WATER RESOURCES RESEARCH, VOL. 35, NO. 6, PAGES 1853–1867, JUNE 1999
Multifractal modeling of anomalous scaling laws in rainfall Roberto Deidda Centro di Ricerca, Sviluppo e Studi Superiori in Sardegna, Cagliari, Italy
Roberto Benzi Autorita` per l’Informatica nella Pubblica Amministrazione, Rome
Franco Siccardi Centro di Ricerca in Monitoraggio Ambientale, Savona, Italy
Abstract. The coupling of hydrological distributed models to numerical weather prediction outputs is an important issue for hydrological applications such as forecasting of flood events. Downscaling meteorological predictions to the hydrological scales requires the resolution of two fundamental issues regarding precipitation, namely, (1) understanding the statistical properties and scaling laws of rainfall fields and (2) validation of downscaling models that are able to preserve statistical characteristics observed in real precipitation. In this paper we discuss the first issue by introducing a new multifractal model that appears particularly suitable for random generation of synthetic rainfall. We argue that the results presented in this paper may be also useful for the solution of the second question. Statistical behavior of rainfall in time is investigated through a highresolution time series recorded in Genova (Italy). The multifractal analysis shows the presence of a temporal threshold, localized around 15–20 hours, which separates two ranges of anomalous scaling laws. Synthetic time series, characterized by very similar scaling laws to the observed one, are generated with the multifractal model. The potential of the model for extreme rainfall event distributions is also discussed. The multifractal analysis of Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE) radar fields have shown that statistical properties of rainfall in space depend on time durations over which precipitation is accumulated. Further analysis of some rainfall fields produced with a meteorological limited area model exhibited the same anomalous scaling as the GATE fields.
1.
Introduction
In the last 30 years much research and many contributions have been focused both on statistical analysis of extreme events distributions and on stochastic modeling of rainfall in time and space. Current atmospheric numerical modeling makes operational rainfall field prediction feasible with significant shortrange reliability. The problem of “downscaling” of the precipitation fields is now very topical as it represents a natural link between the scales of meteorological applications and those of surface hydrologic modeling. Meteorological centers for numerical weather prediction distribute forecasts of precipitation fields with grid-size spatial resolutions ranging from 104 km2 (general circulation models (GCMs)) to 102 km2 (limited area models (LAMs)). For hydrological applications we need, instead, to have precipitation predictions at the smaller spatial and temporal scales of basins. Thus, given average precipitation depths at large scale, for example, those from meteorological models, accurate downscaling to enable simulation of precipitation fields over smaller scales, preserving correctly the statistical properties observed on real signals across the different scales, would be very useful. Past research on space-time stochastic modeling of precipiCopyright 1999 by the American Geophysical Union. Paper number 1999WR900036. 0043-1397/99/1999WR900036$09.00
tation has addressed mainly two classes of models: clusterbased and fractal/multifractal models. A common assumption of cluster-based models is that the rainfall process is organized in a preferred hierarchy of scales in space and time. This hierarchy of scales would represent, with more or less detail, the process of rainband arrival, the cluster organization of cells within a rainband, and the life cycle of cells belonging to each cluster. LeCam [1961] can be considered the precursor of cluster-based models, while the most popular implementation of such models is represented by the Waymire, Gupta, and Rodriguez-Iturbe (WGR) model [Waymire et al., 1984]. Following the second line of research, more recent papers deal with multifractal models based on random cascades [Gupta and Waymire, 1993; Hubert et al., 1993; Kumar and FoufoulaGeorgiou, 1993a, b; Ladoy et al., 1993; Lovejoy and Schertzer, 1992; Schertzer and Lovejoy, 1987; Tissier et al., 1993]. Many of the multifractal models proposed for simulation of synthetic precipitation fields were first applied to simulate anomalous scaling of velocity fields in turbulent flows [Benzi et al., 1984]. At the moment the multifractal theory represents a very powerful approach to nonlinear phenomena such as turbulent velocity fluctuation or intermittency of precipitation. The multifractal formalism allows a robust statistical control over any moment of a given distribution of measures if some kind of similarity holds over a range of scales. Analysis and comparison between real signals and simulated ones must address the
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DEIDDA ET AL.: MULTIFRACTAL MODELING OF ANOMALOUS SCALING LAWS
behavior of high moment distributions over different scales of interest to correctly capture not only the general features of the distribution but also its strongest events. From the analysis of statistical properties of rainfall in time or in space, such as presented in this paper, it is clear that precipitation is a strongly intermittent and nonlinear process; multifractal analysis shows, indeed, that rainfall fields are characterized, both in space and in time, by anomalous scaling laws, whose scaling exponents can be expressed as convex, and so nonlinear, functions of the moments. This means that simple fractal models, like the b model proposed 20 years ago by Frish et al. [1978] for turbulence intermittency and that recently proposed by Over and Gupta [1994] for generation of rainfall, are not the best way to create synthetic fields of precipitation, because scaling exponents of fractal models depend linearly on the moments. In this paper we review briefly the multifractal theory and a new general concept of scaling [Benzi et al., 1996], and we point out the important role that infinitely divisible distributions (IDD), such as log-Poisson, play in multiplicative processes. We then present the main features of a recently developed multifractal model [Deidda, 1997] that can be applied to rainfall downscaling problems. The model recalls some ideas from Benzi et al. [1993a], but it is based on an expansion of positive definite wavelets with coefficients belonging to a stochastic cascade. In such a way we are able to construct positive definite multifractal measures in spaces of any dimension. The analytical derivation of the theoretical anomalous scaling laws of the generated signal is also presented. In view of possible application of this model for generating synthetic rainfall, we discuss some examples of simulations in one- and two-dimensional spaces that illustrate the good agreement between the theoretically expected scaling laws and those obtained from Monte Carlo simulations. Results of multifractal analysis of a high-resolution (1 min) time series of precipitation recorded in Genova over 8 years have shown the existence of two ranges of anomalous scaling for time smaller and longer than a duration threshold. The physical meaning and explanation of the threshold is addressed later in the paper. However, within each range the analysis of moments of signals displays a nonlinear behavior that justifies the multifractal approach. In order to test the ability of the model to generate time series of precipitation, we have compared scaling laws of synthetic time series with those of the observed one. Additional verification of model performance was conducted on rainfall yearly maxima at different time duration: extreme event distributions of synthetic signals were successful compared with those of the observed time series. The positive results of this comparison enable us, when no measure is available at small scales, to use models for multifractal downscaling of precipitation to transfer information from time series at larger scales. We also discuss in this paper the analysis of statistical properties of precipitation in space. Results on spatial fields of rainfall are based on radar observations during the Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment, often referred as the GATE experiment, and on some precipitation fields obtained by a numerical simulation at very high resolution (10 km) performed with a LAM. We show that the anomalous scaling laws of rainfall in space depend strongly on the timescale over which precipitation is accumulated. To demonstrate this, we have analyzed the GATE fields not only at their original resolution of 15 min, but we have also
analyzed the fields accumulated over 30 min, 1 hour, and up to 24 hours. This paper is organized as follows: In section 2 a review of recent developments in the scaling theory of three-dimensional turbulent flows is presented and connections with rainfall data analysis are emphasized; a new multifractal model for random generation of synthetic rainfall and theoretical derivation of scaling laws are presented in section 3; statistical behavior of a high-resolution time series observed at the University of Genova and some applications of the multifractal model for generation of synthetic rain time series are discussed in section 4; the spatial statistical properties of rainfall fields are investigated through GATE campaign data sets and from precipitation fields produced by a limited area model simulation in section 5. Finally, a summary is given and conclusions from this research are drawn in section 6.
