The areal reduction factor (ARF) ? is a key quantity in the design against hydrologic extremes. For a basin of area a and a duration d, ?(a, d, T) is the ratio between the average rainfall intensity in a and d with return period T and the average rainfall intensity at a point for the same d and T. Empirical ARF charts often display scaling behavior. For example, for large (/d) ratios and given T the ARF tends to behave like (/d)-a for some a. Here we obtain scaling properties of the ARF under the condition that space-time rainfall has multifractal scale invariance. The scaling exponents of the ARF are related in a simple way to the multifractal properties of the parent rainfall process. We consider regular and highly elongated basins, quantify the effect of rainfall advection, and investigate the bias from estimating the ARF using sparse rain gauge networks. We also study the effects of departure of rainfall from exact multifractality. The results explain many features of empirical ARF charts, while suggesting dependencies on advection, basin shape, and return period that are difficult to quantify empirically. The theoretical scaling relations may be used to extrapolate the ARF beyond the empirical range of a, d, and T.
 In hydrological risk analysis and design, one is often interested in the rainfall intensity averaged over a region of area a and duration d, with return period T. Plotting such extreme rainfall intensity I(a, d, T) against d for given a and T produces so-called intensity duration area frequency (IDAF) curves. For a ? 0 (precipitation at a point), the IDAF curves reduce to the familiar intensity duration frequency (IDF) curves [see, e.g., Singh, 1992, pp. 905-908]. While various definitions of T are in use, the one that makes best sense for many hydrologic applications and is easiest to handle analytically is the reciprocal of the exceedance rate [e.g., Willems, 2000; Veneziano and Furcolo, 2002a]. Accordingly, I(a, d, T) is the intensity i that satisfies
where I(a, d) is the average intensity in (t, t + d).
 Direct estimation of the IDAF curves from rainfall data requires very long records from spatially dense rain gauge networks or radar, which are seldom available. A common strategy to avoid direct estimation is to express I(a, d, T) as the product of the IDF value I(d, T) and the areal reduction factor (ARF) ?(a, d, T) = . Advantages of this factored approach are that the IDF values can be found using long records from single pluviometric stations (which are available at many locations) and, if the ARF does not vary much in space, the function ?(a, d, T) needs be estimated just once. In the literature, two different types of ARFs are found: the storm centered ARF and the fixed area ARF [Hershfield, 1962; Omolayo, 1993]. The storm centered ARF is associated with rainfall intensity within the isohyets of specific storm events, and does not have a precise return period interpretation. By contrast the fixed area ARF is obtained as the ratio of return period rainfall intensities over a fixed area and at a point. Hence the fixed-area definition of the ARF is better suited for hydrologic risk analysis, and is the one used throughout this paper.
 An alternative to empirical IDF, IDAF and ARF estimation is to use theoretical analysis based on a random process representation of rainfall. Some studies have derived properties of the IDAF curves and the ARFs using nonscaling representations of rainfall. An early attempt in this direction was made by Roche , who developed a theoretical approach to point and areal rainfall based on the correlation structure of intense storms. Rodriguez-Iturbe and Mejia  extended Roche's  approach by assuming that the rainfall field is a zero mean stationary Gaussian process. A different approach to ARF estimation, based on crossing properties of random fields, was proposed by Bacchi and Ranzi . Properties of extremes of random functions were used also by Sivapalan and Blöschl . Finally, Asquith and Famiglietti  derived the ARF as the catchment average of the ratio between the T-year rainfall depths at distance r from the centroid of the storm and at the centroid itself.
 Several other studies have assumed that rainfall intensity has scale invariance and used multifractal analysis to derive scaling properties of the IDF curves and ARFs with a, d and T. De Michele et al.  have argued directly that the annual maximum value of I(a, d) could scale in a self similar or multifractal way with a and d. They then focus on the self similar case and specify the form of I(a, d, T) by reasoning on the limiting behavior when a ? 0 and a or d ? 8. By contrast, Hubert et al.  and Veneziano and Furcolo [2002a] derive scaling properties of the IDF curves with d and T from the condition that rainfall has multifractal scale invariance. In a recent study, Castro et al.  developed a multiftractal approach to explain how the IDAF values scale with a, d and T. Although not explicitly stated, the analysis of Castro et al.  is valid only for large values of T [see Langousis, 2004].
