A comparison of stochastic models for spatial rainfall downscaling

Abstract

[1] We explore the performance of three types of stochastic models used for spatial rainfall downscaling and assess their ability to reproduce the statistics of precipitation fields observed during the GATE radar experiment. We consider a bounded multifractal cascade, an autoregressive linear process passed through a nonlinear static filter (sometimes called a meta‐Gaussian model), and a model based on the presence of individual rainfall cells with power law profile. As test statistics we use the low‐order moments of the amplitude distribution, the distribution of generalized fractal dimensions, the generalized scaling exponents, the slope of the power spectrum, and the properties of the spatial autocorrelation. The results of the analysis indicate that all models provide, on average, a satisfactory representation of the statistical properties of the GATE rainfall fields (including the anomalous scaling behavior), with a slightly better performance of the model based on individual rainfall cells. All models, however, display large scatter in the field‐to‐field comparison with the data. These results indicate that data analysis alone does not allow, at the moment, for preferring one downscaling approach over another.

1. Introduction

[2] Forecasting sudden floods in small mountain catchments and urban areas requires knowledge of the precipitation field down to scales of a few square kilometers and tens of minutes [Castelli, 1995; Ferraris et al., 2002]. Current operational practice relies heavily on the use of limited‐area meteorological models (LAMs) that provide precipitation forecasts on scales of about 100 km² and a few hours. A gap thus exists between the scales resolved by limited‐area meteorological models and the scales required for hydrological applications. (The meteorological models mentioned here are numerical limited‐area models based on parameterizations of turbulent convection and of cloud microphysics, that are routinely employed for regional meteorological forecasts and are usually nested into the output of General Circulation Models. Cloud‐resolving models, on the other hand, deal explicitly with turbulence and microphysics and they can reach much smaller scales, but cannot run over large domains due to computational constraints.)

[3] One option to fill the scale gap and to obtain small‐scale rainfall estimates is based on the use of stochastic models for rainfall downscaling. A typical downscaling procedure is based on the implementation of a stochastic disaggregation algorithm that is capable of generating a small‐scale fluctuating field from a smoother rainfall distribution on larger scales. In principle, this approach provides random precipitation fields that should simultaneously satisfy the large‐scale constraints imposed by meteorological forecasts (e.g., the expected average rainfall intensity) and are consistent with the known statistical properties of the small‐scale rainfall distribution. For example, the power spectrum of a precipitation field produced by a downscaling procedure should smoothly merge, at low wave numbers, with that predicted by meteorological models on larger scales, and it should reproduce, at high wave numbers, the power spectra of rainfall fields measured under similar circumstances.

[4] Note, also, that a rainfall field produced by a downscaling procedure should never be taken as “the” rainfall distribution that is to be expected but rather as one possible realization of it. Repeated application of the downscaling procedure naturally leads to an ensemble of possible realizations of the small‐scale rainfall field and to the concept of ensemble rainfall prediction [Ferraris et al., 2002]. Clearly, stochastic disaggregation is not a substitute for a physically based dynamical model and for a better understanding of rainfall dynamics, but it can be seen as a way to resolve variability at scales smaller than those currently resolved by physical models.

[5] In past years, several stochastic models for rainfall downscaling have been proposed. To our knowledge, disaggregation models can be grouped into three main categories: (1) point processes based on the random positioning of a given number of rainfall cells [Waymire et al., 1984; Rodriguez‐Iturbe et al., 1986; Eagleson et al., 1987; Northrop, 1998; Wheater et al., 2000; Willems, 2001], (2) simple autoregressive processes [Mejia and Rodriguez‐Iturbe, 1974; Bell, 1987; Guillot and Lebel, 1999], sometimes called “meta‐Gaussian” models, and (3) fractal cascades [Lovejoy and Mandelbrot, 1985; Schertzer and Lovejoy, 1987; Gupta and Waymire, 1993; Over and Gupta, 1996; Perica and Foufoula‐Georgiou, 1996; Menabde et al., 1997a, 1997b, 1999; Deidda et al., 1999]. Mixed models, combining some of the above approaches, have also been proposed [Veneziano et al., 1996; Veneziano and Iacobellis, 2002]. A relevant question then concerns the choice of the “best” stochastic model to simulate a given rainfall event. What type of model is more appropriate, and is there a way of choosing an optimal downscaling model? Is there any model that is capable of reproducing all the statistical properties we are interested in, or, by contrast, that fails in reproducing some of the basic properties of precipitation fields? In particular, what models are consistent with the observed anomalous scaling behavior of rainfall fields?

[6] The present paper is devoted to an exploration of some of the above issues. For simplicity, here we focus on the spatial properties of rainfall fields, and defer the study of space‐time rainfall downscaling to future work. Also, we do not attempt to build specific procedures to downscale the large‐scale information provided by meteorological models but we simply check whether the available disaggregation models are capable of reproducing the observed properties of precipitation fields. A different issue, that is left for future work, is to find what is the stochastic model that provides the best downscaling of the predictions of limited‐area meteorological models.

[7] The rest of this paper proceeds as follows. In section 2 we introduce the three main types of disaggregation models. In section 3 we introduce the different test statistics and the corresponding analysis tools. These are used to determine the statistical properties of an ensemble of large‐intensity precipitation fields measured during the GATE experiment (Global Atmospheric Research Program Atlantic Tropical Experiment). In section 4 we study the statistical properties of the fields generated by the disaggregation models and compare them with those of the GATE data. In section 5 we give a summary and conclusions.

