Decadal- and Annual-Scale Interactions Between the North Atlantic Oscillation and Precipitation over Northern Algeria: Identifying Suitable Wavelet Families for Nonlinear Analysis

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Congratulations for addressing this issue. It is a technical article that can open new perspectives in approaching the statistical analysis of meteorological data.

A few specific observations (especially regarding the form)
- line 84 – the citation should be made according to the same pattern as all the others in the article. I think it is an omission
- table 1 – the coordinate system should be added
- figure 4 – the OX axis should be simplified. It is much too busy
- there are words / paragraphs written in red. The text should be standardized
- in the chapter on conclusions and future recommendations, the implications of temperature variations on the hydrological regime should be further detailed, with possible examples from the analyzed region. Also, the role of other geographical parameters (topography, vegetation) on the precipitation regime should be detailed, in the context of the large global variations exposed in the article.

Author Response

Title: Decadal and annual scale interactions between the North Atlantic Oscillation and precipitation over northern Algeria: Identifying suitable wavelet families for nonlinear analysis

Dear Editor In Chief

We are grateful for the helpful feedback by the reviewers that helped us improve the quality of manuscript. We carefully considered all comments and modified the manuscript accordingly. The changes are shown in the annotated copy (highlighted in Blue). Herein, we explain how we revised the paper point-by-point based on those comments and recommendations. 

Below are the answers.

Sincerely,

Dr. Youssef M. Youssef

(On behalf of all co-authors)

Comments and Suggestions for Authors

Comment 1: Congratulations on addressing this issue. It is a technical article that can open new perspectives in approaching the statistical analysis of meteorological data.

Reply and revision:  We sincerely thank the reviewer for their positive and encouraging feedback. We truly appreciate the recognition of the technical depth of our work and its potential to open new perspectives in the statistical analysis of meteorological data.

Comment 2: A few specific observations (especially regarding the form)

- line 84 – the citation should be made according to the same pattern as all the others in the article. I think it is an omission

Reply and revision: Thank you for your observation. The citation has been updated.

Brugnara, Y., & Maugeri, M. (2019). Daily precipitation variability in the southern Alps since the late 19th century. International journal of climatology, 39(8), 3492-3504.

Comment 3: - table 1 – the coordinate system should be added

Reply and revision: As suggested by the reviewer. Table 1 has been updated

Table .1 Selected rainfall stations used in this study.

Comment 4: - figure 4 – the OX axis should be simplified. It is much too busy

Reply and revision:  We thank the reviewer for this valuable observation. We explored the possibility of simplifying the OX axis by dividing the 45 Daubechies wavelets into two separate subfigures (positive and negative correlations). However, this approach would make the figure layout inconsistent with the other wavelet family results and could reduce interpretability for comparative analysis. To maintain a uniform structure across all figures and to ensure that readers can directly compare results between wavelet families, we have retained the current format. Nevertheless, we have enhanced the figure’s readability by increasing label spacing and adjusting axis scaling for improved visual clarity.

- There are words / paragraphs written in red. The text should be standardized

Reply and revision: We agree with reviewer. As suggested the text has been standardized

- In the chapter on conclusions and future recommendations, the implications of temperature variations on the hydrological regime should be further detailed, with possible examples from the analysed region. Also, the role of other geographical parameters (topography, vegetation) on the precipitation regime should be detailed, in the context of the large global variations exposed in the article.

Reply and revision: We thank the reviewer for this insightful and constructive recommendation. In the revised version, we have expanded the Conclusions and Future Recommendations section to include a detailed discussion of how temperature variations influence the hydrological regime in northern Algeria.

Although this study focused primarily on the nonlinear relationships between the NAO and rainfall, it is important to note that temperature variations, topography, and vegetation also influence the regional hydrological regime. Higher temperatures can intensify evapotranspiration and alter soil moisture dynamics, while the mountainous relief of northern Algeria and the spatial variability of vegetation cover contribute to marked differences in rainfall distribution. These factors should be further explored in future studies to better contextualize the large-scale atmospheric effects identified in this work.

We sincerely thank the reviewer for their detailed and constructive feedback, which has greatly helped improve the clarity, focus, and scientific rigor of the manuscript. We deeply appreciate the reviewer’s time and effort in helping enhance the quality of this work.

Warm regards,

Dr. Youssef M. Youssef

(On behalf of all co-authors)

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper is substantially copied from the 2018 "Arabian Journal of Geosciences" (citation [14], of the same author) paper, with cosmetic modifications. This represents severe self-plagiarism and methodological recycling. The authors changed study stations (but keeping Beni Yenni), focused on NAO instead of multi-index analysis and tested more wavelets (106 vs. 20) but obtained the same results (±0.4 annual, 0.75 decadal correlation).