 In this paper, we study the behavior of the IDAF curves and the areal reduction factor ? under exact and approximate multifractality. We also analyze how these quantities depend on basin shape and rainfall advection and quantify the distortions in ARF scaling caused by the common practice of estimating area rainfall from sparse pluviometric networks. Finally, we show how the theoretical results explain many features of empirical ARFs.
 Since in certain limiting cases to be considered later (such as sampling along a line segment) the basin has finite extent but zero area, it is convenient to parameterize the basin through its shape S and largest linear dimension l rather than area a. For example, a basin could have in approximation the shape S of a disc, a square or a rectangle with a given aspect ratio. Together, S and l define the planar geometry of the basin (except for rigid translation, rotation, or reflection). Accordingly, we use the notation I(S, l, d, T) and ?(S, l, d, T) in place of the less descriptive I(a, d, T) and ?(a, d, T).
 In hydrologic applications, the averaging duration d is often related to the response time of the basin, for example the concentration time, but in general the region of interest needs not be a river basin and d needs not refer to the travel time of water particles. While hydrologic extremes remain the main practical focus of our analysis and, consequently, we refer to the geographical region as a “basin,” results hold beyond this application context.
2. IDAF and ARF Scaling Under Multifractal Rainfall
Veneziano and Furcolo [2002a] analyzed the IDF curves when temporal rainfall is a stationary multifractal process. Multifractality means that, for any given duration d and scaling factor r = 1,
where I(t|d) is the mean rainfall intensity in [t, t + d], Br is a nonnegative random variable with mean value 1 whose distribution depends on r, and denotes equality of all n-dimensional distributions (statistical equivalence of the two processes) (see, for example, Gupta and Waymire  or Veneziano ). The distribution of Br, which determines the scaling properties of the rainfall process, can be characterized by the moment-scaling function K(q) = logr(E[Brq]). For example, if Br has lognormal distribution, then K(q) = C1(q2 - q) where C1 = Var[logr(Br)] is a parameter. Equation (2) expresses the property of multifractal scale invariance. Because of stationarity, I(t|d) is also invariant with respect to shifts of the time parameter.
The constants ?1, q1 and q1* in equation (3) can be found from the function K(q) as shown in Figure 1: ?1 is the slope of the tangent to K(q) with Y intercept equal to −1, q1 is the value of q at the point of tangency, and q1* is the order above which the moments of I(t|d) diverge (q1* can be found as the value of q > 1 such that K(q) = q - 1). For example, in the case of a lognormal multifractal process with parameter C1, these constants are ?1 = 2 - C1, q1 = 1/, and q1* = 1/C1.
 Next we extend the analysis of Veneziano and Furcolo [2002a] from point to spatially averaged rainfall. While the extension is in many ways straightforward, new elements to be considered are that the scaling of the IDAF values depends on the linear size l and shape S of the basin and the rainfall advection velocity vad.
2.1. Lagrangian Scaling of Multifractal Space-Time Rainfall
 The case of no advection corresponds to working in a Lagrangian reference that moves with the rainfall field. In that reference, rainfall is assumed to be a stationary random measure with isotropic multifractality in space and time. In analogy with equation (2), this means that for any basin (S, l), duration d and scaling factor r = 1,
where I(t|S, l, d) is the mean rainfall intensity inside the basin (S, l) during [t, t + d] and all other notation is as in equation (2).
 Multifractal scaling applies below some maximum region size lmax and duration dmax, which might represent the size and duration of the largest organized rainfall features at the synoptic scale. It is then convenient to use lmax and dmax to render all length and time variables dimensionless. For example, L = l/lmax and D = d/dmax are the dimensionless basin size and averaging duration. When using such dimensionless quantities, equation (4) becomes
To further discuss the scaling properties of space-time rainfall, we introduce two quantities with the physical dimension of velocity [length/time]. One is the “rainfall evolution rate” vrain = lrain/drain, where lrain and drain are the linear size and lifetime of organized rainfall features such as convective cells, cell clusters or mesoscale precipitation regions. For what follows, it is not important to specify which of these features lrain and drain refer to, because what matters is the ratio vrain = lrain/drain and, under isotropic multifractality, vrain is the same for all such features. In particular, vrain = lmax/dmax.