2. Downscaling Models

[8] We first briefly review the three classes of spatial downscaling models that are available in the literature. As mentioned in the Introduction, disaggregation models can be grouped into three main categories: (1) (multi)fractal cascades, (2) nonlinearly filtered autoregressive processes, and (3) point processes based on the random positioning of a given number of rainfall cells. All these models are characterized by an extremely fast numerical implementation and by a low number of free parameters. In each model, a lower amplitude threshold is also introduced. The threshold fixes the value below which the field produced by the model is set to zero. The introduction of a threshold is motivated by the fact that all the downscaling models considered here generate field values that are always different from zero, while rainfall fields are characterized by large areas of null values. (A cascading model that directly produces null field values is the so‐called β‐model of turbulence [Frisch, 1995]. In this model, each step of the cascade is characterized by a nonnull probability of generating null field values. The original version of the β‐model, however, cannot generate anomalous scaling. Multifractal extensions of the β‐model lead to cascades that have at least two parameters controlling the anomalous scaling behavior and one parameter controlling the percentage of null values.)

2.1. Multifractal Cascades

[9] Cascade processes were introduced in the seventies by Kahane [1974], Peyriere [1974], Mandelbrot [1974], and Kahane and Peyriere [1976] and have been widely used to reproduce the variability of precipitation fields [Schertzer and Lovejoy, 1987; Gupta and Waymire, 1993; Marsan et al., 1996; Perica and Foufoula‐Georgiou, 1996; Menabde et al., 1997a, 1997b, 1999; Deidda et al., 1999].

[10] Standard cascade processes are known to fail to reproduce observed spectral slopes and structure functions [Davis et al., 1994; Menabde et al., 1997b]. For this reason, generalizations of fractal cascades that could reproduce these values have been developed. Here we employ the bounded random cascade developed by [Menabde et al., 1997b, 1999], to whom we refer for details.

[11] The procedure adopted here is a generalization of the multifractal α model [Schertzer and Lovejoy, 1987; Provenzale et al., 1992]. The construction starts from the total rainfall volume, π₀, defined in a square domain of linear size L₀. At the first step, the spatial domain is divided into four daughter cells of size L₁ = L₀/2. These cells are assigned rainfall volumes π₁ = π₀W₁¹, π₂ = π₀W₂¹, π₃ = π₀W₃¹ and π₄ = π₀W₄¹. The weights W_k¹ are realizations of a random variable W to be defined below. At the following step, each first‐generation cell with size L₁ is further divided into four second‐generation cells with size L₂ = L₁/2, and the rainfall volume is redistributed among them.

[12] At the Jth step, the field is composed by 4^J cells with size L₀/2^J. The rainfall volume of the ith cell is given by

where 1 ≤ j ≤ J and k(j) labels the individual cell involved in the jth iteration of the cascade sequence that leads to the ith cell.

[13] The weights W takes only two values, namely,

with probability p and

with probability 1 − p.

[14] In the above construction there are four parameters: w₁, w₂, h and p. The conservation of rainfall volume leads to a relationship between p, w₁ and w₂, and only two of these three parameters are independent. In the following, we fix p = 0.5 and we are left with only two independent “internal” model parameters, namely w₁ and h, plus the external threshold used to generate a percentage of zero values comparable with that of the measured fields.

[15] For a bounded cascade the value of W_k^j depends on the level j in the iteration procedure. Owing to the effect of h, the field becomes smoother as the cascade proceeds and, in the limit of large j, the power spectrum assumes a power law shape with logarithmic slope β = 1 + 2h. A field generated by a bounded cascade correctly reproduces the shape of rainfall power spectra, as pointed out by [Menabde et al., 1997b].

2.2. Autoregressive Processes

[16] The generation of multidimensional random fields by autoregressive stochastic processes has a long history in Hydrology [Mejia and Rodriguez‐Iturbe, 1974; Smith and Freeze, 1979; Bell, 1987; Guillot and Lebel, 1999]. The autoregressive model adopted here generates a two‐dimensional, isotropic, linearly correlated random field by Fourier anti‐transforming a spectrum with assigned amplitude distribution and random Fourier phases, and subsequently applying a nonlinear static transformation to the linearly correlated field (see also [Bouchaud et al., 2000; Toniolo et al., 2002]). To this end, we start with the radially symmetric, power law power spectrum

where A is a normalization constant (fixed for example by requiring that the field has the same integral of the data), k = (k₁² + k₂²)^1/2 is the radial wave number and n is the spectral slope. The Fourier spectrum is then obtained as

where ϕ are random uncorrelated phases, uniformly distributed in [0, 2π]. The spatial fluctuation field is obtained from

The field π_g has a Gaussian amplitude distribution and it is linearly correlated. The linear correlation is completely determined by the second‐order moment, which, in turn, is fixed by the form of the power spectrum.

[17] The following step is to apply a static nonlinear transformation to the field π_g. Here we use an exponential [Bell, 1987; Guillot and Lebel, 1999],

The field π has a non‐Gaussian amplitude distribution, whose shape is fixed by the specific nonlinear transformation adopted. Additionally, the nonlinear transformation (7) introduces phase correlations in the Fourier transform of the field π. Fields obtained with this procedure are sometimes called meta‐Gaussian [Guillot and Lebel, 1999].

[18] The nonlinearly filtered autoregressive process considered here has two parameters, namely, the spectral slope of the linear field, n, and the constant b that enters the nonlinear transformation. Note, also, that the nonlinear field π does not necessarily have a power law spectrum with logarithmic slope n, as spectra are not invariant under a nonlinear transformation such as equation (7) [Balmforth et al., 1999]. As it turns out, in this case the field π still has an approximate power law power spectrum with logarithmic slope n′, function of both n and b. Again, an external threshold is required to generate null field values.