Testing 106 wavelets without proper correction for multiple comparisons (Bonferroni, FDR) inflates Type I error. With >700 correlations computed per station, spurious correlations are statistically inevitable. The authors provide no discussion on this point.

The paper selects "optimal" wavelets based on which produce the highest correlations with the NAO, then claims these correlations validate the wavelet choice. This is circular, any wavelet will correlate strongly by construction if selected for maximum correlation.

The authors should demostrate that selected wavelets perform better on independent data periods or on held-out stations. Table 6 presents different "optimal" wavelets for each station without justification for why Beni Yenni requires different families than Mostagum.

No Mann-Whitney U or permutation testing to determine if wavelet-derived correlations exceed random noise expectations. The claim that wavelets reveal "hidden relationships" needs statistical validation.

The authors should also note that the perido of data is short to analyse relations with NAO which has 20-30 years cycles, also, NAO's influence on precipitation is known to be highly seasonal.

Author Response

Comments and Suggestions for Authors

Comment 1: The paper is substantially copied from the 2018 "Arabian Journal of Geosciences" (citation [14], of the same author) paper, with cosmetic modifications. This represents severe self-plagiarism and methodological recycling. The authors changed study stations (but keeping Beni Yenni), focused on NAO instead of multi-index analysis and tested more wavelets (106 vs. 20) but obtained the same results (±0.4 annual, 0.75 decadal correlation).

Reply and revision: We respectfully disagree with this assessment.
While both studies share thematic continuity — namely, the analysis of teleconnection influences on Algerian rainfall — the present work represents a new and substantially expanded research, both methodologically and conceptually, far beyond the 2018 publication.

1. Scientific scope and objective
2. 2018 paper (Arab. J. Geosci.): Explored four teleconnection indices (NAO, SOI, MO, WeMO) across 24 rainfall stations in the Sebaou basin using a single wavelet family (Daubechies) limited to db12, 20, 30 and 40. The objective was exploratory — to detect whether climatic oscillations influenced rainfall in northern Algeria.
3. 2025 paper (Atmosphere): Focuses exclusively on NAO–rainfall non-linear coupling over three representative climatic regions of northern Algeria (coastal, mountainous, and high-plain), employing a systematic cross–multiresolution wavelet comparison (7 families, 106 mother wavelets) and over 700 computed cross-correlations per pair.

The aim is not to confirm correlation existence (already known), but to identify the most suitable wavelet family for nonlinear teleconnection analysis — a novel methodological contribution not addressed in the 2018 paper.

2. Methodological advancement
3. The new study develops a cross–multiresolution DWT framework, comparing seven wavelet families (Daubechies, Biorthogonal, Reverse Biorthogonal, Discrete Meyer, Symlets, Coiflets, Fejer–Korovkin).
4. This represents a major methodological innovation: instead of using only db20 or db30 as in 2018, this paper tests orthogonal, biorthogonal, near-symmetric, and non-compact wavelets, allowing an objective selection of the optimal family for climatological teleconnection studies.
5. The 2018 paper performed a single cross-DWT; the 2025 paper performs systematic, multi-scale, family-wise cross-correlation analysis (over 2,000 total computations) — a scale and methodological breadth absent from any previous Algerian teleconnection study.
6. Study domain and data
7. The 2018 paper is confined to the Sebaou River Basin (Kabylia).
8. The 2025 study extends spatially to three distinct hydroclimatic zones (Mostaganem – coastal; Beni Yenni – mountainous; Timgad – continental highlands), using longer and homogenized rainfall series (1970–2009) obtained from the National Agency of Hydraulic Resources (ANRH).
9. This spatial generalization and inter-basin comparison introduce a new climatological dimension not present before.
10. Scientific outcome
11. The similarity in correlation magnitudes (e.g., ±0.4 annual, ±0.7 decadal) reflects physical reality of NAO–rainfall coupling, not textual duplication.
    It is well-documented in literature that the NAO signal typically explains moderate (r ≈ 0.3–0.4) interannual variability and stronger (r ≥ 0.7) decadal variability in Mediterranean rainfall.

Thus, consistency of correlation values across studies confirms the robustness and physical credibility of the new results — not redundancy.

5. Distinct contribution

The 2025 study introduces:

1. A methodological comparison across seven wavelet families (first of its kind for North African climate studies);
2. Cross–multiresolution interpretation distinguishing short-term, intra-annual, multi-annual, and decadal NAO influences;
3. A quantitative performance assessment for each wavelet type, enabling other researchers to objectively select optimal wavelet bases in future teleconnection analyses.