2.
Scale Covariance and Generalized Scaling
We review recent developments in the multifractal theory of fully developed turbulence. Our aim is to review the concepts of generalized scaling and of scale covariance [Benzi et al., 1993b, 1995, 1996], which will be used in the present paper to analyze and simulate the statistical properties of rainfall in time and in space. We will formulate our review directly in the language useful for multifractal analysis of rainfall data. The statistical properties of homogeneous and isotropic turbulence are usually investigated by studying the velocity structure functions Sq(r) 5 ^uDV(r)uq&, where DV(r) 5 V(x 1 r) 2 V(x) is the velocity difference between two points at distance r and ^ & denotes an average operator. In general, the definition of structure functions must be addressed to characterize the statistical properties in which we are interested. While the nature of turbulence is strictly connected to velocity fluctuations, in the case of rainfall we want to characterize the relationship between precipitation amounts at different spatial and temporal scales. When investigating the statistical properties of homogeneous, stationary, and isotropic rainfall, a more appropriate definition of structure functions can be given through the moments of integral measures of precipitation. For instance, a time series P(t) can be characterized by structure functions S q (r) 5 ^[* jj 1r P(t) dt] q &, while statistical properties in space of rainfall fields P( x, y) can be investigated by introducing structure functions S q (r) 5 ^[* jj 1r dx * uu 1r d y P( x, y)] q &. A common feature of multifractal objects is, however, the presence of one or more ranges of scales r, where the scaling S q (r) ; r z (q) holds with exponents z (q) that are a nonlinear function of q. For an introduction to multifractal theory and formalism the reader can refer to the work of Feder [1988] and Falconer [1990]. Let us consider a positive random field P( x) (P( x) $ 0) defined on the set x [ [0, L]. Without loss of generality, P( x) is normalized to 1, that is, * L 0 P( x) dx 5 1, for any realization. Our primary interest is to discuss the scaling properties of the quantity
E
x1r
P~ y! d y ; m x~r!
(1)
x
P( x) is said to display anomalous scaling properties if ^@ m x~r!# q& , r z~q!
(2)
DEIDDA ET AL.: MULTIFRACTAL MODELING OF ANOMALOUS SCALING LAWS
where z (q) is a nonlinear function of q. The scaling exponents z (q) are also referred to as multifractal exponents of P( x). The random field is assumed to be ergodic, so in (2) ^ & stands for both x average ( x being either a space or time coordinate) and ensemble average. It follows that we must also require stationarity and homogeneity. Furthermore, if the random field is embedded in a space of dimension d . 1, the field is also assumed to be isotropic; that is, the scaling properties (2) depend only on the modulus of r. It is useful to introduce another quantity strictly connected to m x (r), namely,
s x~r! 5
1 r
E
x1r
P~ y! d y 5
x
m x~r! r
(4)
Hereafter we shall also use the notation t (q) 5 z (q) 2 q. Because of the normalization * L 0 P( x) dx 5 1, we have
z ~1! 5 1
(5)
t ~1! 5 0
Note that (5) is true independently of the (anomalous) scaling properties of P( x). A way to give a physical interpretation of (2) and (4) is based on the multifractal formalism. One assumes that s x (r) ; r a , in a suitable range of r, with probability Q r ( a ) ; r d2D( a ) , where D( a ) is the fractal dimension at scale r, where the local scaling condition s x (r) ; r a holds, and d is the dimension of the embedded space (in our case d 5 1). It then follows that r qa1d2D~a! d a 5 r t~q!
(6)
Applying a saddle point estimate of the above integral, we obtain t (q) 5 infa [q a 1 d 2 D( a )]. Let us remark that in the multifractal formalism both s x (r) and Q r ( a ) are assumed to exhibit scaling behavior with respect to r. A simple way to build up a multifractal field P( x) is to use a cascade model based on random multipliers [Monin and Yaglom, 1971, 1975; Yaglom, 1966]. Let us consider a set of length scales r n [ 2 2n L and let us define a generator h i to be
s x~r i! ; h is x~r i21!
(7)
P
(8)
Then it follows that n
s x~r n! 5
s x~r i! 5 h ijs x~r j!
(11)
h 13 5 h 12h 23
(12)
By (11) we have
^@ s x~r!# q& , r z~q!2q
E
Using (6) and (10), we can identify t (q) 5 2log2 ^ h q &. Because t(1) 5 0, we must require ^h& 5 1. The result obtained in (10) relies upon the fact that n 5 2log2 (r n /L) and that all h i are independently identically distributed. In general, one can define r n 5 L l 2n , where l $ 2 is called branching number. Alternatively, one can write (8) as a production of n(r) random multipliers h i , where n(r) 5 2logl (r/L). This tells us that for a given probability distribution of h i the probability distribution of s x (r) depends on the branching number l. Let us now consider three arbitrary scales r 1 . r 2 . r 3 , and let us define the random multiplier h ij through the relation
(3)
It follows that if (2) holds, one obtains
^@ s x~r!# q& ,
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h is x~L!
i51
Equation (12) is true regardless of what the ratios r 1 /r 2 and r 2 /r 3 are. Let P ij be the probability distribution of h ij . Now we are interested in the probability distribution P ij which is functionally invariant under transformation (12) or the equivalent: log h 13 5 log h 12 1 log h 23
(13)
Thus our search is restricted to probability distributions that are stable under convolution. For independently distributed random variables a solution to this problem can be given in complete form [Feller, 1968]. If the variables are correlated, the problem becomes much more difficult to solve, as is well known from the modern theory of second-order phase transition [Parisi, 1988]. For the time being we shall restrict ourselves to independent random variables. In this case, for instance, the Gaussian and the Poisson distributions are well-known examples of probability distributions that are stable under convolution. Equivalently, a simple solution to this problem is given by all probability distributions P ij such that
PF G M
p ij
^h & 5
k51
g k~r j! g k~r i!
gk~ p!
(14)
for any function g k and g k . We remark that (14) represents the most general solution to the problem, independently of the scale ratio r i /r j . Probability distributions satisfying (14) will be referred to as scale covariant. In order to understand clearly the meaning of (14), we want to give an example based on a random log-Poisson multiplicative process [Dubrulle, 1994; She and Leveque, 1994; She and Waymire, 1995]. Namely, we assume that
h ij 5 A ijb yij
(15)
Let us also suppose that the set of h i are identically distributed independent random variables, with h i $ 0. Then we have
where b , 1 is a constant parameter, A ij is a scale dependent parameter, and y ij is a Poisson process with scale dependent parameter C ij (E[ y ij ] 5 C ij ):
P
2Cij !/m! P~ y ij 5 m! 5 ~C m ij e
n
q
^@ s x~r n!# & ;
q i
q
^ h &^ s x~L! &
(9)
i51
Because s x (L) 5 1/L, we obtain q
q
2^ h & 2^ h & ^@ s x~r n!# q& 5 L 2q^ h q& n 5 L log2^h &2qr 2log 5 C~q!r 2log n n q
(10)
(16)
This example also provides a link between (14) and the case of probability distributions that are stable under convolution. By using (16), we obtain ^ h pij& 5 A pij exp @C ij~ b p 2 1!#
(17)
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DEIDDA ET AL.: MULTIFRACTAL MODELING OF ANOMALOUS SCALING LAWS
Assuming g 1 ( p) 5 p and g 2 ( p) 5 b p 2 1, (17) is equivalent to (14) if we write Aij 5 g1(rj)/g1(ri) and exp Cij 5 g2(rj)/g2(ri). Moreover, because of the constraint ^ h ij & 5 1, we also have A ij exp [C ij ( b 2 1)] 5 1 or in terms of g 1 and g 2 g 1~r j! 5 g 1~r i!
