Rainfall Generation Revisited: Introducing CoSMoS2s and Advancing CopulaBased Intermittent Time Series Modeling
Abstract
What elements should a parsimonious model reproduce at a single scale to precisely simulate rainfall at many scales? We posit these elements are: (a) the probability of dry and linear correlation structure of the wet/dry sequence as a proxy reproducing the distribution of wet/dry spells, and (b) the marginal distribution of nonzero rainfall and its correlation structure. We build a twostate rainfall model, the CoSMoS2s, that explicitly reproduces these elements and is easily applicable at any timescale. Additionally, the paper: (a) introduces the Generalized Exponential () distribution system comprising six flexible distributions with desired properties to describe nonzero rainfall and facilitate time series generation; (b) extends the CoSMoS framework to allow simulations with negative correlations; (c) simplifies the generation of binary sequences with any correlation structure by analytical approximations; (d) introduces the rankbased CoSMoS2s that preserves Spearman's correlations, has an analytical formulation, and is also applicable for infinite variance time series, (e) introduces the copulabased CoSMoS2s enabling intermittent times series generation with nonzero values having the dependence structure of any desired copula, and (f) offers conceptual generalizations for rainfall modeling and beyond, with specific ideas for future improvements and extensions. The CoSMoS2s is tested using four long hourly rainfall records; the simulations reproduce rainfall properties at multiple scales including the wet/dry spells, probability of dry, characteristics of nonzero rainfall, and the behavior of extremes.
Key Points

New flexible system of probability distributions for rainfall intensity

Advanced rainfall generation preserving wet/dry spells, marginal distributions, and copula dependence of nonzero rainfall