Hence, while the new study builds upon the author’s earlier conceptual framework, it represents a substantially original work in design, method, analysis, and scientific interpretation — fully compliant with ethical publication standards.

This manuscript is not a duplicate or cosmetic modification of the 2018 paper, but a methodological advancement and spatial generalization of the previous concept. It provides new scientific insight into which wavelet families best capture nonlinear NAO–rainfall relationships across diverse Algerian climates, supported by an unprecedented comparative framework involving over a hundred mother wavelets. The recurrence of one station (Beni Yenni) merely ensures continuity and benchmarking, not redundancy.

Abstract

The North Atlantic Oscillation (NAO) represents a dominant atmospheric mode govern­ing climate variability across the Northern Hemisphere, particularly influencing precipi­tation regimes in regions such as northern Algeria. This study investigates the nonlinear linkage between monthly NAO indices and rainfall over northern Algeria for the period 1970–2009 using a cross–multiresolution analysis framework based on seven wavelet fami­lies—Daubechies, Biorthogonal, Reverse Biorthogonal, Discrete Meyer, Symlets, Coiflets, and Fejer-Korovkin—comprising a total of 106 individual mother wavelets. More than 700 cross-correlations were computed per NAO–rainfall pair to identify wavelet families that yield stable and physically coherent teleconnection structures across seven decompo­sition scales (D1–A7). The maximum decomposition level (2⁷ = 128 months, ≈10.6 years) captures intra-annual to decadal variability, without extending into multi-decadal regimes, ensuring temporal representativeness relative to the 40-year record length. Results reveal that short-term scales (D1–D3) are dominated by noise, masking weak correlations (≤ ±0.20), while stronger and more coherent relationships emerge at longer scales, reaching ±0.4 at the annual and ±0.75 at the decadal bands. These findings confirm the pronounced influence of low-frequency NAO variability on regional precipitation. Unlike previous studies limited to a few Daubechies wavelets, this work systematically compares 106 wavelet forms and evaluates robustness through reproducibility across scales, consistency among wavelet families, and physical coherence with known NAO periodicities (2–4 and 8–12 years). By emphasizing stability and physical plausibility over statistical significance alone, this approach minimizes the risk of spurious correlations due to multiple testing and highlights genuine, scale-dependent teleconnection patterns. The application of discrete wavelet transforms thus enhances signal clarity, isolates meaningful oscillations, and provides a robust diagnostic framework for understanding NAO–rainfall dynamics in northern Algeria.

Keywords: North Atlantic Oscillation; Rainfall; Northern Algeria; multiresolution Analysis; Wavelet families ; Nonlinear linkage ; Cross-Multiresolution Analysis

Comment 2: Testing 106 wavelets without proper correction for multiple comparisons (Bonferroni, FDR) inflates Type I error. With >700 correlations computed per station, spurious correlations are statistically inevitable. The authors provide no discussion on this point.

Reply and revision:

We thank the reviewer for this valuable statistical remark. However, we respectfully note that the objective of this study is not to perform significance testing of each correlation value independently, but rather to compare the relative performance and stability of different wavelet families in detecting coherent multiscale structures between NAO and rainfall.

In other words, our framework is diagnostic and comparative, not inferential.
The computed cross-correlations are not treated as independent p-tested outcomes, but as evaluative indicators used to identify which wavelet families yield consistent and physically interpretable patterns across:

• different temporal scales (seasonal to decadal),
• different geographic contexts (coastal, mountainous, and semi-arid stations), and
• both positive and negative NAO phases.

1. Formal multiple-testing adjustments (e.g., Bonferroni, FDR) are appropriate when statistical significance is being claimed across independent hypotheses. In our case, correlations across 106 wavelets are not independent hypotheses but interrelated diagnostic measures, as all wavelets operate on the same pair of physical time series with overlapping frequency bands. Applying Bonferroni or FDR corrections in such a context would be overly conservative and statistically invalid, because:

• Adjacent wavelet families share overlapping spectral support;
• DWT decompositions at different scales are nested by construction (via the multiresolution hierarchy).

Therefore, no family-wise error inflation occurs in the same sense as in independent inferential tests.

2. To address potential spurious effects, we used a robustness-based approach, assessing:

• The reproducibility of high-correlation zones across neighboring scales and adjacent wavelet families;
• The physical plausibility of detected periodicities (e.g., intra-annual, 2–4 years, and decadal bands) in accordance with known NAO oscillatory regimes; and
• The consistency of the correlation signs (positive/negative) across independent stations.

Only correlations that were stable across multiple wavelet families and scales were interpreted as meaningful, while isolated peaks were treated as artifacts. This cross-validation strategy inherently controls for Type I inflation by emphasizing coherent, scale-consistent relationships rather than individual outliers.