2.3. Individual Precipitation Cells

[19] The spatial structure of intense rainfall fields is often characterized by strong convective cells within small mesoscale areas [Austin and Houze, 1972]. On the basis of this physical property of precipitation, in past years several point process models based on the presence of individual rain cells have been proposed [Waymire et al., 1984; Rodriguez‐Iturbe et al., 1986; Eagleson et al., 1987; Northrop, 1998; Wheater et al., 2000; Willems, 2001]. In these models, the rain process is modeled by simulating the band structure and the cell patterns, and the arrival times of the cells. Rainfall cells are usually given a Gaussian shape and are spatially distributed with a Neyman‐Scott process [see, e.g., Cowpertwait et al., 2002].

[20] In these models, the use of a Gaussian shape for the individual precipitation cells does not generate well‐defined scaling behavior of the rainfall fields. Here we introduce a simple extension of this construction and use a set of rainfall cells with a power law profile. The centers of the different cells are randomly, independently and uniformly distributed on the plane, and the power law holds up to a small distance from the center. A smooth regularizing core is introduced at the center of the cell. (With finite resolution, and a finite amount of precipitation in each rainfall cell, the size of the smooth core can be set equal to the size of the pixel.) The introduction of a power law profile allows for generating fluctuation fields with well‐defined scaling behavior; see, for example, Murante et al. [1997] for an application of this approach to a different problem.

[21] To build the rainfall fluctuation field, we sum the contributions of a given number of individual rain cells. In each cell, the precipitation intensity is distributed around the center of the cell as

where r = (x₁, x₂), r_j is the center of the jth cell, the symbol ∣ · ∣ indicates the modulus of the vector, and 1 < α < 2.

[22] Operationally, we define R_min ≤ ∣r − r_j∣ ≤ R_max, where R_min marks the transition from the power law profile for ∣r − r_j∣ ≥ R_min to the smooth core at small distance from the center. For ∣r − r_j∣ < R_min we assume constant precipitation. In the following, we shall always take R_min as small as possible, compatible with the chosen resolution. We consider a number density of N_s rain cells, that are randomly distributed in the two‐dimensional plane. The average distance between two cell centers is δ = N_s^−0.5, and, for simplicity, we assume R_max = δ. The total cumulated rainfall associated with a given cell is

where ρ = ∣r − r_j∣. The total rainfall for each cell, Π(R_max), is taken as a random value with either uniform or Gaussian distribution around a mean value. The mean value is fixed by the average rainfall intensity of the field and by chosen the number density of cells.

[23] The relevant model parameters, and the ones that will be varied in the analysis that follows, are the number density of cells, N_s, and the power law exponent α that controls the rainfall distribution in the individual cells. Again, an external threshold is required to obtain a percentage of zero precipitation values comparable with that of the data.

[24] Note that a recent investigation of the shape of rain cells in the GATE data have indicated that cells are definitely non‐Gaussian. In this data set, cells apparently display an approximate exponential profile, rather than a power law profile as assumed here [von Hardenberg et al., 2003]. Point process models based on the presence of exponential‐shaped cells do not lead to theoretical scaling behavior as power law cells do. However, we experimented with random distributions of cells with exponential shape and found that also in this case an approximate scaling behavior is present. The results obtained with the exponential cells are quite close to those obtained with power law cells, and definitely different from those obtained with Gaussian cells. For simplicity, here we show only the results provided by the use of point process models that use cells with power law profile; similar conclusions hold also for cells with exponential shape.

3. Statistical Properties of the GATE Data

[25] As a test case, we focus on a set of radar maps measured during the GATE experiment. As test statistics, we consider the low‐order moments of the amplitude distribution, the set of generalized box‐counting (multifractal) dimensions, the set of generalized scaling exponents obtained from the logarithmic slope of the structure functions, the logarithmic slope of the power spectrum, and the spatial autocorrelation. We opted for using the GATE data set because these fields have been extensively analyzed in the past and provide a well‐studied test case.

[26] The radar maps measured during the GATE experiment were obtained in 1974. One data set, measured from 28 June to 15 July, is composed of 1716 images and is referred to as GATE‐1. A second data set, composed by 1512 images, was measured from 28 July to 15 August and it is called GATE‐2. These data provide an ensemble of individual precipitation fields, p(x₁, x₂), defined on the two‐dimensional plane (x₁, x₂), and discretized on a regular grid. Each radar image is available with space resolution Δx = 4 km. The time interval between different images is Δt = 15 min. In the following, we use the GATE‐1 frames recorded on a grid with total size of 256 × 256 km (thus each image is digitized on a grid of 64 × 64 points).

[27] The set of GATE‐1 fields is characterized by large variability of the average rainfall intensity of the individual fields, 〈p〉, and by significant dependence of the statistical properties on average rainfall intensity [Over and Gupta, 1994; Ferraris et al., 2003]. For this reason, in the comparison with the output of the downscaling models we shall focus on a subset of 272 large‐intensity GATE‐1 fields characterized by average rainfall intensity 〈p〉 ≥ 1 mm/h. Figure 1 shows one example of a large‐intensity GATE field. The properties of these fields are especially interesting because high‐rainfall episodes are the most crucial to reproduce when applying downscaling procedures to operational flood forecast.

Figure 1

One example of the large‐intensity rainfall fields measured during the GATE radar experiment. The rainfall intensity is expressed in millimeters per hour. The average rainfall intensity of this field is 1.2 mm/h.

[28] Before proceeding with the analysis, we acknowledge the fact that radar maps may not be good estimators of true precipitation fields, and their statistical properties can differ from those of rainfall at the ground. This is a well‐known problem whose consideration is beyond the scope of this work. In the present study, we use the GATE radar maps simply to address the question of whether different types of downscaling models are able to reproduce a set of observed statistics.

