the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 27 May 2024
Theoretical Annual Exceedances from Moving Average Drought Indices
Abstract. Numerous drought indices originate from the Standardized Precipitation Index (SPI) and use a moving average structure to quantify drought severity by measuring normalized anomalies in hydroclimate variables. This study examines the theoretical probability of annual exceedances from such a process. To accomplish this, we derive a stochastic model and use it to simulate 10 million years of daily or monthly SPI values in order to determine the distribution of annual exceedance probabilities. We believe this is the first explicit quantification of annual extreme exceedances from a moving average process where the moving average window is proportionally large (5–200 %) relative to the year. The resulting distribution of annual minima follow a Generalized Normal distribution, rather than the Generalized Extreme Value (GEV) distribution, as would be expected from extreme value theory. From a more applied perspective, this study provides the expected annual return periods for the SPI or related drought indices with common accumulation periods (moving window length), ranging from 1 to 24 months. We show that the annual return period differs depending on both the accumulation period and the temporal resolution (daily or monthly). The likelihood of exceeding an SPI threshold in a given year decreases as the accumulation period increases. This study provides clarification and a caution for the use of annual return period terminology (e.g. the 100 year drought) with the SPI and a further caution for comparing annual exceedances across indices with different accumulation periods or resolutions. The study also distinguishes between theoretical values, as calculated here, and real-world exceedance probabilities, where there may be climatological autocorrelation beyond that created by the moving average.
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RC1: 'Comment on egusphere-2024-1430', Anonymous Referee #1, 08 Jul 2024
General comment
The study presents a theoretical analysis of the annual return period of standardized variables (such as SPI) commonly used in drought study. I found the topic of the research of interest, as someone that fully support a better clarity on terms such as ‘100 years drought’ often used in the community without robust statistical support. The paper is well structure and easy to follow. I have, however, two major concerns that make hard for me to recommend the publication of the paper in its current form:
1) The authors generate 10 million years of data based on only two criteria: i) each month is standard normally distributed, and ii) a uniformly weighted backward average. As also stated by the authors, the only factor that generates autocorrelation in this procedure is the moving window, and any other factor causing persistence is ignored. This is a very strong assumption, as the scientific literature is full of studies on the tendency of rainfall to persist in the dry status, clustering of rainfall days, burstiness, etc. My question for the authors is: how much the theoretical time series generated with this approach resemble actual SPI time series? The authors do not provide any evidence that the theoretical values behave like real values, so any conclusion on the statistical behaviour of the theoretical data can be completely meaningless in real conditions, unless the author demonstrate that real and theoretical data are similar (statistically). My statistical background is not good enough to suggest a “validation” strategy (autocorrelograms?), but without this key step the results reported in this study cannot go beyond a mere mathematical exercise not suitable for a scientific publication.
2) The way that SPI (or any other standardized index) is used in real applications is often for the detection of drought events (i.e., consecutive periods with values below a certain threshold). In this context, the analysis of annual minima is not really in line with what is commonly used by practitioners. To refer to one of the examples reported in the manuscript “The idea of experiencing an extreme flash drought at least once every other year…”: no one is going to define as an “extreme drought event” a single isolated anomaly on a 15-day period (which, by the way, is a very unusual time window for anomaly computation). The concept of consecutive time steps under a given threshold is a key factor in defining a drought, and, in this regard, your analysis on the annual minima may have very limited connection to what is commonly used from claims such as “a one in 100 years drought”. A much more interesting analysis would be on the drought periods, and their effective (annual) return period (including the effects of inter-arrival time, etc). I understand that this may diverge too much from the goal of this study, but, at the minimum, the focus on annual minima (rather than event) should be super clear from the start (title, abstract, motivation, etc.) and the caveats in using these results when discussing events should be clarified.
I am aware that the authors have a clear view on the limitations of this study in regards of both topics, as evidenced in some parts of the discussion. However, I still believe that a proper analysis of the base assumptions of the study need to be added before considering valuable the obtained outcomes.
Beyond these two major criticisms, I report some additional comments that I hope will be useful to improve the overall quality of the manuscript.
Title: the focus on annual minima should be clear already in the title.
L12. Same here, exceedance of annual minima…
L17. Something on the extreme clustering should be mentioned here.
L112. There is nothing that backup this claim. As a key factor, a proof of this assumption is absolutely required.
Fig. 1. Would an analogous plot for real SPI-6 values have a similar shape?
L172. L-moment and l-moment are both used. Please use a consistent terminology.
L175. “…is exactly symmetrical…” Is this statement true? It is true that SPI are derived from a standard normal distribution, but as a rescaling of a Gamma (usually, or any other left bounded distribution), SPI shouldn’t be symmetrical. I do not think that this is a problem in your study, but I would be carefully rewording this sentence.