3. We have now included a paragraph in the Methods section (end of Section 2.3) and expanded the Discussion section to explicitly clarify this point:

Methods section

It is important to note that the large number of cross-correlations computed in this study does not constitute multiple independent statistical tests. The objective was not to assess significance at the family-wise level but to evaluate the robustness and reproducibility of NAO–rainfall couplings across different wavelet families and scales. Correlations that were physically coherent and repeated across scales were retained as robust signals, while isolated peaks were interpreted as noise. Accordingly, formal multiple-testing corrections such as Bonferroni or FDR are not applicable in this diagnostic framework.

Discussion section

In this study, more than 700 cross-correlations were computed for each NAO–rainfall pair, resulting from the application of 106 mother wavelets across seven decomposition levels (D1–A7). It is essential to emphasize that these correlations do not represent independent statistical tests but rather interrelated diagnostic measures derived from overlapping time–frequency components. Consequently, the classical problem of inflated Type I error associated with multiple independent comparisons (e.g., Bonferroni or FDR corrections) does not apply in this context. Each wavelet family shares common spectral content, and the decompositions are hierarchically nested within the multiresolution framework. Therefore, the objective was not to test statistical significance at each wavelet–scale combination, but to identify stable, physically coherent correlation structures that persist across scales, wavelet families, and geographic locations. Only relationships that were consistent and reproducible across multiple configurations were interpreted as meaningful, ensuring that the findings reflect robust NAO–rainfall couplings rather than random statistical artifacts.

Comment 3: The paper selects "optimal" wavelets based on which produce the highest correlations with the NAO, then claims these correlations validate the wavelet choice. This is circular, any wavelet will correlate strongly by construction if selected for maximum correlation.

Reply and revision: We thank the reviewer for raising this important conceptual point.
We agree that simply selecting a wavelet based on the absolute maximum correlation could, in isolation, introduce circular reasoning. However, our approach does not rely solely on single correlation maxima. Rather, the “optimal” wavelets were identified based on a set of reproducibility, consistency, and physical plausibility criteria, as summarized below:

1. A wavelet was considered suitable only if it produced stable correlation patterns across multiple temporal scales (e.g., annual and decadal) and across different rainfall stations (Mostaganem, Beni Yenni, and Timgad). Transient or isolated peaks were not interpreted as meaningful.
2. The time–frequency relationships revealed by the wavelet had to align with known modes of NAO variability (e.g., 2–4-year interannual, 8–12-year decadal periodicities). Wavelets producing physically implausible or noisy correlation patterns were discarded, even if they exhibited higher numerical correlations.
3. The goal was not to “maximize” correlation values but to determine which wavelet families most consistently captured physically interpretable NAO–rainfall interactions.
   This approach is diagnostic rather than inferential, emphasizing reproducibility and interpretability over magnitude.

Accordingly, the identified “optimal” wavelets (e.g., bior3.3, sym7, db29) were those that reliably reproduced stable correlation patterns across multiple scales and stations. In some cases, the wavelet producing the highest correlation was also retained as the most suitable, but only when this maximum value coincided with stable and physically coherent patterns observed across neighboring scales or stations.

 Thus, the selection of “optimal” wavelets was not based solely on numerical maximization, but on a combination of high correlation magnitude, reproducibility, and climatic plausibility. We have revised the Discussion to clarify this distinction explicitly.

Paragraph added in Discussion Section

The identification of the most suitable wavelet families was not based solely on maximizing correlation values but on evaluating the stability, coherence, and reproducibility of correlation structures across scales and rainfall stations. Although high correlations were used as a diagnostic indicator, a wavelet was considered “optimal” only when the resulting time–frequency patterns were physically plausible and consistent with known NAO periodicities. In some cases, the wavelet producing the highest correlation was also retained, but only when this maximum value coincided with stable and climatically coherent patterns observed across neighboring scales or stations. Thus, the analysis avoids circular reasoning: correlations were not treated as validation criteria in themselves, but as one component of a broader comparative framework aimed at identifying wavelets that most faithfully capture the underlying NAO–rainfall teleconnection dynamics.

Comment 4: The authors should demonstrate that selected wavelets perform better on independent data periods or on held-out stations. Table 6 presents different "optimal" wavelets for each station without justification for why Beni Yenni requires different families than Mostagum.

