On temporal stochastic modeling of precipitation, nesting models across scales

Advances in Water Resources 63 (2014) 152–166

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Advances in Water Resources journal homepage: www.elsevier.com/locate/advwatres

On temporal stochastic modeling of precipitation, nesting models across scales Athanasios Paschalis ⇑, Peter Molnar, Simone Fatichi, Paolo Burlando Institute of Environmental Engineering, ETH Zurich, Switzerland

a r t i c l e

i n f o

Article history: Received 27 June 2013 Received in revised form 19 November 2013 Accepted 20 November 2013 Available online 28 November 2013 Keywords: Stochastic rainfall models Poisson cluster model Multiplicative Random Cascade Markov chain Alternating renewal process

a b s t r a c t We analyze the performance of composite stochastic models of temporal precipitation which can satisfactorily reproduce precipitation properties across a wide range of temporal scales. The rationale is that a combination of stochastic precipitation models which are most appropriate for specific limited temporal scales leads to better overall performance across a wider range of scales than single models alone. We investigate different model combinations. For the coarse (daily) scale these are models based on Alternating renewal processes, Markov chains, and Poisson cluster models, which are then combined with a microcanonical Multiplicative Random Cascade model to disaggregate precipitation to finer (minute) scales. The composite models were tested on data at four sites in different climates. The results show that model combinations improve the performance in key statistics such as probability distributions of precipitation depth, autocorrelation structure, intermittency, reproduction of extremes, compared to single models. At the same time they remain reasonably parsimonious. No model combination was found to outperform the others at all sites and for all statistics, however we provide insight on the capabilities of specific model combinations. The results for the four different climates are similar, which suggests a degree of generality and wider applicability of the approach. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Stochastic simulation of precipitation time series at a site is demanded for numerous applications where precipitation is a key variable, e.g., watershed hydrology, urban drainage design, natural hazard assessment, agricultural and ecological applications, etc. These requirements have led to the formulation of many stochastic models of temporal precipitation. One of the fundamental problems with existing models is that they are designed to perform best within a limited range of time scales and for selected statistics for which they are calibrated. Often key statistics such as intermittency, correlation structure, extremes, etc., are not satisfactorily reproduced at multiple scales. The premise behind this study is that a simultaneous correct reproduction of key statistics over a wide range of hydrologically relevant time scales from minutes to months is fundamental. We argue that even when simulated precipitation time series are to be used for instance for impact studies at fine resolutions, we should require models at the same time to provide good results at coarser resolutions, e.g., to preserve storm internal variability, intermittency and inter-storm clustering properties, seasonality, etc.

⇑ Corresponding author. E-mail address: [email protected] (A. Paschalis). 0309-1708/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.advwatres.2013.11.006

Early rainfall modeling focused on the simulation of daily precipitation time series. The most common processes adopted in hydrology for stochastic simulation of precipitation were Markov chains [1–5], Alternating renewal processes [6,7] etc. Capabilities of such models have been recently enhanced using the concept of generalized linear models (GLMs) (e.g., [8,9]). Those processes served as precipitation simulation tools for some of the most widely used weather generators (e.g., [10,11]). A common deficiency of these models was that they were generally not able to reproduce higher order statistics and statistics across different temporal scales. Furthermore, they were not designed for temporal scales finer than daily, with some exceptions (e.g., [12]). In order to improve precipitation simulations at fine resolutions and for higher order statistics, new approaches were introduced. Among them point processes have been widely used [13,14]. These models appeared as an ideal candidate for precipitation modeling, since they reproduce a structure of storm arrivals and persistence that mimics the physics of the precipitation process. Developments on the initial ideas led to the implementation of Poisson cluster models based on the concepts of storms, cells, and clustering [15–25]. Their main advantage was that statistics were reasonably reproduced across scales from 1 h to several days, with the exceptions of intermittency and correlation at high resolutions. These models have also been introduced into last generation weather generators (e.g., [26]) and used to solve practical problems in

