), FNWN (Flicker noise plus white noise), PLWN (power law noise and white noise), GGMWN (generalized Gauss-Markov and white noise).

References as follows:

• He, X., Bos, M. S., Montillet, J. P., & Fernandes, R. M. S. Investigation of the noise properties at low frequencies in long GNSS time series. J. Geodesy. 2019, 93, 1271-1282.
• Bos, M., Fernandes, R., Williams, S. and Bastos, L. The noise properties in GPS time series at European stations revis-ited. EGU General Assembly, Geophysical Research Abstracts. 2013, EGU2013-12825.
• Akaike H. A new look at the statistical model identification. IEEE Transactions on Automatic Control. 1974, 19,716-723.
• Schwarz G. Estimating the dimension of a model. The annals of statistics, 1978, 461-464.
• Bozdogan H. Model selection and Akaike's information criterion (AIC): The general theory and its analytical extensions. Psychometrika, 1987, 52, 345-370.
• Wagenmakers, E. J., & Farrell, S. AIC model selection using Akaike weights. Psychon. B. Rev. 2014, 11, 192-196.

 

5. L125 “The simulated station time series range from 2 to 30 years (denoted in the following as 2a, 4a, 6a, 8a, 10a, 12a, 14a, 16a, 18a, 20a, 22a, 24a, 26a, 28a, 30a), with 100 stations simulated for each case”. How the simulation was practically carried out: description or outer sources. L130 “Figure 2. The 30 year-long simulated time series with different noise models”. The situation with the PLWN model is incomprehensible.

Reply: We used Hector software (Bos et al., 2013) and the python appendices to simulate the GNSS time series. For each scene, we set the observation with 365.25*n samples (n mean year/a), e.g., 2a with 730 samples, and 30a with 10957 samples. We have added the description in the revised version.

 

 

 

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

Some of the issues mentioned in the first review were not understood or explained by the authors in the article.

Question 1: In the first question in 1st round review, I know that the purpose of many of the graphs already in the article is to clarify the relationship between the MISSING RATE and the noise model. Aslo, I notice that the authors write in conclusion (3) that MISSING DATA is independent of the noise model. This is consistent with the figure, but conflicts with the title. This is because the title emphasizes the impact of time span and missing data, but missing data is shown by the authors to have no impact.

So the current title is inappropriate. It could be changed to "a relationship study of ... and ... " Something like that.

Question 2:The author lists up to 3 levels of headings, and there is no stringing text between level 1 headings and level 3. This results in the article resembling an experimental report rather than a scholarly work. A correct form would be:
3, Results
This section ...
3.1 Impact of ... and ...
at first, we show the simlation model, and then, ...
3.1.1 simulation model
...

In all, I recognize the workload of this paper from the data processing level. After the above two issues have been adequately corrected, I have no further comments and recommend its acceptance.

 

The language of the article is not well presented, especially the analysis of the diagrams. It is recommended to use short sentences as much as possible for understanding. I would recommend the author to revise the language carefully, it is not only about whether the article could be accepted, but also about your peers' perception of your writing level.

Author Response

Thanks for your comments again.

Question 1:the current title is inappropriate. It could be changed to "a relationship study of ... and ... " Something like that.

Reply: Thank you for your suggestion. We modify the title to “The Relationship of Time Span and Missing Data on the Noise Model Estimation of GNSS Time Series”, we also modified the abstract.

 

Question 2: The author lists up to 3 levels of headings, and there is no stringing text between level 1 headings and level 3. This results in the article resembling an experimental report rather than a scholarly work. A correct form would be:

3. Results

This section ...

3.1 Impact of ... and ...

at first, we show the simulation model, and then, ...

3.1.1 simulation model

Reply: We added content and marked it as red, as follows:

3. Results

This section displays two aspects: 1) the results of simulated time span and missing data on the noise model; 2) impact of length and missing data on the noise model of the GNSS time series and discussion on the influence of these parameters on the estimated noise model.

3.1 Impact of Time Span and Missing data on the Noise Model.

We show the results of the selected noise model using the BIC_tp when varying the length of the simulation time series. Then, we simulate 100 time series with various lengths to discuss the impact of missing data on the noise 

Author Response File: Author Response.docx

Reviewer 3 Report

 Regrettably, the persisting issue remains unchanged as the authors' description has only added complexity to the already problematic aspects of noise description, noise modeling, and its suppression method. It is unclear how the authors can effectively investigate the presence and mitigate the adverse effects of noise when the entity of the noise is unknown! This problem was evident in the initial version and has not been adequately addressed in the resubmitted version. Therefore, despite their efforts, I must reject the paper and strongly recommend that the authors start afresh, ensuring a thorough and comprehensive approach to addressing the challenges associated with noise.

Maintaining a high standard of English language proficiency is important when it comes to remote sensing taxonomy.

Author Response

Dear reviewer, we understand your concern. The various stochastic noise models have been applied to the geodetic time series analysis since the late 90's with the pioneer work. However, we have developed a robust method which is based on the IC. We test the noise selection algorithm with 1/ the test for "false/true positive" 2/ with fitting the spectrum with our selected model. As we have already mentioned this approach has shown its robustness in various studies (Bennett et al., 2008; Bos et al., 2013; He et al., 2019; Bian et al., 2021; Zhai et al., 2022;).

• He, X., Bos, M. S., Montillet, J. P., & Fernandes, R. M. S. (2019). Investigation of the noise properties at low frequencies in long GNSS time series. Journal of Geodesy, 93(9), 1271-1282.
• Zhai, W., Zhu, J., Lin, M., Ma, C., Chen, C., Huang, X., ... & Yan, L. (2022). GNSS Data Processing and Validation of the Altimeter Zenith Wet Delay around the Wanshan Calibration Site. Remote Sensing, 14(24), 6235.
• Bian, Y., Yue, J., Ferreira, V. G., Cong, K., & Cai, D. (2021). Common mode component and its potential effect on GPS-inferred crustal deformations in Greenland. Pure and Applied Geophysics, 178(5), 1805-1823.
• Bennett, R. A. (2008). Instantaneous deformation from continuous GPS: Contributions from quasi-periodic loads. Geophysical Journal International, 174(3), 1052-1064.
• Bos, M. S., Fernandes, R. M. S., Williams, S. D. P., & Bastos, L. (2013). Fast error analysis of continuous GNSS observations with missing data. Journal of Geodesy, 87(4), 351-360.

Author Response File: Author Response.docx