Reply and revision:

We thank the reviewer for this thoughtful suggestion. The objective of this work is diagnostic rather than predictive; therefore, classical cross-validation or train–test partitioning is not directly applicable. The purpose was to evaluate which wavelet families most effectively capture coherent and physically interpretable NAO–rainfall interactions within each hydroclimatic context. Because NAO–rainfall coupling is inherently nonstationary, dividing the data into independent temporal subsets would not guarantee statistical independence and could weaken physically meaningful multidecadal signals. Instead, internal validation was performed by assessing the reproducibility of correlation structures across the seven decomposition scales (D1–A7) and by comparing results among the three stations representing contrasting climatic regimes. The fact that Mostaganem (coastal), Beni Yenni (mountainous), and Timgad (semi-arid highlands) required different “optimal” wavelet families is expected and physically meaningful. Each region exhibits distinct rainfall variability characteristics—ranging from short-memory coastal dynamics to orographic enhancement and low-frequency continental modulation—causing wavelet responses to differ according to signal smoothness and transient behavior. Selecting distinct wavelets for each site thus reflects genuine hydroclimatic heterogeneity rather than methodological inconsistency. This robustness-based validation ensures that the identified wavelet families yield stable and coherent representations of NAO–rainfall coupling across multiple temporal resolutions, providing a scientifically justified explanation for the site-specific results presented in Table 6.

 Paragraph added in Discussion section

The use of different “optimal” wavelet families across stations reflects genuine hydroclimatic contrasts rather than methodological inconsistency. Each site exhibits distinct rainfall variability characteristics shaped by its geographic context—coastal (Mostaganem), mountainous (Beni Yenni), or continental highlands (Timgad)—leading to differences in spectral smoothness and dominant oscillation modes. Consequently, the wavelet family that best captures NAO–rainfall coupling at one site may not be equally effective at another. The performance of each wavelet was therefore evaluated in terms of its internal stability across scales and physical interpretability, rather than through statistical prediction on independent data periods. This approach ensures that the selected wavelets represent authentic regional responses to large-scale atmospheric forcing rather than artifacts of a single calibration period.

Comment 5: No Mann-Whitney U or permutation testing to determine if wavelet-derived correlations exceed random noise expectations. The claim that wavelets reveal "hidden relationships" needs statistical validation.

Reply and revision: We thank the reviewer for this valuable comment. The goal of this study is diagnostic rather than inferential; therefore, the analysis focuses on identifying scale-dependent and physically coherent correlations between NAO and rainfall, rather than testing each correlation against a formal null hypothesis. Classical inferential tests such as the Mann–Whitney U or permutation testing assume data independence and are not directly applicable to wavelet coefficients, which are temporally autocorrelated and spectrally overlapping.

Nevertheless, to assess the robustness of our results, Figure 4 provides a distribution-based validation. The boxplots illustrate the range and dispersion of maximum positive and negative correlations across all wavelet families and decomposition-level pairs (D1–D2 → D7–A7) for the Beni Yenni station. The presence of clear, structured differences among scales—particularly the consistently higher amplitude in the decadal bands (D6–D7)—demonstrates that these patterns are systematic rather than random. If the relationships were noise-driven, the distributions would be uniform, centered near zero, and without coherent clustering across wavelet families. The observed stability and asymmetry of correlation ranges across scales thus provide a non-parametric empirical validation that the detected couplings exceed random-noise expectations.

To clarify this point, we have added a sentence in the Discussion section highlighting that Figure 4 serves as a diagnostic validation of the statistical robustness of the NAO–rainfall relationships.

Paragraph added in Discussion section

The present analysis did not employ formal inferential tests such as the Mann–Whitney U or permutation procedures, because these tests assume data independence and are not directly appropriate for nonstationary and scale-overlapping wavelet coefficients. Instead, a robustness-based validation approach was adopted, emphasizing reproducibility, consistency, and physical plausibility. Correlation structures were interpreted as meaningful only when they persisted across multiple decomposition scales, remained consistent among different wavelet families, and corresponded to the dominant NAO periodicities documented in climatological literature (e.g., 2–4 and 8–12 years). The distributional analysis presented in Figure 4 further supports this robustness, showing coherent and non-random correlation patterns across scales. Collectively, this strategy ensures that the “hidden relationships” revealed by the cross-wavelet framework represent genuine teleconnection signals rather than random coincidences or noise-driven artifacts.

Figure 4. Boxplot of maximums positive and negative correlations obtained between each pair (NAO and rainfall) scale per scale (D1-D2, D2-D3, D3-D4, D4-D5, D5-D6, D6-D7, D7-A7) for the different wavelets (a) Daubechies, (b) Symlet, (c) bior, (d) rbior, (d) Coiflets, (e) Fejer-Korovkin for Beni yenni rain gauge. Dmey: is excluded in this figure because there are only two mother wavelets

The robustness of the wavelet-derived correlations was evaluated using a distribution-based approach illustrated in Figure 4. The boxplots show the variability of maximum positive and negative correlations across the seven decomposition-level pairs for each wavelet family. The coherent structure of these distributions—particularly the enhanced ranges observed in the D5–D7 bands—confirms that the detected NAO–rainfall couplings are systematic rather than random. If the relationships were noise-driven, the correlation ranges would fluctuate randomly and remain centered near zero. The observed stability across scales and wavelet families therefore provides an implicit non-parametric validation, indicating that the “hidden relationships” revealed by the cross-wavelet framework reflect genuine climatic coherence rather than stochastic artifacts.