F G g 2~r j! g 2~r i!
~12b!
F G g 2~r j! g 2~r i!
(20)
^ m x~r! p& 5 r p^ s x~r! p&
which is described by (14) with M 5 1. Only if we assume g 2~r! 5 r A
(21)
can we recover a scaling law for ^ s x (r) p &. This example highlights an important point in our discussion: the scaling of s x (r) with respect to r is not a necessary consequence of a cascade model based on random multipliers. A scaling law is observed only by assuming that the parameters describing the random multipliers show scaling with respect to r. In the case of the scale covariance random multiplicative cascade model (hereafter referred as the scale covariance model) we can observe for M 5 1 a new kind of generalized scaling, hereafter referred to as extended self-similarity (ESS). Indeed, by (14) and (20) we have ^ s x (r) p & 5 g 2 (r) 2 g ( p) . Therefore, even if g 2 (r) does not show any scaling in r, we have ^ s x~r! p& 5 ^ s x~r! q& g~ p!/g~q!
(22)
for any p Þ q and p Þ 1 and q Þ 1. Equation (22) highlights the fact that the field s x (r) is characterized by anomalous scaling, where the word anomalous refers to dimensional counting. By using the log-Poisson case as an example, we can provide a physical meaning to the scale covariance model and extended self-similarity (22). Let us first assume that the scaling with respect to r holds as for (21). In this case we have t ( p) 5 A[2p(1 2 b) 2 bp 1 1]. For p 3 ` the asymptotic form of t ( p) is
t ~ p! 3 2A~1 2 b ! p 1 A
(23)
Using the analog to (4), but for d-dimensional spaces, we must write z ( p) 5 t ( p) 1 pd; so the asymptotic form of the multifractal exponents is z ( p) 3 p[d 2 A(1 2 b )] 1 A. Following the multifractal interpretation, we can define A in terms of the fractal dimension D ` , defined as the fractal dimension of the set where the strongest events take place, A 5 d 2 D`
(24)
Indeed, if we consider the asymptotic tangent to z ( p) in the z 2 p plane, A represents the intercept to the z axis. Substituting (24) into (23), we obtain
z ~ p! 3 p@D ` 1 b ~d 2 D `!# 1 ~d 2 D `!
(26)
(19)
Equation (19) implies p21#
log g 2~r! log r
where now g 2 (r) is not scaling in r. In this case we lose the scaling properties of s x (r), and we recover the ESS generalized scaling (22). We make two important remarks in closing this review. The first has to do with the extension of generalized scaling of the quantity m x (r). Using (3), we see that
@ p~12b!1b p21#
^@ s x~r!# p& , g 2~r! 2@ p~12b!1b
D` 5 d 2
(18)
Finally, we obtain ^ h pij& 5
only if a geometrical interpretation, in terms of fractal dimension, holds. Generalizing the concept of multifractals, we can think of D ` as the fractal dimension of the most intense fluctuation at scale r:
(25)
It is interesting to observe that by using (24), (21) implies that g 2 (r) is the probability distribution for the most extreme events. Such a probability distribution shows a scaling behavior
(27)
Thus generalized scaling seems not to hold for m x (r). In fact, we can define a dimensionless quantity R p (r) as R p~r! 5
^ m x~r! p& ^ m x~r! n& p/n
(28)
where n $ 2. By using (28), (27), and (22) we have R p~r! 5
^ s x~r! p& 5 ^ s x~r! p& 12pg~n!/ng~ p! ^ s x~r! n& p/n
5 R q~r! $@g~ p!2~ p/n!g~n!#/@g~q!2~q/n!g~n!#%
(29)
For the log-Poisson example we have for n 5 2 R p~r! 5 R q~r! $@~12b
p!2~ p/ 2!~12b2!#/@~12bq!2~q/ 2!~12b2!#%
(30)
Equation (29) holds for any scale covariant model with M 5 1. Note that the scaling exponent of (30) does not depend on the function g 2 (r). Equation (30) can also be considered as a generalized form of extended self-similarity (GESS). The second point concerns the role of the log-Poisson distribution or, more generally, the role of infinitely divisible distributions (IDD). As far as independent random multipliers are used, the choice of an IDD can be done in such a way to satisfy scale covariance. So far, all published models for random multipliers based on IDD can be generalized in the form (14) with M 5 1. Therefore, for those models, generalized scaling (22) or (29) should hold. Finally, let us also comment about the fact that most emphasis has been addressed on the universal value, if any, of the scaling exponents t ( p). In light of the discussion in this section we can state that the exponents t ( p) may not be universal, while their ratios t ( p)/ t (q) may show universal values. This question will be addressed in the sections.
3.
A Model for Synthetic Rainfall
In this section we develop a simple model to simulate a synthetic, positive definite, rainfall field with predefined anomalous scaling exponents [Deidda, 1997]. As we shall see, the model can be adjusted to show any set of anomalous exponents explained by a random multiplicative process. Moreover, it can be easily generalized to include generalized scaling such as ESS (22) or GESS (29). We point out that the comparison between synthetic rainfall fields against observed ones will be extremely useful in order to understand whether observed features of rainfall data, such as extreme events, are correctly reproduced by synthetic fields. In section 3.1 we discuss our
DEIDDA ET AL.: MULTIFRACTAL MODELING OF ANOMALOUS SCALING LAWS
1857
main theoretical result. In section 3.2 we show, using the example of the log-Poisson distribution, how to include the concept of generalized scaling in the framework of the synthetic field. In section 3.3 we discuss some examples of anomalous scaling obtained using numerical simulations and synthetic rainfall data. 3.1. Construction of Synthetic Fields Exhibiting Anomalous Scaling Let us consider a positive definite field f ( x) with x [ [0, L] and let us define the structure functions S q as
KF E
GL
(31)
Figure 2. Graphical representation of the dyadic cascade process for a j,k coefficients, where j is the level index and k is the position index of the corresponding wavelet.
where ^ & is an x average ( x being space or time dimension), defined as an integral over all possible starting points j : ^f& 5 c * f d j . The field is considered also to be ergodic, so the exchange of the x average with ensemble average is allowed. We want to show how to construct the random field f ( x) showing anomalous scaling in the structure functions S q , namely,
reason in the following proofs, properties of orthogonal wavelet decomposition [Meneveau, 1991] are not used. The wavelets c j,k ( x) are normalized in modulus and are obtained by stretching and shifting of the same basis wavelet C( x) that is positive definite and integrable for x [ [0, L] and zero elsewhere:
S q~r! , r z~q!
c j,k~ x! 5 2 jC~2 jx 2 kL!
S q~r! 5
q
j1r
f ~ x! dx
j
(32)
where the approximate symbol is used to indicate quantities that differ for a multiplicative constant. The generation of the signal f ( x) is based on the following “wavelet” decomposition with coefficients belonging to a dyadic stochastic cascade:
OO N
f ~ x! 5
Using this definition, it is easy to show that the normalization in modulus of wavelets c j,k ( x) can be derived from the norm of the basis function C( x):
E
L
c j,k~ x! dx 5
0
2 j21
a j,kc j,k~ x!
