A realistic description of land surface/atmosphere interactions in climate and hydrologic studies requires the specification of the rainfall forcing at aggregation scales of 1 hour or less. This is in contrast with the wide availability of daily rainfall observations and with the typically coarse output resolution of climate and numerical weather forecast models. Several methods have been devised to generate hourly or subhourly data from daily or monthly values, which usually rely on statistical regressions determined under the current climate conditions. Here we present a new method for downscaling rainfall in time using theoretically based estimates of rainfall variability at the hourly scale from daily statistics. The method is validated on a wide data set representative of different rainfall regimes and produces approximately unbiased estimates of rainfall variance at the hourly scale when a power law–tailed autocorrelation is assumed for the rainfall process. We further demonstrate how the downscaling method together with a Bartlett-Lewis rainfall stochastic model may be used to generate hourly rainfall sequences that reproduce the observed small-scale variability uniquely from daily statistics. Conclusions of a somewhat general nature are also drawn on the capability of finite memory stochastic models to reproduce the observed rainfall variability at different aggregation scales.
 Hydrologic analyses and land-atmosphere interaction studies in general require the specification of the rainfall forcing at time scales of the order of 1 hour or less [e.g., Berndtsson and Niemczynowicz, 1988; Krajewski et al., 1991; Berne et al., 2004; Rinaldo et al., 2006a, 2006b]. In fact, on one hand the resolution must obviously be a suitably small fraction of the characteristic concentration time of the basin and, on the other, the use of coarse (e.g., daily) rainfall observations averages out short and intense rainfall events, thus resulting in the underestimation of runoff due to the nonlinearity of runoff-generation mechanisms [Philip, 1957; Eagleson, 1978; Marani et al., 1997]. These considerations have relevant implications for flood forecasting based on numerical weather prediction models, for climate change impact studies, and for the evaluation of floods by hydrological modeling using the vast amount of daily rainfall observations available worldwide.
 Current climate and numerical atmospheric circulation models generate predictions which rapidly become unreliable at small temporal and spatial scales [Giorgi and Mearns, 1991; Joubert and Hewitson, 1997; Randall et al., 2007]. A large body of quite recent work has thus been aimed at the development of techniques allowing the disaggregation of rainfall at hydrologically relevant scales.
 In climate change studies this is usually done through statistical models describing observed rainfall characteristics and possibly including their relation with other atmospheric variables (predictors) [Wilby et al., 1998; Prudhomme et al., 2002; Wilby and Harris, 2006]. The statistical models calibrated on the present climate are then used to obtain downscaled rainfall from global or regional climate models outputs under the climate change scenario of interest. Such methods have only been applied to obtain daily rainfall from longer-timescale estimates (e.g., monthly) and postulate that the statistical relations between rainfall and the atmospheric predictors used for downscaling (e.g., large-scale circulation indices) are not affected by the climate change.
 In all cases, whether we are interested in rainfall disaggregation or downscaling, in a climate change or flood frequency analysis context, the crucial step is to assign the small-scale statistical properties of rainfall. In some studies, only the large-scale characteristics are assigned in the calibration of the stochastic rainfall model used, while the small-scale properties are dictated by the specific model structure adopted and do not necessarily reflect actual small-scale rainfall characteristics. In other cases, small-scale statistics may be assigned on the basis of generic information about the climatic characteristics of the area. In climate impact studies the small-scale properties of rainfall may be inferred from relationships established from current climate simulations [e.g., Wilby and Harris, 2006] or derived by making assumptions on the values assumed by model parameters under changed conditions [Burlando and Rosso, 1991].
 It has been noted that, in downscaling and in disaggregation procedures, it is most important to preserve rainfall variability at different timescales, both to address impacts of climate change [Katz, 1996] and to correctly describe hydrologic fluxes at the soil/atmosphere interface [Marani et al., 1997]. Here we focus our attention on rainfall variability as expressed by the temporal variance, σ2(T), as a function the aggregation interval (T). We base our approach on previous analyses of σ2(T), providing theoretically based, analytical expressions in agreement with observations [Marani, 2003, 2005]. Given observations at scale T1, we use such expressions to derive estimates of the values of σ2(T0) for T0 < T1. The method has the distinctive advantage of not requiring ‘external’ information (e.g., from the local climatology) or assumptions on the relationships between small- and large-scale statistics.