 The other quantity is the “response velocity” vres = l/d, where l and d are the maximum linear size of the basin and the duration of rainfall averaging (as stated in the Introduction, d is a duration of interest, which may or may not correspond to the hydrologic response time of the basin). What matters for the analysis that follows is the relative magnitude of vrain and vres, as expressed by the dimensionless “response velocity parameter” ures = vres/vrain = = L/D, where L and D are the normalized quantities introduced above. The parameter ures indicates whether the response of the basin is faster (ures > 1) or slower (ures < 1) than the evolution rate of the rainfall features. For example, if d is set equal to the concentration time of the basin (on the concentration time [see, e.g., Viessman and Lewis, 2003, p. 265]), then vres ranges approximately between 3 and 8 km/h depending mainly on the average slope of the basin; for a detailed analysis, see Langousis . On the other hand, the rainfall evolution rate vrain ranges from about 5 to 20 km/h [see, e.g., Austin and House, 1972; Orlanski, 1975] (see also the review by Langousis ). Hence ures may be either smaller or larger than 1.
 We now return to the multifractal scaling property in equation (5). That property can be represented graphically by noting that rainfall intensity has “Br multifractal scaling” along 45° lines on the [ln(L), ln(D)] plane (see Figure 2a). Each 45° line in Figure 2a is characterized by one value of the response velocity parameter ures = L/D. The line with ures = 1, denoted by U1, corresponds to square regions on the (L, D) plane and has special importance. Below U1 is the “fast response region” where ures > 1 and above U1 is the “slow response region” where ures < 1. The regions where ures » 1 (in practice, ures larger than about 5) and ures « 1 (ures smaller than about 1/5) will be referred to as the very fast and very slow response regions, respectively. In Figure 2 we have made use of rectangles with side lengths L and D to visually illustrate the relative value of these quantities and conditions when ures = L/D is smaller or larger than 1.
 Multifractality is expressed by the scale invariance property in equation (5). It is important to notice that in the right hand side of that equation L and D are stretched by the same factor r (hence also the physical length l = Llmax and physical averaging duration d = Ddmax are both stretched by r). However, more general scaling relations hold in good approximation in the very fast and very slow response regions. When ures » 1, the temporal process I(rt|S, rL, rD) in equation (5) is insensitive to D, provided that D remains much smaller than L [see Veneziano and Furcolo, 2002b]. The basic reason is that for D « L the temporal correlation of rainfall intensity within an interval of duration d = Ddmax is much closer to 1 than the spatial correlation along a spatial segment of length l = Llmax. Using this property and equation (5), one obtains
The scaling factor rD in equation (6) is arbitrary, provided that (rLL)/(rDD) » 1.
 For ures « 1, L is much smaller than D and a similar argument gives the following scaling relation, which is symmetrical to equation (6):
In this case rL is arbitrary, provided that (rLL)/(rDD) « 1.
 The scaling relations in equations (6) and (7) are shown schematically in Figure 2b. What is given along each arrow in Figure 2b is the scaling factor in each transformation of L and D (“1” means that the distribution is unchanged by the transformation). For example, rain gauge records have minimal area coverage; hence scaling in time is of the multifractal type, as given by equation (7) and shown at the top left of Figure 2b. In the region around the U1 line (for ures in the approximate range [1/5, 5]), equations (6) and (7) do not apply. Rather, there are complicated transformations TD,r and TL,r in the directions of the log(D) and log(L) axes, which in combination produce Br multifractal scaling along 45° lines; see dashed triangle in Figure 2b.
 Next we derive the scaling properties of the IDAF curves and ARFs in the very fast and very slow response regions. We start with two limiting basin shapes, a square (or disc) and a line segment, and then discuss the case of general rectangular regions. Section 2.2 assumes no advection, and section 2.3 extends the results to vad ? 0.
