Oceanic Mesoscale Eddy Fitting Using Legendre Polynomial Surface Fitting Model Based on Along-Track Sea Level Anomaly Data

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript presents a valuable study comparing Legendre polynomial surface fitting with bicubic quasi-uniform B-spline surface fitting for sea surface height data. The authors highlight the need for improved direct fitting methods and demonstrate the viability of Legendre polynomial surface fitting. The introduction of a regional stitching methodology to maintain fitting efficacy over large areas is a notable contribution. However, the manuscript could benefit from a brief discussion on specific scenarios where the Legendre method might be preferred and a comparison of computational efficiency between the methods. Overall, the study is well-organized and offers significant insights. I recommend its acceptance with major revisions to address the mentioned points.

 

1. The manuscript's review of previous research, particularly on mesoscale eddies, is somewhat weak. It is recommended to include more recent research findings in this area.

 

2. It is suggested to delete lines 171-173, as they are not suitable for a scientific paper.

 

3. The specific properties of Legendre polynomials and their relevance to the subsequent research content should be further elaborated. It is suggested to add a description of how these properties contribute to the study.

 

4. The formatting of the equations is inconsistent, with some being centered and others right-aligned. This needs to be corrected for uniformity.

 

5. Therefore, for mesoscale eddies characterized by static patterns, data from 9 to 13 days are considered optimal. What causes this phenomenon?

 

6. The y-axis of Figure 7 should be labeled as "frequency count" rather than "frequency."

 

7. Does Figure 10a represent the error, or is it simply the difference between the two?

 

8. Many readers may be interested in using the new eddy identification algorithm proposed in this paper. Are the authors willing to make the code publicly available, perhaps by hosting it on a website for downloading and use by others?

Author Response

Comments 1: [The manuscript presents a valuable study comparing Legendre polynomial surface fitting with bicubic quasi-uniform B-spline surface fitting for sea surface height data. The authors highlight the need for improved direct fitting methods and demonstrate the viability of Legendre polynomial surface fitting. The introduction of a regional stitching methodology to maintain fitting efficacy over large areas is a notable contribution. However, the manuscript could benefit from a brief discussion on specific scenarios where the Legendre method might be preferred and a comparison of computational efficiency between the methods. Overall, the study is well-organized and offers significant insights. I recommend its acceptance with major revisions to address the mentioned points.]

Response 1: 

We appreciate your positive assessment of the value of our study in comparing Legendre polynomial surface fitting with bicubic quasi-uniform B-spline surface fitting for sea surface height data. The study employed a desktop computer featuring an Intel(R) Core(TM) i7-9700 processor and 32.0 GB of RAM. It is important to acknowledge that the runtime of the program may vary based on the configuration of the computer used. However, on the same computer, the time required to complete the fitting using the Legendre method is consistently less than that required by the bicubic quasi-uniform B-spline method. Following multiple repeated experiments, for the area depicted in Figure 6, the time required for Legendre polynomial surface fitting is approximately 74.5% of the time needed for bicubic quasi-uniform B-spline surface fitting. This explanation can be found in lines 467-474 of the revised manuscript. In practical experiments, the accuracy of the Legendre method is comparable to that of the bicubic quasi-uniform B-spline method, indicating that the Legendre method is generally optimal. We have endeavored to identify instances where the Legendre method may exhibit superior accuracy. Thus far, all areas in which the Legendre method has been applied have demonstrated results comparable to those of the quasi-uniform bicubic B-spline method. Our research will persist in exploring this domain further. We agree that more studies would be useful to understand the details of enhancement.

Comments 2: [The manuscript's review of previous research, particularly on mesoscale eddies, is somewhat weak. It is recommended to include more recent research findings in this area.]
Response 2: 

Thank you for identifying the shortcomings and providing constructive suggestions. In response, we have incorporated recent research findings on mesoscale eddies, highlighting their seasonal variations and depth-dependent characteristics. This change can be found in lines 62-92 of the revised manuscript. In the introduction section of the article, we have incorporated pertinent content as outlined below:

Mesoscale eddies are thought to arise through various mechanisms, including direct wind forcing, baroclinic instability, and barotropic instability, with direct wind forcing being a predominant factor [6]. A fundamental characteristic of these eddies is their rotational speed or eddy intensity, which exhibits notable seasonal variability. In the Northern Hemisphere, eddy intensity peaks in spring and reaches a minimum in autumn, whereas the Southern Hemisphere displays an inverse pattern. This variability is especially pronounced within the tropical-subtropical transition zones (15°-30° latitude) in regions such as the western Pacific, northwestern Atlantic, and eastern Indian Ocean, primarily due to rapid changes in baroclinic instability [7]. Most mesoscale eddies in the upper ocean layer, whether cyclonic or anticyclonic, have lifespans shorter than 30 days. Mesoscale eddies in the ocean typically propagate westward at the phase velocity of non-dispersive baroclinic Rossby waves. Cyclonic and anticyclonic eddies tend to exhibit slight deviations towards the polar and equatorial regions, respectively. The majority of these eddies are characterized by nonlinear dynamics [8]. Additionally, the size of cyclonic eddies remains fairly constant with depth, whereas anticyclonic eddies exhibit a slight reduction in size [9]. The influence of these eddies on sea surface wind speeds and heat fluxes is particularly pronounced during the winter season, whereas their impact on precipitation rates is more substantial in the summer. The magnitude of sea surface temperature (SST) anomalies induced by mesoscale eddies is modulated by the strength of the background SST gradient field, with a stronger gradient in winter leading to larger anomalies and a weaker gradient in summer resulting in smaller anomalies [10]. Furthermore, mesoscale eddies play a significant role in the dynamics of monsoon rainfall, influencing the onset and cessation of the rainy season within the Indian monsoon region [11]. These eddies are characterized by horizontal scales ranging from tens to hundreds of kilometers and vertical penetration to depths exceeding a thousand meters below the thermocline [12]. As a result, they play a significant role in the transport of momentum and energy within the ocean [13,14]. Their lifespan typically spans from several days to weeks, with generation and dissipation processes intricately linked to oceanic circulation dynamics and the broader influences of climate change [15]. Governed by the quasi-geostrophic vorticity conservation equation, the motion of mesoscale eddies holds critical importance in altering the spatial distribution characteristics of oceanic elements [16,17,18].

References

6. Baiyang, C.; Lingling, X.; Quanan, Z.; Lei, Z.; Lei, W.; Baoxin, F.; Zipeng, Y. Seasonal variability of mesoscale eddies in the Banda Sea inferred from altimeter data. Acta Oceanol. Sin. 2021, 39, 11-20.
7. Yongcan, Z.; Yue, F.; Shuangwen, S.; Libao, G.; Yang, Y.; Guijun, G. Seasonal variation of mesoscale eddy intensity in the global ocean. Acta Oceanol Sin 2024, 43, 48-58.
8. Yang, Y.; Liang, X.S. The intrinsic nonlinear multiscale interactions among the mean flow, low frequency variability and mesoscale eddies in the Kuroshio region. Science China Earth Sciences 2019, 62, 595-608.
9. Zhang, Y.; Hu, C.; Mcgillicuddy, D.J.; Liu, Y.; Barnes, B.B.; Kourafalou, V.H. Mesoscale eddies in the Gulf of Mexico: A three-dimensional characterization based on global HYCOM. Deep-Sea Research Part II 2024, 215, 105380.
10. Bowen, S.; Baofu, L.; Jingyu, Y.; Yuqi, Z.; Shuo, Z. Seasonal variation of atmospheric coupling with oceanic mesoscale eddies in the North Pacific Subtropical Countercurrent. Acta Oceanol. Sin. 2022, 41, 109-118.
11. Greaser, S.R.; Subrahmanyam, B.; Trott, C.B.; Stork, H.L.R. Interactions Between Mesoscale Eddies and Synoptic Oscillations in the Bay of Bengal During the Strong Monsoon of 2019. Journal of Geophysical Research. Oceans 2020, 125.
12. Duan, Y.; Zhang, H.; Chen, X.; Zhou, M. A Gaussian Function Model of Mesoscale Eddy Temperature Anomalies and Research of Spatial Distribution Characteristics. Remote Sens. 2024, 16.
13. Xu, L.; Gao, M.; Zhang, Y.; Guo, J.; Lv, X.; Zhang, A. Oceanic Mesoscale Eddies Identification Using B-Spline Surface Fitting Model Based on Along-Track SLA Data. In Remote Sensing, 2022; Volume 14.
14. Wyrtki, K.; Magaard, L.; Hager, J. Eddy energy in the oceans. Journal of Geophysical Research (1896-1977) 1976, 81, 2641-2646, doi: https://doi.org/10.1029/JC081i015p02641.
15. Adams, D.K.; Mcgillicuddy, D.J.; Zamudio, L.; Thurnherr, A.M.; Liang, X.; Rouxel, O.; German, C.R.; Mullineaux, L.S. Surface-Generated Mesoscale Eddies Transport Deep-Sea Products from Hydrothermal Vents. Science 2011, 332, 580-583, doi: 10.1126/science.1201066.
16. Zhang, J.; Deng, K.; Nie, T.; Ren, K.; Song, J. Overview on ocean mesoscale eddy detection and identification based on machine learning. Computer Engineering and Science 2021, 43, 2115-2125.
17. Zhang, Z. Submesoscale Dynamic Processes in the South China Sea. Ocean-Land-Atmosphere Research 2024, 3, 45, doi: 10.34133/olar.0045.
18. Weifang, J.; Chujin, L.; Junyang, H.; Qicheng, M.; Haibin, L.; Yuntao, W.; Feilong, L.; Xiaoyan, C.; Xiaohui, L. Modulation Effect of Mesoscale Eddies on Sequential Typhoon-Induced Oceanic Responses in the South China Sea. Remote Sens. 2020, 12, 3059.

