the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 10 Jul 2025
Why does the signal-to-noise paradox exist in seasonal climate predictability?
Abstract. Estimates of the potential predictability limit (PPL) for seasonal climate, typically based on a perfect model framework, sometimes encounter challenges of being paradoxical, as actual skill surpasses the PPL. The signal-to-noise paradox (SNP) gets its name from the use of model-based signal-to-noise ratios to estimate the PPL. Here, we study seasonal climate predictability in the tropical and subtropical regions during the boreal summer (June to September), with a focus on the SNP. We estimate PPL within the perfect model framework, only considering error growth from initial conditions. Signal and noise components display temporal non-orthogonality and a weak association between estimates of PPL and actual prediction skill, contradicting its intended purpose. Moreover, paradoxical regions do not align with significant correlations between signal and noise, indicating that the accurate separation of seasonal forecasts into signal and noise components alone is not sufficient to avoid paradoxes. We have also demonstrated that sub-seasonal components, which are building blocks of seasonal mean, substantially contribute to seasonal anomalies in association with major global predictors. The co-variability between sub-seasonal components and seasonal anomalies is wide-ranging and often skewed compared to observations, thereby influencing seasonal prediction skills and PPL. Therefore, a robust PPL estimation should consider errors from initial conditions and model-related factors such as physics, dynamics, and numerical methods. In this context, we propose a novel method to estimate the PPL of seasonal climate, which can be free from paradoxical situations.
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CC1: 'Comment on egusphere-2025-1683', Youmin Tang, 07 Sep 2025
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AC1: 'Reply on CC1', S.K. Saha, 09 Sep 2025
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We thank you for your comments.
Here are point wise reply:
1. Lack of Clarity in Methods and Logical Exposition The manuscript suffers from poor readability and methodological rigor. Critical details are missing, for example:
Figure 2 presents correlation coefficients between signal and noise components, but it does not specify how these components were extracted from the raw data or model output;
Reply: In equation 3, which is based on Rowell et al. (1995), considers that total variance is the sum of external (i.e. signal) and internal variance (i.e. noise). Lines 104-106 clearly mention, what are signal, noise and their ratio, i.e. signal-to-noise ratio (SNR). Equations 1-3 and related description in lines 106-118 describe, how it can be calculated from model output variable (assumed ‘x’ in equations) . As per equation 3 or 4 signal and noise components are considered to be orthogonal.
In result section, line 197-1998 “This comprehensive analysis is conducted on seasonal (June-July-August-September) averaged rainfall, SST, MSLP, and 2-metre air temperature (land region) data of 1981-2021 …. “. So, here rainfall, SST, MSLP etc are ‘x’ variable in equations 1,2. This is a well known method and majority of predictability estimates are based on this method. We have cited several papers/books for further description (e.g. Rowell et al. (1995); Rowell, 1998; Kang and Shukla, 2006; Westra and Sharma, 2010).Figure 3 abruptly introduces the Niño3.4 index and regional precipitation without clarifying the research objective or the definition of "correlation skill" (e.g., does it refer to the correlation between the ensemble mean forecast and observations?). The role of Niño3.4 (as a predictor or a reference field) remains unexplained.
Reply: ENSO is the dominant mode of global climate variability and the primary large-scale predictor of both the Indian summer monsoon and tropical Pacific variability. Therefore, the Niño3.4 index was included to examine, by default, how the predictability of the predictor (Niño3.4) and the predictand (regional precipitation) compares with the actual skill achieved by the model. In other words, it provides a baseline for assessing the extent to which monsoon predictability arises from ENSO. Since ENSO remains the single most influential predictor of Indian summer monsoon rainfall, its inclusion is both natural and necessary. Therefore, we chose not to elaborate much, why we are using Niño3.4. Having said these, we are open to add in the manuscript briefly reason for using Niño3.4
Such breaks in logical flow and ambiguous methodological descriptions make it impossible for readers to understand the study process or verify the conclusions, severely violating standards of scientific writing.
We thank you for your comments, certainly we can add brief description to make it clearer to the readers.