2.2. IDAF and ARF Scaling: No Rainfall Advection
 In the very slow response region equation (7) holds and recalling equation (3) one finds that, for basins of given shape S,
Notice that equation (3) holds for temporal rainfall (in this case the Euclidean space dimension is n = 1 and the critical moment order is q1*), whereas equation (8) is for space-time regions of any Euclidean dimension n (e.g., n = 2 if rainfall is observed along a line segment and n = 3 if rainfall is observed inside a region with positive area). In the latter case, the order of moment divergence qn* is the value of q > 1 such that K(q) = n(q - 1) (see Figure 1). For example, in the case of lognormal multifractal rainfall, qn* = n/C1. The reason why the right hand side of equation (8) does not contain L is that, for ures « 1, I is insensitive to the size of the basin.
In this case, I is insensitive to the averaging duration. Equations (8) and (9) give asymptotic scaling properties for the IDAF values. Next we use these properties and the time-only results in equation (3) to derive scaling relations for ?(S, L, D, T) = .
 In the very slow response region (for L/D small), equations (3) and (8) give
whereas in the very fast response region (for L/D large), equations (3) and (9) give
Equations (10) and (11) show that ? depends on the response velocity L/D, the return period T and, through n in qn*, the shape of the observation region. Notice that, when the response velocity parameter ures = L/D is small, the ARF does not depend on L and D and is close to 1. If on the other hand ures is high, the ARF becomes a power function of L/D, whose exponent depends on T. These properties correspond to features typically observed in empirical ARFs [see, e.g., NERC, 1975; Koutsoyiannis, 1997; Asquith and Famiglietti, 2000; De Michele et al., 2001].
 Before we include advection, we briefly mention two issues related to the algebraic tail of the rainfall intensity distribution and its implications on how ? behaves for large T. Consider a generic rectangular space-time region (L1 × L2 × D) with L1 = L2. If rainfall is a fully developed multifractal process and q3* < 8, then for L2 > 0 the marginal distribution of I(t|L1, L2, D) has a “q3* upper tail” of the type P[I(t|L1, L2, D) > i] ~ Hence the asymptotic scaling results for square regions (n = 3 in equations (8)–(11)) hold also for general rectangular regions. However, the upper tail of the average rainfall intensity in a rectangular region may include first a range with algebraic q1* behavior, followed by a range with q2* behavior and finally by the extreme q3* tail, as illustrated in Figure 3. A limited q1* tail develops if one of the three dimensions (L1, L2, or D) clearly dominates over the other two and a limited q2* tail develops if one of the three dimensions is much smaller than the other two. For example, a sequence of q1*, q2*, and q3* tail regimes exists if L1 » L2 » D. Similar considerations apply when observing rainfall on a line segment. In this case, if L » D or L « D, there is a nonextreme q1* tail that precedes the extreme q2* tail.
 The second issue is that fully developed multifractal processes like those we have considered up to now have singularities that cannot exist in nature. The singularities are due to oscillations at subobservation scales which, when continued to infinite resolution, cause divergence of the moments of I(t|S, L, D) of order q = qn*. More plausible models of rainfall are multiplicative cascades developed to a finite resolution or, as we shall consider later, “bounded” cascades in which the fluctuations at finer scales have decreasing amplitudes [Menabde et al., 1997]. In either case, the distribution of I(t|S, L, D) does not have an algebraic upper tail and for T ? 8 the IDAF and ARF values do not have (an exact) power law dependence on T.
 The main conclusion from these considerations is that the scaling relation ? ? T with n > 1, which is predicted by theory for T ? 8, may not apply in reality or may occur for return periods that are too large to be of practical interest.
2.3. Effect of Advection
 To our knowledge, the effect of rainfall advection on the IDAF curves and the ARF values has not been previously studied. This effect largely depends on the “advection velocity parameter” uad, defined as the dimensionless ratio uad = vad/vrain between the advection velocity vad and the rainfall evolution rate vrain. As before, we assume that in a Lagrangian reference that tracks the rainfall motion, rainfall intensity satisfies the multifractal scale invariance condition in equation (5). To determine the effect of vad on the IDAF and ARF values, one must find how advection changes the shape and size of the rainfall averaging regions from a Eulerian (fixed) to a Lagrangian (moving) reference frame and then use results from section 2.2 for the Lagrangian regions. Here we do so for rainfall observed at a geographical point, along a line segment, or over a disc. Primed symbols denote quantities in the Lagrangian reference. We work with dimensionless length L and duration D, but analogous relations hold for the unnormalized quantities l and d.