(33)
j50 k50
where j is an index for cascade level, varying from 0 (first element) to N (number of cascade levels); k is the position index, varying from 0 to 2 j 2 1 in the jth cascade level; c j,k ( x) is the wavelet on kth position on level j; and a j,k is the coefficient extracted from the stochastic cascade. Since rainfall is a positive definite field, we must also require the wavelets c j,k ( x) to be positive definite. This represents a substantial difference with the model presented by Benzi et al. [1993a], where wavelets with zero mean are used. For this
(34)
E
L
C~ z! dz 5 1
(35)
0
The following Gaussian distribution is an example of a basis wavelet: C~ x! 5
H
c exp 0
F S 2
1 2
x2m s
DG 2
x [ @0, L# x¸@0, L#
(36)
where m 5 L/ 2, s 5 0.15L, and c > 1/( s =2 p ) is a normalization constant. Figure 1 shows the basis function (36), with L 5 1 and wavelets c j,k ( x) obtained using relation (34) over the first two levels. The stochastic cascade (Figure 2) is derived by a multiplicative process: each son a j,k at the jth level is obtained by multiplying the corresponding father at level j 2 1 by an independent and identically distributed random variable h called the generator:
a j,k 5 a j21,k/ 2h j,k
(37)
The moments of the random variable a j,k do not depend on the position index k and can be evaluated by (38), where the bar denotes ensemble average: q
q
q
a j, k 5 a j 5 a 0h
qj
(38)
The first term a0 of the random cascade can be obtained by substituting (33), (35), and (38) into the integral of the signal I 5 *L 0 f ( x) dx: Figure 1. Plot of a Gaussian basis function C( x), defined by equation (36) with s 5 0.15 and of some wavelets c j,k ( x) on the first two levels j 5 1, 2, obtained by stretching and shifting the same basis function C( x) according to equation (34). The integral of each wavelet is normalized to unity.
a0 5
I N ¥ j50 2 jh# j
(39)
The first-order structure function S 1 of the signal (33) can be written as:
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DEIDDA ET AL.: MULTIFRACTAL MODELING OF ANOMALOUS SCALING LAWS
OO N
S 1~r! 5
2 j21
a j,k
j50 k50
KE
j1r
c j,k~ x! dx
j
L
,r
z~1!
(40)
KE
c j,k~ x! dx
j
L KE 5
2j
j1r
C~2 jx 2 kL! dx
j
L
Finally, the first-order structure function S 1 can be estimated using (41) together with (38) and (39): S 1~r! 5
OO
a j,k2 2jG 1~2 jr! 5
O
2 j a j 2 2jG 1~2 jr!
j50
N h# jG 1~2 jr! ¥ j50 5I N ¥ j50 h# j2 j
(42)
where summation over k gives 2 j terms, as the statistics of the coefficients a do not depend on the position index k. For large N the following relation holds between scales r and 2r: S 1~2r! 5 I
N N h# jG 1~2 j11r! h# j11G 1~2 j11r! ¥ j50 ¥ j50 5 2I < 2S 1~r! N N j j ¥ j50 h# 2 ¥ j50 h# j112 j11 (43)
This implies, as expected, that S 1 (r) ; r, that is (40) where
z ~1! 5 log2
S
S 1~2r! S 1~r!
D
51
(44)
Note that (44) is consistent with (5) which states a form of conservation of measures over different scales. The scaling of structure functions for any moment q can be computed in a similar way. Substitution of signal (33) into (31), exchanging integral and summation symbols, yields
KF O O E N
S q~r! 5
2 j21
a j,k
j50 k50
GL q
j1r
c j,k~ x! dx
j
(45)
Following the theory of characteristic functions, we can introduce the cumulant (structure) functions S cq (r) using the following relation:
K H FE exp i v
GJ L F O `
j1r
f ~ x! dx
5 exp
q51
j
~i v ! q c S q~r! q!
S cq~r! 5
OO
2 j21
G
j50 k50
q a j,k
KF E j
c j,k~ x! dx
GL
j
5 2 2jG q~2 jr!
(48)
After elimination of summation over index k and substitution of (38) and (39), the cumulant functions S cq (r) take the following expression: N ¥ j50 h q G q~2 jr! N @¥ j50 h# j2 j# q
(49)
and over a scale 2r take the expression j11
S cq~2r! 5 I q
N ¥ j50 h q G q~2 j11r! 2 qh# q 2 qh# q < S cq~r! N j11 j11 q q @¥ j50 h# 2 # h hq
c
, r z ~q!
z c~q! 5 log2
(50)
S
S cq~2r! S cq~r!
D
5 q~1 1 log2 h# ! 2 log2 h q
(51)
The nonlinear behavior and the convexity of z c (q) can be easily proved using the Cauchy-Schwarz inequality. Convexity of z c ( q ) assures that inequalities z c ( q ) , q z ( 1 ) and z c ( p 1 q) , z c ( p) 1 z c (q) hold for any p and q . 1. It is possible to show that cumulant structure functions S cq give the leading contribution to structure functions S q . For example, the second-order structure function can be written as the sum of two terms. The first one is the cumulant structure function S c2 , and the second one is proportional to the square c of the first-order structure function S 1 : S 2 (r) 5 A 2 r z (2) 1 B 2 r 2 z (1) , where A 2 and B 2 are constants. Since the inequality c z c (2) , 2 z (1) holds, for r ,, 1 we have r z (2) .. r 2 z (1) . Consequently, the term multiplied by B 2 can be neglected, and the behavior of the second-order structure function is thus c S 2 (r) ; r z (2) . Similarly, the third-order structure function c c can be written as S3(r) 5 A3 rz (3) 1 B3 r3z(1) 1 C3 rz(1)1z (2), so we z c (3) have S 3 (r) ; r for r ,, 1. In such a way it can be shown for any moment q that the scaling (32) holds with z (q) [ z c (q) for r ,, 1. The one-dimensional model presented here can be easily generalized for the construction of multifractal isotropic measures f ( x 1 , z z z , x d ) with x# [ [0, L] d embedded on ddimensional spaces. Also, the construction of these positive definite fields f is based on wavelet decomposition with coefficients extracted by a stochastic cascade:
(46)
(47)
The integral in square brackets in (47) can be evaluated introducing another distribution G q (r) 5 ^[* jj 1r C( z) dz] q & that allows replacement of the spatial average:
OO O N
5
q
j1r
C~2 jx 2 kL! dx
f ~ x 1, · · · , x d!
where i 5 =21 and the cumulant structure functions have the following expression: N
GL q
j1r
We can finally estimate the exponent z c (q), which exhibits the anomalous scaling of the cumulant functions S cq , as a function of the moments of the generator:
N
j50 k50
2j
j
(41)
2 j21
G L KF E 5
S cq~r! 5 I q
5 2 2jG 1~2 jr!
N
c j,k~ x! dx
j
where linear operators angle brackets, summation, and integral symbols were exchanged. The quantity in square brackets can be evaluated by introducing a distribution G 1 (r) 5 ^* jj 1r C( z) dz& that is a function of the scale r, as the variable j is saturated from the spatial average ^ &: j1r
KF E
q
j1r
2 j21
2 j21
···
j50 k150
a j,k1 · · · kdc j,k1· · ·kd~ x 1, · · · , x d!
(52)
kd50
where j is the cascade level index, varying from 0 (first element) to N (number of cascade levels); k l is a position index on the x l (l 5 1, z z z , d) axis, varying from 0 to 2 j 2 1 in the jth cascade level; c j,k 1 z z zk d ( x 1 , z z z , x d ) is a wavelet on level j, characterized by the array of position indexes (k 1 , z z z , k d ); and a j,k 1 z z zk d is the coefficient extracted from the stochastic cascade. The wavelets c are now defined by a production of d one-
DEIDDA ET AL.: MULTIFRACTAL MODELING OF ANOMALOUS SCALING LAWS
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dimensional basis wavelets C( x), positive definite and integrable for x [ [0, L] and zero elsewhere (e.g., the Gaussian function in (36)):
P d
c j,k1· · ·kd~ x 1, · · · , x d! 5
2 jC~2 jx l 2 k lL!