Applicable to a single time scale and reproducing rainfall characteristics at multiple scales
1 Brief Review of Rainfall Models
“Water is the driving force of all nature.” ∼Leonardo da Vinci
Nature cannot escape randomness. In fact, randomness is one of nature's building blocks with mechanistic laws allowing only shortterm prediction of hydrometeorological processes such as rainfall. Yet longterm rainfall modeling is imperative for investigating hydroclimatic variability and natural or humanmade environments. This motivated the development of probabilistic rainfall models that are based on different foundations—different models target reproducing different rainfall characteristics or are developed under different theoretical mandates. The perfect stochastic model would reproduce the joint distributions (of any order and at any scale) describing the process under investigation. Yet this approach, for the general case, might not be technically feasible—at least not in a meaningful way. An operational model should identify and reproduce key properties of a process, be as parsimonious as possible, and still represent adequately the process at multiple scales.
It would be a colossal task to describe here the mechanics of all known stochastic rainfall models or cite the whole literature; this is not the study's scope. Yet some characteristic and popular modeling strategies include those based on point processes, Markov chains (MCs), multifractals, as well as recent ones targeting to reproduce any marginal distribution and correlation structure.
A popular modeling strategy relies on a specialized class of point processes (e.g., Cox & Isham, 1980), the socalled clusterbased models. These models represent point rainfall as storms generated by clusters of rectangular pulses (cells). The most wellknown rectangular pulse models are the NeymanScott (NS; Neyman & Scott, 1958) and BartlettLewis (BL; RodriguezIturbe et al., 1987). Originally, Neyman and Scott (1958) used point processes to model spatial clustering in galaxies. Later, this approach inspired applications in rainfall modeling and resulted in the NS rectangular pulse model (Kavvas & Delleur, 1976, 1981; Le Cam, 1961). In both models, a Poisson distribution specifies epochs of storms. A random number of cells, defined by a Poisson or Geometric distribution, is attributed to each storm event. The rectangular pulses have random intensity and duration, usually exponentially distributed. The main difference between the models is the placement of cell origins relative to storm origins. In the NS model, the time intervals between the storm origin and the birth of individual cells are independent random variables, exponentially distributed. In the BL model, the storm duration is exponentially distributed and the intervals between successive cells are independent. The cells may overlap both within the same storm and with cells of different storms. Many studies explored variations of the NS model (e.g., Cowpertwait, 1991; Cowpertwait et al., 2002; Wheater et al., 2005) and others reparametrized the BL model and randomized the storm arrivals and durations as well as the cell intensity (e.g., Kaczmarska et al., 2014; Onof & Wang, 2020; RodriguezIturbe et al., 1988). These models are typically used to simulate daily or subdaily rainfall time series.
MC models (Markov, 1906) have been extensively used in hydrology in different topics ranging from stochastic reservoir and flood cascade theories (see Pegram, 1971) to simulate rainfall occurrence, or else, wet/dry states. Gabriel and Neumann (1957, 1962) seem to have first proposed an MC probability model of order one for rainfall occurrence, with some properties derived by Gabriel (1959). In the firstorder MC, the daily rainfall probability is conditioned on the wet/dry state of the previous day. The lengths of the alternating wet and dry spells are independent, distributed according to the Geometric distribution. Higherorder MC models, based on wet day probabilities of few consecutive days, can improve the wet/dry clustering; other variations include the hybridorder (Stern & Coe, 1984) and the multistate MC (Haan et al., 1976). Over the past few decades, several stochastic rainfall generators coupled MC models with continuous probability distributions to describe nonzero rainfall (e.g., Exponential distribution in WGEN [Richardson, 1981], Gamma in CLIGEN [Nicks & Gander, 1994], Weibull in ClimGen [Stockle et al., 2001], and Exponential and Gamma in WeaGETS [J. Chen et al., 2012]; see also Wilks and Wilby [1999] for a relevant review). Another modeling strategy for rainfall occurrence uses renewal processes (e.g., FoufoulaGeorgiou & Lettenmaier, 1987; Jones et al., 1972; Quélennec, 1973) assuming independence between wet and dry spells. Distributions such as the logarithmic series, truncated negative binomial distribution, and truncated geometric distribution were used to generate wet/dry spells (Wilks & Wilby, 1999) with other distributions describing nonzero rainfall.
Many researchers, circa mid60s, identified the socalled fractal properties in nature. That is, an “object” can be subdivided into reducedsize copies of the whole in a cascade (Mandelbrot, 1982). Extensions to multifractal theory assume that singlescale fluctuations inform fluctuations at other scales via scale invariance (Grassberger, 1983). The multifractal representation of rainfall has found applications, for example, in predicting rainfall extremes and constructing IDF curves (e.g., Langousis & Veneziano, 2007; Veneziano et al., 2006). In time series or random fields simulation based on multifractals, the scale invariance is reproduced by using random multiplicative cascade processes (Lovejoy & Mandelbrot, 1985; Schertzear & Lovejoy, 1987). Multiplicative cascades, first introduced probably by Yaglom (1966), have been explored and applied in many later studies to disaggregate daily rainfall to finer temporal resolutions studies (e.g., Deidda, 2000; Gaume et al., 2007; Güntner et al., 2001; Menabde et al., 1997; Molnar & Burlando, 2005; Olsson, 1998; Perica & FoufoulaGeorgiou, 1996; Rupp et al., 2009; Serinaldi, 2010). In the context of rainfall simulation, several advances and applications of multiplicative cascade models have been presented over the last decade (see e.g., AguilarFlores et al., 2021; Akrour et al., 2015; Gires et al., 2013; Licznar et al., 2015; Lombardo et al., 2012; Müller & Haberlandt, 2015; MüllerThomy, 2020; Paschalis et al., 2014; and references therein). Finally, several variations of multifractal models exist, such as pulsebased, nonpulsebased using wavelet decompositions, and nonpulsebased using discrete or continuous multiplicative cascades (Flores, 2004).
The previous modeling strategies have dominated the literature for decades; yet other flexible approaches are gaining momentum. Their origins lie in time series generation by autoregressive moving average (ARMA) models tracked in the works of Box and Jenkins (1970), Matalas (1967), Pegram and James (1972), Thomas and Fiering (1962), and of others. Initially, ARMA models were applied for nonintermittent processes such as rivers flows (see also Salas, 1980) generating Gaussian times series with linear dependence. A more general approach, allowing simulations with various dependence structures beyond linear, is based on conditional sampling from copulas (e.g., Joe, 1997). This method was mainly applied in econometrics, yet its potential has been highlighted in hydrology for river flow simulation (Lee & Salas, 2011) while copulabased multisite models for daily rainfall have been suggested by Bárdossy and Pegram (2009) and Serinaldi (2009a, 2009b). Still, the most prominent approach is based on generating and transforming Gaussian time series to match any desired marginal distribution. Such transformations yet alter the Gaussian correlations (see Section 4 for details) with early cumbersome numerical approaches (e.g., Li & Hammond, 1975) focusing on reproducing continuous marginals and shortterm correlations. This strategy was extended and simplified (Papalexiou, 2018; see also an earlier approach in Papalexiou, 2010) in the CoSMoS framework by introducing simple parametric correlation transformation functions (CTFs). This allowed rainfall generation reproducing the probability of dry, any marginal distribution describing nonzero rainfall, and the whole correlation structure of the intermittent process. In general, the onestate rainfall generation based on Gaussian variable transformations has been applied many times in hydrology, typically for spacetime modeling, with zero rainfall corresponding to Gaussian values below a threshold; for example, see Bardossy and Plate (1992), Bell (1987), and Glasbey and Nevison (1997) to mention just a few early works, since the literature on spacetime modeling is vast and outside the scope in this study.
Many of the previous models, to improve simulation of statistical characteristics at multiple time scales were coupled with disaggregation schemes. Such schemes, initially applied to river flows, were based on the ideas of Harms and Cambel (1967) and Matalas (1967), and were followed by works of Stedinger and Vogel (1984), Valencia and Schaake (1972, 1973), and of many others. Later, disaggregation was extended to rainfall, recognizing the challenge to preserve intermittency and multiscale characteristics. The target was to simulate rainfall sequences at a fine scale (e.g., hourly) conditioned on rainfall totals at a larger one (e.g., daily). Many rainfall disaggregation schemes were based on the Glasbey et al. (1995) framework that uses conditional simulation from a pointprocess model. They suggested simulating long rainfall sequences at the desired fine scale and select subsequences matching coarsescale totals, or alternatively, generate sequences iteratively until the coarsescale total is achieved—many variations and extensions followed (e.g., Connolly et al., 1998; Cowpertwait et al., 1996). Recently, the DiPMaC scheme (Papalexiou, Markonis, et al., 2018) enabled disaggregation that explicitly reproduces the probability distributions and correlations at a fine scale, also allowing nonstationary disaggregation reproducing time varying properties. For a more detailed historical evolution of the disaggregation literature see Papalexiou, Markonis, et al. (2018).
This study aims to explore stochastic modeling strategies for intermittent processes and build a rainfall model that is easily applicable at a single scale and reproduces rainfall characteristics at multiple scales. Toward this aim several novelties are introduced merging into the CoSMoS2s model that shows promising results in reproducing wet\dry spells and rainfall characteristics at large range of temporal scales.
The paper does not follow the typical structure but is rather based on the “natural” steps that build the CoSMoS models. The major components of these models identify: (a) the marginal distribution, and (b) the autocorrelation structure (ACS) (or dependence structure) that describe the process under investigation, and (c) the characteristics of a parent process (typically Gaussian but it can have other type of dependence) to be transformed. To help the reader navigate the outline: Section 2 introduces a new system of flexible marginal distributions; Section 3 explains the use of ACS's and their link with autoregressive (AR) models; Section 4 focuses on correlation transformations and extends previous works to negative correlation; Section 5 introduces the theoretical framework of the twostate intermittent model (CoSMoS2s) that couples a binary and a continuous process; Section 6 introduces an analytical approach to simulate binary time series that facilities the operational use of CoSMoS2s; Section 7 offers guidelines for different modeling strategies to simulate intermittent processes, including the onestate approach (CoSMoS1s), and variations of the CoSMoS2s including new analytical rankbased and copulabased alternatives; Section 8 tests the performance of CoSMoS2s in simulating hourly rainfall and its potential to preserve statistical properties at a large range of time scales; Section 9 offers a discussion and conceptual generalizations for rainfall modeling with specific ideas for future improvements and extensions toward building the “ultimate” rainfall model; and Section 10 summarizes and concludes the paper.
2 New System of Probability Distributions
2.1 Probabilistic Behavior of Rainfall
The statistical characteristics of rainfall and of other hydroclimatic processes typically vary, depending on: (a) time and spatial scale, (b) season, (c) location, and (d) the period under investigation for nonstationary cases. For example, rainfall at submonthly scales (daily, hourly, etc.) is intermittent having high probability of dry (probability mass at zero) and continuous positively skewed marginals describing nonzero values. This can also be conceptualized as having a mixedtype (zeroinflated) marginal distribution. Noteworthy, analysis of intermittency at fine temporal scales should be conducted with caution, as rainfall measurement methods (e.g., tipping buckets) can affect the results (Mascaro et al., 2013). At larger time scales, such as annual or interannual, and in most places of the world, rainfall can be described by continuous marginals, typically bellshaped due to the central limit theorem. The same holds for rainfall at different spatial scales; fine scale characteristics (e.g., point measurements) differ from those at large scales (e.g., at 1° × 1° grids), that is why concepts such as the areal reduction factor are so popular in hydrology (e.g., Wright et al., 2014). Seasons typically affect rainfall greatly; for example, Mediterranean regions have dry summers and wet winters (see Papalexiou & Koutsoyiannis, 2016 for a global analysis on seasonal variation of daily rainfall). Seasonal patterns change with location too, for example, the tropics have low seasonal variability while high latitude artic and subarctic regions have distinct seasons. Location, however, may alter rainfall characteristics even in the same latitude regions due to regional weather patterns, presence of mountains, etc. For example, rainfall climatology differs across the United States as eastern parts get more rainfall than the western. The behavior of extremes also varies with location as the tail heaviness shows clear regional patterns (e.g., Nerantzaki & Papalexiou, 2019; Papalexiou, AghaKouchak, & FoufoulaGeorgiou, 2018). Finally, if nonstationarity is assumed, then time varying distributions should be considered, yet nonstationarity (e.g., Serinaldi & Kilsby, 2015) should be used with caution. Clearly, stationarity should also be used with caution since describing a nonstationary process with a stationary model can lead in underestimating the risk of extremes. While the natural variability of rainfall is high and potentially masks deterministic changes in many locations, there are many recent and largescale or global studies indicating changes in the rainfall regime (e.g., Alexander, 2016; Barbero et al., 2017; Markonis et al., 2019; Moustakis et al., 2021; Papalexiou & Montanari, 2019; Prein et al., 2017; Ye et al., 2017).
The previous arguments dictate that probability distributions qualified to describe rainfall in a large range of spatiotemporal scales, different seasons, and regions, must have desired properties such as: (a) consistent domain—that is, (0, ∞) since nonzero rainfall is positive. This excludes distributions with a location parameter as it imposes a lower (or upper) bound different than zero. (b) Flexibility—the probability density function (pdf) of nonzero rainfall might be Jshaped (e.g., at subdaily scales) or bellshaped (e.g., at annual scales), having a thin or heavy right tail. (c) Parsimony—the most parsimonious distribution, flexible enough to match the previous demands, have three parameters, that is, a scale and two shape parameters controlling the left and right tails. Twoparameter simplifications, if adequate, should be preferred where appropriate. For example, the Exponential (), Gamma (), Lognormal (), and Weibull (), distributions have been popular choices; however, the Exponential has fixed shape, the Lognormal allows control on the right tail but not on the left, the Gamma has J and bellshaped densities but always thin tail, and the Weibull has stretchedexponential tails for Jshape densities and hyperexponential tails for bellshaped densities. These limitations restrict such distributions to perform well across a wide range of spatial and temporal scales.
2.2 The Generalized Exponential () System
Known threeparameter distributions with the previous three properties exist, yet their use has been limited with most studies, as previously mentioned, focusing on the , , , and distributions. Possible reasons are: (a) complex expressions are not as favored as simpler two or threeparameter models that include a location parameter; (b) fitting challenges; (c) complicated or not analytical moments and Lmoments expressions, and (d) the need to use simple distributions in stochastic models to facilitate their mathematical formulation. Yet advances in stochastic modeling (Papalexiou, 2018; Papalexiou & Serinaldi, 2020; Papalexiou, Serinaldi, & Porcu, 2021), allow rainfall modeling with any desired distribution and correlation structure. Also, global studies (Papalexiou & Koutsoyiannis, 2012, 2016) analyzing thousands of records indicate that such distributions describe effectively nonzero rainfall. Particularly, the Generalized Gamma (; Equation A1 in Appendix A) introduced by Stacy (1962) and the Burr type XII (; Equation A2) from the Burr (1942) system have been reparametrized and used extensively for rainfall (Papalexiou & Koutsoyiannis, 2012, 2016). The Burr type III (; Equation A3) has also been suggested (Papalexiou, 2018), while a generalization of the Beta of the second kind (; Equation A4) and the Generalized Standard Gompertz (; Equation A5) were introduced in Papalexiou and Serinaldi (2020) to describe rainfall in random fields. The , , and are powertype distributions and can have very heavy tails. The and are of exponential and double exponential form, respectively, and their tails can be heavier (stretched exponential) or thinner (hyper exponential) than the tail of the exponential distribution.
In general, the distribution describes rainfall well; however, (a) its cumulative distribution (cdf) and quantile functions are not analytical, slowing down quantile transformations in time series generation, and (b) its shape parameters do not converge to meaningful values if fitted to heavytailed rainfall. Powertype distributions can describe heavy rainfall, yet some specific shape parameters correspond to infinite variance distributions. This excludes time series generation based on the CoSMoS framework since the ACS cannot be defined. Thus, to better describe rainfall and further advance stochastic modeling we need alternative exponentialform distributions with simple and analytical cdf's, having their moments finite (to guarantee the ACS existence) and being more versatile than the .
The pdf's are, in general, J and bellshaped for and , respectively, while mainly controls to the right tail heaviness (see Figures 1a and 1b for pdf shapes). Exceptions are the and where both shape parameters can affect the density shape. These expressions follow a consistent notation, that is, is a positive scale parameter and and are positive shape parameters controlling mainly the left and right tail, respectively (see notation in Appendix A).
The framework creating these distributions starts with a valid distribution function defined in [0, ∞) and replaces with any increasing function , that is, . Functions can also be formed by distribution functions defined in [0,∞). If is the survival function (sf) of a distribution , then the function has the desired properties of . Thus, the function , with and being valid distribution functions in [0,∞) defines a valid distribution function in [0, ∞).
For example, the emerges by setting to be the Exponential distribution (Equation A6), and the distribution (Equation A2). The inherits the parameters of having one scale and two shape parameters. The uses as the Pareto II () distribution (Equation A7) and as the Weibull () distribution (Equation A8). Information on how each emerges is given in Table 1. Here, we formed a system of poweredexponential distributions with analytical invertible cdf's, having J and bellshaped densities, and with control over the right tail heaviness. All distributions for simplify to the Exponential distribution , and for to the Weibull distribution . This does not imply that all distributions have the same asymptotic behavior when fitted to real data. For example, if we estimate the parameters of these distributions to have the same first three Lmoments (e.g., first Lmoment , Lvariation , and Lskewness ), we note that their predictions for low exceedance probabilities differ (Figure 1c) indicating that some distribution have heavier tails than others.
type  F_{1}(x)  F_{2}(x) 