3.1. Amplitude Distribution

[29] A basic statistical information is the distribution of rainfall intensity. Figure 2 shows the rainfall intensity distribution obtained from the subset of 272 GATE‐1 fields with average rainfall intensity 〈p〉 ≥ 1 mm/h. The intensity distribution has been obtained by averaging the probability density distributions obtained from the individual fields. The upper and lower curves bracket 95% of the values obtained from the 272 individual distributions. As Figure 2 shows, the rainfall intensity distribution has an approximate exponential tail at large intensities.

Figure 2

Rainfall intensity distribution obtained from the subset of 272 GATE‐1 fields with average rainfall intensity 〈p〉 ≥ 1 mm/h. Solid circles indicate the mean distribution obtained by averaging the amplitude distribution of the individual fields in the subset. The upper and lower curves bracket 95% of the values obtained from the individual distributions.

3.2. Generalized Fractal Dimensions

[30] The scaling properties of a random field can be quantified by a set of generalized fractal dimensions, that are estimated by the box‐counting procedure discussed below.

[31] Consider a precipitation field, p(x₁, x₂), discretized on a regular grid. First, we need to define a measure, ε(x₁, x₂), on this field. In the case of precipitation, the measure can be the field itself, i.e., ε = p. We define the integral of the measure, μ_j(λ), as the integral of ε on an area with linear size λ, centered on the point (x₁^j, x₂^j). For a discretized field, this simply corresponds to taking the sum of all values of ε in a square with linear size λ, where λ is a multiple of the grid spacing Δx. We define the partition functions, (λ, q), as

where q ≥ 0. For random fields with scaling properties, in the limit for λ → 0 one has

When dealing with measured data, however, the power law behavior extends only on a finite range of values of λ, and the exponent τ(q) is usually obtained by a least squares fit of log versus log λ in the scaling range. The generalized fractal dimensions, D(q), are then obtained from τ(q) as [Hentschel and Procaccia, 1983]

which is valid for q ≠ 1.

[32] For q = 0, one finds the “fractal,” or box‐counting, dimension D(0). This represents the fractal dimension of the support where the field is different from zero. For q = 2, one finds the correlation dimension, D(2). Fields characterized by D(q) = D(0) for any q are said to possess “simple scaling” or to be “monofractal,” while scaling fields characterized by D(q) < D(q′) for q > q′ are said to possess “anomalous scaling,” to possess “multiscaling” properties, or to be “multifractal.”

[33] By applying the box‐counting method to the individual GATE‐1 fields we obtain estimates of the partition functions (λ, q) for each field and, from these, the set of generalized dimensions. As an example, Figure 3a shows the partition functions for the field shown in Figure 1. An approximate scaling behavior is apparent, which allows for defining the set of generalized dimensions. The box‐counting analysis of the individual GATE‐1 fields in the selected subset indicates the presence of mild anomalous scaling, generally providing D(0) < 2 and a slight decrease of D(q) for increasing q. Figure 3b shows the set of generalized fractal dimensions obtained from an average over the subset of fields with 〈p〉 ≥ 1 mm/h. The logarithmic slopes of the partition functions have been estimated on the range of scales 4 km ≤ λ ≤ 128 km. The upper and lower curves bracket 95% of the values obtained from the analysis of the 272 individual fields. (For simplicity, we show the generalized dimensions only for integer values of the moment order q. The estimate of the dimensions for noninteger values of q is straightforward and the values of the dimensions smoothly interpolate the values obtained for integer q.)

Figure 3

(a) As an example, the generalized partition functions for the field shown in Figure 1. (b) Average generalized fractal dimensions, D(q), obtained from the subset of 272 GATE‐1 fields with 〈p〉 ≥ 1 mm/h, plotted versus the moment order q. The logarithmic slopes of the partition functions have been estimated on the range of scales 4 km ≤ λ ≤ 128 km. Solid circles indicate the mean generalized dimensions obtained by averaging the dimensions of the individual fields in the subset. The upper and lower curves bracket 95% of the values obtained from the 272 individual fields.

[34] Before closing this section, we note that while the number of independent data points used in the box‐counting analysis is large at small λ, it becomes rather small at the largest values of λ considered. For this reason, caution should be exercised when interpreting the results at the coarser resolutions.

3.3. Structure Functions

[35] Structure functions are often employed in the study of turbulence, to investigate the scaling properties of velocities and velocity derivatives [Frisch, 1995]. Here we define the structure functions as

where λ is a space increment (multiple of the grid spacing Δx) and q > 0 is the order of the moment. If the field possesses scaling behavior, then

at small values of λ. The quantities H_l(q), l = 1, 2, are called the scaling exponents. For an isotropic field, the structure functions along x₁ and x₂ are statistically equivalent; thus H₁(q) = H₂(q) = H(q) for an isotropic field. In the case of simple scaling, H(q) = H(q′) for any positive values of q and q′, while anomalous scaling is characterized by H(q) < H(q′) for q > q′. The structure function for q = 2 is simply related to the autocorrelation function and to the power spectrum [Frisch, 1995], and it provides information on the linear correlations present in the field under study.

[36] To obtain estimates of H₁(q) and H₂(q), we have computed the structure functions for the individual GATE‐1 fields and we have performed linear least squares fits of log versus log λ on the range 4 km ≤ λ ≤ 128 km. Figure 4a shows the mean scaling exponents obtained as an average over the values obtained for the 272 fields with intensity larger than 1 mm/h, for the two directions separately. The upper and lower curves bracket 95% of the values obtained from the individual fields in this subset. The generalized scaling exponents decrease for increasing moment order, indicating the presence of anomalous scaling. Because of the approximate isotropy of the fields, in the following we shall consider the average of H₁(q) and H₂(q).

