Analytical multiplicative random cascades (MRCs) are widely used for the temporal disaggregation of coarse-resolution precipitation time series. This class of models applies scaling models 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 in the analytical scaling models. Our work presents a unified MRC modelling framework that allows the cascade generator to depend in a continuous way on the 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 into the model leads to a drastic improvement in 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.

Multi-decadal time series of sub-daily precipitation at hourly or even higher temporal resolutions are necessary for many applications, e.g. assessment of soil erosion

Many disaggregation models have been presented in the past (see, for example,

MRC models are usually implemented with a so-called branching number equal to 2. In such a case, the amount of precipitation at any time step is partitioned into two parts, attributed respectively to the first and second subdivision of this time step. The partition is repeated throughout the cascade levels until the final temporal resolution is achieved

The probabilities

Different models have been proposed for the cascade generator

Besides empirical MRC models, analytical models aim to represent in a synthetic way the dependency of

In this work, we present an analytical MRC modelling framework that allows the cascade generator

To what extent can a continuous index of local precipitation asymmetry describe the way the cascade generator

Is it possible to identify some scaling behaviour with respect to this asymmetry index? And is it possible to propose an analytical relationship to model this scaling behaviour?

What is the added value of including such an asymmetry dependency in the cascade generator

However, introducing precipitation asymmetry dependence is at the expense of model parsimony. In this work, we thus consider different MRC models of different complexities to find, if relevant, a compromise between model performance and parsimony. If the cascade generator is known to depict different types of scaling dependencies, not all are necessarily required to achieve fair model performance. Accounting for temporal scale dependence is widely considered to be beneficial. However, to our knowledge, the corresponding gain in performance is questionable (no improvement in

An additional objective of the present work is to investigate the loss of performance obtained when the cascade generator

The paper is structured as follows. In Sect.

As mentioned previously, the cascade generator

Model A accounts for the dependency on the temporal scale and precipitation intensity.

Model B is a simplification of model A. It disregards the dependency on the temporal scale.

Models A

To account for the seasonality of precipitation characteristics in the region, models are estimated and evaluated on a seasonal basis. Seasons are defined as follows: winter (December, January, February – DJF), spring (March, April, May – MAM), summer (June, July, August – JJA) and autumn (September, October, November – SON).

In models A and B, the dependency on precipitation asymmetry is disregarded, and the distribution of the cascade generator is assumed to be symmetric. The probabilities

The probabilities

In model A,

In models A

To characterise the asymmetry of

Asymmetry index

The following points can be noticed:

A symmetric precipitation sequence, i.e. when

For sequences with central precipitation amounts much larger than the adjacent ones,

The

Figure

Statistical characteristics of the breakdown coefficients

The asymmetry of the precipitation sequence directly translates to an asymmetry of the ECDF of

To quantify the asymmetry between

For asymmetric precipitation sequences, the distribution

In models A

The four models compared and the parameters to be estimated for the scaling sub-models accounted for. The – symbol denotes a configuration where the scaling dependency is disregarded. The total number of parameters per season and station is given in the column “No. of params”.

The introduction of asymmetry in model A, leading to model A

Model B

Table

Whatever the MRC model, the estimation of the different parameters required for the scaling model for

This is illustrated in Fig.

Precipitation data aggregated at six temporal resolutions are considered for model estimation (40, 80, 160, 320, 640 and 1280 min). The 40 min precipitation data are obtained from the 10 min time series available for the stations. The 40 min time series are aggregated to 80 min time series to calculate the observed breakdown coefficients

Note that the 1280 min aggregated temporal scale does not correspond to the daily resolution of 1440 min. Following

As reported in previous works, the precipitation measurement resolution, defined by the precipitation tipping bucket of automatic stations (0.1 mm here), is very likely to introduce artefacts and/or biases in different statistical characteristics of precipitation at fine temporal scales, especially sub-hourly ones, and can impact the empirical distributions of the breakdown coefficients

Following previous works, the breakdown coefficients

The 10 min observational records aggregated to the resolution of 1280 min are disaggregated back to the 40 min resolution using models A, B, A

The performance of a given model is evaluated by its ability to reproduce standard statistical metrics of precipitation, such as the standard deviation of precipitation amounts, the probability of precipitation occurrence and return levels of maximum precipitation amounts for given return periods. In addition, the temporal autocorrelation, wet–dry transition probabilities, and the duration of wet and dry spells are used to assess the ability of the models to reproduce the temporal persistence of precipitation. Evaluations are carried out on a seasonal basis for all available stations and for all temporal scales involved in the generation process (i.e. 40 to 640 min).