1  
2  
3  
4  
5  
6 
One could name these distributions based on the names of and , yet since they were designed to generalize the Exponential distribution, they were named accordingly. The system is not exhausted here and more distributions can be formed. In general, this framework can create infinitely many distributions, beyond the type, such as powertype distributions, or distributions with different support, defined for example in (0, 1) or in . Also note that these distributions are continuous, suited to describe nonzero rainfall. Yet we can conceptualize rainfall as onestate process with mixedtype marginals and this conceptualization can facilitate its stochastic modeling (Papalexiou, 2018). The mixedtype expressions of the cdf, quantile functions, and momentrelated quantities are given in Equations A9–A15 in Appendix A.
2.3 Remarks on Applications
The system offers flexible distributions aiming to describe the whole sample (main body and tail) of skewed variables such as rainfall. This is crucial in stochastic modeling since synthetic time series are used as inputs in many models applied in risk assessment, streamflow prediction, water resources management, crop production, energy production and consumption, building resilience, etc. These models, to produce reliable outputs, need reliable inputs reproducing the marginal and joint properties of observations, since the frequencies in all quantiles can affect the system's response—as the saying goes: garbage in, garbage out. For example, we can feed a hydrologic model with two time series described by the same distribution tail but different densities—the outcome will differ.
A point of caution regards the fitting of such distributions as with great flexibility comes great responsibility. There are more than 20 distribution fitting methods (for a detailed review see Nerantzaki & Papalexiou, 2022); some are easily applicable but this does not imply they should be preferred. All distributions of the system have analytical and invertible cdf's and pdf's. Thus, generic methods such as the maximum likelihood and least squares estimation can be implemented numerically. The method of moments is probably the most popular one, but it should be used with caution especially when higher order moments need to be estimated; there is large uncertainty in estimating such moments in skewed samples. An alternative and more robust method is that of Lmoments (Greenwood et al., 1979; Hosking, 1990; Sillitto, 1951) which has gained popularity over the last decades and been extensively applied in hydrology (e.g., Royston, 1992; Vogel & Fennessey, 1993). While the system does not have analytical Lmoments the method can be applied numerically (for details see Zaghloul et al., 2020).
Another point is whether such threeparameter distributions perform well in reproducing the behavior of extremes when compared to classical methods. Two methods dominating the analysis of extremes use block maxima (coupled typically with the Generalized Extreme Value () distribution) and peak over threshold (POT) values (coupled typically with Generalized Pareto ( distribution); yet an alternative method can use the whole sample (see e.g., Salas et al., 2020). We performed here a toymodel Monte Carlo experiment comparing returnlevel estimates from different methods that does not verify the superior performance of the classical methods in general (see Section S1 and Figure S1 in Supporting Information S1). Samples from three popular distributions (, , and ) are generated and the 100 and 500 yr return levels are estimated by fitting the to block maxima, the to POT maxima, and the and to the whole sample. The and estimates are almost unbiased in all cases but with very large variance. In contrast, the or estimates show very low variance but can be biased depending on the distribution that generated the sample. For example, for thintailed samples the heavytailed overestimated return levels, while for samples (heavy tail) the (thinner tail than ) underestimated return levels. This toymodel experiment shows that if the fitted distribution is consistent with the underline tail, then it can describe extremes accurately.
In general, identifying the tail type or quantifying the tailheaviness is not trivial and many methods have been invented and tested (see e.g., El Adlouni et al., 2008; Embrechts et al., 1997; Langousis et al., 2016; Nerantzaki & Papalexiou, 2019; Serinaldi, 2013; Smith, 1987; Wietzke et al., 2020). Tailtype identification or tailheaviness estimates can be more robust if informed by global or regional studies (e.g., Papalexiou et al., 2013; Rajulapati et al., 2020; Serinaldi & Kilsby, 2014). For example, Papalexiou, AghaKouchak, and FoufoulaGeorgiou (2018) used regional tail estimates to fix the tail parameter in the and distributions before fitting them to the whole sample; this approach was used to assess hourly precipitation depths at large return periods. Additionally, many recent studies have used ordinary events (most of the sample) to assess extremes indicating better performance at high return periods (e.g., Marani & Ignaccolo, 2015; Marra et al., 2018; Zorzetto et al., 2016). Finally, the block and POT maxima methods are not free of limitations. For example, convergence to these liming laws is not guaranteed and in many cases, the estimated parameters indicate that extremes have an upper bound which might lead in underestimating risk (e.g., Moccia et al., 2021).
3 Autocorrelation Structures (ACS)
A stochastic process, in simple terms, is a collection of random variables (rv's) typically associated to each other. If is such a collection of rv's in time, with being an indexed set (e.g., ), then any form of association among the rv's, loosely speaking, implies that information on some rv's (e.g., on their values) can provide information on some of the others. This underlines the importance of this association. However, quantifying association is not easy and there is more than a centurylong research on different association measures (see Särndal, 1974 for an early comparative study). Probably the most popular association measure is the Pearson correlation coefficient quantifying linear dependence. If rv's are associated nonlinearly then quantifying correlations using can be misleading (see e.g., Altman & Krzywinski, 2015). These issues led to different correlation measures. For instance, rankbased measures were introduced by Kendall (1938) and Spearman (1904) (suitable for linear and nonlinear dependencies), and later Linfoot (1957) formed the informational coefficient of correlation, an entropybased measure that generalizes the Pearson correlation and is invariant under rv transformations.
A temporal ACS is a mathematical parametric formula describing parsimoniously how the rv's of a stochastic process are correlated with each other in time. For example, the ACS can describe the linear dependence, as expressed by the lag Pearson correlation coefficient for all possible lags in a stationary process. This is convenient since a parametric ACS ( is a parameter vector) can be reproduced by AR models up to any desired lag. The model's parameters reproducing a desired positive definite are given by , where and is a matrix with (e.g., Box et al., 2008). The notion of the ACS should not be limited in expressing linear dependence, since we could also use it to express rank correlations (or any other correlation measure) at different lags given that a model could reproduce such ACS's. For instance, we could reproduce a rankbased ACS expressing the decay of Spearman's rho with time using Gaussian or other multidimensional copulas (e.g., copula).
Reproducing an observed (empirical) ACS expressing linear or rankbased dependencies should not be the target per se. The target is mimicking a natural process as precisely as possible and there is no guarantee that simple correlation measures can express complex dependence patterns. For instance, we can construct copulas having the same Pearson correlation and marginals but different dependence structures. Yet reproducing linear dependencies to model natural processes seems to provide a useful approximation of “reality”.
The number of parametric ACS's in literature is large and many date back in the sixties and seventies (see e.g., the list of ACS's given by Buell, 1972). These early works were followed by efforts to define theoretical requirements of valid ACS's such as positive definiteness (e.g., Franke et al., 1988; Julian & Thiebaux, 1975). Later on in the nineties, as Gneiting (1999) remarks, advances in global analysis systems led to a quest for flexible parametric ACS's and points out the work of Gaspari and Cohn (1999) for a detailed mathematical treatise on correlation functions.
These ACS's have a scale parameter and a shape parameter (for the W ACS ). The asymptotic behavior, or else the rate at which the ACS approaches zero for increasing lags, is controlled by the parameter and affects the longterm temporal dynamics. The asymptotic behavior, demonstrated in a loglog plot fixing the lag1 correlation to the same value (Figure 2), shows that the W ACS is concave (powered exponential), the PII asymptotically straight line (power type), and the GL (for large values) convex (slower decay than power type). The PII and GL for simplify to the Markovian ACS . These ACS's can be generalized by adding a third parameter; for example, the can be replaced by in PII and GL or use a survival function of the distribution system instead of the W ACS. Yet parsimony should be sought, and twoparameter ACS's should always be preferred over threeparameter ones given adequate fit.
Different parametrizations of the ACS's suggested here, or their special cases and generalizations, have appeared several times in literature. For example, a special case of the W ACS is given in Buell (1972) while in other works (mainly as space correlation) is mentioned as the powered exponential (e.g., Gneiting, 2013). Martin and Walker (1997) on their work on power law correlations mentioned the as a valid isotropic ACS with longrange dependence (LRD) for 0 . Gneiting (2000) generalized this powertype ACS proposing the and gave the permissible parameter space. This ACS (named Cauchy by Gneiting), based on the framework suggested in Papalexiou (2018) coincides with the Burr type XII survival function and was also used to simulate 10 s rainfall in Papalexiou et al. (2011). The motivation for these powertype ACS's was to generalize the LRD correlations of the celebrated fractional Gaussian noise (fGn) process (Mandelbrot & Wallis, 1968), given by , with the parameter controlling the correlation strength (see Beran, 1994; Graves et al., 2017). Note yet that LRD is elusive and should be used with caution; shortterm strong correlations can provide a false impression of LRD (Markonis et al., 2018).
4 Correlation Transformations and Extensions to Negative Space
It was observed early on that the linear correlation between two rv's and following the bivariate Gaussian distribution is maximum, and any nonlinear transformation applied to and/or will decrease it. In mathematical terms, this states that if then , where denotes the linear correlation between the subscripted rv's. This is the maximal property of the bivariate Gaussian correlation and its proof is linked with Lancaster (1957) with first results dating back to Gebelein (1941) and Maung (1941). In fact, these concepts exist in the works of Karl Pearson related to contingency theory and contingency tables (see the “On the theory of contingency and its relation to association and normal correlation”, Pearson, 1904). Interestingly, Lancaster (1958) refers to Hirschfeld (1935) who “…sought for transformations of the marginal variables that would yield linear least squares regression lines. He found that these variables maximized the coefficients of correlation”.
This property was exploited to generate time series having desired marginal distributions and linear autocorrelations. An early application is given by Conner (1971) in his Ph.D. thesis titled “Pseudorandom number generators having specified probability density functions and autocorrelations”. The same year Rowland and Holmes (1971) presented similar techniques in a report for the U.S. Army Missile Command citing Conner's thesis. Li and Hammond (1975) published probably the first paper with a clear demonstration, with Hammond being the supervisor of Conner's thesis. In a nutshell, the key idea was generating time series with appropriately inflated autocorrelations and Gaussian marginal distributions, which when transformed, resulted in time series with desired properties. These early works focused on simple cases such as generating time series with continuous marginals and preserving observed autocorrelations up to a few lags. Analytical expressions linking and exist for the uniform (Baum, 1957) and the Lognormal (Matalas, 1967) distributions, while asymptotic approximations based on HermiteChebyshev polynomial expansions, were used for other distributions (Lancaster, 1957; van der Geest, 1998).
The same approach can be extended to transform negative correlations; this allows one to use parametric ACS's having negative values (e.g., holeeffect ACS's or those suggested in Equation 12). To achieve desired negative correlations, in contrast to positive ones, the Gaussian correlations must be deflated. Also processes with nonGaussian marginals typically have a lower limit for negative correlations larger than −1. To assist the nonfamiliar reader with this framework and its extension to negative correlations we graphically demonstrate the mapping of a Gaussian rv to one having Bernoulli, continuous and mixedtype marginal (see Figures 3a, 3d, and 3g respectively). The corresponding CTF's (Figures 3b, 3e, and 3h) show the inflation for positive correlations, the lower negative correlation limit that can be achieved for these marginals, as well as, the rapid deflation for negative correlations. Once the CTF's are defined they can be used to estimate the parentGaussian ACS's (Figures 3c, 3f, and 3i) for any desired target ACS.
Note that this modeling approach was applied in multivariate simulation using parametric crossCTFs (Papalexiou, 2018), in the DiPMaC disaggregation scheme and for nonstationary simulation (Papalexiou, Markonis, et al., 2018), and more recently, for static and Lagrangian spatiotemporal random fields in Papalexiou and Serinaldi (2020), Papalexiou, Serinaldi, & Porcu (2021), respectively. The same approach is modified here to couple processes with Bernoulli and continuous marginal distributions to improve simulation of intermittent processes and reproduce rainfall characteristics at a large range of scales.
5 TwoState Intermittent Rainfall
Let be a mixedtype rv, with probability mass at zero, describing intermittent rainfall (zeros and nonzero values), a discrete (Bernoulli) rv describing dry and wet states, and a continuous rv describing nonzero values. Then denotes a stationary stochastic process with mixedtype marginals, and and processes with Bernoulli and continuous marginals, respectively; for brevity, hereafter, , , and denote either processes or rv's depending on the context. Also, for clarity, the terms “intermittent process”, “binary process”, and “continuous process” refer to the , , and processes, respectively. Similarly, their ACS's or generated time series of these processes will be labeled as “intermittent”, “binary”, and “continuous”.
The previous fact can affect profoundly the stochastic modeling of intermittent processes. We demonstrate this by simulating an intermittent process resembling, for example, rainfall, with: , continuous marginal the (gray line Figures 4e and 4j), and (red line Figures 4d and 4i). Based on these characteristics , , and . We simulate this process in two different ways.
Case 1:.We generate binary times series (Figure 4a) assuming the binary process has ACS (blue line Figure 4d) and . From Equation 16, given the and , we find the (green line; Figure 4d) and generate time series (Figure 4b) having the marginal. The two time series are multiplied forming the intermittent time series (Figure 4c) that preserves the target ACS (red dots; Figure 4d) and marginal distribution (Figure 4e).
Case 2:.We repeat the previous steps assuming now the binary process (Figure 4f) has a stronger ACS (blue line; Figure 4i). We reestimate the (green line; Figure 4i) and regenerate time series from the process (Figure 4g). The two new time series are combined to give the new intermittent time series (Figure 4h).
The two intermittent time series (Figures 4c and 4h) have the same linear ACS, probability dry, and marginal distribution for nonzero values, and yet, they emerge by multiplying very different processes. Comparing the binary time series (Figures 4a and 4f) we observe the effects of the stronger ACS in Case 2: expressed as longer and more frequent dry spells. Likewise, the difference between the continuous time series (Figures 4b and 4g) is evident with stronger clustering of low/high values in Case 1:. Interestingly, in Case 2:, to match the target ACS (red line; Figure 4i) the continuous (green line; Figure 4i) takes negative correlations. Differences between the two time series are better demonstrated by comparing the probabilities of wet and dry spells (Figures 5a and 5b); Case 2: time series, given the stronger binary ACS, has larger probability for long wet and long dry spells. This affects the scaling of probability zero with the stronger binary ACS resulting in slower decrease with scale (Figure 5c). In contrast to the scaling of probability zero, the weaker ACS of the continuous process in Case 2: contributes to faster decay of distributional shape measures (here Lskewness , and Lkurtosis Figure 5d).
6 Binary Time Series Generation Made Simple
The onevariable integral in Equation 17 facilitates the numerical integration and will be used to form a readily applicable solution.
Investigation showed that this oneparameter version performs equally well. For example, Figure 6a depicts the integralestimated points (Equation 18) for a binary process with and the fitted . The concavity of the function depends on , and thus, is a function of . An easytoapply solution requires one to know the parameter in for any value of . The steps to identify a function are: (a) set to a specific value; (b) use Equation 18 to estimate points for a few values; (c) fit the to estimate the ; (d) repeat step 1–3 for a large number of values. This process results in a set of points (see dots in Figure 6b) that are used to identify the function .
The performance of , with given by Equation 20, is compared to integralestimated points from Equation 18 (Figure 6c).