Figure 4

(a) Average scaling exponents obtained as a mean over the subset of 272 fields with intensity larger than 1 mm/h, for the two directions separately. Solid circles refer to the x₁ direction, and open circles refer to the x₂ direction. The upper and lower curves bracket 95% of the values obtained from the individual fields in the subset. Solid lines refer to the x₁ direction, and dashed lines refer to the x₂ direction. (b) Average unidimensional spectra for the subset of 272 GATE‐1 fields with average intensity larger than 1 mm/h, for the two directions separately. Solid circles refer to the x₁ direction, and open circles refer to the x₂ direction. The upper and lower curves bracket 95% of the spectra of the individual fields in the selected subset. Solid lines refer to the x₁ direction, and dashed lines refer to the x₂ direction. (c) Average unidimensional spatial autocorrelation functions for the subset of 272 GATE‐1 fields with average intensity larger than 1 mm/h, for the two directions separately. Solid circles refer to the x₁ direction, and open circles refer to the x₂ direction. The upper and lower curves bracket 95% of the spectra of the individual fields in the selected subset. Solid lines refer to the x₁ direction, and dashed lines refer to the x₂ direction.

3.4. Power Spectra

[37] Estimates of power spectra provide a standard characterization of random fields. Here we consider the unidimensional spectra, P₁(k₁) and P₂(k₂), along the two spatial directions x₁ and x₂. Here k_l = m_l/L, l = 1, 2, is the inverse of the wavelength, L = 256 km and 1 ≤ m_l ≤ 32 is an integer. Figure 4b shows the two average unidimensional spectra for the subset of 272 GATE‐1 fields with average intensity larger than 1 mm/h. The upper and lower curves bracket 95% of the spectra of the individual fields in the subset.

[38] The average spectra of the 272 large‐intensity fields display a broad‐band appearance with an approximate power law shape at large scales. If we fit a power law to the low‐wave number portion of the spectra, , we obtain β₁ ≈ 1.16 and β₂ ≈ 1.24 for this subset of the GATE‐1 data. The spectral exponents have been computed by a linear least squares fit of log P versus log k for wavelengths larger than 12 km. The exponents obtained by fitting the spectra of the individual fields range between 0.82 and 1.63. Given the approximate isotropy of the spectra, in the following we shall consider the average of the two unidirectional spectra.

3.5. Spatial Autocorrelation

[39] The spatial autocorrelation provides the same information of the power spectrum, as well as of the second‐order structure function. However, it is useful to estimate the shape of the autocorrelation function for the GATE data, and use it to estimate the correlation length of the rainfall fields. In Figure 4c we show the two average one‐dimensional autocorrelation functions, C₁(Δ) = 〈p(x₁, x₂)p(x₁ + Δ, x₂)〉/σ₁² and C₂(Δ) = 〈p(x₁, x₂)p(x₁, x₂ + Δ)〉/σ₂², where σ₁² and σ₂² are the (one‐dimensional) field variances for the rainfall fields in the selected GATE‐1 subset. The functions have been obtained as an average over the 272 fields selected for the analysis. The upper and lower curves bracket 95% of the autocorrelation functions obtained from the individual fields. The spatial correlation lengths along x₁ and x₂ are theoretically defined as δ_i = ∫₀^∞C_i(Δ) dΔ. These, however, cannot be estimated directly from the data as the autocorrelations do not decay to zero rapidly enough (as illustrated by the fact that the power spectrum does not saturate to a constant at low wave numbers, see Figure 4b). An heuristic way to estimate the correlation length is to assume an exponential shape for the autocorrelation, C_i(Δ) ≈ exp(−Δ/δ_i), and estimate the value of δ by solving the transcendental equation

where L_c is the maximum lag for which the autocorrelation can be computed. Typically, L_c = L/2 where L = 256 km for the GATE data. With this method, we get the estimates 〈δ₁〉 ≈ 9.5 km and 〈δ₂〉 ≈ 13 km for the average correlation lengths along x₁ and x₂, for the subset of large‐intensity GATE fields with 〈p〉 > 1 mm/h. (A simpler way to estimate the correlation length is to take the value of the lag for which the autocorrelation function becomes 1/e. This method provides average values of the correlation length that are equivalent to those reported in the text but that are affected by larger scatter.) In the following analysis, we shall use the values of the correlation lengths averaged over the two spatial directions.

4. Comparison Between Models and Data

[40] To compare the output of the downscaling models with the GATE‐1 data we need to optimize the model parameters on a (possibly small) set of test statistics. This is done by minimizing the squared differences between the model statistics and the data, thus determining the “best fit” parameter values for each model. Since the numerical implementation of the downscaling models is very fast and there are only three free parameters for each model, we opted for a “brute force” approach where we run the downscaling procedures for a very large set of parameter values and select those values that provide the closest fit to the data.

[41] Note that the procedure adopted here is different from the standard downscaling approach. In an operational downscaling procedure, at least some of the model parameters are fixed by the output of the meteorological model, or by the type of synoptic conditions (e.g., stratiform vs. convective precipitation). In the present case, we do not attempt to relate the model parameters to the meteorological situation or rainfall characteristics, and we simply try to determine, for each GATE field, the model parameters that provide the best agreement with the test statistics. This approach allows for answering some of the questions listed in the introduction.

