the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Accounting for Precipitation Asymmetry in a Multiplicative Random Cascades Disaggregation Model
Kaltrina Maloku
Benoit Hingray
Guillaume Evin
Abstract. Analytical Multiplicative Random Cascades (MRCs) are widely used for the temporal disaggregation of coarse-resolution precipitation time series. This class of models applies simple scaling laws to represent the dependence of the cascade generator on the temporal scale and the precipitation intensity. Although determinant, the dependence on the external precipitation pattern is usually disregarded. Our work presents a unified MRC modelling framework that allows the cascade generator to depend in a continuous way on temporal scale, precipitation intensity and a so-called precipitation asymmetry index.
Different MRC configurations are compared for 81 locations in Switzerland with contrasted climates. The added value of the dependence of the MRC on the temporal scale appears to be unclear, unlike what was suggested in previous works. Introducing the precipitation asymmetry dependence in the model leads to a drastic improvement of model performance for all statistics related to precipitation temporal persistence (wet/dry transition probabilities, lag-n autocorrelation coefficients, lengths of dry/wet spells). Accounting for precipitation asymmetry seems to solve this important limitation of previous MRCs.
The model configuration that only accounts for the dependence on precipitation intensity and asymmetry is highly parsimonious, with only five parameters, and provides adequate performances for all locations, seasons and temporal resolutions. The spatial coherency of the parameter estimates indicates a real potential for regionalisation and for further application to any location in Switzerland.
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Kaltrina Maloku et al.
Status: final response (author comments only)
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RC1: 'Comment on egusphere-2023-544', Anonymous Referee #1, 12 May 2023
egusphere-2023-544
Accounting for Precipitation Asymmetry in a Multiplicative Random Cascades Disaggregation Model
OVERVIEW
This paper develops variants of models that use multiplicative random cascades (MRC) to disaggregate precipitation from quasi-daily to 40-minute time steps. The models begin by closely following those proposed by Rupp et al. (2009) but add an important feature: explicitly accounting for temporal asymmetry in the breakdown coefficients. Four models are calibrated and evaluated with data from many stations across Switzerland. Incorporating asymmetry leads to substantial improvement with regards to statistics related to persistence.
GENERAL COMMENTS
This paper is well-written, the methods are clearly and sufficiently described, the results are adequately discussed, and the analysis supports the conclusions (mostly). The inclusion of asymmetry is handled in a relatively parsimonious manner and the improvements in some performance metrics are clear (notably the length of wet spells and lag-1 autocorrelation).
However, based on some exploratory analysis in Rupp et al. (2009), I do have a lingering doubt related to the apparent asymmetry and the need to explicitly account for it fully.
Asymmetry has been considered before although differently. For example, Olsson (1998) and Güntner et al. (2001) developed the distributions of the breakdown coefficients separately for time intervals that start or end a rainfall sequence or are within a rainfall sequence. They showed that starting/ending intervals had distinctly asymmetric distributions. Rupp et al. (2009) showed that models that did not explicitly incorporate asymmetry did not generate asymmetry at the time steps at which the rainfall was simulated. However, when they resampled their synthetic rainfall at an interval of the same duration but offset in time by small amount, asymmetry in the breakdown coefficients was introduced. The breakdown coefficients from the resampled series were remarkably like the breakdown coefficients from the observed data (see their Figure 16).
Rupp et al. (2009) concluded that at least some of apparent asymmetry in the breakdown coefficients arises from imposing a discrete, regularly timed sampling interval to an irregularly timed phenomenon. To what degree, then, are the authors simply reproducing an artifact of sampling by incorporating asymmetry explicitly into their models? I think this issue needs discussion.
I recommend this paper be published after minor revisions.
LINE-BY-LINE COMMENTS
33-35: Yes, many types and variations of disaggregation models exist. Although it would be excessive to describe them all here, I suggest referencing one or two review papers.
157: Winter should be defined.
169: The text claims to be referring to Figure 1d but I think it should be Figure 1b.