2. Downscaling Method
Vanmarke  shows that, given a stationary stochastic process i(t) (here the instantaneous rainfall intensity) and its aggregation over a time interval of duration T, h(ti, T) = (cumulative rainfall depth), the variance of h(ti, T) may be expressed as
where ρi(τ) is the autocorrelation function of i(t). Marani [2003, 2005] examines the theoretical properties of (1) and, among other things, determines the limits of possibly scaling regimes [Mandelbrot and Wallis, 1968], which are theoretically found not to extend indefinitely to smaller and smaller aggregation timescales. On the contrary the scaling regime, if existent, is found to be valid only asymptotically for large values of T (greater than 20–80 hours). An inner regime is identified for small T's (smaller than 10–15 min), where σ2(T) ∝ T2, together with a transition regime linking the inner and the asymptotic regimes. This theoretical picture is confirmed by analyses of a large number of high-resolution rainfall observations from a wide variety of climates [Marani, 2005]. Here we use these previous theoretical and observational analyses to construct a method to estimate the variance at the hourly scale from daily observations. Following Marani , closed-form expressions for σ2(T) under different assumptions can be derived. A common autocorrelation model for rainfall has an exponential form of the type ρi(τ) = exp(τ/I), where I is the integral scale of the process. This process is by definition characterized by a “finite memory” (finite integral scale), belongs to the “Brownian domain of attraction” [Mandelbrot and Wallis, 1968] and the variance of the aggregated process is obtained from (1) as
The fast decay of the autocorrelation function suggests that this model may be suitable to describe rainfall regimes in which events are temporally sparse and independent from one another, e.g., because of local and short-lived convective storms.
 Of an opposite nature are rainfall regimes in which events are clustered and exhibit significant statistical dependence, as in the case of large-scale perturbations producing series of related events. In this case the covariance model may exhibit a more persistent memory and the autocorrelation function may thus be characterized by a slowly decaying tail. Marani  shows that an autocorrelation model of this type, capturing the statistical characteristics of observed rainfall at many sites, is
where it should be α < 1 for the process to have an infinite memory. The following variance for the aggregated process may be derived from equation (1):
 A third simple assumption for the dependence of the variance from the aggregation scale is a power law form (echoing the definition of a scaling process [Mandelbrot and Wallis, 1968]):
 Previous work [Marani, 2005] extensively tests the forms (2), (4) and (5) on several high-resolution rainfall series, noting that (4) best interprets observations in most cases in the interval of aggregations between 15 min and 96 hours. We now use these analytical variance models to develop a method to estimate the hourly variance on the basis of observed daily values. In order to evaluate the accuracy of the estimation, we use hourly and subhourly rainfall observations from different climates (see Table 1), which are separately processed according to the month of collection so as to ensure stationarity. The resolution of the data is then reduced to 24 hours and the daily values thus obtained are further aggregated to obtain time series with resolutions of two, three and four days. For each month in the year we then evaluate the sample temporal variance for the 1- to 4-day resolutions and use them to fit equations (2), (4) and (5), by minimization of the root mean square error. The resulting fits are finally used to obtain estimates of the variance at the hourly scale. Such estimates can be validated by comparison with the sample variance obtained from the original hourly or subhourly data.
Table 1. Data Set Used in the Validation of the Downscaling Modela
Sources are the Natural Environmental Research Council (NERC, United Kingdom), the European Research Project AquaTerra (505428-GOCE), the National Climatic Data Center (NCDC, United States), and the Istituto Veneto di Scienze Lettere ed Arti (Italy, www.istitutoveneto.it). The rain gauges are all of the tipping bucket type.