Select the characteristics of the target binary process, that is, the and the ACS .

Find the value for the desired from Equation 20.

Estimate the parent Gaussian ACS from Equation 19.

Generate Gaussian time series using an AR model of sufficiently large order to reproduce .

Transform the Gaussian values to binary by setting where .
For demonstration, we generate binary time series having the three different ACS's shown in Figure 2 and different levels of probability zero. The first, has and the W ACS (Figures 7a and 7b); the second, has and the PII ACS (Figures 7c and 7d); and the third, has and the GL ACS (Figures 7e and 7f). The scale parameter in each of these ACS's was estimated so that the lag1 autocorrelation is , that is, the same value in all ACS's. A comparison between the target theoretical ACS's (solid blue lines) with the empirical ones (blue dots) of the simulated time series shows their agreement (Figures 7b, 7d, and 7f). This scheme makes the generation of binary time series with any positive definite ACS possible even in a spreadsheet environment (see CoSMoS.xlsx in Supporting Information S2).
7 Rainfall Generation Strategies
7.1 CoSMoS1s  OneState Generation
This section summarizes the rainfall generation method in Papalexiou (2018) and this model will be referred as CoSMoS1s. Conceptually, rainfall is considered as a onestate process described by a single linear ACS and a mixedtype marginal distribution . Thus, CoSMoS1s reproduces the observed ACS of the intermittent time series, the probability of zero, and the marginal distribution of nonzero values.

Use the observed time series to estimate the probability of zero , fit a parametric probability distribution to nonzero values, and fit a parametric ACS .

Estimate the CTF for the desired and (for details see Papalexiou, 2018), and the Gaussian ACS as .

Generate standard Gaussian time series using an AR model that reproduces the .