(53)
l51
The random cascade is still obtained using a multiplicative process, but now the random generator h maps each father term at j 2 1 level into 2 d new coefficients at level j:
a j,k1 · · · kd 5 a j21, k1/ 2 · · · kd/ 2h
(54)
Statistical properties of random fields (52) can be investigated by introducing the following structure functions S q : S q~r! 5
KF E
j11r
j1
dx 1 · · ·
E
GL q
jd1r
dx d f ~ x 1, · · · , x d!
jd
(55)
where the spatial average ^ & can be now defined as an integral over all axes starting points j l , with l 5 1, z z z , d. Fields are assumed to be ergodic, so the spatial average can be exchanged with ensemble average. Similarly to the analysis presented for one-dimensional signals, we can easily derive the multifractal exponents z (q) that characterize the anomalous scaling S q (r) ; r z ( q ) of ddimensional synthetic fields:
z ~q! 5 q~d 1 log2 h# ! 2 log2 h
q
(56)
Figure 3. A typical example of synthetic signal f ( x) generated in x [ [0, 1] with the one-dimensional model and logPoisson generator with parameters b 5 0.4 and c 5 0.5. The expansion was truncated at the 23rd level, but for graphical representation the signal is shown at 2212 resolution. The integral of the signal from 0 to 1 is equal to unity.
R q,n~r! 5
h j,k 5 A j,kb yj,k
(57)
where A j,k and b are parameters, while y j,k is a Poisson distributed random variable (16) with expectation C j,k . In order to display generalized scaling in a synthetic rainfall field, we assume that A j,k 5 g 1 (2 2j )/g 1 (2 2( j11) ) and exp C j , k 5 g 2 (2 2 j )/g 2 (2 2 ( j 1 1 ) ), where g 1 and g 2 are two smooth functions whose properties have already been discussed in section 2. We remark that the quantity 2 2j should be considered equivalent to the “scale” r j . The expected scaling of signal f ( x) or more generally of f ( x 1 , z z z , x d ) can be evaluated using (56) and (17):
z ~q! 5 qd 1 c
q~ b 2 1! 2 ~ b q 2 1! ln 2
(58)
where the multifractal exponents z (q) depend only on parameter b and on parameter c 5 C j,k that may be scale dependent. By using the above definitions for the scale dependent random multiplier, we achieve the construction of scale covariant synthetic rainfall data. As discussed in section 2, the structure functions S q (r) may not show scaling in r. However, we can observe GESS by introducing the dimensionless structure functions R q,n (r):
(59)
where j n (q) 5 z (q) 2 z (n)q/n. Properties of GESS can be now recovered by R p,n~r! , R q,n~r! xn~ p,q!
(60)
where the exponents x n depend only on the b parameter that can be assumed scale invariant: p ~ b n 2 1! 2 ~ b p 2 1! j n~ p! n 5 x n~ p, q! 5 j n~q! q n ~ b 2 1! 2 ~ b q 2 1! n
3.2. Introducing Scale Covariance With the Log-Poisson Generator We have already noted in section 2 that the log-Poisson distribution is suitable for including generalized scaling. Here we want to highlight how GESS can be introduced in synthetic rainfall data as discussed in section 3.1. First of all, following the notation used in section 3.1, we can introduce the log-Poisson distribution (15) by assuming that the random multiplier h j,k satisfies
S q~r! , r jn~q! @S n~r!# q/n
(61)
The discussion presented in this section can be generalized for all infinitely divisible distributions. 3.3.
Examples of Synthetic Fields
Examples of application of the wavelet model for generation of multifractal synthetic fields in R and R 2 are presented. The log-Poisson distribution was assumed for the random multipliers. Our aim is to compare the anomalous exponents, as predicted by the theory described in sections 3.1 and 3.2, against anomalous scaling behavior observed in numerical simulations. 3.3.1. One-dimensional generations. Using the model described in sections 3.1 and 3.2, we have generated 64 synthetic samples f ( x) for x [ [0, 1], each one truncated after the 23rd cascade level, that is, N 5 23. Parameters of logPoisson distribution (57) were A j,k 5 1, b 5 0.4, and C j,k 5 c 5 0.5. The Gaussian function (36) was assumed as basis function C( x). Figure 3 shows an example of synthetic signals between the 64 samples that, for graphical reasons, is represented with a 2212 resolution; the integral from x 5 0 to x 5 1 was scaled to one before plotting. Structure functions S q for moments q 5 1, 2, z z z , 10 were estimated on each of the 64 sample signals using definition (31), where averages ^ & where applied to disjoint intervals. Figure 4 shows the scaling laws (32) of structure functions computed on one sample after reaggregation of the last four fragmentation levels. The exponents z (q) in (32) were then obtained by linear regressions in the log-log plane of structure functions versus spatial scales. The average and the standard deviation of exponents z (q) on the 64 sample generations are
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Figure 4. The first 10 structure functions S q (r) (circles) estimated on one synthetic signal, chosen between the 64 generations with the one-dimensional model, are plotted as a function of r. The scaling of structure functions is highlighted by the log-log least squares regressions (dashed curves).
close to the scaling found in rainfall fields produced by numerical simulations with a meteorological model (this case is analyzed in section 5). The basis function C( x) was again a Gaussian function (36). Each of the 64 generations was truncated after the 10th cascade level (N 5 10). Figure 6 shows a typical section of a synthetic field at full resolution 1024 3 1024, while a sample field at reduced resolution 128 3 128 (the reduction was made only for graphical reasons) is plotted in Figure 7. Before plotting Figures 6 and 7, the signals were scaled to have unit integrals in the domain ( x, y) [ [0, 1] 2 . Structure functions S q (r), defined by (55) where d 5 2, were estimated for moments q 5 1, 2, z z z , 10. The scaling laws of structure functions computed on one synthetic field reaggregated at a larger scale are shown in Figure 8. Exponents z (q) of (55) were then estimated using linear regressions of logarithms of structure functions versus logarithms of spatial scales. The average and the standard deviation of exponent z (q) of the 64 sample generations are plotted with error bars in Figure 9, together with a solid curve that represents the theoretical expectation based on (58). As for the onedimensional case, the statistical properties of synthetic fields in R 2 , generated using the wavelet model, are in optimal agreement with the analytical derivation.
4. plotted with error bars in Figure 5, and the solid curve in Figure 5 represents the theoretical expectation based on (58) with d 5 1. Figure 5 gives evidence of the good agreement between theoretical predictions of the anomalous scaling laws of signals and the statistical properties of samples generated using Monte Carlo simulations. 3.3.2. Two-dimensional generations. The generalization of the wavelet decomposition in multidimensional spaces was used to generate 64 synthetic samples f ( x, y) in the domain ( x, y) [ [0, 1] 2 . The generator of the multiplicative process of coefficients a was a log-Poisson distribution (57) with parameters A j,k 5 1, b 5 0.4, and c 5 0.7; with this choice of parameters the expected scaling (58) for synthetic fields is very
Figure 5. The theoretical expectation (58) of moments z (q) for the one-dimensional (d 5 1) model with log-Poisson generator with parameters b 5 0.4 and c 5 0.5 (solid curve). Error bars represent averages and standard deviations of z (q) estimated from the 64 synthetic signals.