Ashover (United Kingdom)
Aqua Terra project
Aqua Terra project
Lafayette (Louisiana, United States)
Lebanon (Indiana, United States)
Lyneham (United Kingdom)
Matilija (California, United States)
Saint Leo (Florida, United States)
 The comparison of observed and estimated values of the variance as a function of the aggregation interval (see Figure 1 for a representative subset of the months and locations explored) shows marked differences in the estimation accuracy obtained using the different forms of σ2(T) (2), (4) and (5). In particular, results are in line with the finding of Marani  that the assumption of a power law–tailed autocorrelation produces a closer agreement with observations than a fast decaying exponential autocovariance or a power law form of the variance σ2(T) (e.g., also compare with Smithers et al. [2002, Figures 2–3]). Equation (5) when fit to 1-, 2-, 3- and 4-day resolution data (circles in Figure 1) tends to consistently overestimate the hourly variance, whereas use of (2) mostly produces an underestimation of actual values. Equation (4) was fitted by first fixing the parameter ε to a small constant value (i.e., independent of the month and of the location). Experiments on the data set available showed that assuming ε ≤ 1 hour produced approximately constant and satisfactory results, suggesting that the exponential portion of the autocorrelation function in (4) needs to be confined to very short temporal scales in order to obtain physically consistent results. This procedure reduces the number of free parameters to two, as for equation (2) and (5), thus also allowing a fair comparison among the methods. Fit of α and σi to match observed 1-, 2-, 3- and 4-day resolution data produces remarkably accurate estimates of the hourly variance in most cases (see Figure 1 for sample results). Interesting exceptions (e.g., see Figure 1c, June) are constituted by rainfall at sites and during periods in which events are short and scarcely correlated in time, e.g., during summer months at Gibraltar or Matilija (both characterized by dry summers with short and intense showers). In these cases the exponential correlation assumption yields the best estimates, coherently with physical interpretation.
 These observations are confirmed by scatterplots of observed (σo2) and estimated (σe2) hourly variance values (Figure 2) and by inspection of the frequency distribution of the relative estimation error (e = (σo2 − σe2)/σ2o, Figure 3). Both comparisons are performed by pooling together estimates for all sites and all months and show how indeed an infinite memory, power law–tailed autocorrelation function best describes most rainfall series, producing approximately unbiased estimates. The power law–tailed autocorrelation assumption yields relatively less accurate estimates (see Figure 3b) only in the few mentioned cases of particularly short and uncorrelated rainfall events.
4. Downscaling Rainfall Using a Point-Process Model
 As mentioned in section 1, it is desirable for climate change impact studies and for hydrologic applications, to be able to generate rainfall time series at the hourly (or shorter) scale on the basis of observations at the daily (or longer) scale. One way to achieve this is to use a stochastic model of rainfall, e.g., calibrated on daily rainfall observations, and to generate synthetic rainfall series at the desired time resolution. This procedure relies on the assumption that not only the stochastic model reproduces the calibration statistics at the daily scale, but that it also incorporates just the right correlation structure in order for the observed rainfall variability to be preserved also at the small scale (as established by equation (1)).
 Here we explore whether a commonly used Bartlett-Lewis rainfall model based on clustered Poisson processes [Rodriguez-Iturbe et al., 1988] possesses the correct correlation structure to infer hourly scale rainfall properties upon calibration at the daily scale. The Bartlett-Lewis model represents the arrival of rainfall events and of rain cells within events through a marked Poisson process. The arrival of storm events is described through a Poisson process with rate λ, while the duration of storms is assumed to be exponentially distributed with mean γ. Rain cells are then generated within each storm through a second Poisson process with parameter β. Cells are characterized by exponentially distributed rainfall intensity (with mean μ) and duration (with mean η). In order to incorporate the possibility of widely varying cell durations, the mean storm duration is in turn assumed to be a gamma-distributed random variable with parameters ν and α. Rodriguez-Iturbe et al.  find the analytical expressions of the first and second order moments of rainfall depth for different aggregation scales as a function of the six model parameters (λ, γ, β, μ, ν, and α).
 We used such expressions to calibrate the model for each month and observational site by assigning the observed values of the mean, the variance, the dry fraction and the autocovariance with lag = 1 day for the daily aggregation, and the variance and the dry fraction for the twice-daily aggregation. This reference calibration set will in the following be called SET1.
 The generation of hourly synthetic values based on the SET1 calibration fails to reproduce the observed variance at small scales in most cases (e.g., see Figure 4 for sample representative results). This may be explained by noting that models based on Poisson processes by construction do not exhibit a power law–tailed autocorrelation function. They are thus unable to reproduce observed statistics, which were seen to be coherent with an infinite memory assumption. It is worth noting that any stochastic model of rainfall exhibiting a finite memory (e.g., Neyman-Scott or models based on Markov chains), independently from its detailed structure, is expected to similarly misrepresent observed rainfall variability over different scales of aggregation. Such limitations are thus expected to affect a large class of stochastic rainfall models widely in use.