2. Conceptual Confusion and Misuse of Terminology The manuscript contains inaccuracies in fundamental scientific expression:
It conflates the concepts of "Analysis of Variance (ANOVA)" and "Signal-to-Noise Ratio." ANOVA is fundamentally a statistical significance testing method, while the manuscript merely borrows its variance decomposition framework to estimate the signal-to-noise ratio without clarifying the distinction, which may mislead readers;
Reply: We do not fully agree with this statement. ANOVA is not solely a tool for statistical significance testing; in practice, it is widely applied for variance decomposition and uncertainty analysis (e.g. Rowell et al. (1995); Rowell, 1998; Kang and Shukla, 2006; Saha et al., 2016a, 2016b ). In fact, the calculation of the ‘signal-to-noise’ ratio is a well-established application that makes direct use of the ANOVA framework.
The term "sub-seasonal scale" is poorly defined. Although filtering bands (e.g., synoptic scale, MJO) are mentioned later, the temporal scope is not specified in key derivations, undermining the core argument that "sub-seasonal components are building blocks of seasonal predictions."
Reply: “Sub-seasonal scale”, i.e. synoptic (~2-7 days), super-synoptic (~10-20 days) and MISOs (~-20-60 days) are commonly used in literature. Section 2.3.3, in method describes comprehensively how co-variability between sub-seasonal and seasonal components are calculated. Line 142 “The sub-seasonal components (e.g., precipitation events on hourly, synoptic, or MISOs timescales) are the building blocks ...” clearly mentioned what we mean by sub-seasonal components. In the same section, we describe method to get these band explicitly in line 190-193: “Fourier analysis is used to construct a smooth annual cycle and total anomaly, while a Lanczos band-pass filter (Duchon, 1979) is applied to filter daily time series data into synoptic (2-5 days), super-synoptic (10-20 days), and Monsoon Intra-Seasonal Oscillations/Madden-Julian Oscillation(MISO/MJO) (20-60 days) bands.” However, we can add one or two lines in the beginning of method section 2.3.3 about sub-seasonal scale, which will make easy to understand for readers.
3. Internal Contradictions in the Theoretical Framework The manuscript contains fundamental academic flaws:
Conflict Between Orthogonality Assumption and Physical Reality: The derivation of Equation (11) assumes orthogonality (zero covariance) between different components. However, significant nonlinear interactions exist across scales (e.g., synoptic, MJO, external forcing) in the climate system, making this assumption physically unrealistic;
Reply: Strict orthogonality between components may not fully hold in the climate system due to non-linear cross-scale interactions (e.g., synoptic variability, MJO, external forcing). However, band-pass filtering minimizes covariance between frequency bands, reducing cross-scale leakage and allowing the total variance to be approximated as the sum of sub-seasonal components. Thus, the orthogonality assumption serves as a practical statistical simplification that enables robust variance partitioning and comparison across time scales, and it is widely adopted in climate diagnostics and prediction studies.
Logical Inconsistency Paradox: The orthogonality assumption directly contradicts the manuscript's central argument that "covariability between sub-seasonal components and seasonal anomalies is a source of error." A theoretical framework based on an unreliable assumption cannot effectively demonstrate the effects of non-orthogonality, rendering the overall argument untenable.
Reply : This interpretation is not correct. Here we argue that, since sub-seasonal components are the building blocks of the seasonal mean, their unrealistic contributions or systematically biased contributions reduces seasonal prediction skill and hence the predictability (evident in Figure 7a,b). The use of the orthogonality assumption is a methodological simplification to partition variance across time scales; it does not imply the absence of physical co-variability. The clear separation between lower- and higher-skill ensemble members (Figure 7), along with the larger biases in their covariance (particularly in the MISO bands), demonstrates that the orthogonality assumption is both useful and practical for understanding their influence on seasonal variability and predictability. Nevertheless, in the perfect model framework, it is assumed that only the initial error contributes to the noise/ensemble spread and hence the predictability, which overlook other sources of error such as physical processes, numerical methods etc. Moreover, it has no binding relationship with observations, thus often giving rise to paradoxical situation .
Conclusion Due to issues such as unclear presentation, conceptual confusion, and logical paradoxes, the conclusions of this manuscript are unreliable, and we recommend rejection.
We hope that we are able to address the concerns/comments. As mentioned in the point wise reply, brief discussion in some part of the manuscript may make it easier to follow for readers.
Citation: https://doi.org/10.5194/egusphere-2025-1683-AC1 -
CC2: 'Reply on AC1', Youmin Tang, 09 Sep 2025
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The article only explains how signal and noise variance are defined and calculated. Since variance itself is not the actual component, it is unclear how the signal and noise are extracted from the data. The concept and defintion are totally different between the variance and the variable itself.