2.3.1. Observation of Rainfall at a Point
 As shown in Figure 4a, when averaging advected rain during a period D, the averaging segment in the Lagrangian reference has length D' = = D. Next we will show that the Lagrangian return period T' that corresponds to the Eulerian T is T' = T.
 Consider rainfall intensity at a fixed geographical point, averaged in an interval of duration d. In T units of time, there are n = T/d such intervals; hence I(d, T) is the intensity i such that P[I(d) > i] = 1/n = d/T. In the case when uad > 0, I(d) is the average of the Lagrangian rainfall intensity field over a segment of length d' = d (see Figure 4a). In T units of time, the sampling point in the Lagrangian reference covers a segment of length T' = nd' = = T. Therefore I(d, T|uad) = I(d, T|uad = 0).
 As a consequence of this analysis, if for uad = 0 the IDF value varies with D and T as D-aTβ, then for uad ? 0 the IDF values must be multiplied by (1 + uad2). Since a > β (see equation (3)), this factor is smaller than 1.
2.3.2. Averaging Rainfall Along a Line Segment
 Suppose now that rainfall is observed along a line segment of length L parallel to the y axis during a period D. In this case the Lagrangian space-time averaging region is a parallelogram with side lengths L' = L and D' = D; see Figure 4b. The Lagrangian return period is T' = T, as in the case of sampling at a point. To understand the implications of these transformations on the IDAF curves and the ARF values, we consider the limiting cases when ures » 1 (very fast basin response relative to the rainfall evolution rate) and ures « 1 (very slow basin response) and denote by uad,x and uad,y the components of the normalized advection velocity vector in the x and y directions, respectively.
 For ures » 1, the Lagrangian observation parallelogram is highly elongated in the spatial direction and is approximated well by a rectangle with side lengths L' = L and D' = D, as shown in Figure 5a. For ures « 1 the parallelogram is highly elongated in the temporal direction and is approximated well by a rectangle with side lengths L' = L/ and D' = D; see Figure 5b. In summary, the parameter transformations from a Eulerian to a Lagrangian coordinate system are
Notice that, when sampling along a line segment, the effective parameters depend not only on the magnitude but also on the direction of rainfall advection relative to the sampling line.
2.3.3. Averaging Rainfall Over a Disc
 When sampling over a disc of diameter L, the direction of rainfall advection does not matter; see Figure 4c. In this case the effect of advection is to change L, D and T to L', D', and T' given by
Notice that when ures « 1 the averaging Lagrangian region is approximated as a cylinder with circular basis of area a' = a/. Scaling results for IDAF and ARF including advection are obtained by replacing L, D, and T on the right hand sides of equations (8)–(11) with the expressions for the effective parameters L', D' and T' in equations (12) and (13). The final results are summarized in Table 1 for very elongated basins (approximated as line segments) and Table 2 for regular basins (approximated as discs).
Table 1. Scaling of the IDAF Curves and the ARFs for Very Elongated Basins of Dimensionless Length La
Very Slow Response Region: ures 1
Very Fast Response Region: ures 1
The effect of advection is included through the parameter uad ? 0.
I(t|L, D) rDD), rL, rD = 1 (no dependence on L)
I(t|L, D) rDD), rL, rD = 1 (no dependence on D)
i(L, D, T, uad) ? (no dependence on L)
i(L, D, T, uad) ? (no dependence on D)
?(L, D, T, uad)
?(L, D, T, uad) ?
?(L, D, T, uad) ?
Table 2. Scaling of the IDAF Curves and the ARFs for Regularly Shaped Basins, Assimilated to Discs of Dimensionless Diametera
Very Slow Response Region: ures 1
Very Fast Response Region ures 1
The effect of advection is included through the parameter uad.