172: Model B has 5 parameters, not three. I_0 and I_1 should be included in the count of parameters.
204: Why use lower-case z for the asymmetry index here and below when it was previously upper-case?
275-279 & 385-398: Licznar et al. (2011) explore in some detail the artifacts arising from measurement resolution. They present a method of adding small amounts of random noise to discretized observations in an attempt to extract the underlying distribution of W at low intensities. It is at least worth a mention even if the authors don’t want to take that approach.
Table 1: As I stated above Model B has a total of 5 parameters when parameters I_0 and I_1 are included. Similarly, Model B+ has 7 parameters.
Figure 1: “SON” should be defined. I assume it is Sep – Nov? Also, the caption says “winter”, whereas SON would be autumn.
Figure 3: Panel (a) takes effort to interpret. I have a few comments:
- What is “x”? Is it W? W_1? W+? For clarity, please replace x by what is represent.
- An ECDF should go from 0 to 1 but it is not obvious that each individual curve does that. For example, the Z =1 curve appears to have a value F(x) ~= 0.5 at x = 1, but does the Z = 1 curve jump to a value of F(x) = 1 very close to x = 1?
- Lastly, although plotting the ECDF is convenient in that several curves can be plotted in one panel, I think it would be much easier to interpret histograms of W for various classes of Z. Notable differences between winter and summer might also be more obvious.
Figure 6: Consider using a log-log scale.
MINOR EDITS
229: Replace “confronted” with “compared”.
474: “…reveals actually not obvious…” Typo?
REFERENCES
Güntner, A., J. Olsson, A. Calver, and B. Gannon (2001), Cascade-based disaggregation of continuous rainfall time series: The influence of climate, Hydrol. Earth Syst. Sci.,5, 145–164.
Licznar, P., Schmitt, T. G., & Rupp, D. E. (2011). Distributions of microcanonical cascade weights of rainfall at small timescales. Acta Geophysica, 59, 1013-1043.
Olsson, J. (1998), Evaluation of a scaling cascade model for temporal rain-fall disaggregation, Hydrol. Earth Syst. Sci.,2, 19–30.
Rupp, D. E., Keim, R. F., Ossiander, M., Brugnach, M., & Selker, J. S. (2009). Time scale and intensity dependency in multiplicative cascades for temporal rainfall disaggregation. Water Resources Research, 45(7).
Citation: https://doi.org/10.5194/egusphere-2023-544-RC1 -
RC2: 'Comment on egusphere-2023-544', Anonymous Referee #2, 12 May 2023
This was the first time I was involved as a reviewer for this manuscript. The manuscript introduces an asymmetry extension for the analytical MRC. The manuscript is well-written and I enjoyed reading it. I have a few moderate and a number of minor comments, which are stated below. My overall recommendation is a moderate revision. Since I had to choose between ‘minor’ and ‘major’ I went for ‘major’ to provide the authors enough time to solve the open issues.
Moderate comments:
The authors compare their results with a very narrow field of the latest developments (here: analytic solvable micro-canonical cascade models). However, the results can be interpreted from a larger perspective (at least micro-canonical cascade models, better: cascade models in general). For example the scale-dependency (model A vs. B, bounded vs. unbounded model) and position-dependency. The discussion would benefit from it and the reader would be provided with a broader perspective on the scientific field. It is also important because some findings (which are new for the analytic solvable MRC) are quite common to apply for other cascade models.