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The article tries to discuss and analyze the paradox, but the purpose of using Nino3.4 to predict precipitation remains unclear. What is the intention behind comparing it with dynamic models? Is it to demonstrate whether the actual or potential forecast skill of dynamic models is higher or lower, reasonable or unreasonable? The objective is not clearly stated. Moreover, can using Nino3.4 to predict precipitation effectively achieve these goals? Would the forecast skill be reliable? Was the forecast skill mentioned in the article derived from training or test data? Similarly, were other modes affecting precipitation in the Indian region, such as IOD, considered?
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Rowell (1995) never defined signal variance and noise variance using ANOVA. While they did mention ANOVA, it was only used for statistical testing. The authors should revisit Rowell (1995) to better understand the content. ANOVA has exactly defintion in statistics, which should be followed to avoide unnecessary confusion.
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I do not understand the meaning of the statement: "The use of the orthogonality assumption is a methodological simplification to partition variance across time scales; it does not imply the absence of physical co-variability." Do physical and mathematical co-variability have different interpretations? In my opinion, if two quantities are physically related, they cannot be assumed to be orthogonal in mathematics. Additionally, I do not comprehend the authors' claim that "sub-seasonal components are the building blocks of the seasonal mean." Following this logic, all time scales would be sources of error, since hourly components are the building blocks of the daily mean, and daily components are the building blocks of the weekly mean, and so on.
- So I have to feel sorry to decline this work again. The topic is interesting that is the reason why I agreed with reviewing it. Unfortunately I do not learn more from this work. To my understanding, the paradox should be from the "defintion" of potential predictability. The ratio of signal to noise may not well represent the potential predictability. If authors wish to work this problem, I suggest them to seek other measures to quantify the potential predictabilty.
- Due to busy schedule, I won't post the comments again. Sorry.
Citation: https://doi.org/10.5194/egusphere-2025-1683-CC2 -
AC2: 'Reply on CC2', S.K. Saha, 17 Sep 2025
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We thank you for taking your time and giving your comments, which are useful for improving the manuscript. Here are point wise clarifications.
1. The article only explains how signal and noise variance are defined and calculated. Since variance itself is not the actual component, it is unclear how the signal and noise are extracted from the data. The concept and defintion are totally different between the variance and the variable itself.
Reply: There are numerous paper on how signal and noise components are extracted from model data and some of them are cited here (e.g. Kang and Shukla, 2006; Scaife et al., 2014; Saha et al., 2016a; Scaife and Smith, 2018; Weisheimer et al., 2018 and many more). While inter-ensemble spread is considered as noise/internal component, the ensemble mean is the signal/external component (equation 1 and 2 respectively in our manuscript). How signal and noise are extracted from data is clearly mentioned in lines 104-108 of the manuscript, section 2.3.1.
2. The article tries to discuss and analyze the paradox, but the purpose of using Nino3.4 to predict precipitation remains unclear. What is the intention behind comparing it with dynamic models? Is it to demonstrate whether the actual or potential forecast skill of dynamic models is higher or lower, reasonable or unreasonable? The objective is not clearly stated. Moreover, can using Nino3.4 to predict precipitation effectively achieve these goals? Would the forecast skill be reliable? Was the forecast skill mentioned in the article derived from training or test data? Similarly, were other modes affecting precipitation in the Indian region, such as IOD, considered?
Reply: The idea is to asses prediction skill of not only predictants (i.e. ISMR, PACR), but also the fidelity in simulating global predictors (e.g. ENSO) and their teleconnections. Figure 9 shows multiple correlations involving major global predictors (Niño3.4, IOD, PDO, AMO) and sub-seasonal components.
3. Rowell (1995) never defined signal variance and noise variance using ANOVA. While they did mention ANOVA, it was only used for statistical testing. The authors should revisit Rowell (1995) to better understand the content. ANOVA has exactly defintion in statistics, which should be followed to avoide unnecessary confusion.
Reply: Please look into page no 699 of Rowell et al. (1995). https://rmets.onlinelibrary.wiley.com/doi/epdf/10.1002/qj.49712152311
,which mention “The approach we use to estimate the components of variance closely follows an ‘analysis of variance’ methodology ...”