Apply to transform the Gaussian time series into the desired intermittent one; where is the mixedtype quantile in Equation A10 (see also Figures 3g–3i).
7.2 CoSMoS2s  TwoState Generation
This strategy assumes rainfall as a twostate process, that is, a binary process describes the wet/dry sequence and a “hypothetical” one with a continuous marginal describes nonzero rainfall. This model, named CoSMoS2s, reproduces the linear ACS's of the binary and continuous processes, the probability zero, and the marginal distribution of nonzero values.
 1.
Create the binary time series from the observed one by replacing positive rainfall values with 1.
 2.
Use the binary time series to estimate the probability zero , and fit a parametric ACS to the empirical . Generate synthetic binary time series reproducing and as described in Section 6 (see also Figures 3a–3c).
 3.
Fit a parametric ACS to describe the continuous process. The sample could be estimated by Equation 16, solving for , and using the binary and intermittent . Alternatively, it might be better to estimate explicitly by defining a conditional correlation coefficient as:
(22)where is th value in the time series; the time series length; and are sets of values for ; is the sample size of or since it is equal; and and are mean and standard deviation estimates for the sets indicated by the subscripts.  4.
Fit a parametric probability distribution to nonzero values and generate synthetic time series reproducing the desired and . Note that the procedure here is the same as in the onestep simulation. The difference is that, instead of the mixedtype quantile , the continuous is used to estimate the CTF , and hence the parentGaussian ACS , and transform the Gaussian time series (see also Figures 3d–3f).
7.3 RankedBased CoSMoS2s
A variation of the previous twostate generation, that will not be evaluated in detail here, is a twostate generation based on rankbased (Spearman's) correlations. The motivation for this is the analytical solution and potentially competent performance. The author posits that reproducing accurately the wet/dry spells markedly contributes to reproducing rainfall characteristics across a large range of scales. Pearson's and Spearman's correlations coincide for a Binary process; hence, if explicitly reproducing the binary linear ACS leads in reproducing the wet/dryspells, then using the Spearman's ACS will have the same results. Thus, binary time series generation with a desired Spearman's ACS is the same as in Section 7.2.
This implies that the parentGaussian ACS for a desired Spearman's ACS is estimated as . In practice, the sample can be estimated by fitting a parametric ACS to Spearman's correlations estimated from the positivevalue samples and defined in Section 7.2. The rest of the steps are the same, that is, the parentGaussian time series with the desired is transformed to one with a desired marginal by . Summarizing, the twostate rankbased generation reproduces the Spearman's ACS's of the binary and continuous processes and is essentially analytical as there is no need to solve any integrals.
Such an approach might be useful for processes where the marginal distribution has infinite variance (e.g., powertype marginals with infinite variance). In this case, the ACS of the parent Gaussian process that will reproduce the observed Pearson ACS cannot be defined. In contrast, the Gaussian ACS that reproduces the observed Spearman's ACS can be easily estimated by Equation 23. Cleary, a similar approach can focus on reproducing Kendall's ACS.
7.4 CopulaBased CoSMoS2s
The framework of CoSMoS2s can be extended to generate intermittent times series with nonzero values having the dependence structure of any desired copula. This extends past schemes generating copulabased time series with continuous marginals and thus not being applicable for intermittent processes such rainfall, wind speed, etc.
Briefly, an dimensional copula is a function satisfying specific conditions and acting as a joint cdf of uniform rv's in . Due to Sklar (1973), copulas are used to connect rv's and form their joint distribution. For example, a 2dimensional joint cdf of and is formed as where and are arbitrary marginals. For time series generation, copulas have been mainly used in the field of econometrics. Some characteristic studies and applications on copulabased time series include the Darsow et al. (1992) on copulas and Markov processes, X. Chen and Fan (2006) on semiparametric time series models, Hofert (2008) and McNeil (2008) on sampling from Archimedean copulas, Ibragimov (2009) on higherorder Markov processes, Lee and Salas (2011) on generating annual flows, Ibragimov and Lentzas (2017) on copulas and long memory, and many more; see also the monographs of Joe (1997, 2014) and Nelsen (2006).
Since this equation provides the cdf of it can be used to generate random values as , where is a random number sampled from a uniform distribution in . This formula is applied recursively to generate uniform time series having the dependence structure of the copula . The copulabased uniform time series can be transformed to time series with any continuous marginal by or to binary time series with any by .
Next, we apply this strategy by generating copulabased intermittent time series by coupling copulabased binary and continuous time series. We show two characteristic cases using copulas with different tail dependence.
Case 1.Clayton copula. We demonstrate intermittent time series simulation by coupling binary and continuous processes both generated based on the Clayton () copula given by:
Case 2.Gumbel copula. We repeat the previous demonstration, keeping the same and marginal, but now the binary and continuous time series are generated based on the Gumbel () copula given by:
An important advantage of the copulabased CoSMoS2s vs. the onestate approach is that the nonzero values of the intermittent time series preserve exactly the copula dependence. To clarify, in the onestate approach, intermittency would be introduced by applying a mixedtype quantile (see Equation A10) to a single copulabased uniform time series. However, in this case, the nonzero values will not preserve exactly the copula dependence structure. Uniform values are mapped to zero, while those are mapped to nonzero values using the target marginal's quantile (see Equation A10). Thus, this mapping reproduces the copula dependence for which does not coincide with the complete copula dependence expressed for (see e.g., the characteristic Clayton copula structure in Figure 8b where low and high values have very different dependence).
The previous demonstration shows the potential to also generate copulabased binary series. This approach, at this stage, is experimental, and its operational use needs further research to identify how a copula should be calibrated to a binary time series. For example, two copulas with different tail dependence, even if calibrated to have the same ACS (a measure easily estimated from the observed binary series), could lead to very different behavior of wet/dry spells. The best bet is to focus on the profile of wet/dry spells. Thus, one should identify how the copula dependence parameter (for given probability zero) affects the occurrence probabilities of the wet/dry spells; this could be exploited to calibrate the copula parameter to match the observed wet/dry spells' profile. The “secret life” of copulabased binary time series and their link with transition matrices and the behavior of wet/dry spells will be the topic of a future communication. Until more is revealed on this topic, one could generate copulabased intermittent time series by coupling binary time series that reproduce the observed ACS (Section 6), with continuous copulabased time series where the copula is calibrated to nonzero rainfall using any of the available fitting methods (see e.g., Joe, 1997, 2014; Nelsen, 2006).
7.5 Operational Use and Parsimony
The previous sections focused on modeling strategies; here, we focus on the operational use and calibration mainly of CoSMoS2s (Section 7.2), yet most of the points made are valid for all CoSMoS variations.
7.5.1 Seasonality
Rainfall (and most of hydroclimatic processes) exhibit seasonal variation. Typically, we assume that statistical characteristics do not change within each season and calibrate the model parameters in a seasonal basis. The question how many seasons we should use is not simple to answer. The optimal number of seasons and their individual lengths within a year can be subjective; one can apply clustering algorithms to estimate number and lengths of seasons, but estimates depend on the objective functions used in such algorithms. A common “outofthebox” approach is to assume each month as a different season and calibrate the model monthly. Once the model parameters are estimated for each month (or for each season in general), the model is applied continuously by switching its parameters cyclically. To clarify, the AR model generating the Gaussian values runs continuously for , where is the desired time series length, yet the parameters depend on the value of . For example, if is running within hours of January then the calibrated parameters for January are used, once reaches the first hour of February the parameters switch to those calibrated for February, etc. The same approach can be applied for all CoSMoS variants (e.g., the copula parameter can change monthly).
7.5.2 Choice of Marginal
Potentially, many different marginals can be used to describe nonzero rainfall (as in other hydroclimatic variables). It is up to the user to assess which one describes the data well. Distributions with one scale and two shape parameters (see Section 2) offer enough flexibility for most cases; distributions with more than three parameters should be used with caution. The user, however, might choose simpler forms, for instance, marginals with one scale and one shape parameter given adequate performance. Similarly, any fitting method can be applied but the method of moments should be used with caution for skewed samples and when higherorder moments are involved in parameter estimation. A relevant point regards whether we need to change all the parameters of the marginal in every season. For instance, if there is high uncertainty in the parameter controlling the heaviness of the tail (which is typically the case), then we can fix its value for all seasons with a value estimated from the whole sample or informed by a regional analysis.
7.5.3 Choice of ACS
The comments made for selecting and calibrating the marginal distributions apply also for selecting and calibrating the ACS's. It up to the user to decide whether a twoparameter ACS is needed for both the binary and the continuous process or if all of their parameters need to change seasonally. For example, the shape parameter of the ACS could be fixed for all seasons and allow changes only in the scale parameter to better match the shortterm dependence. Of course, the copulabased CoSMoS does not reproduce a linear ACS but can be calibrated to match, for example, the lag1 dependence.
7.5.4 Order of AR
The calibrated ACS's are reproduced by AR models of order . The rule of thumb is that the order should be large enough to reproduce the ACS; a precise answer cannot be given. For example, one could choose so that the ACS value at is close to zero (e.g., less than 0.05). It should be clear that all the parameters of the AR model are derived analytically from the ACS (see Section 3); thus, an AR model fitted to a twoparameter ACS, no matter of the order , is always a twoparameter model.
7.5.5 Parsimony