A. Paschalis et al. / Advances in Water Resources 63 (2014) 152–166

real-world applications (e.g., [27]). Another class of models that was found to give promising results for rainfall simulation at various temporal scales, even though often used for rainfall simulation at the daily scale only, is the Hidden Markov class of models (e.g., [28–30]). The concept of self similarity, widely known as fractal or scaling behavior [31,32] combined with the quest for rainfall stochastic models that perform well across scales has led to an additional category of precipitation modeling. Self similarity has been identified in precipitation time series and precipitation spatial fields (e.g., [33–36]). The concept of scale invariance has its origins in the study of turbulence [37,38] and it allows one to link statistical properties of a certain process (e.g., precipitation) across scales. The extension of the self similarity concept into multi-scaling behavior, widely known as multifractals [39,40] has found many interesting applications in rainfall simulation (e.g., [41–44]). One of the modeling tools for the simulation of multifractals, Multiplicative Random Cascades (MRC), has been widely used for precipitation disaggregation and downscaling [45–51]. Despite the large number of available approaches and models mentioned above, none is free of problems when multiple scales and statistics are considered. Markov chains, Alternating renewal processes, and GLMs serve as reliable simulation tools for precipitation only for coarse aggregation scales and are often poor in reproducing extremes. Poisson cluster models have been found to reproduce well the statistics only for the scales at which they are calibrated (typically between one hour and a few days). The self-similarity assumption and multifractal behavior have several limitations, posing doubts on the generality and validity of the approach. Temporal scaling regimes were found to be limited [52,53] and scaling relations themselves imperfect [54]. In the MRC it was shown that weights do not follow iid (independent and identically distributed) assumptions required by the model [55,56], all of which pose restrictions on the global scaling behavior of precipitation. Significant effort has been undertaken to modify and improve the original models. For example Katz and Parlange [12] applied a chain dependent process for hourly precipitation. Recently, Cowpertwait et al. [57] introduced modifications to a Poisson cluster model in order to overcome the common problem of inadequate representation of small scale (sub-hourly) variability. Other authors provided several variants of the initial MRC model in order to take into account dependencies of the MRC weights (e.g., on scale, rainfall intensity) [45,54,58–60]. However, these approaches have the tendency to become increasingly difficult to parameterize. In summary, a single stochastic model which reproduces precipitation statistics across a wide range of temporal scales of hydrological interest is difficult to find. An alternative approach which we explore in this study is to combine several types of stochastic precipitation models at the scale for which they perform well. The nesting of an internal and external model is expected to strongly enhance the performance of the precipitation generator across scales, without requiring excessive parameterization if the nested models are appropriately chosen. Although most studies in the literature have focused on improving a specific model type, there are some attempts at model combinations. Menabde and Sivapalan [61] successfully applied a combination of an Alternating renewal process with a bounded MRC in order to reproduce rainfall extremes at fine temporal scales. Fatichi et al. [26] combined an autoregressive model with a Poisson cluster model to introduce into the latter the interannual variability that it failed to capture. Veneziano and Iacobellis [62] obtained encouraging results combining an external model of conventional renewal type with an internal model based on the iterated random pulse process. Furthermore, Rodriguez-Iturbe et al. [15], Onof and Wheater [63] and Gyasi-Agyei and Willgoose [64]

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improved the fine-scale properties of the Poisson cluster models by perturbing their realizations with an independent multiplicative noise stochastic process called jitter. Finally, Koutsoyiannis [65] developed a model-free theoretical framework in order to couple different stochastic models. The novelty in our work is to show that for stochastic modeling of temporal precipitation a composite model – consisting of a point-process or Markov chain as an external model and a nested Multiplicative Random Cascade as an internal model – performs better across a wider range of scales than the individual models alone and at the same time remains reasonably easy and parsimonious to calibrate. In order to make our point we extensively investigate and compare different model types and structures for a wide range of statistics and temporal scales. We then apply the composite models to four sites representing very different climatological regions around the world to verify their performance. 2. Data Data from four meteorological stations with long records of reliable high resolution precipitation were used. Each station belongs to a different climatological region of the world (Fig. 1 and Table 1). Two of the stations are located in Europe. The first station, Zurich (Switzerland), is representative of a temperate continental climate with distinct seasonality and a mean annual precipitation of about 1130 mm. The gauge has a heated tipping bucket recording mechanism, and its temporal resolution is 10 min. Precipitation in Zurich mainly consists of stratiform events during the cold season and intense convective events during the warm season [66]. Data from this station have been used previously in several studies [45,56,67]. The second station, Florence (Italy), is representative of a Mediterranean climate with a less pronounced seasonality, dry summers, and mean annual precipitation of about 800 mm. The tipping bucket gauge in Florence records with a temporal resolution of 5 min. Data from this station have also been extensively analyzed in the past [68–70,62,71]. Outside of Europe we chose two stations with contrasting precipitation regimes. Lucky Hills (Arizona, USA) is representative of a semiarid climate in southwestern USA. It is located within the Agricultural Research Service (ARS) Walnut Gulch Experimental Watershed [72]. Precipitation at Lucky Hills has a pronounced seasonality with a wet monsoon season in the summer and rare stratiform events during the dry season. Mean annual precipitation is about 340 mm. The gauge is a weighing gauge with a temporal resolution of 1 min. Finally, the station Mount Cook (New Zealand) is representative of the oceanic climate of southern New Zealand with a uniform distribution of precipitation throughout the year, accumulating an average of about 3900 mm per year. This high precipitation total is the result of the orographic enhancement imposed by the southern Alps which leads to considerable precipitation amounts in the entire southwestern area of New Zealand. The gauge is a tipping bucket with a temporal resolution of 10 min. Data were obtained from the National Climatic Database operated by NIWA (National Institute of Water and Atmospheric Research). Stronger diurnal patterns of precipitation intensity are only present in the Mt. Cook and Lucky Hills stations (Fig. 1). These diurnal patterns are mostly due to convective afternoon rain in the summer, JJA for Lucky Hills and DJF for Mt. Cook. 3. Methods The stochastic models which are used as building blocks for the composite model are presented in this section (Fig. 2). The composite model consists of an external model which captures the

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0

J FMAMJ J ASOND

200 100 0

J FMAMJ J ASOND

Florence Depth [mm]

50

Zurich Depth [mm]

Depth [mm]

Lucky Hills 100

200 Lucky Hills

100 0

0.6

J FMAMJ J ASOND

Mt Cook

Zurich

DJF

MAM

JJA

SON

Florence

Intensity [mmh−1]

0.4 0.2 0

0.4

0.2

Depth [mm]

Mt Cook 1000

0

0

5

10 15 Hour

20

0

5

10 15 Hour

20

500 0

J FMAMJ JASOND

Fig. 1. Locations of the stations and their seasonal mean diurnal cycle.