Comments 3: [It is suggested to delete lines 171-173, as they are not suitable for a scientific paper.] Response 3: We agree with your assessment., and these sentences have been deleted. Comments 4: [The specific properties of Legendre polynomials and their relevance to the subsequent research content should be further elaborated. It is suggested to add a description of how these properties contribute to the study.] Response 4: 

Your suggestion really means a lot to us. We have incorporated an explanation on how the orthogonality and recursive formula of Legendre polynomials can facilitate and promote research, and this explanation has been included immediately following the properties section of Legendre polynomials. Orthogonality ensures the independence of each coefficient, thereby preventing errors from interfering with one another. Furthermore, orthogonality often leads to a sparse or diagonalized coefficient matrix, which significantly improves computational efficiency. Orthogonality mitigates the risk of ill-conditioned matrix equations and enhances the accuracy of the results. The use of recursive relationships enhances the computational efficiency of higher-order Legendre polynomials. By utilizing the initial polynomial, subsequent higher-order polynomials can be computed iteratively through the recursive relationship, obviating the need for solving explicit expressions for each polynomial. This approach not only simplifies the computation but also reduces the accumulation of numerical errors that are common in direct calculations of high-order polynomials. This change can be found in lines 250-260 of the revised manuscript.

Comments 5: [The formatting of the equations is inconsistent, with some being centered and others right-aligned. This needs to be corrected for uniformity.] Response 5: Thank you for identifying these inconsistencies. The format has been duly adjusted to ensure uniformity. Comments 6: [Therefore, for mesoscale eddies characterized by static patterns, data from 9 to 13 days are considered optimal. What causes this phenomenon?] Response 6: Thank you for your inquiry. Utilizing data for fewer than 9 days may lead to notable fitting errors due to the limited number of observation points. Conversely, employing data spanning more than 13 days could yield distorted fitting outcomes attributable to the substantial time gap between the earliest and latest observations, as well as errors arising from the motion of eddies. This change can be found in lines 358-360 of the revised manuscript. Comments 7: [The y-axis of Figure 7 should be labeled as "frequency count" rather than "frequency."] Response 7: Corrected. Comments 8: [Does Figure 10a represent the error, or is it simply the difference between the two?] Response 8: The focus here should be on highlighting the distinctions between the two rather than emphasizing any discrepancies. We have made a more accurate modification accordingly. This change can be found in the title of Figure 10a in the revised manuscript. Comments 9: [Many readers may be interested in using the new eddy identification algorithm proposed in this paper. Are the authors willing to make the code publicly available, perhaps by hosting it on a website for downloading and use by others?] Response 9: Thank you for your suggestion. We do not plan to make the code available on the website. However, if any readers express interest in our research and contact us via email, we will be pleased to provide all relevant data and code upon request.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

“Surface fitting model based on along track SLA data” authored by Kong et al.