Section 4.1 The authors highlight the improvement by introducing the asymmetry in comparison to model A and model B. From my understanding neither model A nor model B takes into account the position-dependency of the current time step to disaggregate. To my knowledge the latest references on micro-canonical cascade models all take into account position dependency (so starting, enclosed, ending, isolated position classes depending on the wetness state of: {Rt-1, Rt, Rt+1}). As mentioned before, for analytic solvable MRC it is maybe not common to take into account the positions/patterns from the coarser scale, but it is common for MRC in general. Here, the asymmetry can be interpreted as an extension of the position-dependency, since it takes into account the intensity of the surrounding time steps rather than the wetness state only (so real vs. boolean). So it is not surprising that A+ and B+ outperform A and B, respectively, but A and B do not represent the state-of-science. I recommend to add a position-dependent cascade model to evaluate the added value of the asymmetry in comparison to the wetness state. Even if both approaches result in similar statistics, the asymmetry would have the benefit of being more parameter parsimonious. It is important here to show the reader the clear benefit of the introduced model approach. Also, the seasonal classification is not common for all cascade models and more common for the pulse models (NSRP & BLRP). The authors show the seasonal variations of parameters in Fig. 8, but I’m still curious how this would affect the results. How would the results look if there is one parameter set applied for the disaggregation of the whole time series?
Specific comments:
L2 The term ‘simple scaling law’ can be confusing. What does ‘simple’ refer to? Linear? Please clarify.
L5 ‘…is usually disregarded.’ I would add the following extension to this sentence: ‘…or taken into account in a simplified way.’ (or similar), since there are possibilities out there taking into account the wetness state of the surrounding time steps.
L176 The term ‘shadow breakdown coefficient’ sounds spectacular, but it is not clear what ‘shadow’ exactly refers to? From my understanding it takes into account the position-dependency as well as the rainfall amounts, because higher rainfall amounts would cause more shadow. However, this name should be introduced/defined so that other authors know when to use it.
Section 2.2 When introducing Zt the authors could state the intended application briefly and refer to Sec. 2.4 with the detailed description: It only affects p01 and p10, px remains unaffected.
Fig. 3c For very low and very high values of z tipping points can be identified. How can it be explained? By the measuring resolution of the measuring instrument, leading to minimum values of e.g. 0.1mm?
Table 1-caption. Please add the information that the number of parameters is not taking into account any seasonal variation. So four seasons would lead to 4*parameter number mentioned in the table.
L271-274 The description is valid and does not be changed. Nevertheless I’m curious why the authors stop the disaggregation procedure at 40mins and don’t go all the way to 10min? Did the scaling behaviour change for finer resolution (often scale invariance hold for ~1d-> ~1h)?
Section 2.6 Maybe I’ve just not seen it: Which distribution function as used to estimate the return periods analysed in Fig. 6?
L379-384 The scale-dependency often plays a minor role if the scaling behaviour is linear, which is often the case for the analysed range of resolution in this study. I suggest to add a figure on scaling behaviour (the typical Mq (Moments of order q=1,2,3,..)-temporal resolution-plot) to verify the finding that A not necessarily outperforms B. Other cascade models apply scale-invariance already for this range of temporal resolutions (e.g. Günther et al, 2001)
Fig. 5a) ‘Standard deviation’ – of what?
L376-378 This is maybe true for analytical developments, but for non-analytical approaches position-dependencies are most often taken into account. This information should be added here for the sake of completeness.
L408-420 The persistence/intermittency is a weakness of micro-canonical cascade models. Müller-Thomy (2020) has introduced an extended position-dependency that improves the autocorrelation for all lags. Here, lag-1 and lag-2 are studied, results for other lags are not shown. Are results similar for all lags? Would the involvement of the extended position-dependency (would be an extended asymmetry approach) also an (additional) improvement for the analytical MRC?
Fig. 9 Which temporal resolution is shown here?
References:
Güntner, A., Olsson, J., Calver, A., and Gannon, B. (2001): Cascade-based disaggregation of continuous rainfall time series: The influence of climate, Hydrol. Earth Syst. Sci., 5(2), 145–164.
Müller-Thomy, H. (2020): Temporal rainfall disaggregation: Possibilities to improve the autocorrelation, Hydrology and Earth System Sciences, 24, 169-188.
Citation: https://doi.org/10.5194/egusphere-2023-544-RC2
Kaltrina Maloku et al.
Kaltrina Maloku et al.
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