4.1. Average Generalized Dimensions

[42] Rainfall fields are characterized by a high degree of intermittency, which is, in turn, reflected into their approximate anomalous scaling behavior. Thus reproducing the anomalous scaling of rainfall fields is an important requirement that downscaling models should satisfy. In the following, we check whether the three classes of downscaling models discussed above are capable of reproducing the average anomalous scaling behavior of the GATE data. (The generalized dimensions for some of the fractal cascades and for the individual rainfall cells can be evaluated analytically [Frisch, 1995; Murante et al., 1997]. Here, however, we prefer to estimate the dimensions numerically, in order to have the same discretization effects and the same numerical procedure for the models and the data.) To this end, we minimize the squared differences between the generalized dimensions of the model outputs and the average generalized dimensions of the subset of GATE fields with intensity larger than 1 mm/h (shown in Figure 3b). For each model and for each set of parameter values, we generate 100 independent realizations of the downscaling model, compute the generalized dimensions for each realization, and average the dimensions over the set of realizations. By varying the model parameters, we minimize the quantity

where D_model(q) are the average generalized dimensions of the model for a given set of parameters, D_GATE(q) are the average generalized dimensions of the large‐intensity GATE data, and 0 ≤ q ≤ 8. Figures 5a–5c show the generalized dimensions of the optimal models together with the generalized dimensions of the large‐intensity GATE fields. The upper and lower curves bracket 95% of the values obtained from the individual fields in the selected GATE subset. Table 1 reports the parameter values used in the three models.

Figure 5

Average generalized dimensions of the optimized models (open circles) together with the average generalized dimensions of the large‐intensity GATE‐1 fields (solid circles). The upper and lower curves bracket 95% of the generalized dimension estimates of the individual fields in the GATE‐1 subset. Results (a) for the multifractal cascade, (b) for the nonlinearly filtered autoregressive process, and (c) for the model based on individual precipitation cells.

Table 1. Values of the Model Parameters Used to Reproduce the Set of Average Generalized Dimensions of the Large‐Intensity GATE Fields

Model	First Parameter	Second Parameter	Intensity Threshold, mm/h
Multifractal cascade	w1 = 0.3	h = 0.2	1.5
Autoregressive process	n = −1.1	b = 1	1.1
Individual rain cells	Ns = 24	α = 1.2	1.5

[43] Figure 5 shows that all the downscaling models are able to reproduce the average generalized dimension estimates obtained from the analysis of the large‐intensity GATE fields. This shows that multifractal cascades are not the sole option to generate anomalous scaling behavior: also the nonlinearly filtered autoregressive process and the model based on the presence of individual precipitation cells with non‐Gaussian profile display multifractal behavior.

4.2. Comparison of Individual Fields

[44] As shown by Over and Gupta [1994] and Ferraris et al. [2003], the properties of the GATE data display significant dependence on the average rainfall intensity, as well as strong field‐to‐field variability. For this reason, looking into the behavior of the individual GATE fields provides a further check of the model performance.

[45] In the following, we compare model outputs and data for each individual GATE field with rainfall intensity larger than 1 mm/h. We run the downscaling models for several different parameter values and determine those that provide the best fit to each individual field. We compute the statistical properties of the model outputs, and compare them with the statistics of the data field on which the model has been optimized. For each model and for each statistical measure, this procedure provides 272 couples of values.

[46] In comparing the model output with the individual GATE fields it is important to optimize the models on a small set of robust and physically meaningful statistical measures. In the following, we use the average rainfall intensity of each field to normalize the model output; this guarantees that the model and the data have the same total rainfall volume. We then minimize the squared differences between the model and the data on three basic statistical measures: the fractal dimension D(0), the variance σ² and the kurtosis K. The choice of D(0) is motivated by the fact that the fractal dimension provides relevant information on the support of the rainfall field and characterizes its “intermittency” properties. The variance and the kurtosis, or equivalently the variance and the skewness, are basic low‐order statistical measures that need to be reproduced by the models. Here we have opted for optimizing the kurtosis; the results obtained by optimizing the skweness S instead of the kurtosis are completely equivalent. (One could have chosen another set of test statistics for the optimization procedure. For example, we could have chosen the value of D(2) or the spectral exponent β, which are related to the spatial correlation of the rainfall fields. However, the area of rainy regions and its scaling with varying resolution, and the low‐order moments of the amplitude distribution, σ², S and K, are basic quantities that one needs to get right in the downscaling procedure. Thus we opted for optimizing the models on these quantities and for checking the model performance on the field correlations.)

4.2.1. Moments of the Amplitude Distribution

[47] To compare the statistics of the data with those of the model outputs, we start by considering the four lower moments of the amplitude distribution of the individual fields. In addition to the average, 〈p〉, which is used to normalize the model output, we consider the variance σ², the skewness S, and the kurtosis K, defined as

Figures 6a–6c, 7a–7c, and 8a–8c show the variance, the skewness, and the kurtosis, respectively, of the model outputs versus those of the data on which the models have been optimized. Here and in the following figures, a perfect agreement between model and data would appear as a set of values lying on a line passing through the origin and with slope a = 1. Instead, some scatter is evident, for all the models.

Figure 6

Variance of the model output versus that of the data for each individual GATE‐1 field with intensity larger than 1 mm/h. The models have been optimized on the fractal dimension D(0), the variance σ² and the kurtosis K. (a) Multifractal cascade, (b) nonlinearly filtered filtered autoregressive process, and (c) model based on individual rainfall cells.

Figure 7

Skewness of the model output versus that of the data for each individual GATE‐1 field with intensity larger than 1 mm/h. The models have been optimized on D(0), σ² and K. (a) Multifractal cascade, (b) nonlinearly filtered autoregressive process, and (c) model based on individual rainfall cells.

Figure 8

Kurtosis of the model output versus that of the data for each individual GATE‐1 field with intensity larger than 1 mm/h. The models have been optimized on D(0), σ² and K. (a) Multifractal cascade, (b) nonlinearly filtered autoregressive process, and (c) model based on individual rainfall cells.

[48] To provide a quantitative comparison between the model and the data, for each statistical measure we compute: (1) The correlation coefficient, r², between the model values and the data. The value of r² provides information on the scatter of the model values. (2) The slope, a, of the regression line obtained by a linear least squares fit of the model values versus the data. The line has been constrained to pass through the origin. The closest to one is the slope of the line, the better is the agreement between the model and the data.