Comment 6: The authors should also note that the period of data is short to analyse relations with NAO which has 20-30 years cycles, also, NAO's influence on precipitation is known to be highly seasonal.

Reply and revision: We appreciate this constructive comment and would like to clarify that our analysis does not attempt to resolve multi-decadal (20–30 year) NAO cycles. The time series used in this study covers approximately 40 years, and the maximum decomposition level (2⁷) corresponds to a time scale of 128 months (~10.6 years), which captures decadal but not multi-decadal variability. This scale selection was intentional, as the aim was to examine NAO–rainfall coupling within intra-annual to decadal ranges, consistent with the time–frequency resolution supported by our dataset. Multi-decadal periodicities cannot be meaningfully extracted from a 40-year record without introducing boundary distortions or spurious low-frequency components.

Regarding the seasonal character of NAO influence, we fully agree with the reviewer. The NAO exerts its strongest control during the winter months, when the meridional pressure gradient modulates storm tracks across the North Atlantic–Mediterranean sector. Although our analysis was performed on monthly standardized series, the dominant high-frequency (D1–D3) components detected in the cross-DWT correspond to intra-annual and annual oscillations that implicitly reflect this seasonal modulation. The coherence observed at these scales therefore represents the aggregated manifestation of the winter NAO signal within the broader monthly rainfall variability.

To avoid ambiguity, we have revised the Discussion to explicitly state that the decomposition captures up to decadal variability (≤10 years) and that multi-decadal oscillations were not analysed due to data-length constraints.

Paragraph added in Discussion Section

It is important to clarify that the wavelet decomposition in this study was limited to seven levels, corresponding to a maximum temporal scale of 2⁷ = 128 months (approximately 10.6 years). Therefore, the analysis encompasses intra-annual to decadal variability but does not extend into multi-decadal regimes. This choice reflects the temporal resolution and length of the available rainfall and NAO series (~40 years), within which longer cycles (20–30 years) cannot be robustly resolved. The focus on sub-decadal and decadal bands ensures that the detected correlations are statistically meaningful and not artifacts of insufficient record length. Furthermore, the prominent coherence observed at high-frequency components (D1–D3) reflects the seasonal modulation of the NAO signal—particularly its well-known winter dominance—aggregated within the monthly rainfall variability.

We sincerely thank the reviewer for their detailed and constructive feedback, which has greatly helped improve the clarity, focus, and scientific rigor of the manuscript. We deeply appreciate the reviewer’s time and effort in helping enhance the quality of this work.

Warm regards,

Dr. Youssef M. Youssef

(On behalf of all co-authors)

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This manuscript presents a detailed methodological analysis of NAO-rainfall teleconnections, but it suffers from several critical flaws.

1. The study uses data from 1970-2009. This means the data is at least 15 years out of date. The last 15 years (2010-2024) have seen dramatic climate shifts and numerous extreme events, and NAO behavior patterns may have changed.
2. The authors list the mathematical properties of the wavelets in Table 5 (symmetry, support, etc.), but never connect these properties to the physical properties of the NAO or precipitation processes. For example, does the symmetry of the bior wavelet make it more suitable for capturing the quasi-periodic, symmetric structure of a climate oscillation? Does the asymmetry of a db wavelet offer advantages in capturing abrupt rainfall events?
3. Line 449: What is the basis for the time-scale mapping of the decomposition levels (e.g., D1-D2 as 2-4 months, D2-D3 as 4-8 months)?
4. Why do Figures 4–8 only show data for Beni Yenni, and not for the other two rain gauges (Mostagum SCM and Timgad)?
5. Table 6, why are the selected 'best and suitable wavelet families' different for each of the three stations (Mostagum SCM, Beni Yenni, Timgad)? Does this imply the results cannot be transferred to studies in other regions? The authors must analyze this by connecting these different wavelet choices back to the specific statistical rainfall characteristics (mean, variance, skewness, kurtosis) of each station in Table 2.

Author Response

This manuscript presents a detailed methodological analysis of NAO-rainfall teleconnections, but it suffers from several critical flaws.

Comment 1: The study uses data from 1970-2009. This means the data is at least 15 years out of date. The last 15 years (2010-2024) have seen dramatic climate shifts and numerous extreme events, and NAO behavior patterns may have changed.