Table 1 Analyzed precipitation stations in this study. H is station altitude, P is mean annual precipitation and Dt is the temporal resolution of the gauge. Station

Climate

H (m)

P (mm/yr)

Dt (min)

Data

Zurich (Switzerland) Florence (Italy) Lucky Hills (Arizona, USA) Mt Cook (New Zealand)

Temperate continental Mediterranean Semi-arid Oceanic

556 50 1372 765

1130 800 340 3900

10 5 1 10

1981–2009 1962–1986 1962–2012 2000–2012

distributions of precipitation depth, extreme event statistics, autocorrelation structure, and probability of zero precipitation. One common assumption behind all the modeling combinations we used is the stationarity of the rainfall process. None of the models presented here can take into account the cyclostationary effect of the diurnal cycle. Although this is a limitation in certain cases, we are of the opinion that for the majority of hydrological/agricultural/ecological applications, the effects induced by the daily cycle of precipitation can be considered small. For the cases where the diurnal pattern of precipitation is considered essential, different modeling techniques should be used (e.g., [12]). 3.1. External models Fig. 2. Schematic representation of the composite model.

processes connected with storm arrivals (timing and intensity) and inter-storm periods. The finer time scales reached by the external model are on the order of hours–days. The nested internal model is then used to capture the processes which describe rainfall variability within a storm by disaggregating the output of the external model to finer time scales on the order of minutes. Although the different combinations of external and internal models are calibrated independently at their appropriate time scales, the performance of the composite model is tested jointly across all simulated temporal scales. This means that the composite model is used to simulate precipitation at the finest time scale of interest and then simulated precipitation series are aggregated back to any other chosen temporal scale, where relevant statistics are computed and compared. The main statistical features we aim to reproduce at the different time scales are probability

For the external model we used three approaches with different levels of complexity: the Alternating renewal process, Markov chain, and the Neyman Scott rectangular pulse model which belongs to the so-called Poisson cluster model category. 3.1.1. Alternating renewal process The Alternating renewal process (ARP) describes the precipitation process as a sequence of wet and dry runs [73–75]. The durations of the runs are assumed to follow a prescribed probability distribution. The main assumption is that the wet and dry runs are independent, identically distributed (iid) and mutually independent. A constant intensity from a prescribed probability distribution is assigned to each day. These uniform intensities are also iid and independent of the duration of the wet run. Several distributions have been used in the literature and the choice is always subjective and dependent on the analyzed dataset (e.g., [73,74,76]). Here, we chose an exponential distribution for the duration of the dry and wet runs [7] and the two parameter gamma

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distribution for precipitation intensity. The two parameter gamma distribution is a specific case of the generalized gamma distribution which has been found to yield very good results for daily precipitation depths worldwide [77]. However, we acknowledge that daily precipitation series possess heavy tails, a behavior which the two parameter gamma distribution cannot capture. When a good simulation of daily extremes is essential, potential candidates for the daily precipitation depths may include the Weibull, Pareto distribution, etc. The ARP model has 4 parameters to be estimated (Table 2). The estimation is based on the maximum likelihood method. Dry and wet spells and their respective daily depths are extracted from the rainfall records and their respective probability distributions are fitted by maximizing their likelihood functions.

3.1.2. Markov chain The Markov chain (MC) describes the precipitation occurrence process in time through transition probabilities between wet and dry states. The two-state first order MC simulates a binary [0–1] process which corresponds to the dry (d) and wet (w) states with the probability transition matrix,

M¼

pw;w

pw;d

pd;w

pd;d

! ð1Þ

where pi;j is the probability of transitioning from state i to state j in the successive time step. A precipitation depth is assigned for each time step that the MC stays in the wet state. Precipitation depth is assumed iid and has a prescribed probability distribution, which we take to be a two parameter gamma distribution identical to the ARP. The probability transition matrix M is dependent on the selected time step of the MC model and has to be estimated from data at the same time scale. In this study the selected time step was approximately 1 day (1280 min). Equally, the distribution of rainfall depth is related to this time step and has to be estimated at the same time resolution. The fit of the rainfall depth distribution was conducted by the maximum likelihood method [78]. Altogether the MC model has 4 parameters (Table 2), two parameters of the gamma distribution for rainfall depths and two parameters of the probability transition matrix pw;w and pd;d , which are related to the remaining probabilities pw;w þ pw;d ¼ 1 and pd;w þ pd;d ¼ 1. Stochastic models based on the Markov chain process have been widely used in hydrology, especially for daily precipitation, due to their simplicity and parsimonious parametrization (e.g., [4,79,?,80]).

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3.1.3. Poisson cluster model Poisson cluster models have gained popularity in hydrology since they allow a more intuitive mechanistic description of the precipitation process by including storms consisting of several cells which may superpose. Generally, Poisson cluster models can be divided into two different categories, the Neyman Scott rectangular pulse (NSRP) and the Bartlett Lewis rectangular pulse (BLRP) [81,23]. Both models simulate the precipitation process as a superposition of storms and rectangular rain cells with uniform intensity during their lifetime. These models have significant similarities in the clustering structure, however they differ in the way rain cells are distributed in time. In the BLRP model, both the storm arrivals and cell arrivals follow a Poisson process. In the NSRP model, storm arrivals also follow a Poisson process but within each storm a random number of cells is drawn and spaced out from the storm origin according to a waiting time that has a specific probability distribution. In both models cell depths and durations have prescribed probability distributions. For this work we chose the six parameter NSRP model [22]. Even though several variants of the Poisson cluster models exist which allow randomization of storm types, we chose the simpler 6 parameter version of the model for two reasons: the parsimony of the model structure, and because the efficiency of the Poisson cluster models with a more complex structure has been challenged (e.g., [81]). It was recently shown that the advantage of a Poisson cluster model with randomized storm parameters [27] was minimal when compared with the simpler 6 parameter NSRP model used in this study [82]. Analytically the model is built as follows: (i) storms arrive as a Poisson process with rate k; (ii) each storm generates a number of cells which follows a geometric distribution with mean value lc ; (iii) the waiting time of cell origins after the origin of the storm follows an exponential distribution with parameter b; (iv) the lifetime (duration) of each cell follows an exponential distribution with parameter g; (v) cell intensities follow a two parameter gamma distribution with scale and shape parameters a and h respectively (Table 2). Parameter estimation is based on the numerical minimization of an objective function which is the standardized mean square difference between observed and analytically derived statistics [22]. The statistics which are taken into account are mean, coefficient of variation, lag-one autocorrelation, probability of no rain, and the conditional transition probabilities pd;d and pw;w . The model was calibrated at four time scales, i.e., T ¼ 1; 6; 24; 72 h. The numerical minimization was conducted with a multi-start downhill simplex algorithm similar to the approach of [24] for