[49] In addition to these two quantities, we have estimated the uncertainty in the value of the regression slope by adopting a “jackknife” procedure [Kottegoda and Rosso, 1997]. In this approach, we select at random half of the 272 couples and compute the regression line from the selected couples only. We repeat this procedure a large number of times (here 1000) and obtain a distribution of values of the regression slopes. The spread of this distribution is a measure of the presence of outliers and, more generally, of the robustness of the regression estimates. Table 2 reports, for the three downscaling models, the correlation between the model and the data, the slope of the regression line, and the minimum and maximum regression slopes, a_min and a_max, obtained with the jackknife procedure. In general, all the models reproduce quite well the variance, the skewness and the kurtosis of the data: the correlation coefficients are larger than 0.8, the regression slopes are close to one, and the spread in the jackknife slopes is quite small.

Table 2. Values of the Correlation Coefficient r², Regression Slope a, and Minimum and Maximum Regression Slopes, a_min and a_max, Obtained by the Jackknife Procedure

Test Statistics	r^2	a	amin	amax
Multifractal Cascade Model
σ^2	0.93	0.93	0.84	1.00
S	0.94	0.87	0.81	0.93
K	0.97	0.99	0.88	1.06
D(0)	0.50	0.97	0.96	0.98
D(2)	0.57	1.10	1.07	1.12
D(4)	0.66	1.11	1.08	1.14
H(2)	0.36	0.83	0.73	0.90
H(4)	0.29	0.74	0.64	0.84
β	0.27	0.81	0.75	0.85
δ	0.44	0.88	0.79	0.99
Autoregressive Process Model
σ^2	0.86	1.13	0.93	1.38
S	0.80	0.92	0.86	1.00
K	0.98	1.03	0.93	1.08
D(0)	0.55	0.96	0.95	0.98
D(2)	0.71	0.98	0.96	1.00
D(4)	0.79	0.97	0.95	0.99
H(2)	0.30	1.30	1.21	1.40
H(4)	0.27	1.41	1.29	1.57
β	0.20	1.19	1.14	1.25
δ	0.49	1.16	1.07	1.28
Individual Rainfall Cells Model
σ^2	0.89	0.91	0.76	1.11
S	0.91	0.89	0.84	0.95
K	0.97	1.03	0.94	1.11
D(0)	0.16	0.95	0.93	0.96
D(2)	0.59	1.01	0.99	1.03
D(4)	0.70	1.01	0.98	1.03
H(2)	0.28	1.08	0.99	1.18
H(4)	0.23	0.92	0.82	1.02
β	0.13	1.16	1.08	1.21
δ	0.34	0.98	0.87	1.11

4.2.2. Generalized Fractal Dimensions

[50] Next, we consider the generalized dimensions. The models have been optimized on D(0), σ² and K, and we want to see whether they are able to reproduce, with the same parameter values, the set of generalized dimensions. Figures 9a–9c, 10a–10c and 11a–11c show the values of the generalized dimensions D(0), D(2) and D(4) of the model outputs versus those of the data. Table 2 reports the correlations coefficient, the slope of the regression line, and the minimum and maximum jackknife regression slopes. The autoregressive process and the model based on the presence of individual rain cells furnish values of D(0) that are less variable that those of the data, and tend to slightly underestimate D(0) for the fields with large D(0). In addition, the correlation coefficients between all the models and the data are much lower for D(0) than for the moments of the rainfall intensity distribution and for the higher‐order generalized dimensions. Conversely, the slopes of the regression lines remain close to one for all the models.

Figure 9

Fractal dimension, D(0), of the model output versus that of the data for each individual GATE‐1 field with intensity larger than 1 mm/h. The models have been optimized on D(0), σ² and K. (a) Multifractal cascade, (b) nonlinearly filtered autoregressive process, and (c) model based on individual rainfall cells.

Figure 10

Correlation dimension, D(2), of the model output versus that of the data for each individual GATE‐1 field with intensity larger than 1 mm/h. The models have been optimized on D(0), σ² and K. (a) Multifractal cascade, (b) nonlinearly filtered autoregressive process, and (c) model based on individual rainfall cells.

Figure 11

Generalized fractal dimension D(4) of the model output versus that of the data for each individual GATE‐1 field with intensity larger than 1 mm/h. The models have been optimized on D(0), σ² and K. (a) Multifractal cascade, (b) nonlinearly filtered autoregressive process, and (c) model based on individual rainfall cells.

4.2.3. Scaling and Spectral Exponents, Correlation Length

[51] Figures 12a–12c and 13a–13c show the generalized scaling exponents H(2) and H(4) of the models versus those of the data, and Figures 14a–14c show the spectral exponent β, obtained as the logarithmic slope of the power spectrum. Table 2 reports the correlation coefficients, the slopes of the regression lines and the minimum and maximum slopes given by the jackknife procedure. As before, the model parameters have been optimized on D(0), σ² and K.

Figure 12

Scaling exponent H(2) of the model output versus that of the data for each individual GATE‐1 field with intensity larger than 1 mm/h. The models have been optimized on D(0), σ² and K. (a) Multifractal cascade, (b) nonlinearly filtered autoregressive process, and (c) model based on individual rainfall cells.

Figure 13

Scaling exponent H(4) of the model output versus that of the data for each individual GATE‐1 field with intensity larger than 1 mm/h. The models have been optimized on D(0), σ² and K. (a) Multifractal cascade, (b) nonlinearly filtered autoregressive process, and (c) model based on individual rainfall cells.

Figure 14

Logarithmic spectral slope, β, of the model output versus that of the data for each individual GATE‐1 field with intensity larger than 1 mm/h. The models have been optimized on D(0), σ² and K. (a) Multifractal cascade, (b) nonlinearly filtered autoregressive process, and (c) model based on individual rainfall cells.