I(t|L, D) rDD), rL, rD = 1 (no dependence on L)
I(t|L, D) rDD), rL, rD = 1 (no dependence on D)
i(L, D, T, uad) ? (no dependence on L)
i(L, D, T, uad) ? (no dependence on D)
?(L, D, T, uad)
?(L, D, T, uad) ?
?(L, D, T, uad) ?
Tables 1 and 2 show that advection does not change the asymptotic algebraic behaviors of the ARF with D, L, and T. However, advection affects the prefactors of those asymptotic relations. To appreciate the practical importance of this effect, consider typical ranges of the velocity parameters vrain and vad. As mentioned earlier, vrain varies approximately from 5 to 20 km/h. The advection velocity vad usually takes values between 30 and 50 km/h at small scales (a few kilometers) and between 20 and 40 km/h at large scales (100 or more kilometers) [see, e.g., Martin and Schreiner, 1981; Kawamura et al., 1996; Deidda, 2000] (see also the review by Langousis ). One concludes that uad varies from 0 to almost 5, and thus the effect of advection may be as large as a factor of 2 on the ARF.
2.4. Numerical Validation
 We conclude this section by numerically validating the theoretical results on the ARF, first for vad = 0 and then for vad ? 0. For the case without advection we use a binary cascade representation of rainfall in two spatial dimensions plus time. The model has lognormal generator Br and moment-scaling function K(q) = C1(q2 - q) with parameter C1 = 0.1. The outer scale of multifractal behavior is fixed to 29 cascade cells in each spatial and temporal direction. Hence dmax = lmax = 29. Simulation is limited to the parallelepiped with spatial dimensions 26 × 26 and temporal dimension 29, and the basin is assumed to be a square with side length at most 25 cells. Similarly temporal averaging is over at most 25 cells. This means that L and D range from 2-9 to 2-4. The 26 × 26 × 29 parallelepiped might represent the rainy season of one year. While this is a highly idealized representation of rainfall, it should suffice for the purpose of validating the theoretical results.
 Numerical estimation of the ARF requires calculation of average rainfall intensities at the catchment and rain gauge scales, the latter assimilated to a point. To obtain these averages, the cascade is generated down to unit space-time cells and then differently “dressed” to produce areal average and point values, as described below.
 Denote by Ib(x, y, t) the “bare” rainfall intensity in the unit tile centered at (x, y, t). This is the rainfall intensity obtained at level 9 of the cascade construction procedure. The actual (“dressed”) rainfall intensity at that unit scale is obtained as
where the subscripts b and d stand for bare and dressed, respectively, and Z3 is the dressing factor for the three-dimensional cascade.
 Consider now a rain gauge inside this tile, for example, at location (x, y). During the unit time interval centered at t, the average rainfall intensity ID,1(x, y, t) measured by the rain gauge is
Equation (15) is analogous to equation (14), except that Z1 is the dressing factor of a one-dimensional cascade with the same cascade generator as the three-dimensional cascade.
 For each simulated season, one can numerically estimate the ARF values for different L and D as the ratio between the maximum average rainfall inside the basin and at a point. Figure 6a shows the iso-ARF lines obtained by averaging the rainfall maxima for different L and D over 10 independently simulated seasons. These values have a return period T of about one season (or 1 year).
 In Figure 6a one observes that the isolines are essentially straight with a 45° slope, as predicted by theory. Also the theoretical scaling relation for large L/D (with exponent ?1 = 0.532), is very closely matched by the simulation results (see Figure 6b). An equally good correspondence between simulation and theoretical results has been obtained using more general beta-lognormal cascades, which are able to represent the alternation of rainy and dry space-time regions [Langousis, 2004].
 The validation of results with nonzero advection is computationally more demanding because the simulation region must be large enough to include the slanted rainfall observation region in the Lagrangian reference. To reduce the numerical effort, we consider the case when rainfall is observed along a line segment of length L and advection is parallel to that segment. The rainfall model is a two-dimensional (one space dimension plus one time dimension) binary lognormal cascade and simulation is in a 210 × 210 Lagrangian region. Hence dmax = lmax = 210. Except for the different size of the simulation region and the lower dimensions (2 rather than 3), the rainfall model is identical to that for no advection.
