the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 02 Feb 2026
Beyond trends and cycles: rainfall as a sequence of irregular regimes
Abstract. Rainfall is an oscillatory rather than purely stochastic signal, whose variability reflects alternating hydrological regimes rather than long-term trends. Recognizing this regime-based nature marks a conceptual shift in the way climatology interprets rainfall variability. At the monthly to multiannual scale, precipitation evolves through irregular wet, dry, and stationary phases whose duration and intensity vary over time. Although trend analyses, anomaly-based metrics, and spectral methods may at times suggest contrasting interpretations – each being sensitive to different aspects of the signal – they capture only partial views of a shared underlying variability. Framing precipitation as a sequence of irregular regimes offers a unifying perspective that helps reconcile these approaches and clarifies how rainfall fluctuations actually unfold. Using the Po River basin (Northern Italy) as an illustrative case, we show that Fourier and wavelet analyses confirm the intermittent character of rainfall oscillations, with regular periodicities emerging only at limited intervals. The Cumulative Deviation from Normal (CDN), computed as the cumulative sum of standardized monthly precipitation (SPI1), provides a simple yet physically consistent framework to visualize these irregular regimes and to quantify the resulting changes in water availability driven by cumulative surplus or deficit.
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RC1: 'Comment on egusphere-2026-292', Anonymous Referee #1, 05 Mar 2026
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The manuscript "Beyond trends and cycles: rainfall as a sequence of irregular regimes" by Di Paola et al., describes an interesting approach to studying variability in a hydrologic time series. However, it seems a bit overemphatic in designating the use of Cumulative Deviation from Normal applied to SPI as "a paradigm shift" (see Section 2 title line 51). The statistical tools used are well-known and quite accessible to any hydrologist. I think the approach being presented is more of an analysis step-by-step guide, piecing together "standard statistical procedures" (see the "code, data... " section).Furthermore, I do not fully understand what is changing here with respect to typical hydrologic analysis methods. What is the contrast between "trends and cycles" and "sequence of irregular regimes"? Stochastic hydrology has long viewed time series as realizations of stochastic processes that show both some degree of systematic fluctuations and irregularity. The ideas presented here, as I understand them, do not really move beyond this point of view. If I have misunderstood, I think the the average reader need some further explanation and illustration of the novelty of the approach. If I have not misunderstood, I ld focus much more on the methodology than on a change of paradigm.Regarding the methodology, I think it also needs better explanations and illustration of what can be ascertained with CDMs of SPI values that cannot be determined with other methodologies. For example, it would be nice to see a few SPI_k traces to compare with Figure 2b and see what type of information may be shared between the two.Overall, this may be a useful contribution to the literature, but needs some major revisions to do so.Detailed comments:lines 38-43. It is not clear how this paragraph connects to the earlier and later ones. Do the authors suggest this as an explanation of the implications of lines 34-37? I don't think one can conclude that trend studies not being in agreement necessarily stems from the listed shortcomings of trend or frequency analysis approaches. Many other things may play a role. Spatial variability, different time series lengths, different methods, etc. Please clarify.44-50. similar comment as above: not clear what the authors are trying to say here. I expected a description of the gap that they are trying to fill, but I could not find compelling information suggesting that we need an analysis method based on CDM and SPI values defined by a QQ transform to eliminate the skewness. I suggest there should be a more structured discussion as to what is missing in our analysis toolbox that the authors are trying to produce.51-61 CDM needs to be introduced clearly at the beginning of this section, before the authors can start discussing its applications. I suggest using one equation to define it.81-82 and 100-103 no need to define the acronym, CDM, again.94-98 It would be useful to see a geometric interpretation of the definition of Delta. My take is that Delta is, ultimately, what the Authors suggest to be the main analysis tool related to CDM traces as it materializes the average slope over sub-intervals. Is this correct? I think the Authors could stress a little more effectively how one is to use the CDM in their framework.98-99 This sentence is difficult to decipher. What do the authors mean by "oscillatory-phase" behaviour? Related to the phase of a spectral decomposition? Related to an amplitude of oscillation? Please expand and clarify.139-141 I strongly believe the authors should make their code publicly available to enable reproducibility. Not sharing codes significantly slows down scientific advancements and leads to limited use of published results. Ultimately, code sharing is in the best interest of the community and of the authors.ReplyCitation: https://doi.org/
10.5194/egusphere-2026-292-RC1 -
CC1: 'Reply on RC1', Arianna Di Paola, 16 Mar 2026
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We thank the reviewer for the insightful comments, which prompted us to clarify aspects of our contribution that were not sufficiently explicit in the manuscript. We hope to have the opportunity to revise the manuscript according to the following considerations.