[52] A first comment is that all models have a very small correlation coefficient with the data, due to the large scatter present for these statistics. However, all models have slopes of the regression lines that are close to one. This indicates that, on average, all the models can furnish estimates of the spectral and scaling exponents that are in the range of the data, but the field‐to‐field agreement is poor due to the large scatter that affects individual field values. In general, the model based on individual rain cells provide a slightly better fit to the scaling and spectral exponents of the data, while the autoregressive model and the multifractal cascade tend respectively to overestimate or underestimate the scaling and spectral exponents. (It is important to recall that an unbounded multifractal cascade severely fail in reproducing the spectral and scaling exponents of the rainfall data. For this reason, in this comparison we did not consider the standard unbounded version of multifractal cascades, which is known a priori to be unable to provide a good representation of rainfall fields [Menabde et al., 1997b].)

[53] Figures 15a–15c show the estimates of the correlation length of the model versus those of the data. In general, all models provide, again with significant scatter, results that are consistent with the data.

Figure 15

Correlation length, δ, of the model output versus that of the data for each individual GATE‐1 field with intensity larger than 1 mm/h. The models have been optimized on D(0), σ² and K. (a) Multifractal cascade, (b) nonlinearly filtered autoregressive process, and (c) model based on individual rainfall cells.

4.3. Discussion

[54] In optimizing a downscaling model to a set of individual fields, one should take into account several different requirements. One is the ease and speed of numerical implementation. All the models considered here are easy to implement and fast to run: from this point of view, they are equivalent.

[55] A second requirement is the capability of the models to reproduce the statistics on which they have been optimized (“in‐sample” estimates).

[56] A third, more important request is that the models optimized on some statistical quantities are able to reproduce also other statistics of the data (“out‐of‐sample” estimates).

[57] Figure 16 provides a summary of the results discussed in this work. Figure 16 shows the values of the regression slopes for the different statistics, for the three downscaling models. The slope of the regression line indicates that the model based on the individual rainfall cells is able to reproduce the behavior of the GATE data, albeit with large scatter, and it leads to a regression slope of about one for all the statistics. Thus this model provides good results for both in‐sample and out‐of‐sample estimates. The bounded multifractal cascade and the filtered autoregressive process produce regression slopes that are somehow different from one for the scaling and spectral exponents. On the other hand, the autoregressive process and the point process based on individual rain cells do not reproduce well the value of D(0) for large‐intensity fields.

Figure 16

Values of the regression slopes for the different test statistics for the three downscaling models considered in the text. Solid triangles are for the multifractal cascade, open triangles are for the individual rainfall cells, and solid circles are for the nonlinearly filtered autoregressive process.

[58] As a word of caution, we also note that all the models have low correlation coefficients for the scaling and spectral exponents, and that significant scatter is generally present between the model and the data. The presence of such a large scatter in the field‐to‐field comparison suggests that, at the present level of description, the differences between the different models are probably of limited relevance.

[59] An obvious question concerns the performance of the different models when they are optimized on different test statistics. When the optimization is performed on the set of generalized dimensions, we found that all the models tend to provide values of the variance, skewness and kurtosis that are quite different from those of the data. In addition, no improvement in the agreement of the scaling exponents is obtained. Since σ², S and K are basic quantities that one needs to get right in the downscaling procedure, optimizing solely on the generalized dimensions of the individual fields does not seem to be a good strategy.

[60] We also mention that the values of D(0) generated by the downscaling models tend to be far off the observed values, unless the models are explicitly optimized on this quantity (as it has been done here). Note that for the models considered here the introduction of a threshold is a necessary ingredient to obtain values D(0) < 2.

[61] Of course, one could optimize the model parameters on all the statistical measures considered. Because of the small number of model parameters, however, this procedure could lead to overdetermination of the parameter values. For the case of the GATE fields, optimizing on the whole set of statistical quantities does not lead to a significant improvement in the overall agreement between the models and the data. In addition, optimizing on too many statistical quantities is not an easy task in operational conditions, where only limited knowledge of the basic field statistics is usually available.

5. Conclusions

[62] In this work we have used a number of simple statistical measures to evaluate the performance of different types of spatial rainfall downscaling models. This approach allows for objectively comparing the different models and testing their behavior in a quantitative way.

[63] All models used here are characterized by a small number of free parameters and by fast numerical implementation. We have optimized the models on the variance σ², the kurtosis K and the fractal dimension D(0) of the data. We have compared the properties of the optimized model outputs with those of measured fields, considering both in‐sample and out‐of‐sample estimates. In general, the best downscaling procedures should be able to reproduce also test statistics on which they have not been explicitly optimized (i.e., to provide good out‐of‐sample estimates).

[64] In general, all models are able to approximately reproduce the statistics of the GATE rainfall fields, indicating that these models are essentially equivalent when evaluated from a performance point of view. When the models are optimized on σ², K and D(0), a small difference between the models emerge, as both the multifractal cascade and the filtered autoregressive process provide average values of the scaling and spectral exponents that are somewhat different from those of the data. Conversely, the model based on individual rainfall cells is able to reproduce, albeit with significant scatter, the scaling and spectral exponents of the data.

[65] The scatter between the model outputs and the data is, however, quite large in general. This indicates that future efforts should be devoted to reduce this scatter, and to refine the models in order to provide a closer field‐to‐field agreement between the downscaled rain fields and the data. Future work shall also consider extensions to space‐time rainfall downscaling and to the problem of linking the model parameters to large‐scale atmospheric conditions.

Acknowledgments

[66] We are grateful to Franco Siccardi for useful comments on this work. This work was sponsored by a grant CNR GNDCI on rainfall downscaling and on applied research in Meteohydrology. We greatly benefited of the detailed comments from two anonymous reviewers on a previous version of this work.