Data processing method of sea level anomaly is meaningful for better understanding oceanic dynamics. The traditional processing data may bring large uncertainties, authors utilized Legendre polynomial surface fitting method to explore grid and along-track SLA data to diminish the errors. It’s an interesting work but needs some revisions.

Major revisions:

1.      CMEMS has L4 products of SLA, authors should compare the Legendre polynomial surface fitting method derived SLA with the L4 products. And show the mean bias and standard deviations of these two products.

2.      In the introduction, the description of mesoscale eddies seems irrelevant to the subject. The SLA is not only important to the mesoscale eddies but also to the large-scale current, submesoscale process and so on. So the introduction should include the importance, calculation, and the observation methods of SSH/SLA, rather than mesoscale eddies.

Minor Revisions,

1.      In the title, SLA should give a full name

2.      Usually, no citations in the abstract

3.      Line 171-172, the sentences seem faulty.

4.      Line 343, “hover” should be “over”?

5.      All the variables should be in formula format

6.      Figures 6, 10, 11 should have the same colormap.

Comments on the Quality of English Language

Could be improved after minor revisions.

Author Response

Comments 1: [Data processing method of sea level anomaly is meaningful for better understanding oceanic dynamics. The traditional processing data may bring large uncertainties, authors utilized Legendre polynomial surface fitting method to explore grid and along-track SLA data to diminish the errors. It’s an interesting work but needs some revisions.] Response 1: We feel great thanks for your insightful comments and for recognizing the significance of our work on the data processing method of sea level anomaly. We also acknowledge your suggestion that our manuscript requires some revisions, and we have addressed your specific suggestions in the subsequent section to enhance the clarity and robustness of our study. Comments 2: [CMEMS has L4 products of SLA, authors should compare the Legendre polynomial surface fitting method derived SLA with the L4 products. And show the mean bias and standard deviations of these two products.] Response 2: We sincerely thank you for your constructive comments. Mean deviation and standard deviation have been included in two specific areas concerning the comparison with L4 products. We have incorporated the following pertinent content: Within the depicted region in Figure 9, the mean bias for these two products is 0.34, while the standard deviation is 2.98. This change can be found in lines 533-534 of the revised manuscript. And within the depicted region in Figure 11, the mean bias for these two products is 0.41, while the standard deviation is 13.23. This change can be found in lines 565-566 of the revised manuscript. Comments 3: [In the introduction, the description of mesoscale eddies seems irrelevant to the subject. The SLA is not only important to the mesoscale eddies but also to the large-scale current, submesoscale process and so on. So the introduction should include the importance, calculation, and the observation methods of SSH/SLA, rather than mesoscale eddies.] Response 3: 

Thank you for your suggestion. In response, we have integrated the significance, calculation methods, and observational approaches concerning SSH and SLA into the introduction section. It is essential to introduce SLA data, given its relevance to the research topic. Moreover, the mesoscale eddies serve as the central physical phenomenon under investigation and merits retention in our study. This change can be found in lines 30-59 of the revised manuscript. In the introduction section of the article, we have incorporated pertinent content as outlined below:

Sea Level Anomaly (SLA) denotes the deviation of sea level from its long-term average and is a crucial metric for analyzing ocean dynamic processes and climate change. SLA data are essential for understanding ocean circulation patterns, which significantly influence the global climate system. By examining SLA variations, researchers can track the oscillations of ocean currents and their role in global heat distribution. As global temperatures rise, the resultant melting of glaciers and ice shelves contributes to sea level rise, with SLA data providing direct evidence of these phenomena. Long-term SLA records are invaluable for identifying trends in sea level rise, which is vital for assessing and mitigating the long-term effects of climate change [1].

The computation of SLA involves several steps. Initially, the actual Sea Surface Height (SSH) is measured using satellite altimetry. Subsequently, the Mean Sea Surface Height (MSSH) is derived from long-term SSH data. SLA is then calculated as the difference between SSH and MSSH. When integrating data from multiple satellite missions, it is imperative to perform data fusion and cross-calibration to ensure consistency.