Table 2 Summary of model parameters. Model

Parameters

Description

External Alternating renewal process (ARP)

kw ; kd

Parameters of the exponential distribution of the dry and wet spell durations

Markov chain (MC)

aARP ; bARP pw;w ; pd;d

Gamma distribution parameters for the depth accumulations Probability transition matrix terms

Neyman Scott (NSRP)

aMC ; bMC k; lc ; b; g; a; h

Gamma distribution parameters for the depth accumulations Model parameters as described in Cowpertwait [22]

Internal MRC model A

al ; bl ; ar ; br

Model parameters controlling the dependence of the probability of zero of the MRC generator on the temporal scale and intensity Model parameters controlling the dependence of the a parameter of the Beta distribution on the temporal scale

a0 ; H MRC model B

al ; bl ; ar ; br a0 ; H; c0 ; c1 ; c2

Model parameters controlling the dependence of the probability of zero of the MRC generator on the temporal scale and intensity Model parameters controlling the dependence of the a parameter of the Beta distribution on the temporal scale and intensity

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the spatio-temporal version of the model. The analytical equations of model statistics can be found in Cowpertwait et al. [21] and Cowpertwait [22]. Even though the NSRP model is used at the (almost) daily time scale, we chose to also use finer temporal scales for its calibration. The reason is that some of the model parameters (i.e. duration of the rain cells) are mainly influenced by fine time scales e.g., hourly. Calibration of the NSRP model only at time scales coarser or equal than daily would most likely provide much poorer results than those presented in this paper.

−1

10

−2

P(X>x)

10

−3

10

−4

10

3.2. Internal models The internal model which disaggregates the time series generated by the external model down to the scale of the order of minutes is based on a multiplicative redistribution of mass from coarse to fine scales – Multiplicative Random Cascade (MRC) model [39,40,83]. MRCs have been shown to be a very useful disaggregation/ downscaling tool, especially for time scales ranging from one day to few minutes [34,45,48,84]. The key parameter of a MRC is the cascade generator W which is a random variable that divides mass from a single interval at one time scale into two intervals (in a dyadic cascade with branching number b ¼ 2) in the next finer time scale. The precipitation depth at a scale n and time i in the cascade is then,

Rn ðDin Þ ¼ R0

n Y

n

W j ðiÞ for i ¼ 1; 2 . . . ; b ;

ð2Þ

0.2 h 21.3 h β lognormal MRC B (microcanonical)

−5

10

−6

10

−1

10

10

0

1

10 depth (mm)

10

2

Fig. 3. Generated exceedance probabilities for the disaggregation of the daily time series for the Zurich station during spring (MAM), according to a canonical (blognormal) and a microcanonical (MRC B) model for the 10 min and the 1280 min aggregation interval. Markers correspond to observations and lines to simulated series.

parameter (symmetric) Beta distribution with probability distribution function (pdf),

j¼1

where R0 is the initial precipitation depth at the coarse scale. The subscript j represents the path of weights from R0 to Rn ðDin Þ. In this formulation the cascade generator W represents the fraction of redistributed mass in every branching step. Since in our analysis we are interested in preserving mass exactly within the internal model, we restrict the pair of W to sum up to 1 in every branching step, which is known as the micro-canonical MRC. When W is not constrained at every branching step and is only required to have an expected value EðWÞ ¼ 1=b then we have a canonical (or macro-canonical) cascade which preserves mass on the average [59,85,86]. An additional reason to select the micro-canonical version of the MRC model is outlined below. To illustrate this, a disaggregation of the ‘‘daily’’ (1280 min) precipitation time series for the Zurich station for the Spring season (MAM) is performed with a microcanonical MRC model and a canonical MRC model and more specifically the b-lognormal model [40,45,56] (Fig. 3). Although the canonical b-lognormal model is capable of reproducing well the heavy tails of the fine scale (10 min) precipitation, aggregating the time series back to the original temporal scale (1280 min) distorts the distribution of the depths. This is because the aggregated series at the 1280 min temporal scale corresponds to the so-called dressed quantities [31], and its distribution is distorted due to the distribution of the dressing factor. Further investigations on this issue are beyond the scope of this study, but the interested reader can refer to (e.g., [31,87–89] and references therein). Because precipitation is an intermittent process, it is further necessary to introduce the probability that the cascade generator W ¼ 0,

PðWðiÞ ¼ 0 or Wði þ 1Þ ¼ 0Þ ¼ p0 ;