Calibrating or building a model is partly art, in the sense that many choices depend on the “eye of the beholder”. Yet it seems that seeking parsimony in models, due to Occam's philosophical razor, has become a “naïve” panacea. As far as the author knows, there was never evidence that the principle of parsimony is itself an irrefutable scientific result. Any model, at least theoretically, can become parsimonious, for example, by assuming fixed parameter values, and this does not imply it is a good model. Building a model, selecting its components, or comparing models based on the principle of parsimony is not straightforward. Parsimony is inevitably linked with specific conditions and is meaningful if assessed based on the desired model outputs. For instance, (a) if two models with different parameter number reproduce the same desired characteristics then the more parsimonious one may be considered better; (b) if two models with the same number of parameters reproduce the same desired characteristics but one of them reproduces additional features then it may be considered better. But if two models with the same parameter number reproduce equal number of desired but different characteristics then which one is the better? If two models with different parameter number reproduce some desired properties but the one with extra parameters reproduces extra desired properties, then which one is better? Model selection is a scientific field itself (see e.g., the review of Nerantzaki & Papalexiou, 2022), but the fact is that such questions cannot be easily answered and rely on defining the desired model outputs and the characteristics that are assumed more important than others. In a nutshell, a general goal would be to achieve the desired model output with the minimum number of parameters.
Regarding CoSMoS models, their structure allows one to select their components and control the number of parameters used. For example, as a general case, it was suggested that a twoparameter ACS and a threeparameter marginal per season offer enough flexibility for hydroclimatic processes. Yet it is up to the user to decide if more parsimonious versions achieve the desired outputs. For example, one can use a Markovian ACS (one parameter) for both the binary and continuous processes, the Exponential distribution (one parameter), and four seasons, ending up with a twelveparameter model. For two and threeparameter ACS's and marginals, respectively, and assuming monthly seasonality, the model will end up with 72 parameters. As aforementioned, various approaches can be applied to reduce the number of parameters if this is desired (e.g., minimum number of seasons, sinusoidal variation of the marginal's scale parameter, fixedtail parameter informed by global or regional studies, a single ACS for the binary and a single ACS of the continuous processes across seasons, etc).
8 Rainfall Simulation in Action
8.1 Simulating and Assessing Hourly Rainfall Time Series
We compare the CoSMoS1s and 2s models using four long records of hourly rainfall. These simulations demonstrate the modeling strategies and their aim is not to identify the most parsimonious version of these models. In the main text, we present results for a centurylong hourly rainfall record from the Philadelphia airport station (Figure 9a); the results for all stations are given in Section S2 in Supporting Information S1. The observed time series is analyzed monthly. The distribution is used to describe the nonzero values for each month (Figure 9d, Figure S3 in Supporting Information S1) and the ACS to represent the empirical autocorrelations of the binary, continuous, and intermittent processes (Figure 9g, Figure S6 in Supporting Information S1).
The CoSMoS1s simulated time series (Figure 9b), as expected, preserves the target marginal distributions (Figure 9e, Figure S4 in Supporting Information S1). However, the simulated ACS of the binary and continuous processes, in general, do not match the target ones (Figure 9h, Figure S7 in Supporting Information S1). The autocorrelation of the binary process is underestimated as the simulated one is weaker than the observed (target). In contrast, the autocorrelation of the continuous process seems in agreement with the observed (except for October and December), yet it is not explicitly preserved in CoSMoS1s. The CoSMoS2s simulated time series (Figure 9c) apart from preserving the target marginal distributions (Figure 9f, Figure S5 in Supporting Information S1) reproduces explicitly the ACS of the binary and continuous processes (blue and green dots, respectively, in Figure 9i and Figure S8 in Supporting Information S1). However, the intermittent ACS in many months appears weaker. The previous observations are valid to all stations simulations; see the Figures in Section S2 in Supporting Information S1. To avoid any misunderstanding, we stress that the models are applied in a continuous basis as described in Section 7.5, thus, the hourly simulated time series (Figures 9b and 9c) are across all months and years.
Differences between the two simulated time series are better demonstrated by comparing the probabilities of wet and dry spells (Figures 10a and 10b). The CoSMoS2s simulated time series, as expected given the stronger binary ACS, has larger probability for long wet and long dry spells. In general, this approach reproduces accurately the distribution of the wet and dry spells (Figures 10a and 10b). The better representation of wet/dry spells also affects the scaling of probability zero with the stronger binary ACS resulting in slower decrease with scale (Figure 10c). Similarly, when the two simulated time series are aggregated at larger time scales the marginal distributions of nonzero values differ especially at large scales. For example, this is demonstrated by comparing distributional shape measures (here Lskewness and Lkurtosis see Figure 10d). The decrease of and with time scale is faster in the CoSMoS2s simulation and matches the observed scaling (Figure 10d). Additionally, the box plots of nonzero rainfall values at different temporal scales (Figure 10e) reveal the superior performance of CoSMoS2s, especially in scales ranging from 2 hr up to 2 days. The same evidence is found in the simulations for all stations studied (see the Figures in Section S2 in Supporting Information S1).
8.2 Assessment of Extremes at Multiple Scales
Any CoSMoS model variant, by definition, reproduces the fitted marginal distribution to nonzero values at the scale it is calibrated from (here at the hourly but it can be at any time scale). If the fitted marginal describes the behavior of rainfall well, then it reproduces the tail properties too and thus the behavior of extremes. This highlights the importance of selecting an appropriate distribution to describe nonzero rainfall. However, there are two points of caution: (a) a distribution might appear to describe the observations well, but this does not guarantee that its tail precisely reproduces extremes. This is because, fitting methods tend to reproduce properties of the main body of observations while the precise asymptotic tail behavior (or tail type) cannot be easily assessed, and (b) reproducing accurately extremes at a single time scale does not imply the same accuracy at coarser time scales. The structure of wet/dry spells or the strength of the ACS, as was previously shown, affects the properties of the process at larger scales and thus its extremes too. For example, a very strong ACS (or strong upper tail dependence) in nonzero rainfall leads to clustering of high values which in turn leads to larger extremes at larger time scales compared to the case of a weak ACS—the same holds for a process with longer wet spells or a binary process with strong ACS.
Here, we compare and assess the performance of CoSMoS1s and 2s in reproducing the observed annual maxima at scales ranging from 1 hr up to 14 days. The annual maxima (from observations and simulations) at each scale are extracted by accumulating the hourly values over a sliding window of duration equal to the one of the investigated scale and picking the maximum of each year (e.g., Papalexiou et al., 2016; van Montfort, 1990). Another common approach accumulates values over nonoverlapping windows leading in underestimating annual maxima and should be avoided. Once the observed annual maxima (at each scale) are identified a distribution is fitted and its 95% confidence interval (CI) is estimated (gray regions in Figure 11). We can assume that simulated annual maxima spotted within the 95% CI at each scale are reproduced well. Note that the 95% CI is constructed assuming that the fitted to the observed maxima is the true one. Clearly, the true underlying distribution is not known; thus, the actual 95% CI can be even broader since the observed maxima might have emerged by a different than the fitted one. Thus, this assessment does not favor the models.
Even so the performance of the CoSMoS2s shows that the empirical distributions of annual maxima at all scales are within the 95% CI and very close to the empirical distribution of the observed maxima (Figure 11); the same holds for all stations simulated (see Figures in Section S2 in Supporting Information S1). CoSMoS1s seems to predict larger exceedance probabilities than those observed or simulated by CoSMoS2s mainly at scales ranging from 4 hr to 2 days. Its deviations in simulating annual maxima at these scales match also the deviations shown for the scaling of probability dry, Lratios, and the box plots of nonzero values (Figures 10c–10e) that are clear at these scales. The same comments are valid for the simulations in the other three stations (see Figures in Section S2 in Supporting Information S1). The fact is that improving the structure of wet/dry spells in CoSMoS2s led to improving the simulation of extremes at every time scale tested. This is an important improvement considering that the model is calibrated at a single scale using the whole sample of values.
9 Insights and Future Quests
9.1 One and TwoState Comparison
Comparing the CoSMoS1s and 2s rainfall generation poses intriguing discussion points. Reproducing the linear ACS of the intermittent process does not explicitly reproduce the ACS's of the binary and continuous processes. This affects the length of wet/dry spells and the correlations within wet spells. The theoretical analysis shows that an intermittent ACS can result from different binary and continuous processes combinations. Thus, reproducing the intermittent ACS does not imply matching the ACS's of the binary and continuous processes. The level of matching, or, when and if reproducing the intermittent ACS is sufficient, has not been studied. The author speculates that for larger time scales (e.g., daily) it provides decent results; however, for fine scales (e.g., hourly, or subhourly) the correlations of wet/dry spells and those within wet spells can be accurately described only by individual ACS's and thus the CoSMoS2s performs better.
An intriguing question is why reproducing explicitly the ACS's of the binary and continuous processes does not reproduce exactly the intermittent ACS. For several months there are clear deviations between the observed and simulated intermittent ACS's. The theoretical relationship (Equation 16) linking the intermittent ACS with the binary and continuous ACS's is always valid given that the binary and continuous processes are independent. Nature yet does not produce a binary and a continuous time series to directly test this hypothesis. We can assess, however, the crosscorrelation between binary and intermittent times series () in observations and simulations (Figure 12a). The monthly analysis in the four investigated stations (48 points in Figure 12a) shows that in most cases the observed is higher than the simulated (points above the diagonal). If this explains deviations in observed ( and simulated () intermittent ACS's then deviations will be larger for larger . To assess this argument we define an error measure as and compare it with the observed in each month and station (48 points in Figure 12b). These results indeed verify that larger error corresponds to larger . Additionally, we performed a Monte Carlo simulation of binary and continuous time series with different levels of crosscorrelation and estimated the error between the simulated intermittent ACS and the one predicted by Equation 16. As expected, there is no error when the time series are independent, but it gets larger as the crosscorrelation increases (Figure 12c).
