the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 14 Apr 2026
Scale dependence of precipitation structure using Tweedie Poisson–Gamma scaling: an Estonian case study from radar composites and gauges
Abstract. We characterize precipitation structure over Estonia and the surrounding region using Tweedie Poisson–Gamma scaling, with Tweedie power as a descriptor of aggregation-dependent mean–variance behaviour in zero-inflated, heavily-positive-skewed precipitation. The study extends existing Tweedie precipitation analysis to the joint examination of temporal aggregation, seasonality, spatial variability, and observing-system differences. The analysis uses a 1 km, 5 min radar composite derived from the two Estonian C-band radars together with an OTT Pluvio2 L gauge network over September 2020 to August 2024. For accumulation lengths from sub-hour to daily windows, is estimated from block-wise mean and variance statistics using ordinary least squares, with availability filtering to handle missing radar timestamps and matched window sampling for radar–gauge comparison.
Across all data sources, p increases with accumulation length, showing that temporal aggregation changes precipitation mean–variance scaling. Seasonal separation is also clear, with generally highest in summer and lowest in winter, and winter showing the strongest increase with accumulation. Spatially, the all-year radar fields show strong scale dependence but only weak geographical contrasts at fixed accumulation length, whereas seasonal maps show clearer heterogeneity at longer windows. Radar-based at station locations is generally higher than gauge-based estimates, and the magnitude and spread of the differences depend on accumulation length and gauge temporal resolution. These results show that should not be used as a fixed precipitation parameter transferable across durations, seasons, space, or products. Instead, it provides a scale-aware benchmark for evaluating precipitation consistency in generated, corrected, or forecast products, such as quantitative precipitation estimation, downscaling, and nowcasting.
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RC1: 'Comment on egusphere-2026-2031', Anonymous Referee #1, 11 May 2026
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-2031/egusphere-2026-2031-RC1-supplement.pdfCitation: https://doi.org/
10.5194/egusphere-2026-2031-RC1 -
RC2: 'Comment on egusphere-2026-2031', Anonymous Referee #2, 16 Jul 2026
Note: this referee comment is independent from the other referee comment as I did not read what the other referee wrote.
Summary:
This article demonstrates the use of the Tweedie power-p, which provides a relationship between the mean and variance of precipitation amount. The statistical modelling of precipitation with a Tweedie distribution is equivalent to the Poisson-Gamma distribution when 1<p<2 and represents an alternative to the Markov-Gamma statistical modelling of precipitation.
Overall, I find the idea and approach interesting, but I would like to see more concrete applications and a deeper analysis of what Tweedie power-p is expected for different accumulation periods and precipitation regimes. I thus recommend major revisions.
Major revision 1:
The motivation (the why) of this study is not completely clear from the Introduction only. Section 5.2 in Discussion helps to better understand the motivation. However, from Results in Section 4, I am left thinking “so what?”. To make the paper more concrete, I recommend adding at least one numerical experiment demonstrating an application of the Tweedie power-p to assess the statistical realism of precipitation products. For example, the Tweedie power-p for different spatial or temporal scales could be assessed for one or several statistical or artificial intelligence-based precipitation downscaling methods.
Major revision 2:
Results show that Tweedie power-p varies depending on accumulation length, radar VS gauge observations, locations and seasons, but it is not clear what causes p to vary. It is not clear whether the difference between the Tweedie power-p for radar and gauge is due to observation errors, to spatial aggregation (for the radar) or to representativeness error. A deeper analysis of the behavior of the Tweedie power-p using a different (not necessarily better) statistical model of precipitation would provide some insight. For example, using the Markov-Gamma statistical model and aggregating to different interval lengths, you could compute the Tweedie power-p and see if it is constant or varies with aggregation length. A more advanced spatio-temporal model of precipitation could also be used to study the effect of spatial aggregation and compare it with radar precipitation. Finally, a model of observation error could be used to assess the impact of ground clutter or radio-frequency interference on the Tweedie power-p.
Minor comments
I have also several minor recommendations to enhance the clarity of the paper.
- l. 27 : “systems” -> “weather systems”
- l. 33: “rainfall statistics” -> “rainfall statistics at a particular location” (note that these statistics do not account for the spatial structure of precipitation)
- Eq. 1: why only show the wet case? “where the wet or dry probability only depends on the previous state W_{t-1} = i: P(W_t = j|W_{t-1}=i) = p_{ij}, where I and j are wet-dry indicators.
- Eq. 2: Define parameters k and theta.
- l. 50-54: clarify if \mu_t, p and \phi are constants over time
- Define zero-inflated, positively skewed totals…
- How do we find if the total precipitation amounts are zero given a mean \mu_t and a variance Var(Y_t), i.e. can you sample the probability distribution? I would suggest to present the Poisson-Gamma distribution for 1<p<2 first and mention that the Tweedie distribution generalizes it for other p’s.
- l. 57: The parameter \lambda_t is not defined, what range of values can it take?
- l. 57: What represents X_{t,i}, what are the parameters, are they all the same for all t and i?
- (Introduction) Why do we care about that the Tweedie power p varies “across space, time, and observing systems”? What are the concrete implications? Can you make a stronger point for applications (see major revision 1)? I don’t see how this study will be useful for “the evaluation of precipitation statistics in radar products, nowcasting systems, and other high-resolution earth system models”. I suggest a slight rewrite by bringing some of the points presented in Discussion (5.2) as well as introducing the application example from major revision 1.