ð3Þ

where i; i þ 1 correspond to an arbitrary pair of complementary weights in the cascade development with branching number b ¼ 2. In order to achieve mass conservation, the positive part of the generator W þ , has to be bounded in ½0; 1. Here, we assumed that the positive part of the generator can be described by a single

f ðW þ Þ ¼

1 a1 a1 W þ ð1  W þ Þ ; Bða; aÞ

ð4Þ

where Bða; aÞ is the Beta function and a is a parameter related to the variance of W (e.g., [45]). The estimation of the a parameter of the Beta distribution was carried out using the method of moments,

a¼

1  0:5; 8VarðW þ Þ

ð5Þ

where W þ values were estimated from the data as the ratio between precipitation depths at two successive (embedded) temporal scales, if both precipitation depths are positive and not equal (i.e. 0 < W þ < 1). In the original formulation of the MRC it is assumed that the cascade generator W is independent and identically distributed (iid) across the entire range of scales. However, it has been shown that the probability distribution of W, in both its parts f ðW þ Þ and p0 , is dependent on the aggregation scale of the precipitation process [45,61]. Paschalis et al. [56] following the ideas of Cârsteanu and Foufoula-Georgiou [55] identified a temporal dependence structure in W using a large dataset in Switzerland. Dependencies of the parameters of the cascade generator (in a canonical MRC) with the mean intensity of the process were also identified [54]. Consequently, these relations were included in recent variants of the MRC models (both canonical and micro-canonical) to better fit the precipitation statistics across temporal scales from minutes to days [59,60]. To explore the effect of scale dependence we used two variants of the micro-canonical MRC model parameterized by Rupp et al. [59]. Models with lower complexity, even though they are more parsimonious and thus more attractive, yielded substantially worse results and are not reported here. 3.2.1. MRC model A In the MRC model A, p0 is dependent on the temporal aggregation scale and on the average intensity of precipitation at the next

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(coarser) aggregation scale. The parameter a of the Beta distribution of W þ is dependent only on the temporal aggregation scale. The probability of a non-zero weight at the cascade development is parameterized as [59]

Px ðI; sÞ ¼ 1  p0 ðI; sÞ ¼

   1 logðIÞ  l pffiffiffi ; 1 þ erf 2 2r

ð6Þ

where s is the temporal scale of the cascade, I is the precipitation intensity of the coarser scale, and erf is the error function defined Rx 2 as erfðxÞ ¼ p2ffiffipffi 0 et . The parameters l and r depend linearly on the logarithm of the temporal scale of the cascade,

l ¼ al logðsÞ þ bl ;

ð7Þ

r ¼ ar logðsÞ þ br :

ð8Þ

The parameter of the Beta distribution is only dependent on the temporal scale, s,

aðsÞ ¼ a0 sH :

ð9Þ

The model has 6 parameters which were estimated for all the stations for temporal scales from 10 min to approximately 1 day (Table2). The estimation of the parameters was carried out using least square fitting for the Eqs. (6)–(9). 3.2.2. MRC model B In the MRC model B both p0 and the parameter a of the Beta distribution are dependent on the temporal aggregation scale and average intensity of precipitation at the next (coarser) aggregation scale [59]. The parametrization of the probability of non-zero weights in the cascade is as in Eq. (6). The a parameter of the Beta distribution is conditioned both on the temporal scale and precipitation intensity at the coarser scale, 2

log½a ðIÞ ¼ c0 þ c1 logðIÞ þ c2 ½logðIÞ ;

ð10Þ

and

aðI; sÞ ¼ a ðIÞa0 sH :

ð11Þ

The estimation of the polynomial coefficients ci for lower intensities (typically <1 mm h1) can be problematic due to precision artifacts introduced by the tipping mechanism of the gauges. For this reason, the fitting procedure of Eq. (10) was restricted to temporal scales coarser than 20 min. In some cases the second order polynomial introduced in Eq. (11) could be simplified to a first order polynomial or even to a constant, however the above parametrization is kept for consistency with Rupp et al. [59]. The model has 9 parameters which were estimated for all the stations for temporal scales from 10 min to approximately 1 day (Table 2). Similarly to MRC model A, the estimation of the parameters was carried out using least square fitting for the Eqs. (6)–(8), (10) and (11). 3.3. Temporal scales and validation statistics The time scale at which we combine the external and internal models is derived from results of previous research and from our own analyses which show that the ARP and MC models (at least in their most parsimonious form used in this study, see [12] for an exception) are most appropriate for temporal scales up to 1 day. The NSRP model can produce reliable results for scales down to one hour, especially when calibrated at sub-daily scales (e.g., [26,23]), which could be another possible temporal scale of nesting for the NSRP model. However, recent findings using spectral analysis for a number of stations in Switzerland [90,91] suggest that the (spatio-temporal) NSRP model fails to capture adequately the

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small scales of precipitation (<10 h). These results, supported by our current findings, suggest that nesting a Poisson cluster model at the hourly time scale with a MRC could be less optimal than nesting at the daily scale. At the same time previous studies have shown that MRC models are particularly suitable for scales from 1 day to a few minutes [45,54,92]. Based on these considerations and on a branching number b ¼ 2 for the MRC model, we decided to use the external models to simulate continuous precipitation time series at time scales T ¼ 1280 min (21.3 h) and the internal MRC models disaggregate these from T ¼ 1280 down to T ¼ 10 min. The time scale of the temporal nesting has to be the same for all the combinations, in order to achieve a fair comparison. A summary of parameters of all models can be found in Table 2. In order to take into account diverse precipitation triggering mechanisms during the year and ensure as far as possible stationarity of the precipitation record, the analysis for all of the stations was conducted on a seasonal basis. A sample of 50 realizations for a 10-year period with temporal resolution 10 min was simulated for all stations. While some of the results are analyzed across a wide range of temporal scales, the comparison among composite models is assessed at three specific temporal scales. The highest resolution T ¼ 10 min (0.2 h) roughly matches the time scale at which most of the records were collected (see Table 1). Rainfall variability at such fine resolutions can be crucial for many applications, for example analyzing the response of urban catchments or steep mountain basins with very low runoff concentration times that are usually prone to flash floods. The second time scale is T ¼ 160 min (2.7 h) which corresponds roughly to the mean duration of high intensity convective storms. The third time scale is T ¼ 1280 min (21.3 h) which is the temporal scale at which we combined the external and internal models, and which is representative of stratiform precipitation events. At this time scale we only validate the performance of the external stochastic model. It should be noted that in our analysis we consider the observed precipitation time series as a realization of a stationary process, on a seasonal basis. The reason for this choice is to have a fair comparison with the output of the models which are as well realizations of stationary processes. In case of nonstationarities, such as a strong diurnal pattern, biases on the estimated statistics can occur (e.g., apparent autocorrelations, higher skewness due to the daily cycle, etc.). The efficiency of each model combination was validated for its ability to reproduce the probability distributions of precipitation depth at different temporal scales, extremes, autocorrelation structure, and probability of zero precipitation.