 In analogy with equations (14) and (15), the dressed measures ID,2(y, t) and ID,1(y, t) needed to calculate the ARF are obtained as
Empirical estimates of the ARF have been obtained by averaging extreme rainfalls over 20 independently simulated seasons. Figure 7 compares results for vad = 0 (no advection, lower curve), vad = 4 space units per unit time, and vad = 8 space units per unit time. The theoretical effects of advection can be found from Table 1 for the case uad,x = 0. For C1 = 0.1, one obtains ?1 = 2 - C1 = 0.532 and q1 = 1/ = 3.162. Moreover, for ures » 1, T finite and L ? 0, Table 1 gives the advection correction factor (1 + uad2)0.108. Hence the curves for uad = 4 and the uad = 8 in Figure 7 should be shifted upward by 0.44 and 0.65, respectively, relative to the uad = 0 case. The numerical results agree very well with these theoretical predictions.
3. Deviations From Multifractality
 Stationary multifractal fields result from cascade constructions in which nonnegative fluctuations Yj(x, y, t) at different scales sj = s0r-j are multiplied. Here s0 > 0 and r > 1 are constants and j = 1, 2,... is the cascade level. A necessary condition for scale invariance is that the fluctuations Yj(x, y, t) be statistically identical to Y(rjx, rjy, rjt), where Y(x, y, t) is some nonnegative mean-1 stationary process called the generator of the cascade.
 The construction of a bounded cascade is identical to that of a multifractal cascade, except that the standard deviation of the generator sy (or some other dispersion measure like the C1 coefficient) decreases as the cascade level j increases. To illustrate the effect on the ARF, we assume that rainfall is a lognormal bounded cascade in space and time, with multiplicity 2 in all three coordinate directions and a generator Bj that varies with the cascade level j as
Figure 8b shows how the ARF varies with L and D. Figure 8b should be compared with Figure 6a, which displays similar results under multifractality. Notice that sB(0) = 0.385 corresponds to C1 = 0.1; hence the two cascade models have the same variability at the largest scale (see Figure 8a). Relative to Figure 6a, the contour lines in Figure 8b are displaced upward because the bounded-cascade process is smoother than the multifractal process. Therefore, for small L, spatially averaged rainfalls in the bounded cascade are nearly identical to point rainfalls. A second important effect is that the contour lines in Figure 8b are not straight, reflecting lack of scale invariance of the bounded cascade. In particular, for large L and small D the lines are very flat (since further reducing D does not affect much the rainfall averages) and their slope increases toward 1 as L decreases or D increases.
4. Effect of Sparse Spatial Sampling
 When the ARF is estimated from rain gauge measurements, as is typically done in practice, the rainfall intensity in a region is estimated as the average (or weighted average) of rain gauge measurements at points inside the region. Unless the rain gauge spacing varies proportionally to the size L of the region, this operation destroys any scaling property the ARFs might have.
 Sparse point sampling can be easily simulated. Suppose that the resolution of the cascade simulation is such that at most one rain gauge site falls inside each cascade tile. Then the only difference with the procedure described in section 2.4 is that one must multiply the “bare” rainfall intensity in the cascade tile that hosts a rain gauge by the one-dimensional random dressing factor Z1 instead of the three-dimensional factor Z3.
 To illustrate the effect of sparse spatial sampling, we use again the three-dimensional lognormal cascade model of Figure 6a. Figure 9 shows ARF results when the rain gauge stations are arranged on a regular square grid with a density of 1 station per four cascade tiles. Comparison with Figure 6a shows that for large L the ARF values are not influenced by sparse sampling. However, significant differences are evident for small L. In the limiting case when L equals the interstation distance (this happens here for log2(L) = 1), the spatially averaged rainfall is estimated as the rainfall at the only station inside the region. This is why the ARF in Figure 9 is identically 1 along the lower boundary. The contour lines of the ARF, which have a 45° slope for large L, must necessarily bend to remain above this horizontal ARF = 1 line. It is emphasized that in this case the curvature of the ARF contour lines is due to lack of scaling of the observation grid not lack of scaling of the rainfall field. Hence the differences between Figures 6a and 9 reflect bias due to sparse sampling.