Overall, we fully agree that our work is based on statistical methods well known in the scientific community. As correctly pointed out by the reviewer, we are not proposing a new method to add to the analysis toolbox, but rather a new analytical recipe: a coherent and operationally accessible way of combining existing tools. The novelty lies in the approach and in what this recipe produces. By computing the CDN as the cumulative sum of SPI1, we obtain a single continuous trajectory that simultaneously encodes the information of all SPIₖ aggregates — as formally demonstrated in Appendix A. The CDN provides a full overview of overlapping wetting and drying phases occurring at different timescales, their duration, and their nesting across scales. We note that we now explicitly refer to wetting and drying phases rather than wet and dry conditions — a terminological distinction that will be reflected in the revised manuscript. This is not a trivial property: to our knowledge, no existing method allows simultaneous clear visualization of multi-scale wetting and drying phases. A single SPIₖ, regardless of the chosen k, does not carry this property. Moreover, choosing a specific k is inherently arbitrary and structurally limiting — it means examining one temporal scale while remaining blind to all others and, especially, to their nesting.
To further clarify the concept of nested and overlapping regimes, we are considering enriching panel (b) of Figure 1 with explicit annotations identifying wetting, drying, and stationary phases at multiple timescales directly on the CDN trajectory. We believe this visual addition will make the regime-based interpretation immediately accessible to a broad readership: [ *** see NEW_FIG1_b.png in attached zip file ***]
These nested and overlapping wetting and drying phases of varying duration lead directly to the contrast between "trends and cycles" and "aperiodic regime sequences". By trends we refer to monotonic directional changes over a specified period; by cycles we refer to regular periodicities. Consider a signal comprising a stationary phase lasting 9 years, followed by a wetting phase of 3 years, then a drying phase of 4 years: neither linear trends nor spectral methods would return operationally useful results on a signal having neither a monotonic direction nor regular cycles. The CDN makes such sequences immediately visible and quantifiable.
Again, while simple methods such as linear trends and anomaly-based metrics are inadequate to reveal aperiodic regimes, complex tools capable of doing so remain inaccessible to most applied practitioners. This gap is not theoretical — it is documented by the very literature cited in our manuscript (Vicente-Serrano et al., 2025; Beranová et al., 2025; Luppichini and Bini, 2025; Doane-Solomon et al., 2025), where recent publications in high-profile journals continue to rely on linear trends and anomaly-based metrics as default analytical frameworks. The gap between what theory has reached and what practitioners routinely implement is real and documented. The CDN contributes to bridging this gap.
With that provided, while we acknowledge that further improvements are required to clearer explanation, we respectfully maintain that "paradigm shift" is appropriate — referring specifically to a methodological and operational paradigm shift, not a theoretical one (it will be stressed in the revised manuscript).
We will revise the text accordingly, and we address the specific expository gaps identified in the detailed comments below.
Anonymous Referee #1, 05 Mar 2026
The manuscript "Beyond trends and cycles: rainfall as a sequence of irregular regimes" by Di Paola et al., describes an interesting approach to studying variability in a hydrologic time series. However, it seems a bit overemphatic in designating the use of Cumulative Deviation from Normal applied to SPI as "a paradigm shift" (see Section 2 title line 51). The statistical tools used are well-known and quite accessible to any hydrologist. I think the approach being presented is more of an analysis step-by-step guide, piecing together "standard statistical procedures" (see the "code, data... " section).