4. Results 4.1. NSRP as a benchmark We first present results of the Poisson cluster model alone as a benchmark for the results obtained by the composite models. The NSRP model is chosen as a benchmark since it is the only model presented here that has been widely used as a ‘‘multi-scale’’ simulation tool. The other external models, at least in their most parsimonious form, are not capable of reproducing fine scale temporal statistics (less than daily). In Fig. 4 the performance of the NSRP model is summarized for the Zurich station. In general the model efficiency in reproducing precipitation statistics for temporal scales coarser than hourly is excellent, however significant departures are found for the subhourly scale. Due to the simplified representation of the precipitation process as a superposition of rectangular rain cells, small-scale

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(a) 100

(b)

0.2 h 2.7 h 21.3 h

0.8 −2

0.6 ACF

P(X>x)

10

−4

10

0.2 h 2.7 h 21.3 h 85.3 h 213.3 h

0.4 0.2 0

−6

10 −1 10

(c)

0

10

1

10 depth (mm)

−0.2 0

2

10

5

15

20

Wet durations

(d) 100

1

10 lag

Observed Modelled −1

10

0.6

P(X>x)

P(r=0)

0.8

−2

10

0.4 −3

10 0.2

0

−4

50 100 aggregation interval [h]

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variability cannot be simulated. The finest temporal scale of fluctuations is associated with the distribution of the rain cell lifetime that is typically not short enough in the model to properly characterize the true precipitation process at this scale. The simulated precipitation time series severely underestimate the occurrence of 10-min rainfall depths greater than 1 mm (Fig. 4(a)). The simulated series are very smooth at this temporal scale, leading to very high autocorrelations (Fig. 4(b)). Another typical problem that is associated with the NSRP model is its inability to successfully reproduce the probability of zero precipitation (e.g., [18]). In our example, the probability of no rain is slightly overestimated for large scales of aggregation (Fig. 4(c)), even though it is one of the statistics that are explicitly considered in the calibration procedure. Intermittency is connected to wet and dry spell durations through clustering of the precipitation process. As a wet period we define here the duration for which precipitation accumulated at the selected temporal resolution stays continuously above zero. This notation is somewhat different from the definition of storm duration (e.g., [93]). The distribution of wet spell durations at the studied temporal aggregation scales is shown in Fig. 4(d). Similarly to the other statistics, the performance for time scales above 1 h is very good, however for the 10-min scale there is a significant overestimation of short wet spell durations by the model. The NSRP model is not able to adequately generate gaps within storms and short rainfall pulses for structural reasons, despite the fact that the transition probabilities from dry to wet states were also taken into account in the calibration procedure.

Fig. 5. Power spectrum for the simulated Neyman Scott model for spring season (MAM) for the Zurich station. The dashed line is the mean spectrum of the simulated realizations shown by thin gray lines.

The correlation structure across temporal scales was also examined with spectral analysis. Fig. 5 shows the observed and simulated power spectra for the Zurich station. The strong autocorrelation at high temporal resolutions in the model results

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in a steeper spectral decay for higher frequencies. The divergence between observed and simulated series starts roughly at temporal scales of about 5 h, which could be considered a reasonable scale for nesting NSRP with other stochastic models. In summary the NSRP model alone as a benchmark performs very well for scales above 1 h, but not for scales below 1 h. In the next step we investigate if and how different composite models improve the high resolution performance. 4.2. Composite model results Different composite model combinations are evaluated for the same statistics analyzed for the NSRP model. The combinations we show are Markov chain–MRC model B (Fig. 6), Neymann–Scott NSRP–MRC model B (Fig. 7) and the Alternating renewal process model–MRC model B (Fig. 8). Combinations of the external models with MRC model A, gave systematically worse results and are discussed only in the model and station intercomparison Section 4.3. We recall that the behavior of the composite models at temporal scales greater than 1 day is solely dependent on the external model while the behavior at finer temporal scales depends on both the external and the internal model. One of the most important improvements, and common to all the composite models, is a substantial enhancement in the reproduction of the probability distribution for the high resolution precipitation depths. Another common feature to all of the models is the improvement in the representation of the probability of zero precipitation. The combination that uses the NSRP model as the

