the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 19 Jul 2024
Inverse Differential Equation Modeling of ENSO Prediction Based on Memory Kernel Functions
Abstract. The ENSO system is a complex climate pattern that is crucial in global climate systems and plays a key role in climate prediction. Both statistical methods and numerical models have been dedicated to achieving accurate predictions of ENSO variations; however, there is still a considerable gap in practical applications. Therefore, we proposed a memory kernel function-based approach to solve the inverse problem of ENSO time-varying systems. We attempted to establish differential equations by constructing memory vectors composed of multiple initial values to describe the evolutionary characteristics of this complex system. Unlike traditional inverse problem-solving methods, our research scheme delved into the inherent properties exhibited by ENSO, such as memory and periodicity, and embedded these properties as specific targets in differential equations. By leveraging the flexibility of evolutionary algorithms to solve mathematical problems, we achieved targeted modeling of the ENSO system. The results demonstrate that this scheme overcomes the limitations of traditional differential equations with a single initial value and extends these equations to memory vector equations based on multiple initial values. This not only enhances our ability to describe the evolutionary laws of complex systems but also improves the timeliness and reliability of ENSO predictions, achieving encouraging results.
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RC1: 'Comment on egusphere-2024-2181', Anonymous Referee #1, 19 Sep 2024
This paper presents a novel approach to ENSO prediction using inverse differential equation modeling based on memory kernel functions (MKFs). The authors utilize wavelet analysis to detect periodic and nonperiodic features in sea surface temperature (SST) data, constructing both periodic and inertial MKFs to capture system dynamics. They employ evolutionary algorithms, transfer entropy, and correlation coefficients to estimate model parameters and evaluate links between observable variables.In general, the methodology is innovative and shows potential for improving ENSO prediction accuracy. However, there are significant issues with paper structure, clarity, and presentation of results that need to be addressed. I suggest a major revision to address the following concerns:ÂMajor comments:
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The introduction lacks sufficient background on ENSO. As a primary subject of the study, ENSO's specific characteristics and importance should be more thoroughly introduced.
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The paper would benefit from a more comprehensive literature review and additional citations. In particular, recent advancements in ENSO prediction methods should be cited to provide a stronger context for this work. This will help readers understand how the proposed method compares to and builds upon existing approaches.
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The theoretical foundation for using MKFs needs more explanation. How do these functions specifically relate to ENSO dynamics? A clearer link between the mathematical formulation and physical processes would enhance the paper's impact.
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The comparison with existing ENSO prediction methods is lacking. How does this approach compare to current state-of-the-art models in terms of accuracy and computational efficiency?
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The discussion of results (Section 4) is quite brief. A more in-depth analysis of the implications of these findings for ENSO prediction and climate modeling, in general, would be valuable.
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The overall structure of the paper is disorganized, particularly in the Results section. The "Correlation coefficient" subsection should not be presented as a standalone section on par with other subsections. This structural issue makes it difficult to follow the logical flow of the analysis and results.
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There is a discrepancy between the conclusion and the results presented. Specifically, the conclusion states that the method "improves the precision and dependability of differential equation models," but this claim is not clearly demonstrated or quantified in the results section. The authors need to either provide evidence for this claim in the results or remove it from the conclusion.
Minor comments:-
Figure 1 is difficult to interpret. Consider adding more detailed captions and explanations in the text.
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The methodology section could benefit from a flow chart or schematic diagram to illustrate the overall modeling approach.
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Some technical terms (e.g., "inertial MKF", and "ODE-MKF") are not well defined upon first use. Providing clear definitions of all key terms will be better.
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In line 151, there should be a space between "Nino3.4" and "was".
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After Equation 4, a comma should be added, and there should not be a space before "where".
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Equation 4 requires more explanation. The time unit of t and the time step should be specified.
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The labeling and presentation of Figures 2-3 are unclear and confusing, particularly the x-axis labels showing "Year/Month". This lack of clarity in crucial figures hinders the reader's understanding of the results and makes it difficult to interpret the data presented.
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In Section 3.2, the units of time-related values are unclear throughout. It is not apparent what Δt represents or what time units are being used. This lack of clarity makes it difficult to interpret the results and understand the temporal scales of the analysis.
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"ff" in line 288 is not explained, is it a typo?
Citation: https://doi.org/10.5194/egusphere-2024-2181-RC1 - AC1: 'Reply on RC1', Qianrong Ma, 08 Nov 2024
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RC2: 'Comment on egusphere-2024-2181', Anonymous Referee #2, 06 Oct 2024
This paper performs data-driven predictions of ENSO by learning a function of past values. Unfortunately, the paper as it stands has severe problems, and I cannot recommend publication. Namely, (a) the method is described very vaguely; (b) I am unclear about whether this method can actually be applied for forecasting or is just a regression from SST to the ENSO index; (c) I am unclear about what novelty this paper has over past ones from the same group; (d) there is no comparison to other methods in the literature; (e) the verification of the method is inadequate; and (f) the overall paper is quite unclear.Â
Major issues:
1. The authors state on lines 182 and 188 that their functions take as input *observed* SST from previous and current months. Unless this was misstated and the authors meant the *predicted* SST (or observed if the lead time is less than the lag of SST one wants to use), this means that the method cannot actually be applied as a forecast method without the use of future information, since it can only be applied up to a lead time of the minimum lag of observed SST taken as input (in this case it would only be able to predict for one step, since it needs the current observed SST). If this is true, then this is not a forecast method, but rather a regression model mapping from observed SST to the ENSO index.