We clarify that we refer specifically to a methodological and operational paradigm shift — not a theoretical one. This distinction is now explicit in the revised manuscript.Let us also recall that precipitation analyses are routinely conducted not only by hydrologists and physicists, but also by agronomists, water resource managers, and policymakers, who often lack the mathematical background required to implement more sophisticated methods.
Furthermore, I do not fully understand what is changing here with respect to typical hydrologic analysis methods. What is the contrast between "trends and cycles" and "sequence of irregular regimes"?
We hope that the detailed response provided in our general comment above has thoroughly addressed this point. Stochastic hydrology has long viewed time series as realizations of stochastic processes that show both some degree of systematic fluctuations and irregularity. The ideas presented here, as I understand them, do not really move beyond this point of view. If I have misunderstood, I think the the average reader need some further explanation and illustration of the novelty of the approach. If I have not misunderstood, I ld focus much more on the methodology than on a change of paradigm.
We hope that the detailed response provided in our general comment above has thoroughly addressed this concern. Having a theoretical framework that acknowledges irregularity is not equivalent to having an operational tool that describes, visualizes, and quantifies it in an accessible way. This is the gap the CDN concurs to fill — not by introducing new methods, but by combining a well-known method (SPI) into a new analytical recipe whose output, to our knowledge, cannot be obtained by any single existing approach.
Regarding the methodology, I think it also needs better explanations and illustration of what can be ascertained with CDMs of SPI values that cannot be determined with other methodologies. For example, it would be nice to see a few SPI_k traces to compare with Figure 2b and see what type of information may be shared between the two.
We welcome this suggestion and have revised Figure 2 accordingly, focusing on a shorter time window to improve readability and adding the corresponding SPIₖ traces alongside the CDN and its local regression slopes:
[ *** see NEW_FIG2.png in attached zip file ***]
SPIₖ traces and the corresponding colored areas convey the same directional information — wetting or drying phases — but the colored areas are considerably more readable: by representing the sign and magnitude of the local slope as filled areas, phases become immediately identifiable without the need to track an noise curve across the zero line. Note that the moving-window regression shown here is one possible way to enrich the visual reading of the CDN, not a prescriptive analytical step — the CDN trajectory itself remains the core diagnostic tool.
Overall, this may be a useful contribution to the literature but needs some major revisions to do so.
Detailed comments:
lines 38-43. It is not clear how this paragraph connects to the earlier and later ones. Do the authors suggest this as an explanation of the implications of lines 34-37? I don't think one can conclude that trend studies not being in agreement necessarily stems from the listed shortcomings of trend or frequency analysis approaches. Many other things may play a role. Spatial variability, different time series lengths, different methods, etc. Please clarify.
We agree that the contrasting results reported in the literature stem from multiple causes, including spatial variability, differences in record length, and methodological choices. The paragraph intends to highlight that methodological inadequacy is among these causes — not the sole explanation. We will revise the text to make this partial causal link explicit, avoiding any implication of exclusivity.
44-50. similar comment as above: not clear what the authors are trying to say here. I expected a description of the gap that they are trying to fill, but I could not find compelling information suggesting that we need an analysis method based on CDM and SPI values defined by a QQ transform to eliminate the skewness. I suggest there should be a more structured discussion as to what is missing in our analysis toolbox that the authors are trying to produce.
Regarding the gap: we agree that it was not articulated with sufficient clarity. The gap we identify is not a missing method in the theoretical toolbox, but the absence of an operational recipe that is simultaneously adequate and accessible. Simple methods — linear trends and anomaly-based metrics — are accessible but inadequate to reveal aperiodic regimes. Complex methods capable of profiling aperiodic regimes, their duration, and their overlapping and nested structure — if they exist at all — require substantial mathematical expertise that most applied practitioners lack. The CDN fills precisely the space between these two extremes. We will revise the text to make this structure explicit, including a clearer closing statement that connects the identified gap directly to the proposed approach.
51-61 CDM needs to be introduced clearly at the beginning of this section, before the authors can start discussing its applications. I suggest using one equation to define it.