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external model performs worse than the others for this statistic, since errors at scales coarser than 1 day cannot be improved upon. Nevertheless, the performance is improved with respect to NSRP alone. The probability distributions of the wet spell durations are significantly better represented at small scales even though, in all of the cases, there is a systematic underestimation of the tails for the 10-min aggregation interval. This means that the composite models fall short in capturing long wet spells observed at this resolution. The positive effect of the MRC models at temporal resolutions below 1 day is evident. Overall, the performance is better at the 10-min resolution than at coarser aggregations. For example, the autocorrelations at intermediate scales of a few hours are not perfectly reproduced. Even though they are almost perfect at the 10-min scale. The underlying reason can be found in the parameterizations of the microcanonical MRCs that are not always sufficient to capture the entire variability/correlation of rainfall. In this case, heavier parameterizations could improve the simulations, but would go against the philosophy of a parsimonious and generic model. Generally, the performance at high resolutions increases with the MRC model complexity. As shown in Fig. 9 the reproduction of the tail of the distribution is much better when the MRC model B is used. This is also reflected in the better reproduction of the annual extremes (Fig. 10). For all the other statistics, both versions of the MRC model (A & B) comparably improve the performance with respect to the NSRP model. In terms of statistics at temporal scales greater than 1 day, the more complex the external model is, the better it performs. The

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results of the composite models based on the NSRP model (Fig. 7) are typically better than the performance of the model based on Alternating renewal processes and Markov chains (Figs. 8 and 6). Common problems associated with the latter models are the systematic underestimation of the tail of the distribution of accumulation depths for temporal scales beyond one day. Some of those problems could have been solved using more complex distributions of daily depths [77,94]. However, this would have also increased the parameterization of the models. Furthermore, the choice of the distribution could become station-dependent limiting its generality. An important statistical property which stochastic precipitation models should be able to reproduce are extremes. A critical test for models is their ability to capture the depth-duration-frequency relationships for annual maxima. These are shown for durations D ¼10 min, 1 h and 1 day for the NSRP Model only, the combination of NSRP–MRC model B and the Markov chain–MRC model B combination in Fig. 11. The performance of the composite model is very good for all three durations, and significantly improves the single NSRP model which underestimates the extremes at the 10-min scale and overestimates the daily maxima. The excellent performance of the composite model is not a trivial result because annual maxima for short durations are determined by the independent behavior of both the external and internal models. 4.3. Model and station inter-comparison After the illustrative presentation of the various model combinations for the Zurich station, we seek a model and station intercomparison and synthesis in this section. The questions raised

are (i) if all composite models perform equally well; and (ii) if this performance is dependent on the station itself. The overall goal is to identify the general patterns of the efficiency of each model combination that may serve as a selection criterion. We chose 4 basic statistics to illustrate model performance: variance (Fig. 12), skewness (Fig. 13), probability of no rain (Fig. 14) and lag-one autocorrelation (Fig. 15). The statistics in each figure are estimated at three temporal aggregation scales (10-min, 2.7 h and 21.3 h), for all 6 composite model combinations at all four stations. The three temporal aggregation scales are shown as a bounding polygon within which all simulations lie. The analysis was conducted on a seasonal basis, and the results we show are for Spring (MAM) only, which in terms of performance was an average season. Results for the other seasons are similar and they do not affect the conclusions of this study [82]. First of all, the mean value was almost perfectly preserved for all the modeling approaches, and thus is not shown here. The variance (Fig. 12) is practically perfectly reproduced by all composite models, at all stations and temporal scales because it is used directly or indirectly in the calibration of both external and internal models. Larger differences are evident in the asymmetry of the precipitation distributions measured by the skewness coefficient (Fig. 13). Generally, the efficiency of all models is relatively good. The scatter around the perfect agreement line reveals some of the common features between all the models. For all the stations the ability of the combination that employs the MRC model B as the disaggregation model performs better than the equivalent combination that uses the MRC model A. This is a strong indication that a parametrization of the cascade generator dependent both on the temporal

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scale and the precipitation intensity is an improvement. No significant differences between the various stations could be further identified, also because of the sensitivity of the coefficient of skewness to outliers. The probability of zero precipitation is well reproduced by all models although patterns can be identified (Fig. 14). The combination of Markov chain and any MRC model give almost perfect results for all the analyzed temporal scales and stations. The

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combinations of NSRP and MRC models, in most of the aggregation intervals overestimate slightly the probability of no rainfall. The overestimation of the zero probability associated with the use of the NSRP model [18,16,95] is expected to be higher for aggregation intervals larger than 1 day. The combination of the ARP and MRC models gives exactly the opposite result, systematically underestimating the probability of zero precipitation. The reason is that the exponential distribution that was selected for parametrizing dry durations does not capture well the tails of the dry periods. A selection of a more heavy tailed distribution would yield better results, but at the same time increase the number of parameters. For the

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Fig. 13. Observed vs Simulated skewness coefficient for precipitation for three temporal aggregation scales and all 6 composite model combinations (markers) at the 4 analyzed stations (different colors) for (MAM). The aggregation scales are shown as a bounding polygon with different colors within which all simulations lie. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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relation coefficient (Fig. 15). At the daily scale, where we can validate the performance of the external models, the NSRP model outperforms the other two models for all the stations. Moreover, at Mt. Cook the performance is overall not good at this scale for any model. The Markov chain and the Alternating renewal process systematically underestimate the autocorrelation function due to the assumption of the iid daily rainfall depths. Another option we explored but not reported here, was to assume a uniform intensity across the wet spells for the ARP model. Such an option led to a substantial overestimation of the autocorrelation for the daily scale. At the intermediate scale (2.7 h) all the model combinations underestimate the lag-one correlation coefficient. The underestimation is more dependent on the station rather than on the model. For the highest resolution (10-min) the models do reasonably well, with a tendency to a slight underestimation which is also station dependent. Overall the largest differences in the reproduction of the lagone correlation between models is found at Mt. Cook. This is the only station with a notable difference between the internal disaggregation models MRC A and B for this statistic. In all cases, MRC model B outperforms MRC model A. Despite the large uncertainty of the results, due to the limited number of stations that are used in this study, we provide some general guidelines for the model selection. Regarding the MRC models, it was found that a parametrization that introduces dependencies of the distribution of the cascade generator W in both scale and intensity is needed. Various parametric forms of such relationships can be found in the literature (e.g., [54,59,60]), and most probably the optimal choice will depend on the analyzed station. The choice of the external model should be driven by the specific application for which precipitation is simulated. For instance, MC models lead to a better representation of the probability of precipitation occurrence, and may be preferable in drought or water resources management studies. The NSRP model represents better the correlation properties of precipitation, and may be preferable for flood risk analyses. Since data from only 4 stations have been used, any generalization on the applicability of each model combination would be premature. Future analysis on more extensive datasets will potentially reveal if an optimal model combination exists for different climates.