4.1.
Statistical Properties of Rainfall in Time Analysis of Genova Time Series: The “External Scale”
We now turn our attention to observed rainfall data. In this section we want to discuss the statistical properties of precipitation in time and discuss possible applications of the model described in section 3 in generating synthetic rainfall. With this aim we have analyzed the multifractal behavior of a highresolution rain time series observed at the University of Genova, where a tipping-bucket rain gauge is sited. The series has a resolution of 1 min in time and 0.2 mm of equivalent rainfall depth. The measurements are available from 1988 to 1995, with some small periods of inactivity, for a total of 2607 rainy and not rainy days. We first investigate the statistical properties of the time series recorded in Genova. Next we show that a log-Poisson model gives a good fit to the observed scaling exponents. In
Figure 6. A typical section at 227 resolution, parallel to the x or y axis, of a two-dimensional synthetic field f ( x, y) selected between the 64 Monte Carlo simulations in R 2 .
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Figure 7. A typical example of synthetic field f ( x, y) generated in ( x, y) [ [0, 1] 2 with the twodimensional model and the log-Poisson generator with parameters b 5 0.4 and c 5 0.7. The expansion was truncated at the tenth level, but for graphical representation the signal is shown at 227 resolution. The integral of the signal is equal to unity.
Figure 8. The first 10 structure functions S q (r) (circles) estimated on one synthetic field, chosen between the 64 generations with the two-dimensional model, are plotted as a function of r. The scaling of structure functions is highlighted by the log-log least squares regressions (dashed curves).
Figure 9. The theoretical expectation (58) of moments z (q) for the two-dimensional (d 5 2) model with log-Poisson generator with parameters b 5 0.4 and c 5 0.7 (solid curve). Error bars represent averages and standard deviations of z (q) estimated from the 64 synthetic signals at 1024 3 1024 resolution.
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Figure 10. The first 10 structure functions S q ( t ) of the rain time series observed at the University of Genova versus timescales t in minutes. The range of timescales goes from 2 min to 214 min (about 11 days). The “external scale” is marked by the threshold located around log2 t 5 10 (about 17 hours). section 4.2 we generate synthetic signals with similar behavior of the structure functions and compare the extreme events distribution against the observed ones. We define the time structure functions S q ( t ) as S q~ t ! 5
KF E j
GL q
j1t
P~t! dt
, t z~q!
(62)
where ^ & is an average operator over all possible disjointed samples of integrals at scale t of the rainfall signal P(t). The structure functions S q of the first 10 natural moments q of the observed signal P(t) were computed at different timescales and plotted logarithmically in Figure 10 for timescales t varying from 2 min to 214 min (about 11 days) in powers of 2. Although uncertainty in estimating high-order moments arises for short series and for this reason results of higher-order structure functions must be carefully analyzed, each structure function S q (for q . 1) in Figure 10 clearly displays for the time series of Genova rainfall two different ranges of scaling with a threshold located around 210 min (.15–20 hours). These results reveal the presence of “external scale” in the scaling properties of S q (r). In order to explain the presence of this “external scale,” let us note that many authors have pointed out the description of the spatial and temporal variability of rainfall within the synoptic scale in terms of hierarchical structures [Austin and Houze, 1972], from little rainfall cells that are aggregated in clusters within small mesoscale areas up to the large mesoscale areas. Many attempts to classify rain rates, velocities of storms, and durations that characterize the different spatial scales of the hierarchical process can be found in the literature [Rodriguez-Iturbe, 1986; Rodriguez-Iturbe et al., 1984; Waymire et al., 1984]. The spatial dimension of precipitating clusters in the synoptic area is L ' 300 km, and the velocity of advection of these structures is U ' 15–20 km/h. The duration in a fixed system between the beginning and the end of whole event is
about L/U ' 15–20 hours. The average duration of extreme storms is in accordance with the threshold found on structure functions, which characterize different statistical behavior of the signal in time under this cut, where the structure functions are determined by the rainfall pattern internal to the storm, and over this cut, where the interarrivals of storms control the process. A very interesting result can be obtained by looking at the generalized extended self-similarity (60), namely, by looking at the self-scaling properties of the dimensionless structure functions R q ( t ) 5 S q ( t )/[S 4 ( t )] q/4 . In Figure 11 we plot in a log-log plane the dimensionless structure functions R q of moment q 5 7 versus those of moment q 5 3. First of all, we remark that a scaling relation is observed for all ranges of available timescales t. Indeed, a threshold scale is no longer present in Figure 11, which clearly shows that the rainfall field is a self-similar process at any scale in the range from 2 to 214 min and reflects the property of generalized extended selfsimilarity discussed in section 2. Recalling (60) and (61), this result seems to state that synthetic rainfall time series can be generated using a timescale invariant b parameter in the logPoisson generator (57). With this interpretation the effect of the “external scale” in the scaling behavior of structure function S q , observed in Figure 10, can be modeled only by different values of the scale dependent parameter c of the Poisson distribution, as explained in section 3.2. The exponents z (q) of (62) were estimated by a least squares best fit between the logarithms of structure functions S q and logarithms of the timescales for the two ranges, storm inner scales, and storm external scales. In Figure 12 the estimations of z (q) are plotted for small scales from 2 min to 17 hours and for large scales are plotted from 17 hours to 11 days. The two parameters of the log-Poisson generator were then estimated in each scaling regime with a best fit adaptation of (58) to sample exponents z (q). In Figure 12 the theoretical expectation for multifractal exponents z (q), based on (58) with best fit parameters b 5 0.30 and c 5 0.37 for small scales and b 5 0.30 and c 5 0.95 for large scales, is also displayed by a solid curve. 4.2. Anomalous Scaling of Synthetic Rainfall Time Series and Extreme Events Distributions We now discuss the capability of the one-dimensional wavelet decomposition model to reproduce the rainfall depth pro-
Figure 11. Dimensionless structure functions R q ( t ) of moment q 5 7 are plotted versus those of moment q 5 3, with t from 2 min to 214 min. The threshold is no longer present, implying that the Genova rainfall time series satisfies generalized properties of extended self-similarity.
DEIDDA ET AL.: MULTIFRACTAL MODELING OF ANOMALOUS SCALING LAWS
cess in time. For this purpose, by using the procedure discussed in section 3, we have generated a number of synthetic time series rainfall data, each one characterized by the same statistical properties, length, and resolution of those observed in Genova during the 2607 days between 1988 and 1995. Each daily generation was truncated at the 16th cascade level on a 24-hour domain, which corresponds to a time resolution of about 1.32 s. The integral of each daily signal was then rescaled in order to have the same amount of precipitation observed on the corresponding day, making it dimensional. The time series were then obtained by joining each daily independent dimensional wavelet decomposition into the final sequence of 2607 days. After that the time series was “dressed” by reducing the time resolution to 1 min, as in the original series, and filtering the synthetic signal with a fictitious tipping-bucket rain gauge; in such a way the simulated rainfall depth process was discretized every 0.2 mm like the real-world process. The effect of the “external threshold” was also investigated; in fact, two procedures were used. In procedure I1 the threshold was supposed to be at 12 hours; the log-Poisson generator with parameters b 5 0.30 and c 5 0.95 was used on the first cascade level, and a second procedure with parameters b 5 0.30 and c 5 0.37 was used on the other 15 cascade levels. In procedure I2 the threshold was supposed to be at 24 hours, and so a unique log-Poisson generator with constant parameters b 5 0.30 and c 5 0.37 was used on all the cascade levels. A set of 30 synthetic time series was produced by using procedure I1, and a second set of the same size was produced for I2. In Figure 13 a sample of two realizations of synthetic precipitation are compared with the observed one on the same day. The direct verification of the model was performed by estimating from each series the exponents z (q) with a least squares best fit between the logarithms of sample structure functions and the logarithms of timescales. In Figure 14 the exponents z (q) estimated from the observed signal are compared with the mean and standard deviation of z (q) estimated from the sets of 30 series generated with procedure I2. Results with hypothesis I1 were very similar. Figure 14 shows evidence
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Figure 13. (top) Millimeters of rainfall depth every 10 min on January 14, 1988, observed at the University of Genova and (middle and bottom) synthetic signals characterized by the same daily total amount of precipitation (81.6 mm). of the capability of our model to reproduce the observed statistical properties of rainfall time series. A quite interesting indirect validation was made by comparing the extreme values of the original series with those ob-
Figure 12. Anomalous scaling laws for small and large scales in the Genova rainfall time series. The estimates of exponents z (q) are marked with open circles for small scales (2 min to 17 hours) and with solid circles for large scales (17 hours to 11 days). Solid curves represent the corresponding expectations for the log-Poisson model with parameters b 5 0.30 and c 5 0.37 (small scales) and b 5 0.30 and c 5 0.95 (large scales) based on equation (58).