- l. 113: Can you explain why the 10 minutes and the hourly time series are not consistent, are they measured with independent instruments?
- How is the block length L selected in Table 1? How are the results sensitive to different choices?
- Figure 3: Explain in Methods what the Poisson and Gamma limit represents/how they are computed.
- Figure 4: the x-axis is logarithmic, are the Gamma and Tweedie distribution also using a logarithmic x-axis? The match between the histogram and the distributions is not very convincing…
- Figure 5: suggest to also add Gamma and Tweedie distribution fit for each season
- Figure 6: Radials artifacts (radio-frequency interference?) can be seen in the radar images…
- A comment about that should be added in the radar data description.
- The high values of these artifacts might alter the analysis of radar data; it would be best to exclude these radials when computing statistics.
- Figure 6: The range-dependence of the Tweedie-p parameters should be commented on.
- Are ground clutter artifacts affecting the precipitation values at lower range?
- Do you have under-detection of shallow precipitation (e.g. snow) at longer ranges?
- How the larger binning size at the longest range affects the results?
- Figure 6: The sub-title over each sub-figure is too small. It should be removed.
- Figure 7: How are the patterns robust to finite sampling? E.g. are the patterns similar from year to year?
- Figure 7: I don’t really get what insight someone would get by looking at a map of Tweedie-p exponents. Figure 7 could be condensed by selecting only one accumulation length.
- Table 2 and 3: Definition of Delta p should be included in the caption. Actually, I would recommend to remove section 4.4 altogether and replace it with another application (see major revision 1).
- Figure A1: I suggest to also add Tweedie distribution fit for each instrument
Citation: https://doi.org/10.5194/egusphere-2026-2031-RC2 -
RC3: 'Comment on egusphere-2026-2031', Anonymous Referee #3, 28 Jul 2026
This is an interesting and well-written manuscript. The authors should be commended. I am commenting on the parts of the manuscript on which I am qualified to comment (e.g., I know nothing of Estonian weather, etc.).
Some comments:
- A lot of the existing work on working with Tweedie distributions with rainfall is not mentioned, and this is required to situate this manuscript in the literature. For instance:
- Dunn (2004): the first paper I know of that uses Tweedie distributions with rainfall (at least in any substantive way).
- Other papers by Hasan et al. (2010, 2012, 2015) are also probably relevant, or at least worth mentioning, as they cover different scales too.
- The method for estimating p (e.g., by regressing log Var(y)) on log μ) is presented as new work, but it is not new work. The first mention I know of using this method to estimate p is found in Dunn & Smyth (2005) (and it is also mentioned in their textbook: Dunn & Smyth (2018)).
- The authors should also mentioned that finding the MLEs of p is possible, as described in Dunn & Smyth (2018) and Dunn & Smyth (2005)). The authors should mentioned why the presumably-superior MLE method was not used even though it is available (presumably, because it is much slower than the method presented, and/or the authors did not use the R package: https://cran.r-project.org/web/packages/tweedie/index.html).
- The authors do not mention what software they used, which they definitely should.
- Many of the figures do not present well in grey-scale (e.g., in Fig. 4 the two lines are indistinguishable; the two shading used in Fig. 5). They authors shoud consider this when drafting their manuscript.
- The subtle range of colours in Fig. 7 can be hard to distuinguish; would the authors consider adding some contour lines, even a small number of contour lines?
- Some of the figures are very busy, containing a lot of information that is hard to differentiate (e.g., Fig. 2). I don't know if it is possible or sensible, but perhaps consider two side-by-side plots that may make the information easier to read.
- Is the use of three decimal places of p actually suitable (e.g., Tables 2 and 3)? Consider two decimal places. The differences between, say, p = 1.41 and p = 1.42 is almost indistinguishable, so I doubt any would notice or care about the third decimal places.
Minor issues:
- Line 35: "...sequence; for example, a first-order..."
- Line 40: "...framework, originating from..."
- Line 43: Rather than "On the amount side", consider "For rainfall amounts".
- Line 174: I'm not sure what is meant by "representative of products"
- Line 207: x-axis rather than x-axis.
References
Dunn, P. K. (2004). Occurrence and quantity of precipitation can be modelled simultaneously. International Journal of Climatology, 24(10), 1231-1239.
Hasan, M. M., & Dunn, P. K. (2010). Two Tweedie distributions that are near‐optimal for modelling monthly rainfall in Australia. International Journal of Climatology, 31(9), 1389-1397.
Hasan, M. M., & Dunn, P. K. (2012). Understanding the effect of climatology on monthly rainfall amounts in Australia using Tweedie GLMs. International Journal of Climatology, 32(7).
Hasan, M. M., & Dunn, P. K. (2015). Seasonal rainfall totals of Australian stations can be modelled with distributions from the Tweedie family. Int. J. Climatol, 35, 3093-3101.
Dunn, P. K., & Smyth, G. K. (2005). Series evaluation of Tweedie exponential dispersion model densities. Statistics and Computing, 15(4), 267-280.
Dunn, P. K., & Smyth, G. K. (2018). Generalized linear models with examples in R (Vol. 53, p. 16). New York: Springer.
Citation: https://doi.org/10.5194/egusphere-2026-2031-RC3 - A lot of the existing work on working with Tweedie distributions with rainfall is not mentioned, and this is required to situate this manuscript in the literature. For instance:
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