Different evaluation criteria are used. For most evaluation metrics, the variability between generated scenarios is small to very small. In this case, the evaluation criterion is the absolute error between the simulated and the observed metric, averaged over the different scenarios (and possibly temporal scales and seasons). To assess the performance of a given model across multiple sites, single-site performances are averaged over the different stations. We refer to this performance criterion as the mean absolute error (MAE). For some metrics, the percentage absolute relative error is considered (dividing for each station, season and temporal scale the absolute error by the observed value of the metric), giving the mean absolute percentage error

For precipitation maxima, the variability between scenarios is often large. In this case, we apply the CASE evaluation framework proposed by

“good” if the observed metric is inside the 90 % limits of simulated metrics,

“fair” if the observed metric is outside the 90 % limits of simulated metrics but within the 99.7 % limits of simulated metrics or absolute relative difference between the observed metric and the average simulated metrics is 5 % or less, and

“bad” otherwise.

The four models are applied to 81 stations of the Swiss meteorological observation network, a relatively dense network, with high-quality observational data (Fig.

Map of Switzerland and gauge locations.

If large precipitation amounts can be observed during winter and spring due to long stratiform and orographic precipitation events, intense precipitation events are often observed in summer and fall due to the topographically triggered convective events

We first focus on results obtained for the target 40 min temporal resolution, namely for a set of standard statistics (Sect.

Simulated values obtained for a set of standard statistics at a resolution of 40 min are compared to observed ones in Fig.

Observed versus simulated statistics for each considered model at a 40 min temporal resolution for different metrics. Each triangle represents a site and a season. The triangles for the simulated metrics correspond to the median of the 30 statistics obtained from the corresponding 30 simulated scenarios. MAE values over all sites and seasons are indicated in the bottom-right corner, and the lowest MAE obtained over the four models is indicated in bold. Pearson's autocorrelation coefficients are estimated using the function

Differences between models are more important for statistics related to precipitation persistence and intermittency. The best-performing model depends on the statistic, but, whatever the statistic, the performance of the model is always drastically improved when precipitation asymmetry is accounted for (see model A

Some differences are also noticed depending on whether or not the dependency on the temporal scale is taken into account (i.e. model B vs. model A and model B

The ability of the models to simulate relevant return levels for the 5- and 20-year return periods at the 40 min temporal resolution is presented in Fig.

Observed versus simulated return levels at the 40 min temporal resolution for

Results are very similar from one model to the other. The percentage of sites with good performance is slightly higher for models B and B

All models are able to reproduce the large variety of 5- and 20-year return levels observed in Switzerland between sites and seasons. They also all perform rather well for both return periods. For most sites and seasons, however, return levels are slightly underestimated. At the 40 min resolution, the underestimation is stronger in spring and summer for some sites (Fig.

Figure

Mean absolute percentage error (MAPE) as a function of the temporal aggregation level and season for lag-1 autocorrelation, mean length of wet spells, and 5- and 20-year return levels. Each boxplot summarises the single-site performances obtained for the 81 stations.

At 40 min resolution, accounting for asymmetry significantly improves the reproduction of all statistics related to precipitation persistence and intermittency, but it does not influence the reproduction of standard deviation, wet step probability and return levels. Similar results are obtained for the intermediate temporal resolutions.

As already mentioned previously, results can also depend on the season. Results for spring and summer are often similar and are in contrast with the results obtained for winter and autumn (Fig.

As highlighted in many previous works, the distribution of the breakdown coefficients

According to the results shown in Sect.

Two issues must be put into perspective with the preceding statements. The first one is related to the recording precision, which induces a relatively high frequency of

Note that these conclusions were also obtained in auxiliary analyses we carried out on jittered high-resolution precipitation data (not shown). Random perturbations were added to the original observed 10 min time series before the model estimation procedure, as in

Another issue concerns the parameter

The statistics related to precipitation persistence have always been found to be significantly underestimated by analytical MRC models

The dependence on precipitation asymmetry, acknowledged for a long time, was never accounted for in a fully analytical MRC framework. This study fills this methodological gap. We show that the dependency on the asymmetry can be modelled with a two-part analytical scaling sub-model, where the dry probability asymmetry ratio

Accounting for precipitation asymmetry seems to be of crucial importance to achieve the generation of a time series with relevant properties. This is also suggested by the results of some auxiliary evaluations described below. As shown previously in Sect.

Figures S12 and S13 demonstrate that the dependency of ECDF of BDCs on

Besides, the statistics of a precipitation time series, e.g. the standard statistics and the ECDF of

As mentioned previously, the asymmetry of the cascade generator is mainly disregarded in analytical MRCs, but it is considered for a long time in empirical ones by estimation of the cascade generator for different external pattern classes. Conditioning the parameterisation of any analytical model on external pattern classes is also possible. Is our continuous asymmetry approach of interest when compared to a class-conditioned approach? To address this question, we considered two more models, namely A position and B position (Ap and Bp). These two models are based on models A and B respectively, but they are estimated by conditioning the estimation on four external pattern classes, i.e. the four position classes (starting, enclosed, ending, isolated) considered in

In order to apply a MRC model in locations where only daily data are available, the possibility to develop a robust regional model for model parameters is necessary. The smaller the number of parameters, the more robust and easier the regionalisation is expected to be. In this regard, model B

Maps of Switzerland showing spatial and seasonal variations of the estimated parameters for model B

Another important factor that can jeopardise the success of parameter regionalisation is the spatial variability of parameters. In the case of no evident relationship to some geographical features (e.g. topography), smooth spatial variations of parameters often help achieve robust regionalisation. The spatial variability of the five parameters of model B

Additionally, the maps presented in Fig.