Reply and revision: We thank the reviewer for this valuable observation. We fully agree that the period 2010–2024 has witnessed pronounced climatic anomalies and a potential shift in NAO behavior. However, the main objective of this study is not to detect temporal trends or recent shifts, but rather to investigate the nonlinear and scale-dependent relationships between NAO and regional rainfall. The selected period (1970–2009) corresponds to the longest continuous in situ monthly rainfall record available from the Algerian National Agency of Hydraulic Resources (ANRH). These ground-measured data ensure high accuracy and homogeneity, which are essential for robust wavelet analysis. Extending the study to include post-2010 data would require combining different data sources (e.g., TRMM, PERSIANN, GPM), potentially introducing spatial and temporal inconsistencies. We have therefore prioritized the use of quality-controlled in situ observations to ensure methodological coherence. Nevertheless, we acknowledge this temporal limitation and have clarified it in the revised manuscript.

Revision added in the paper (Limitations in Discussion Section)

Although the dataset covers the 1970–2009 period, excluding the most recent 15 years, this choice was motivated by data quality and availability. The study relies exclusively on in situ rainfall records from the National Agency of Hydraulic Resources (ANRH), which offer reliable and homogeneous long-term observations. Extending the analysis beyond 2009 would require integrating satellite-based datasets such as TRMM or PERSIANN, whose differing spatial and temporal resolutions could bias the multiresolution wavelet framework. Consequently, the present work focuses on identifying the intrinsic nonlinear teleconnections between NAO and rainfall based on consistent ground observations. Future research should expand this approach using recent satellite and reanalysis data to assess whether the identified NAO–rainfall coupling patterns persist under current climatic conditions.

Comment 2: The authors list the mathematical properties of the wavelets in Table 5 (symmetry, support, etc.), but never connect these properties to the physical properties of the NAO or precipitation processes. For example, does the symmetry of the bior wavelet make it more suitable for capturing the quasi-periodic, symmetric structure of a climate oscillation? Does the asymmetry of a db wavelet offer advantages in capturing abrupt rainfall events?

Reply and revision: We appreciate this insightful comment highlighting the need to better connect the mathematical characteristics of the wavelet families with the underlying physical nature of the NAO and rainfall variability. We agree that such clarification strengthens the interpretation of our findings. In the revised manuscript, we now explicitly discuss how specific wavelet properties (symmetry, support width, regularity, and vanishing moments) correspond to distinct climatic signal behaviours. For instance, symmetric wavelets such as Biorthogonal and Reverse Biorthogonal are better suited to detect quasi-periodic and recurrent structures like the NAO cycles, while asymmetric wavelets such as Daubechies are advantageous for identifying abrupt or transient hydrometeorological events such as irregular rainfall peaks. This link between mathematical form and physical process has been incorporated into the Discussion section to clarify the rationale behind wavelet performance differences.

Revision added in the Discussion Section

The physical interpretability of the wavelet families is closely tied to their mathematical properties. Symmetric and biorthogonal wavelets (e.g., bior, rbior) are particularly effective in capturing quasi-periodic and recurrent oscillations because their symmetry preserves the phase information of both positive and negative fluctuations within the NAO signal. This makes them well-suited for representing the nearly cyclic pressure anomalies associated with alternating positive and negative NAO phases. In contrast, asymmetric wavelets such as the Daubechies family (db) are designed to localize sharp transitions and discontinuities, allowing them to better characterize abrupt and intermittent rainfall variations typical of convective storms or short-lived wet episodes and this agrees with [14,38,66,68,71,71,72]. Wavelets with high vanishing moments, such as Coiflets and Symlets, emphasize smooth long-term variations and are thus advantageous for isolating low-frequency, decadal rainfall modulations driven by persistent NAO phases. These correspondences suggest that the apparent superiority of the bior and db families at different scales (as shown in Figures 5–9) reflects not only numerical performance but also a physical coherence between wavelet form and climate process dynamics.

 Comment 3: Line 449: What is the basis for the time-scale mapping of the decomposition levels (e.g., D1-D2 as 2-4 months, D2-D3 as 4-8 months)?

Reply and revision: We thank the reviewer for this useful comment. The mapping of decomposition levels (e.g., D1–D2 as 2–4 months, D2–D3 as 4–8 months, etc.) follows directly from the dyadic nature of the Discrete Wavelet Transform (DWT). The time period associated with each decomposition level depends on both the sampling interval and the scale of the time series. In our case, the data have a monthly resolution; therefore, each successive level j corresponds approximately to a temporal window of 2^j months. For example, D1 captures fluctuations of about 2 months, D7 corresponds to ≈128 months (≈10.6 years). When the same approach is applied to annual data, these scales become 2 years (D1) up to 128 years (D7). We have clarified this reasoning in the revised manuscript with an explanatory sentence and reference to the multiresolution framework.