We agree that a formal equation defining the CDN should appear early in Section 2, and we will add it. However, we respectfully maintain that the equation is most effectively introduced after a brief motivational narrative explaining why SPI1 — rather than raw precipitation — is the appropriate variable to accumulate. The positively skewed distribution of monthly precipitation and the resulting numerical drift of arithmetic-mean-based cumulative curves are prerequisites for understanding why the CDN is defined as it is. Introducing the equation before this context would risk presenting a formal definition without the physical motivation that makes it meaningful. The equation will therefore appear at the first naturally appropriate point in the narrative, immediately following the introduction of SPI1
81-82 and 100-103 no need to define the acronym, CDM, again.
We agree
94-98 It would be useful to see a geometric interpretation of the definition of Delta. My take is that Delta is, ultimately, what the Authors suggest to be the main analysis tool related to CDM traces as it materializes the average slope over sub-intervals. Is this correct? I think the Authors could stress a little more effectively how one is to use the CDM in their framework.
The reviewer's geometric interpretation of Δ is correct: it represents the cumulative change along the CDN over a window of width W, geometrically equivalent to the slope of the linear regression fitted to the CDN within that window, where W is a simple scalar. However, we clarify that Δ is not the primary analytical tool — the CDN trajectory itself is. The CDN provides the continuous visual framework from which regime phases are identified directly by eye: ascending segments indicate wetting, descending segments indicate drying, and near-horizontal segments indicate stationary conditions. Δ is a derived quantitative descriptor that allows one to measure the magnitude of a phase — how much water was gained or lost over a given period — rather than to identify it. It is one possible way to enrich the reading of the CDN, not the core of the perspective. Many other approaches could be developed depending on the operational needs of the user.
A concrete example of how the CDN is used in practice is illustrated in both the old and revised Figure 2, focused on the Po basin. Consider the extreme drought of 2023: the CDN reaches its absolute minimum in March 2023. The alignment between this minimum and the minimum of both the 36-month local slope and the SPI36 trace (panel b) identifies 36 months as the dominant timescale of that drought event: scales shorter than 36 months had already changed the phase to wetting, while longer scales had not yet reached their minimum. We can confidently state that the 2023 event resulted from a precipitation deficit of approximately 1000 mm accumulating over 36 months. This diagnostic reading emerges directly and immediately from the CDN, without requiring any additional analysis.
We will revise the text to make this hierarchy explicit and to provide a clearer operational description of how the CDN is used in practice
98-99 This sentence is difficult to decipher. What do the authors mean by "oscillatory-phase" behaviour? Related to the phase of a spectral decomposition? Related to an amplitude of oscillation? Please expand and clarify.
We acknowledge that the expression 'oscillatory-phase behaviour' is ambiguous and may suggest a connection to spectral decomposition or regular oscillation amplitudes, which is not our intent. We refer simply to the directional state of the CDN at a given time — whether the system is in a wetting, drying, or stationary phase. This confusion is symptomatic of a broader terminological issue that we have identified in our manuscript: the term 'oscillatory', used in several places, inadvertently implies regularity. As noted in our general response, we will replace it systematically with 'aperiodic regime alternations' or equivalent phrasing throughout the revised manuscript, which will also resolve the ambiguity in this specific sentence.
139-141 I strongly believe the authors should make their code publicly available to enable reproducibility. Not sharing codes significantly slows down scientific advancements and leads to limited use of published results. Ultimately, code sharing is in the best interest of the community and of the authors.
We thank the reviewer for this important recommendation and fully agree that code sharing is in the best interest of the community. Input data are openly available through the ARCIS project. The analyses are based on the DroughtScan Python library (https://github.com/PyDipa/DroughtScan) developed by the authors, which is publicly available on GitHub and periodically updated. The revised Figure 2 will be available as a dedicated function within this library shortly after the revised manuscript is submitted. A stable release of DroughtScan will subsequently be deposited on PyPI and, together with the ready-to-use input data, on Zenodo upon final acceptance. We acknowledge the reviewer's point and will update the 'Code and Data Availability' section accordingly.
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CC1: 'Reply on RC1', Arianna Di Paola, 16 Mar 2026
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