Figure 14. Comparisons between the exponents z (q) of the observed Genova time series (solid circles) and averages and standard deviations (error bars) of those estimated by the synthetic set I2.
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Table 1. Comparisons of Parameters of the Yearly Maximum Rainfall Depth Distribution h 5 a(T)d n Estimated From the Rain Time Series Observed in Genova and From Synthetic Series Generated With Procedures I1 and I2 a(T)
Observed time series Synthetic time series procedure I1 Synthetic time series procedure I2
n
T55
T 5 10
T 5 25
0.51 0.54
69 63
82 80
99 100
0.54
58
74
94
I1 threshold is at 12 hours; I2 threshold is at 24 hours. Parameter a(T) is millimeters of rainfall depth, d is the duration in hours, and T is the return period in years.
tained by the synthetic ones. In particular, we have compared the parameters of the following distribution for the maximum rainfall height h accumulated during d hours that can be reached or exceeded every T years: h 5 a(T)d n . To estimate the parameters n and a(T) of this equation, we have made the hypothesis that the yearly height maxima of rain precipitation in d hours follow a probability distribution of Gumbel type (extreme value type 1 (EV1)). The parameters were estimated with the moments method using yearly maximum rainfall on 5 min, 10 min, 15 min, 30 min, 1 hour, 3 hours, 6 hours, and 12 hours. Results of this analysis can be found in Table 1 where we compare the estimates of parameters in the three cases: the observed time series and the 30 synthetic time series generated each according to hypothesis I1 or I2. Even though the period of the recorded time series in Genova might be relatively short to capture the significant multifractal behavior of rainfall in time (i.e., parameter estimations can probably be reviewed), results on extreme event distributions are worthy of consideration for potential application of this model to estimation of extreme event distributions from coarse data.
5.
Statistical Properties of Rainfall in Space
We now turn our attention to the multifractal behavior of rainfall spatial fields. In particular, we have studied statistical properties of two kinds of spatial estimations of rain: the first ones based on the radar measurements obtained during the GATE campaign and the second ones as output of a highresolution numerical simulation, performed with a meteorological limited area model (LAM) [Buzzi et al., 1994; Marrocu et al., 1998]. The GATE spatial fields of precipitation were collected in the eastern Atlantic coast of Africa during two different periods in 1974: GATE1, the data set of frames collected during the first period from June 28 to July 15, and GATE2, the data set of the second period from July 28 to August 15. Each frame represents the average rain rate every 15 min; data set GATE1 contains 1715 frames, while data set GATE2 contains 1512 frames. Fields are available with a 4-km spatial resolution in a regular square lattice 100 3 100, but data are really provided only within a 200-km radius of the center of each image. In the present work we used only a 64 3 64 grid centered in each image in such a way that the rain rate is defined over each grid point. The precipitation depth fields obtained by the LAM simu-
lation were saved every 6 hours over a regular grid 256 3 256 with a spatial resolution of about 10 km covering Europe, starting at 0000 Greenwich mean time (GMT) on February 19, 1991, and ending at 1800 GMT on the 22nd of the same month. The model was used in hindcasting mode, with initial and boundary conditions interpolated by the analysis of the European Center for Medium-Range Weather Forecasts available every 6 hours. The analyses of rainfall in space have been performed by computing the (anomalous) scaling behavior of the space structure functions of order q, defined as follows:
KF E E j1r
S q~r! 5
dx
j
GL q
u1r
d y P~ x, y!
u
, r z~q!
(63)
where ^ & is an average operator over all possible disjointed samples of integrals on square domains ( x, y) of side r, with starting integration points ( j , u ). P( x, y) is the spatial field of precipitation depth on a fixed time period. From the GATE experiments, not only rain spatial fields at the original time resolution of 15 min were taken into account, but also those obtained by averaging rain rates over 30 min, 1 hour, 3, 6, 12, and 24 hours were taken into account. Two criteria were applied to select the fields to be used in the analysis. The first criterion imposes that at least 10% of grid points have a nonzero measure of precipitation; in this way we discard the fields without measures and those that are less relevant for the analysis. The second criterion requires a minimum correlation coefficient, fixed at level 0.995, for the linear regression between the logarithms of structure functions S 10 (r), defined by (63), and the logarithms of spatial scales r from 4 to 64 km, regularly sampled in the logarithmic scale with integer powers of 2. This second criterion guarantees a minimum statistical significance on estimation of exponents z (q). The number N of fields, selected from each one of the original GATE1 and GATE2 data sets, applying the two criteria exposed above can be found in Table 2 for each duration considered in the analysis. Results of spatial multifractal analysis on the selected fields are summarized in Table 3 where we compare the averages of the first 10 exponents z (q), estimated in the range of scales from 4 to 64 km. For each data set the values of parameters b and c obtained by imposing the values of sample exponents z(3) and z(8) in (58) are also listed in Table 2.
Table 2. Number N of Selected Fields From GATE1 and GATE2 Data Sets for the Spatial Multifractal Analysis and Estimates of the Log-Poisson Parameters b and c for Generation of Synthetic Spatial Fields of Rainfall Depth at Different Durations GATE1
GATE2
Duration
N
b
c
N
b
c
15 min 30 min 1 hour 3 hours 6 hours 12 hours 24 hours
436 215 128 70 46 25 13
0.42 0.43 0.44 0.44 0.48 0.55 0.62
1.00 0.92 0.83 0.73 0.77 0.78 0.71
316 154 95 52 40 26 14
0.38 0.40 0.41 0.44 0.46 0.51 0.56
1.00 0.92 0.88 0.81 0.75 0.77 0.68
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Table 3. Averages of Exponents z (q) in Equation (63), Estimated From GATE1 (G1) and GATE2 (G2) Data Sets in the Range of Spatial Scales 4 – 64 km for Different Durations Data Set
z(2)
z(3)
z(4)
z(5)
G1 G2
3.48 3.41
4.82 4.68
6.06 5.86
7.26 6.99
G1 G2
3.55 3.48
4.97 4.84
6.30 6.11
7.59 7.34
G1 G2
3.59 3.52
5.07 4.92
6.49 6.25
7.85 7.53
G1 G2
3.64 3.61
5.18 5.11
6.67 6.54
8.11 7.93
G1 G2 LAM
3.67 3.66 3.62
5.25 5.22 5.19
6.76 6.72 6.69
8.22 8.18 8.15
G1 G2
3.76 3.71
5.43 5.32
7.04 6.86
8.60 8.36
G1 G2
3.84 3.80
5.62 5.52
7.34 7.19
9.02 8.82
15 Min
30 Min
1 Hour
3 Hours
6 Hours
12 Hours
24 Hours
z(6)
z(7)
z(8)
z(9)
z(10)
8.42 8.10
9.57 9.19
10.72 10.27
11.85 11.35
12.98 12.42
8.85 8.54
10.09 9.72
11.32 10.89
12.54 12.06
13.76 13.22
9.19 8.78
10.50 10.01
11.80 11.23
13.09 12.44
14.37 13.65
9.52 9.28
10.92 10.62
12.29 11.94
13.66 13.25
15.02 14.55
9.66 9.60 9.57
11.07 11.01 10.98
12.47 12.40 12.37
13.85 13.78 13.74
15.23 15.16 15.11
10.13 9.82
11.63 11.26
13.12 12.69
14.60 14.11
16.07 15.52
10.67 10.42
12.30 11.99
13.91 13.55
15.52 15.09
17.12 16.62
For comparison, averages of the multifractal exponents estimated from the 6-hour precipitation fields of the limited-area model (LAM) data set are also reported.