In theory, we could modify the model and generate crosscorrelated binary and continuous processes to reproduce exactly the intermittent ACS too; for example, Equation 16 probably could be generalized for correlated processes; Goodman (1960) offers a complex variance expression for the correlated case. However, this will complicate the model's operational use (largescale application, implementation speed, extension to multisite simulation, etc.) with no clear benefits. CoSMoS2s reproduces the structure of wet/dry spells, and the probability dry, distribution of nonzero values and behavior of extremes at a large range to time scales and it is not clear what aspects will be further improved if the intermittent ACS is additionally preserved. This point deserves further investigation as this mismatch could be an artifact due to nonstationarities in storm's behavior (e.g., lower values in the beginning and end of a storm), remaining seasonality within each month, or other causes. In any case, there is always a trade off in building models to achieve operational functionality.
9.2 Reproducing Characteristics at Multiple Scales
CoSMoS2s is more accurate than CoSMoS1s (and more complex as it uses an additional parametric ACS) since it is calibrated to reproduce the ACS's of the binary and continuous processes. This improves its performance in reproducing the wet/dry spells and the correlations within wet spells; in turn, this leads to mimicking the process more precisely at multiple scales (see Figure 10).
However, an appealing question is whether reproducing any form of ACS (linear or nonlinear) can lead to exact multiscale representation. Reproducing the ACS acts as proxy in reproducing the joint distribution of the process. A valid postulate would be that precise reproduction of the joint distribution at a single scale would lead to precise reproduction of statistical properties at all larger scales. Yet the level that the true joint distribution of rainfall can be approximated by reproducing ACS's and marginal distributions needs further investigation. For example, the binary process in CoSMoS2s clearly contributes to improving the simulation at multiple scales. Theoretically, a binary process can be fully characterized by a transition matrix that defines the occurrence probability of any binary sequence. This is ideal but not parsimonious; a transition matrix estimated from data can have thousands of parameters. Whether transition matrices emerging from copulabased binary processes, as generated here, match observed transition matrices is unexplored and this will be the topic of a future communication.
The popular technique for reproducing characteristics at multiple scales was based on disaggregation schemes. Yet such schemes, are not always easily applicable and still have limitations. Ideally, competent simulation at all scales should be achieved by simulating at a single scale which entails capturing the “soul” of the process. We can exploit yet the improved performance of CoSMoS2s and use it as a disaggregation kernel in the DiPMaC scheme (Papalexiou, Markonis, et al., 2018) to further enhance its performance. DiPMaC is constrained to reproduce the intermittent ACS at a fine scale and match totals at a larger scale. Yet DiPMaC is not constrained in reproducing wet/dry spells and thus its performance could be markedly improved by coupling it with CoSMoS2s.
9.3 One, Two, Three… MultiState Processes
The twostate approach can be generalized to simulate processes conceptualized as having multiple states, where each state is expressed by a specific occurrence probability. For example, a binary process can model two states (here dry as 0 and wet as 1); for rainfall the drystate value coincides with the actual rainfall value. However, generalizing, we can use a process with discrete marginal to describe, for example, the probabilities , , , of three states indicated by arbitrary values (e.g., 1, 2, 3). Then each state can be simulated by another process (e.g., with continuous marginals). Technically, an state process, generalizing the binary simulation framework (Section 6), can be simulated by transforming a parent Gaussian (or nonGaussian) process. For example, a threestate process can be formed by mapping Gaussian values as where is the score corresponding to the probability indicated by the subscript. This approach could improve performance, for example, of the CoSMoS1s suggested in (Papalexiou & Serinaldi, 2020) for processes having a mixedtype marginal with probability masses at two points, that is, at a minimum and a maximum. Such variables appear frequently in nature and include cloud cover, water depth in natural reservoirs, etc.
9.4 Advancing Algorithmic Implementation
A model with more parameters, in general, reproduces explicitly more features. Yet more parameters increase complexity and jeopardize parsimony. Modeling explicitly the binary and continuous processes adds two parameters (if a twoparameter binary ACS is used). However, in practice, this approach facilitates simulation since analytical approximations bypass the need for numerical estimation of correlation transformation integrals. The characteristics of the binary process are now assessed by analytical equations (Section 6). Similar approximations can be formed for the continuous process. The correlation decrease caused to Gaussian variables when transformed to follow desired marginals (see Section 4) depends only on the shape parameters of the target distribution (location and scale parameters have no effect since their transformation effect is linear). Thus, if the target marginal has only one shape parameter , we can create parametric (or nonparametric) interpolation functions and (following Section 6). Such functions readily provide the CTF (Equation 12) parameters and for any value of the target distribution's shape parameter . If the target distribution has two shape parameters and , it is still technically feasible to form bivariate functions (surfaces) and to estimate the CTF parameters. For example, the parameters and are estimated in a grid of ) points and bivariate interpolation is used to assess and for any desired . Clearly, this procedure must be repeated for different distributions, yet it is applicable for any distribution, and a “library” for popular distributions can be created. This approach can also be applied for the CoSMoS1s model; however, the probability zero is an additional “shape” parameter affecting the correlation decrease. Thus, it is laborious to form functions and in a threedimensional grid comprising a huge number of points. In fact, this technique was applied a few years ago to create interpolation functions and to simulate intermittent rainfall with oneshapeparameter marginals (see the CoSMoS.xlsx file in Supporting Information S2).
9.5 Extending to SpaceTime Simulation
A main quest would be to extend the CoSMoS2s and potentially its rank and copulabased variants for spacetime rainfall modeling. Advances in generating intermittent static (frozen) random fields (RF's; Papalexiou & Serinaldi, 2020), and the introduction of locally varying velocity and anisotropy capabilities (Papalexiou, Serinaldi, & Porcu, 2021), could potentially be enhanced. Formulating a twostate approach for spacetime modeling would allow one to explicitly simulate the spatiotemporal (linear or rank) correlation structures (STCS) of the binary and continuous processes. This may lead to improved modeling of wet/dry regions, and better representation of storm cells; this would be manifested also in more accurate spatiotemporal scaling descriptions of the process. However, coupling the binary and continuous spacetime processes is not trivial as in the univariate case. If the two processes are independent of each other, unrealistic space patterns are formed. A first exploration shows that storm cells are not smoothed out naturally, or else, the transition from wet to dry regions seems artificial. This could be tackled by imposing some form of dependence between the binary and continuous processes. Finally, another option for spacetime modeling could be a threestep approach that builds on: (a) a univariate binary process to model longterm clustering of wet/dry fields, (b) a spatiotemporal binary process to reproduce the shortterm spatiotemporal structure of wet/dry regions, and (c) a continuous spatiotemporal process to imitate spatiotemporal structure of wet regions or storm cells.
9.6 Further Explorations
The analysis indicates that rainfall properties are better reproduced at multiple scales by accurately simulating the wet/dry spells, or else the intermittency and the correlations of wet clusters. This was accomplished by reproducing explicitly the ACS's of the binary and continuous processes. Yet such a framework can include many variations. For example, more methods should be explored to generate intermittency. Such methods include: generating binary sequences using transition matrices; reproducing explicitly the distribution of wet and dry spells (potentially as a bivariate process with discrete marginals); or reproducing linear or nonlinear binary ACS's based on different copulas. The same holds for simulating the continuous process, for example, using not only Gaussian and tcopulas but others such as the Clayton, Gumbel, etc. These options and their combinations will be investigated in a followup study.
10 Conclusions
The Holy Grail in rainfall modeling is a consistent representation of rainfall at all scales. This implies that any characteristic in observed time series, at any scale, should be reproduced in the simulated ones. Yet a competent modeling strategy cannot aim in reproducing explicitly too many characteristics, but rather identify those basic ones, keep them as few as possible, and still adequately represent the process.
Modeling essential rainfall characteristics at a single scale could lead in simulating well rainfall at many scales. The author deems that these characteristics are: (a) the probability of dry, (b) the probability distribution describing nonzero values, and (c) the dependence structure. There are different ways to combine and reproduce such features, mainly because there are different dependence structures and methods to introduce intermittency. Thus, identifying the form of these components to design easily applicable and accurate models reproducing multiscale properties is not trivial. Here, we compare two conceptually different methods to generate rainfall (or intermittent processes in general). Both preserve the probability of dry and marginal distribution of nonzero values. The first (CoSMoS1s), introduced in Papalexiou (2018), treats rainfall as onestate process, and reproduces the intermittent linear ACS, that is, the autocorrelations derived from the complete observed time series (including zero and nonzero values). Technically, the intermittency is introduced in one step by transforming Gaussian time series using a mixedtype quantile. The second (CoSMoS2s), introduced here, considers rainfall as a twostate process. It couples a binary and a “hypothetical” continuous process reproducing explicitly their linear (or rank/copula based) ACS's. The intermittency, here, is modeled by the binary process.