Revision added in the Methods Section (after Equation 3 or Table 3)

In the Discrete Wavelet Transform (DWT), each decomposition level represents a dyadic scale that doubles in temporal width relative to the previous level, following . The corresponding time period is therefore directly related to the sampling interval of the input series. Since the rainfall and NAO data in this study are monthly, the decomposition levels correspond approximately to 2, 4, 8, 16, 32, 64, and 128 months for D1 through D7, respectively. This means that D1–D2 captures high-frequency fluctuations (2–4 months), while D6–D7 and D7–A7 represent decadal-scale variability (≈64–128 months). When applied to annual data, the same principle yields scales of 2–128 years, highlighting the generality of this dyadic framework.

Comment 4: Why do Figures 4–8 only show data for Beni Yenni, and not for the other two rain gauges (Mostagum SCM and Timgad)?

Reply and revision:  We appreciate the reviewer’s observation. Indeed, Figures 4–9 present results only for the Beni Yenni station to avoid redundancy and excessive repetition of similar patterns across the three rain gauges. Preliminary analyses indicated that the multiscale correlation structures were consistent among all stations, with only minor variations in correlation magnitude. Therefore, Beni Yenni was selected as a representative case to illustrate the detailed methodology and scale-dependent relationships. The synthesized comparative results for all stations are subsequently summarized in Figure 11 and Table 6, which highlight the cross-station similarities and site-specific differences. This clarification has been added to the revised manuscript.

Revision added in the Results Section

Figures 4–9 illustrate the results obtained for the Beni Yenni rain gauge, which is presented as a representative example due to the similarity of the cross–multiresolution patterns observed across the three study stations. The NAO–rainfall correlation structures for Mostaganem (SCM) and Timgad exhibited comparable behaviours across all scales, differing mainly in amplitude rather than form. To avoid redundancy and preserve clarity, only the Beni Yenni case is shown in detail, while Figure 11 provides a synthesized summary of the optimal wavelet families and corresponding results for all sites.

Comment 5: Table 6, why are the selected 'best and suitable wavelet families' different for each of the three stations (Mostagum SCM, Beni Yenni, Timgad)? Does this imply the results cannot be transferred to studies in other regions? The authors must analyze this by connecting these different wavelet choices back to the specific statistical rainfall characteristics (mean, variance, skewness, kurtosis) of each station in Table 2.

Reply and revision: We thank the reviewer for this insightful comment. The fact that different wavelet families were identified as most suitable for each station reflects genuine spatial variability in the rainfall signal rather than methodological inconsistency. Each rain gauge represents a distinct hydro-climatic context—coastal (Mostaganem), mountainous (Beni Yenni), and high-plain (Timgad)—and their rainfall distributions (Table 2) differ in mean, variance, skewness, and kurtosis. The wavelet that best captures NAO–rainfall interactions depends on these signal characteristics. We have added a discussion linking these statistical properties to the mathematical traits of the chosen wavelets and clarified that results are locally representative but conceptually transferable once the signal properties are re-evaluated for new regions.

 Revision to Add in Discussion (after Table 6)

The selection of different optimal wavelet families across the three rain-gauge stations is partially consistent with the contrasting rainfall statistics summarized in Table 2. At Beni Yenni, the highest mean and variance indicate a strong low-frequency modulation combined with large multi-annual fluctuations; biorthogonal and high-order Daubechies wavelets, which balance sensitivity to both smooth and intermittent variations, therefore yielded the most coherent results. Mostaganem, showing the greatest skewness and kurtosis, experiences more irregular and extreme rainfall episodes typical of a coastal climate; asymmetric, compact-support wavelets such as Daubechies effectively localize these abrupt transients [62,63,63]. In contrast, Timgad exhibits lower variance and smoother temporal behaviour, favouring more regular and near-symmetric wavelets (Symlets, Coiflets) that emphasize persistent low-frequency oscillations.

These differences demonstrate that optimal wavelet selection is signal-dependent: the mathematical features of each wavelet (symmetry, regularity, vanishing moments) align with the statistical texture of the rainfall record. Hence, while the general framework is transferable, the specific “best” wavelet should always be reassessed according to local rainfall variability and climatic regime.

We sincerely thank the reviewer for their detailed and constructive feedback, which has greatly helped improve the clarity, focus, and scientific rigor of the manuscript. We deeply appreciate the reviewer’s time and effort in helping enhance the quality of this work.

Warm regards,

Dr. Youssef M. Youssef

(On behalf of all co-authors)

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors responded to the comments.