4. I do not understand the meaning of the statement: "The use of the orthogonality assumption is a methodological simplification to partition variance across time scales; it does not imply the absence of physical co-variability." Do physical and mathematical co-variability have different interpretations? In my opinion, if two quantities are physically related, they cannot be assumed to be orthogonal in mathematics. Additionally, I do not comprehend the authors' claim that "sub-seasonal components are the building blocks of the seasonal mean." Following this logic, all time scales would be sources of error, since hourly components are the building blocks of the daily mean, and daily components are the building blocks of the weekly mean, and so on.
Reply: The argument why we are using assumption of orthogonality and not the actual one, lies on the fact that it is challenging (if not impossible) in a non-linear system to separate individual components.
Sub-seasonal components of the monsoon particularly have clear preferred band. Some of the band are more vigorous in terms of their spatial scale, strength than the others. In terms of their contribution to the mean and variability/predictability also varies. While MISOs have very large spatial structure and strong sub-seasonal variability, their contribution to year-to-year monsoon rainfall variability is minimum (weak negative correlation). So, clearly, we are not talking here about hourly/daily events but some known and prominent sub-seasonal variability/bands, which shape the seasonal monsoon rainfall of a year. Here are literatures, cited in support of our arguments (Saha et al., 2019; Borah et al., 2020). Some important papers in the similar lines but not cited here are.
Goswami, B. N., & Ajayamohan, R. S. (2001). Intraseasonal oscillation and interannual variability of the Indian summer monsoon. Journal Climate, 14, 1180–1198.
Webster, P. J., Magaa, V. O., Palmer, T. N., Shukla, J., Tomas, R. A., Yanai, M., & Yasunari, T. (1998). Monsoons: Processesp, redictability, and the prospects for prediction. J. Geophys. Res., 103 , 14451 – 14510.
Krishnamurthy, V., Achuthavarier, D. Intraseasonal oscillations of the monsoon circulation over South Asia. Clim Dyn 38, 2335–2353 (2012). https://doi.org/10.1007/s00382-011-1153-7
5. So I have to feel sorry to decline this work again. The topic is interesting that is the reason why I agreed with reviewing it. Unfortunately I do not learn more from this work. To my understanding, the paradox should be from the "defintion" of potential predictability. The ratio of signal to noise may not well represent the potential predictability. If authors wish to work this problem, I suggest them to seek other measures to quantify the potential predictabilty.
Reply: We wish, if you could have read the full manuscript. The main content of the manuscript is the following:
i) Perfect model framework is used to estimate potential predictability of seasonal anomaly, which often shows paradoxical behaviour. ‘Analysis of variance’ framework is used for calculating ‘signal’ and ‘noise’ components using 52-ensemble member re-forecasts.
ii) Here we argue that ‘perfect model framework’ is not adequate, as the error growth is not from only initial condition errors but also from other sources, like physics, numerical scheme etc. We demonstrated that sub-seasonal component, which is part of the physics, adds error (biased contribution) in the seasonal forecast anomaly (i.e. Figure 7). However, ‘perfect model framework’ assumes, ensemble spread solely attributed to initial condition error. Consequently, true limit of predictability is not known. So, here our argument matches with your point of view that the method of estimating PPL based on perfect model framework is inadequate. We have already mentioned it in lines 337-344, in the last para of section 3.3
iii) Finally we propose a method for estimating PPL, which is free from paradox (section 3.4). Therefore, we believe the rationale provided for rejection does not fully capture the merits of the manuscript
Citation: https://doi.org/10.5194/egusphere-2025-1683-AC2
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CC2: 'Reply on AC1', Youmin Tang, 09 Sep 2025
reply
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AC1: 'Reply on CC1', S.K. Saha, 09 Sep 2025
reply
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RC1: 'Comment on egusphere-2025-1683', Anonymous Referee #1, 22 Sep 2025
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The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1683/egusphere-2025-1683-RC1-supplement.pdf
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CC3: 'Comment on egusphere-2025-1683', BN GOSWAMI, 02 Oct 2025
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With my own interest in Indian monsoon prediction and predictability, I read the manuscript with interest. The manuscript addresses an important and timely issue concerning the potential predictability limit (PPL) of the seasonal monsoon and the inherent challenges associated with the ‘perfect model’ framework. The question of how far seasonal climate, particularly Indian summer monsoon rainfall, can be predicted remains a subject of active debate. The study is highly relevant to the climate dynamics and prediction community, with potential implications for both advancing scientific understanding and improving operational forecasting. Overall, the manuscript is well-structured, presents new insights, and uses a comprehensive set of model simulations. I am happy to recommend the manuscript for publication after minor revision.