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highest resolution scale, the largest discrepancies in the probability of no rain was found for the wettest station (Mt Cook). For this dataset the parameterization of the probability of zero rain p0 in the MRC models with intensity and scale (Eq. 6) was not satisfactory. A correct representation of the autocorrelation function is critical for all composite models. We present only the lag-one autocor-

The topic of this study was to compare and combine methods for stochastic modeling of precipitation time series. We set out to overcome common problems of the most widely used stochastic precipitation models and propose an alternative strategy for modeling temporal precipitation. The methodology that is presented here is to combine different stochastic models across the range of temporal scales where they perform best. Traditional modeling techniques, even if they are developed in order to reproduce multi-scale statistics, such as Poisson cluster models and Multiplicative Random Cascades, suffer from serious restrictions. Inherent structural problems in these models do not allow robust simulation across a wide range of scales. Combining different techniques at temporal scales where they have been found to perform well can enhance the applicability of a model with a minor increase in the number of parameters. We examined six different model combinations in this study. All of them consisted of two stochastic models – an external model for coarse scales (Alternating renewal process, Markov chain, Poisson cluster model of the Neyman Scott type) and an internal disaggregation model for fine scales (two versions of the microcanonical Multiplicative Random Cascade). The composite models were

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tested on high resolution precipitation data at four sites in different climates. As a first result we showed that the NSRP model by itself was not able to reproduce satisfactorily statistics at the highest (10-min) resolution, in particular, the tails of the rainfall depth distributions, the probability of no rain, wet/dry spell durations and serial correlation structure. Adding a MRC model to disaggregate simulated precipitation from the daily scale vastly improved these statistics. Good performance at the 10-min scale was also shown for the other composite models. The added value of the nested internal model was most evident in the analysis of extremes for different durations, where for example we showed that the MC-MRC-B combination reproduced almost perfectly the depth-duration-frequency relations for different aggregation times, while the NSRP model alone had significant bias for durations (time scales) which were not used in its calibration. Second, the results do not give clear evidence for which composite model is best overall. Each of the model combinations has its own advantages, mainly based on the properties of the external model that influence the fine scale statistics through the MRC. For instance the MC-MRC model combinations were best for representing intermittency at all scales, while the NSRP-MRC models were on the average best for autocorrelation. The choice of the appropriate modeling approach should be based on the most significant statistics that need to be preserved in the application. The most evident result of the comparison was between the two variants of the MRC models. For all the stations, the more complex structure of the MRC model systematically provided better results. Therefore, dependence of the cascade weights both on scale and intensity was found to be significant for all the climates. Third, autocorrelation is difficult to reproduce across all relevant scales. At the minute to hourly temporal aggregation scales lag-one autocorrelation was underestimated by practically all model combinations at all stations. This suggests a problem with the discrete MRC which does not reproduce the autocorrelation structure of rainfall accurately. To overcome this issue, different disaggregation procedures can be tested as alternative options in future research. However, the fact that the composite models perform reasonably well for stations from such different climates strengthens the argument that the proposed methodology has general validity and potentially wide applicability. Acknowledgments We would like to thank the anonymous referees and the editor, Andrea Rinaldo, for their helpful comments that improved the manuscript. Precipitation data for Switzerland were provided by MeteoSwiss, the Federal Office of Meteorology and Climatology. Funding for this research was provided by the Swiss National Science Foundation Grant 20021-120310. References [1] Todorovic P, Yevjevich V. Stochastic process of precipitation. Technical report, Colo. State Univ. Fort Collins, Fort Collins, Colorado; 1969. [2] Todorovic P, Woolhiser DA. A stochastic model of n-day precipitation. J Appl Meteorol 1975;14:17–24. http://dx.doi.org/10.1175/1520-0450(1975)014<0017: ASMODP>2.0.CO;2. [3] Chin E. Modeling daily precipitation occurrence process with Markov chain. Water Resour Res 1977;13:949–56. http://dx.doi.org/10.1029/WR013i006p00949. [4] Katz R. Precipitation as a chain-dependent process. J Appl Meteorol 1977;16:671–6. http://dx.doi.org/10.1175/1520-0450(1977)016<0671: PAACDP>2.0.CO;2. [5] Foufoula-Georgiou E, Lettenmaier D. A Markov renewal model for rainfall occurrences. Water Resour Res 1987;23:875–84. http://dx.doi.org/10.1029/ WR023i005p00875. [6] Schmitt F, Vannitsem S, Barbosa A. Modeling of rainfall time series using twostate renewal processes and multifractals. J Geophys Res 1998;103:23,181–93. http://dx.doi.org/10.1029/98JD02071.

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