A framework for building probability distributions is introduced and applied to form a new system of distribution suitable for rainfall. The Generalized Exponential distributions type 1–6 ( − ) are flexible, comprising a scale and two shape parameters to control the left and right tails; have analytical quantile expressions that facilitate fitting, and fast quantile transformations applicable in time series generation. In contrast to powertype distributions, distributions have always finite variance securing the existence of correlation and CTFs.

The use of CTFs expressed by simple parametric functions (Papalexiou, 2018) is extended to negative correlations. There is a lower limit of negative correlation a process with nonGaussian marginals can reach which is larger than −1 and can be easily estimated using the fitted CTF.

Simulating binary times series having any linear (or Spearman's) ACS is simplified with analytical approximations.

Theoretical analysis shows that the ACS of intermittent rainfall (including zero and nonzero values) can emerge using different combinations of binary and continuous processes. The gain in parsimony in CoSMoS1s seems to be balanced by accuracy loss in reproducing the wetdry spells.

The CoSMoS2s model is introduced which simulated accurately the wet/dry spells and the correlations within wet spells. This improved rainfall simulation at multiple scales.

A rankbased CoSMoS2s is proposed that reproduces the Spearman's rank correlations of the binary and continuous process. This variant is “analytical” and does not require numerical integrations.

A copulabased CoSMoS2s model is introduced that enables generation of intermittent times series with nonzero values having the dependence structure of any desired copula.

Many extensions of the twostate approach are suggested and can spark further research. Such extensions include multistate processes, variations in simulating intermittency and wet clusters, as well as spacetime generalizations.
No doubt, the list of available rainfall models keeps piling up—none is perfect. This study attempts to advance rainfall modeling by building an accurate and easily applicable model at a single scale which reproduces rainfall properties at multiple scales. Yet the endeavor for the “ultimate” model remains.
Acknowledgments
I am grateful to Geoff Pegram, the two anonymous Reviewers, and the Associate Editor for their constructive remarks that helped to improve the original manuscript. I also thank Sofia Nerantzaki for spotting early references of rainfall models, and Francesco Serinaldi and Giuseppe Mascaro for discussing with me several points during the revision. This work was supported by the project “Investigation of Terrestrial HydrologicAl Cycle” (ITHACA) funded by the Czech Science Foundation (Grant: 2233266M). The support of the Natural Sciences and Engineering Research Council of Canada is also acknowledged (NSERC Discovery Grant: RGPIN201906894). The CoSMoS R package (Papalexiou, Serinaldi, Strnad, et al., 2021), originally resealed in April 2019, is available at CRAN (R Core Team, 2021); see also https://cran.rproject.org/web/packages/CoSMoS/vignettes/vignette.html.
Conflict of Interest
The author declares no conflicts of interest relevant to this study.
Appendix A: Notation and Additional Equations
1 Notation
Probability distributions are abbreviated by using script letters followed by the parameters within parentheses. The parameters are omitted for brevity when appropriate. Distributions mentioned in this study include: the Exponential, ; Weibull, ; Pareto II, ; Burr type III, ; Burr type XII, ; Generalized Standard Gompertz, ; Generalized Gamma, ; and the Generalized Exponential type 1–6, to . In all expressions is a positive scale parameter and a shape parameter. When more than one shape parameters exist, they are subscripted, for example, and denote positive shape parameters controlling the left and right tail, respectively.
2 Additional Equations
Open Research
Data Availability Statement
The author used fourhourly rainfall records (database codes: 366889, 310301, 097847, 237976) from the data set DSI3240 archived at the National Climatic Data Center (NCDC) and found at https://doi.org/10.5065/YP2DXA17.