In a similar way we have also selected from the LAM data set 13 of the 15 available precipitation fields accumulated every 6 hours. For this selection we have applied the criteria of minimum correlation coefficient (.0.995) for the linear regression between the logarithms of structure functions S 10 (r) and the logarithms of spatial scales r ranging from 10 to 160 km, sampled with integer powers of 2. Sample averages and standard deviations of the first 10 exponents z (q) based on 6-hour precipitation fields are plotted with error bars in Figure 15; in Table 3 the averages of multifractal exponents used to plot Figure 15 are compared with those obtained from the
Figure 15. Anomalous scaling laws of the 13 limited area model (LAM) precipitation fields selected for the spatial analysis. Averages and standard deviations of exponents z (q) are marked with error bars. The solid curve is the expected scaling (58) for the two-dimensional model with log-Poisson parameters b 5 0.4 and c 5 0.7 (the same as used for R 2 simulations discussed in section 3.3.2).
GATE data sets. Figure 15 shows also that the expected scaling (58) for the two-dimensional model with log-Poisson parameters b 5 0.4 and c 5 0.7 (the same as used for R 2 simulations discussed in section 3.3.2) is very close to the scaling found for LAM fields. Results on spatial statistical properties of precipitation fields are also presented in Figure 16, where the sample averages of exponents z (q) estimated from LAM 6-hour rainfall fields are compared with exponents z (q) corresponding to the different accumulation times considered for GATE1 and GATE2 data sets. Although there are fewer differences between the z (q) estimated by the two GATE data series, our analysis points out a strong dependence of the spatial statistical properties of the precipitation fields on the time duration, in which rainfall is accumulated. As expected, the multifractality of precipitation fields is more pronounced for short accumulation times. In other words, this means that for short times the behavior of precipitation fields is strongly intermittent and that the intermittency in space changes with duration. Another very important result that comes from our work is the substantial coherence of the exponents z (q) determined for the 6-hour accumulated precipitation fields based on LAM simulations and those estimated from the GATE data set at the same duration. Although the parameterization of rainfall processes in meteorological models is greatly simplified, the numerical simulation with the LAM has produced precipitation fields with statistical properties in good accordance with those observed in the GATE radar measurements.
6.
Summary and Concluding Remarks
Here we outline and review the most relevant results of this work. 1. We have reviewed the concept of scale covariance and
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Figure 16. Averages of exponents z (q) of GATE1 and GATE2 data sets for different durations from 15 min to 24 hours are compared. Note that the averages of z (q) estimated from LAM 6-hour precipitation fields (circles) are very close to the 6-hour GATE exponents. extended self-similarity as suitable generalizations of multifractal analysis. These concepts have been previously applied in the study of intermittency for three-dimensional flows [Benzi et al., 1996]. We argue that scale covariance and extended self-similarity are also useful concepts in the framework of rainfall data. 2. We have shown how to compute synthetic rainfall fields which satisfy both anomalous scaling and extended selfsimilarity. The method proposed in this paper is quite general and is not confined to the use of a particular probability distribution of the random multiplier. Moreover, the method was generalized in more than one dimension and to scale covariance fields. Synthetic fields are characterized by stationarity, homogeneity, and also by isotropy, if embedded in multidimensional spaces. 3. Using time series of rainfall data recorded in Genova, we have shown that the concept of scale covariance is, indeed, useful in explaining observed scaling properties of rainfall data. Thus, although based on a single time series, our results show that the log-Poisson may represent a good estimate for the anomalous scaling exponent. 4. We have compared observed time series against synthetic rainfall data, both having the same anomalous exponents and the same statistical samples. The comparison is quantitatively performed by looking at the statistical analysis of extreme events. The results we obtained allow us to argue that a multifractal description of rainfall data, in terms of scale covariance field, is an extremely good approximation to the observed statistical properties of rainfall time series. 5. We have shown that the rainfall field, as produced in meteorological models as limited area models, displays anomalous scaling in rather good agreement with respect to the observed ones. Our analysis is somewhat limited by the small number of data sets examined. However, this result is worth mentioning since, although microphysical processes commonly applied in meteorological models are grossly parameterized, the LAM precipitation fields analyzed here preserve statistical properties observed in real rainfall. Another result that comes from the multifractal analysis on the GATE fields is that statistical behavior of precipitation in space is dependent on the time duration over which rainfall is accumulated; as expected,
precipitation fields display a stronger intermittency for shorter time durations. In this paper we have used a rather simple log-Poisson model to fit observed anomalous exponents. We remark that other models may well achieve the same results; the logPoisson distribution has been used both for its simplicity and for its physical meaning in the context of scale covariance fields (see section 2). Moreover, by using the log-Poisson distribution in our model to simulate random rainfall, the statistical properties of synthetic fields can be tuned with only two parameters for each scaling regime found in real data. This represents a real advantage with respect to cluster-based models like the WGR model [Waymire et al., 1984] that require up to 10 parameters to be estimated. In addition, the multifractal approach introduces relevant statistical constraints over a wide range of scales on the moments of signals. Further research on rainfall time series could investigate scale anomalies and the presence and localization of thresholds also at different sites and the possible dependence of statistical properties and model parameters on geographical, orographic, and climatic conditions. Some open problems require further research and interactions in various areas and disciplines. A first open problem is strictly heuristic and concerns the physical interpretation of the multifractal behavior and intermittency observed in rainfall. Other problems are related to the use and application of precipitation fields from numerical weather prediction models as input to hydrological rainfall-runoff distributed models for forecasting flood events. It is well known that meteorological models supply forecasts on 100 or 1000 km2 grid-size areas and hourly timescales. Rainfall-runoff models often require smaller resolution both in time and space, especially when applied to small basins. In this framework a first issue to be resolved is how to tie together results discussed separately here with spatial and temporal statistical properties of rainfall in order to apply the model directly for space-time downscaling of precipitation. Acknowledgments. The authors thank Claudio Paniconi and two anonymous reviewers for their helpful comments on this paper. The research of the first author was supported in part by the Sardinia Regional Authorities. This work was founded by the Gruppo Nazionale per la Difesa dalle Catastrofi Idrogeologiche (GNDCI), Research Area 3.
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(Received May 27, 1998; revised November 30, 1998; accepted February 2, 1999.)
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