Strength:
1) In the perfect model framework, signal and noise component of a parameter is estimated, assuming that ensemble spread is due to initial error, which is more appropriate for short-range forecast (e.g. weather), but may not hold true for long-range forecast (i.e. seasonal), where noise/error introduced by slowly varying boundary conditions are also important. Therefore, estimates of PPL for seasonal climate in this framework may not represent the true limit, which is all about paradox here.
2) Figure 7 shows an important aspect: how the internal variability could contribute to the prediction skill/predictability of ISMR. However noise component is fully attributed to initial error in ‘perfect model’ assumption.
4) Although signal and noise are estimated under the assumption of orthogonality, it is clear that this assumption does not always hold.
3) Interestingly, proposed method of estimating PPL and ANOVA based PPL are similar in their maximum predictability of rainfall over tropical Pacific region and more importantly free from paradox.
Weakness:
- PPLs are model dependent, improvements in model likely to increase the limit. Longer observations may be required to estimate actual PPL.
Suggestions:
- For readers from other domain, some basic discussions, like what is the basic premise/hypothesis of the ‘perfect model’ framework while using ensemble forecast is required.
- The estimate of PPL by using a seasonal prediction model from a large ensemble of hindcasts by choosing the ensemble mean of ‘best’ initial conditions may be acceptable. However, the manuscript does not provide a discussion on how to realise the PPL in operational framework. It is possible that growth of ‘initial error’ may never allow the model to achieve the PPL. Even if we knew what are the ‘best initial conditions’ in the ensemble, it would lead to overfitting and unreliable forecast. A discussion on how the PPL could be achieved either by tradition methods or by a deep learning/AI model trained on the large ensemble of hindcast experiments would significantly enhance the quality of the manuscript.
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This manuscript aims to explore the paradox in estimating Potential Predictability (PPL) in seasonal climate forecasts. However, significant issues in its methodology, logical coherence, and theoretical foundation render it unsuitable for publication at this stage. The specific concerns are as follows:
1. Lack of Clarity in Methods and Logical Exposition
The manuscript suffers from poor readability and methodological rigor. Critical details are missing, for example:
Figure 2 presents correlation coefficients between signal and noise components, but it does not specify how these components were extracted from the raw data or model output;
Figure 3 abruptly introduces the Niño3.4 index and regional precipitation without clarifying the research objective or the definition of "correlation skill" (e.g., does it refer to the correlation between the ensemble mean forecast and observations?). The role of Niño3.4 (as a predictor or a reference field) remains unexplained.
Such breaks in logical flow and ambiguous methodological descriptions make it impossible for readers to understand the study process or verify the conclusions, severely violating standards of scientific writing.
2. Conceptual Confusion and Misuse of Terminology
The manuscript contains inaccuracies in fundamental scientific expression:
It conflates the concepts of "Analysis of Variance (ANOVA)" and "Signal-to-Noise Ratio." ANOVA is fundamentally a statistical significance testing method, while the manuscript merely borrows its variance decomposition framework to estimate the signal-to-noise ratio without clarifying the distinction, which may mislead readers;
The term "sub-seasonal scale" is poorly defined. Although filtering bands (e.g., synoptic scale, MJO) are mentioned later, the temporal scope is not specified in key derivations, undermining the core argument that "sub-seasonal components are building blocks of seasonal predictions."
3. Internal Contradictions in the Theoretical Framework
The manuscript contains fundamental academic flaws:
Conflict Between Orthogonality Assumption and Physical Reality: The derivation of Equation (11) assumes orthogonality (zero covariance) between different components. However, significant nonlinear interactions exist across scales (e.g., synoptic, MJO, external forcing) in the climate system, making this assumption physically unrealistic;
Logical Inconsistency Paradox: The orthogonality assumption directly contradicts the manuscript's central argument that "covariability between sub-seasonal components and seasonal anomalies is a source of error." A theoretical framework based on an unreliable assumption cannot effectively demonstrate the effects of non-orthogonality, rendering the overall argument untenable.
Conclusion
Due to issues such as unclear presentation, conceptual confusion, and logical paradoxes, the conclusions of this manuscript are unreliable, and we recommend rejection.