 However, in most applications, one is interested in preserving the actual rainfall variability over a well-defined interval of aggregation scales, rather than over all possible timescales. In this case it may still be possible, by appropriate calibration, to reproduce the shape of the observational σ2(T) within the range of scales of interest even with a finite memory model. In order to do so in a downscaling framework one needs an estimate of the small-scale variance to be added to the set of parameters used in the stochastic model calibration.
 To explore this possibility we have used the downscaling procedure described above, using the power law–tailed autocorrelation assumption (4) to estimate the value of σ2(T = 1 hour) from 1-, 2-, 3- and 4-day resolution data, and added the hourly variance to the list of calibration parameters of SET1. This new set of parameters for stochastic downscaling is here called SET2. We then generated long series of hourly rainfall using the SET2-calibrated stochastic model and constructed the σ2(T) curve from the synthetic data. Comparisons with observations (Figure 4) show that indeed the stochastic model can now correctly reproduce the actual variance at all scales between 1 hour and 4 days. This is true for all months and sites for which the downscaling procedure produced accurate estimates of the hourly variance. In the few remaining cases (e.g., the summer months for Matilija and Gibraltar) the resulting synthetic statistics were not as close to observed ones, but were still overall satisfactory.
 It is important to evaluate the impact of the use of the downscaled variance on the probability distribution of rainfall values, rather than just on the variance itself. An important test regards the evaluation of the probability distribution of hourly rainfall depth, which is of greatest importance in hydrologic applications. We thus compared (Figure 5) the cumulated probability of exceedance of observed hourly rainfall and of synthetic hourly rainfall generated from calibrations using SET1 (without hourly variance) and SET2 (with downscaled hourly variance) parameter sets. In order to characterize the benefit of including the downscaled hourly variance in the calibration parameter set, one needs to account for possible limitations of the stochastic rainfall model in reproducing the observed rainfall distribution. To this end, we also calibrated the stochastic rainfall model using the observed hourly variance (and identified this case as SET3): The evaluation of SET3 results allows to verify whether a good agreement with observations is found when the value of the hourly variance is exactly known.
 We find that the use of the downscaled hourly variance produces probability distributions which are much closer to observed ones than those obtained when just the daily variance is included (see Figure 5 for sample representative results). Furthermore, the results obtained from the SET2 case are very similar to those obtained when the actual hourly variance is known (SET3), further supporting the effectiveness of the downscaling procedure developed.
5. Summary and Conclusions
 We have introduced a new, theoretically based method for producing approximately unbiased estimates of the hourly variance from daily observations, which assumes the autocorrelation function of the continuous rainfall process to be power law–tailed.
 The agreement between the downscaled and the observed values of the variance for a wide set of rainfall regimes supports the notion that the variance of rainfall time series does not exhibit scaling over the range of scales of practical interest.
 The fact that a power law–tailed autocovariance structure best interprets the observed variability of rainfall over a wide range of scales indicates that stochastic models of rainfall exhibiting a finite memory (i.e., many of the models currently widely used, including nonclustered Poisson models, the Bartlett-Lewis model, the Neyman-Scott model and models based on Markov chains), cannot in general reproduce actual rainfall statistics over large ranges of aggregation scales.
 On the contrary, we showed that the new variance downscaling procedure described here allows the introduction of small-scale information into a finite memory stochastic model (otherwise calibrated on daily statistics) and to reproduce the observed rainfall variance over a wide range of scales of typical interest (1 hour to several days). Tests of the downscaling procedure included the reproduction of the observed hourly variance as well as of the overall hourly rainfall probability distribution and the resulting level of performance is comparable to what is obtained when the actual value of the hourly variance is known.
 The effectiveness of the downscaling procedure developed relies on the selection of the correct nature of the autocorrelation (finite or infinite memory). In almost all cases the infinite memory assumption best reproduced observations. The few cases in which the exponential correlation assumption yielded the most realistic description of rainfall variability corresponded to rainfall regimes characterized by isolated and uncorrelated events e.g., typical of summer convective showers in a maritime climate. Further developments of the model might use external climatic information (e.g., linked to typical storm and interstorm durations from climatology) to identify the cases in which a finite memory assumption is more appropriate and may improve the downscaling estimates.
 This research was funded by the Aquaterra EU integrated project (contract 505428, GOCE). The authors thank NERC (UK) and NOAA-NCDC (USA) for making the data available.