SLA observations predominantly utilize satellite altimetry technology. Satellite altimeters ascertain sea level height by emitting microwave pulses toward the Earth's surface and measuring the time delay of the reflected signals. Additionally, tide gauges installed along coastlines provide precise local sea level data, which complement satellite altimeter measurements. By synthesizing these observational techniques and employing advanced data processing methods, researchers can generate high-precision and high-resolution SLA datasets, thereby furnishing a reliable foundation for oceanographic and climate research [2].

Peng et al. (2020) examined the accuracy improvement of high-frequency satellite altimeter sea level anomaly (SLA) data by mitigating sea state bias (SSB) and correlation errors within the 1 Hz frequency range [3]. Their study evaluated the precision of 20 Hz SLA estimations using Jason-1/2/3 and Sentinel-3A data within a 100-kilometer radius of the Australian coastline. The findings demonstrated that composite SSB correction significantly enhances data accuracy. Zhang et al. (2020) reviewed methodologies for identifying mesoscale eddies using satellite altimeter data, including methods based on closed SLA contour lines, Okubo-Weiss number, Winding Angle, and flow vectors [4]. Yu et al. (2021) analyzed the spatial distribution and temporal changes of SLA in the South China Sea from 1993 to 2017 using satellite altimeter data. Their analysis employed techniques such as linear regression, Winters exponential smoothing, and empirical mode decomposition to elucidate the spatiotemporal characteristics of sea level changes in the region [5].

References

1. , K.S.; D., N.S.; Matthias, P.; S., S.J.; Klaus, A.; L., C.H.; Michael, S.; Pepijn, B.; G., L.S.A.; Michael, C. Impacts of Variations in Caspian Sea Surface Area on Catchment‐Scale and Large‐Scale Climate. Journal of Geophysical Research: Atmospheres 2021, 126.
2. Climate Modeling; Studies from Russian Academy of Science in the Area of Climate Modeling Described (Simulation of the Spatiotemporal Variability of the World Ocean Sea Surface Hight by the INM Climate Models). Global Warming Focus 2016.
3. Peng, F.; Deng, X. Improving precision of high-rate altimeter sea level anomalies by removing the sea state bias and intra-1-Hz covariant error. Remote Sens. Environ. 2020, 251, 112081.
4. Zhang, Y.; Wang, N.; Zhou, L.; Liu, K.; Wang, H. The Surface and Three-dimensional Characteristics of Mesoscale Eddies: A Review. Advances in Earth Science, 2020, 35(6): 568-580, doi: 10.11867/j.issn.1001-8166.2020.050
5. Yu, R.; Xu, H.; Liu, B. Analysis of Spatial and Temporal Variation of Sea Level in South China Sea Based on Satellite Altimeter Data (In Chinese). Journal of Ocean Technology, 2021

Comments 4: [In the title, SLA should give a full name.] Response 4: As suggested by the reviewer, we have corrected it. Comments 5: [Usually, no citations in the abstract.] Response 5: Corrected. Comments 6: [Line 171-172, the sentences seem faulty.] Response 6: We are truly sorry for the careless mistake we made. Thank you for your suggestion, and these sentences will be deleted. Comments 7: [Line 343, “hover” should be “over”?] Response 7: Thank you for highlighting the ambiguity. The term "hover" has been revised to "are", reflecting an error rate of approximately 1%. This change can be found in line 410 of the revised manuscript. Comments 8: [All the variables should be in formula format.] Response 8: Corrected. Comments 9: [Figures 6, 10, 11 should have the same colormap.] Response 9: Corrected.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors provided detailed answers to my questions. Regarding the article's code, they stated it will be shared upon email request from readers. I understand and accept their decision, and hope they honor their promise to share the code when requested.

Author Response

Comments 1: [The authors provided detailed answers to my questions. Regarding the article's code, they stated it will be shared upon email request from readers. I understand and accept their decision, and hope they honor their promise to share the code when requested.]

Response 1: 

Thank you for acknowledging our response, and we appreciate your understanding and support. We are committed to fulfilling our promise. Should any readers express interest in our research and reach out via email, we will gladly provide all relevant data and code upon request.

Author Response File: Author Response.pdf