These results suggest the importance of seasonal stratification for model parameter estimation. The interest of seasonal stratification is even more obvious when simulations are carried out with a model where parameter dependency on season was ignored. This is illustrated in Figs. S8 and S9, where the seasonal performances of all models are presented for a configuration where one single set of parameters has been estimated for all seasons together. The performance of the models without seasonal stratification degrades a lot for a number of statistics, especially in winter and summer seasons.

An opportunity for model improvement concerns the scaling sub-model for the no-dry subdivisions probability

Following previous works,

The misrepresentation of

According to many previous works, the distribution of the breakdown coefficients (BDCs) used in MRC models depends on a number of factors, including the temporal scale, precipitation intensity and the external pattern of the local precipitation sequence.

In the present study, we compare different MRC models both with and without the scaling dependency on the temporal scales and with and without the scaling dependency on the external pattern of precipitation. Conversely to the scaling dependency on precipitation intensity and asymmetry, the scaling dependency on temporal scales is not obvious, and its added value in terms of model performance is less clear than what was suggested in previous works. Moreover, accounting for a dependency on the temporal scales drastically increases the number of parameters to be estimated (six more in the present case), which is especially expected to make the model much less robust and less appropriate for further regionalisation works.

The dependency on the external pattern of precipitation is shown to be important. In previous studies, it was mainly accounted for with empirical models where the BDC distribution was conditioned on different external pattern classes. To our knowledge, although determinant, it was never accounted for in an analytical scaling framework, which also accounts for temporal scale and intensity dependencies. Our work presents a unified analytical MRC modelling framework that allows the cascade generator to depend in a continuous way on the temporal scale, precipitation intensity and precipitation asymmetry. The continuous dependency of the cascade generator on the precipitation asymmetry index, which is introduced here, allows for the interpretation of the presented asymmetry sub-model as an extension of the position dependency approach already considered in several previous works. This sub-model could be easily assimilated in other multiplicative cascade models, either micro-canonical or canonical ones.

Initially, we demonstrate the feasibility of characterising the external precipitation pattern with a hidden BDC, the so-called precipitation asymmetry index. We show that the larger the deviation of this index from 0.5 (the index value for a symmetrical precipitation configuration), the larger the asymmetry of the distribution of the BDCs. The relationships with this asymmetry index are modelled with two scaling sub-models. The first sub-model represents the dry probability asymmetry ratio

Accounting for precipitation asymmetry in the cascade generator preserves the good performances of MRCs concerning statistics related to precipitation distribution (standard deviations and precipitation extremes) and improves other aspects of the disaggregated precipitation time series. For the Swiss context considered in our work, we outline the following.

Accounting for precipitation asymmetry leads to significant model improvements for all statistics related to the temporal persistence and intermittency of precipitation, which are known to be difficult to simulate with standard MRC models.

The statistical distribution of BDCs is expected to strongly depend on the external pattern of precipitation. This dependency is well (respectively, is not) reproduced when precipitation asymmetry is included (respectively, not included) in the MRC.

The statistics of a precipitation time series are not expected to change when the time series is offset in time by a small time duration. This offset independence property is well (respectively, is not) satisfied when precipitation asymmetry is included (respectively, not included) in the MRC.

Among the four different MRC models considered here, the one that accounts only for precipitation intensity and asymmetry seems promising. It performs very well for all considered statistics, for all seasons and for all temporal resolutions. It is, moreover, very parsimonious, with only five parameters. The five parameters are almost all independent from each other and can be estimated in a robust way, which avoids equifinality issues

The parameters of the scaling models of each MRC model are estimated as follows:

The precipitation data used in this study are maintained by the Swiss Federal Office of Meteorology and Climatology, MeteoSwiss

The open-source code with routines allowing the fitting and disaggregation of precipitation data based on the four MRC models presented in this study is available as an R package. It can be installed via Zenodo:

The supplement related to this article is available online at:

Funding acquisition: BH. Data acquisition: KM. Experimental design: KM, BH and GE. Script development: KM and GE. Model calibration, simulations and analyses: KM. Figure preparation: KM. Paper redaction: KM, BH and GE.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We thank the editor, Hannes Müller-Thomy, and one anonymous reviewer for their constructive comments that helped us to improve the quality of this paper.

This research has been supported by the Bundesamt für Energie (grant no. SI/502150-01) and the Bundesamt für Umwelt (grant no. SI/502150-01), through the Extreme floods in Switzerland project.

This paper was edited by Nadav Peleg and reviewed by Hannes Müller-Thomy and one anonymous referee.