2. The description of the genetic algorithm is very unclear. The authors say that a genetic algorithm was used to pick the functions comprising the model and their parameters. First of all, the actual set of functions that can possibly be selected, as well as what the parameters represent, is not stated anywhere. The authors say that "a subset of the parameter vectors was selected based on their fitness for crossover and mutation", but the selection criteria for this subset is not stated. The authors then state that "[p]arts of the two parameter vectors were combined through crossover operations, and genes were randomly altered through mutation operations to generate new parameter vectors", but don't state how the crossover nor mutation operations work.
3. What is the novelty of this paper over a similar paper from the same group, Ma et al. (2023)? That paper seems to use the same method and applies it to ENSO prediction. Similarly, that paper is quite vague about the algorithm.
4. The validation of this method is insufficient. The authors only evaluate it on two different initialization dates. More initial dates should be considered, and the skill measures averaged over these. Moreover, the authors do not look at how the skill decays with lead time, which is crucial.
5. I did not understand the point of the whole section on transfer entropy, or actually how it was applied. Later the authors use the autocorrelation to determine relevant timescales, which seems to be sufficient.
6. There is no comparison of skill to other methods, either dynamical or statistical/machine learning, to see if this one is competitive. For a recent benchmark, see Ham et al (2019).
7. I don't understand what "inertial memory" means, or what Figure 6 is showing.Other issues:
1. What numerical method is used to integrate the model? Presumably a time-step of 1 month is used?
2. Eq. 1 (and 2) is written as a regular ordinary differential equation, but if it a function of previous points in its own history then it should be written as a delay differential equation.
3. The axes of Figure 1 are not labelled.
4. Why do Figure 2, 3 and 6 have the curve in the test period go from red to purple? Is there some significance to this?
5. In Figure 2, what is the difference between panel a and c?
6. Why do the "high-frequency" components in Figures 2 and 6 actually look low-frequency?
7. "Through the characterization of memory using the tanh function": this was never described. Similarly, the authors say that "[t]he high-frequency component primarily comprised retrospective initial values that included certain nonlinear terms", which is also not described.
8. "These equations were designed to capture the dynamic behavior of climate systems, where the future state depends not only on the present conditions but also on a series of past states." This is not true; in fact, the climate is a Markovian system. However, a partially observed Markovian system (e.g., just the ENSO index) will act in a non-Markovian way. This can be understood through the Mori–Zwanzig formalism or Takens' embedding theorem; see, e.g., Levine & Stuart (2022).
9. In the introduction, "short-term forecasts" is used, as far as I can tell, to refer to forecasts on timescales shorter than those involved in climate projections (i.e., decadal and longer). This is not standard terminology. I would suggest to be more specific, e.g., refer to seasonal or interannual forecasts.
10. The reference for "Predictability of El Niño over the past 148 years" has the wrong year: it should be 2004, not 2024.
11. "combining the strengths of dynamic and statistical approaches has emerged as a promising strategy": there is a great deal of work in this area. See for example Bach et al. (2024) and the references therein.
12. There has also been a great deal of work on time-series forecasting methods that make use of memory, such as recurrent neural networks and reservoir computers. Some of this previous work should be cited. See for example Pathak et al. (2018).
13. The authors claim that numerical models are "powerful for long-term projections but often less effective for short-term predictions involving rapid changes and smaller scales". I don't understand this claim. In fact, numerical models have, until very recently, outperformed data-driven methods for medium-range weather forecasting (3 to 10 days lead time). Data-driven methods have generally been more competitive for subseasonal-to-seasonal prediction (two weeks to several months).
14. Why in Eq. 1 do the authors use F and for Eq. 2 they use F_a?
15. On line 178, the authors write MKF(sin t , X). I assume MKF(t , X) was meant?
16. ENSO should be defined and explained in the introduction.References
– Bach, E., Krishnamurthy, V., Mote, S., Shukla, J., Sharma, A. S., Kalnay, E., & Ghil, M. (2024). Improved subseasonal prediction of South Asian monsoon rainfall using data-driven forecasts of oscillatory modes. Proceedings of the National Academy of Sciences, 121(15), e2312573121. https://doi.org/10.1073/pnas.2312573121
– Ham, Y.-G., Kim, J.-H., & Luo, J.-J. (2019). Deep learning for multi-year ENSO forecasts. Nature, 573(7775), 568–572. https://doi.org/10.1038/s41586-019-1559-7
– Levine, M., & Stuart, A. (2022). A framework for machine learning of model error in dynamical systems. Communications of the American Mathematical Society, 2(07), 283–344. https://doi.org/10.1090/cams/10
– Ma, Q., Sun, Y., Wan, S., Gu, Y., Bai, Y., & Mu, J. (2023). An ENSO Prediction Model Based on Backtracking Multiple Initial Values: Ordinary Differential Equations–Memory Kernel Function. Remote Sensing, 15(15). https://doi.org/10.3390/rs15153767
– Pathak, J., Hunt, B., Girvan, M., Lu, Z., & Ott, E. (2018). Model-Free Prediction of Large Spatiotemporally Chaotic Systems from Data: A Reservoir Computing Approach. Physical Review Letters, 120(2), 024102. https://doi.org/10.1103/PhysRevLett.120.024102Citation: https://doi.org/10.5194/egusphere-2024-2181-RC2 - AC2: 'Reply on RC2', Qianrong Ma, 08 Nov 2024
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