L183. How did you define a good fitting based on the AIC? Which values? Significance?
L191. This section is a little confusing. As currently stated, it may give the impression that a “normal” distribution is followed. However, in my understanding, the 3-parameter lognormal distribution is still a distribution designed to reproduce extremes, it is only not part of the GEV family. In the current form, it seems that the data do not follow the behaviour of extreme values but that of “normal” values, but this is not the case. I think the section need to be reworded to better clarify what does it mean (in practice) that the data follow the 3-parameter lognormal distribution rathe than the GEV. Most of the readers of the papers may not be expert in statistics and may come up with a wrong conclusion.
L212. The log transformation is commonly deployed to use normal distribution on extreme values, so this is coherent with the extreme nature of annual minima.
L215-216. It is not clear if these studies used the generalized normal for extreme values, similar to the ones analysed here.
L232. It would be useful to report an example for the GEV too. Also, I don’t see any AIC values reported. How did you use AIC to evaluate the goodness of fit? Are all the fittings statistically significant?
L283. Stagge et al. (2016)
L288-295. Albeit true, this effect may be amplified by the particular method used to generate the data. In real SPI time series, some effect of persistency will be present even in SPI-1 or SPI-3, otherwise the concept of “drought event” would not be possible in such time series.
L307. For which SPI value? -2?
L311-313. This is true only if annual minima are analysed. If events are analysed (consecutive periods under a certain threshold), then the daily or monthly time scale should only have minimal effects.
L323. This should read “Discussion”.
L344. Again, how can you confidently claim that those are minimal deviation, as they are an integral part of how precipitation behaves.
L351. Berman (1964).
L365. A figure on the extreme clustering is needed to better show this concept and its effects. In general, this part seems really useful but very poorly presented and discussed.
L371-372. This is true for long accumulation periods only.
L375. Again, a reference to extreme clustering is made but without support of data showing the effect to the readers.
L379. There is a typo.
L383. Again, the focus on annual minima is not clear here. Annual exceedance probability can be computed for any metrics derived from SPI time series.
L400. It should be observatory (also in other sections of the text).
L418. …period).
L419-420. This is related to a common understanding of how probability works, and it has very little to do with the results of your study. Similarly, someone could argue that saying that a minimum annual SPI < -2 has a certain occurrence probability (as reported in this study) is completely different than the probability of a certain drought event (with a given severity) to occur.
Citation: https://doi.org/10.5194/egusphere-2024-1430-RC1 -
RC2: 'Comment on egusphere-2024-1430', Anonymous Referee #2, 25 Jul 2024
The manuscript „Theoretical Annual Exceedances from Moving Average Drought Indices“ applies a statistical model for approximating return periods for moving averages processes of standardized drought indices. The study highlights the problem of the terminology of return periods in case of drought indices and provides theoretical values for these return periods by a simulation study. I think the study provides a novel contribution scientific literature, especially for the application (and misuse) of standardized drought indices. Nevertheless, I have some major concerns, which hopefully can be addressed by the authors, as they make an important point about the use of drought indices, which is worth publishing.
First, I find it problematic, that the temporal autocorrelation of the drought index is outlined, but then a distribution is fitted to the annual minima series, where the temporal correlation is not considered prior to fitting the distribution. In my opinion the study would benefit, if a statistical sound estimation procedure is outlined, which considers the temporal autocorrelation before fitting an extreme value distribution. This does not mean, to actually change the entire scope of the study, but to provide a better approach for many applications.
Second, I think the Introduction would benefit of highlighting, where in scientific literature this pitfall of annual return periods for drought indices, actually can be identified. As drought events are mainly characterized over intensity, duration, I think it is important to underline, where return periods of annual SPI values are estimated to better frame the scope of the study.
Finally, the simulation study could be set up in a more realistic scenario, which includes sample sizes that are currently common for e.g. precipitation, streamflow or groundwater time series. In this context, also the uncertainty of these moving averages can be quantified. I think it is not the best solution to provide values for these return periods, but to outline a more robust statistical estimation procedure.
Minor comments:
Line 63: Maybe this sentence could be split up.
Line 181: Which method was then actually used (MLE or l-moments)? I believe there should be no difference in the estimates.
Line 185: Please specify the actual used exceedance probabilites somewhere in the methods section.
Line 220: Rootzen, 1986).
Line 305: Is this not only an effect of sample size (and the temporal autocorrelation)? As I have 365 chances of the SPI value falling below -2 (daily data), or 12 chances of a fixed window falling below -2 (monthly data)?
Line 349: Missing bracket.
Citation: https://doi.org/10.5194/egusphere-2024-1430-RC2
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