5. Interpretation of Empirical Areal Reduction Factors
 To conclude, we examine features of empirical ARFs in the light of previous model-based results. For this purpose, we use the ARF data of the Natural Environmental Research Council [NERC, 1975]. The data comprise ARF estimates from thirteen basins in the United Kingdom, with areas ranging from 10 to 18,000 km2 and durations from 2 min to 25 days; see Table 3. According to NERC, these ARFs refer to rainfall events with return periods of 2–3 years.
NERC  interpolated and extrapolated the original ARF values to produce charts that cover a wider range of catchment areas (from 1 to 30,000 km2) and averaging durations (from 1 min to 25 days). The interpolated values fit well the original data for some but not all combinations of a and d. To more faithfully reflect the original data, we have reinterpolated the original values in Table 3 using first-order triangulation. The results are shown in Figure 10.
 One may distinguish four regions in Figure 10 where the ARF contour lines have different behaviors. Region 1 displays simple scaling of the ARF with a and d. Specifically, the ARF is constant for d ? (for d ? l, considering that basin shape is essentially independent of basin size). This agrees with results obtained in sections 2.1 and 2.2 under the assumption that rainfall is multifractal in space and time.
 In region 2 the contour lines become flatter as a or d decreases. This is also what happens if, at small space-time scales, rainfall behaves like a bounded cascade; see section 3. In region 3 the contour lines have higher curvature and become nearly parallel to the d axis for small a and d. Sparse spatial sampling produces a similar effect, as l approaches the interstation distance; see section 4. Finally, in region 4 where d/ is large, the contour lines are more widely spaced than under exact multifractality. Langousis  has shown that this feature could be due to high lacunarity of the rain support at synoptic and mesoscales. However this is only a tentative conclusion, since the behavior of the ARF in region 4 is poorly constrained by the data.
 Next we show how the results in Figure 10 can be quantitatively reproduced. To reduce the computational effort, we focus on the region with a in the range 30–1000 km2 and d in the range 15 min to 6 hours. This includes the various subregions mentioned above except region 1, where the ARF behaves consistently with multifractal cascades and need no further confirmation.
 For regions 2 and 3, we use a bounded Lognormal (LN) cascade representation of rainfall in two spatial dimensions plus time. The cascade has multiplicity 2 in all directions. The cascade generator Bj satisfies equation (17), where s(j)2 varies with the cascade level j according to Figure 11a. Figure 11a approximates the empirical findings of Menabde and Sivapalan  for temporal rainfall. For durations longer than those of Menabde and Sivapalan  we assume that the distribution of the generator is the same as that at the largest scale available.
 The numerical simulation procedure is the same as in sections 2 and 3, with tiles at the highest resolution representing space-time regions of area 1 km2 and duration hours. To model sparse spatial sampling, we calculate area intensities as averages at geographical points with regular spacing and a density of 1 rain gauge per 4 km2. This is comparable to the average density in the NERC data [see NERC, 1975, vol. IV, p. 24].
Figure 11b shows ARF results averaged over 10 independently simulated seasons. The contour lines in regions 2 and 3 are in good agreement with Figure 10. Notice in particular the high curvature in region 3, which is caused primarily by sparse sampling. In region 4 the agreement is not as good. As noted above, better agreement in this region can be achieved through the inclusion of large-scale lacunarity [Langousis, 2004]. However, in region 4 the ARF is close to 1. Hence its accurate determination is not critical in practice and the simple bounded lognormal model illustrated in Figure 11 should suffice.
 We have analyzed the scaling properties of the areal reduction factor (ARF) under the condition that space-time rainfall has exact or approximate multifractal scale invariance. We have considered regular and highly elongated basins, quantified the effect of rainfall advection, and investigated the bias when estimating the ARF from sparse rain gauge networks.
 We have found that under perfect multifractality the ARF has asymptotic scaling behaviors with L/D and T, where L is the largest linear size of the region of rainfall averaging, D is the duration of averaging, and T is the return period. Specifically, ARF ~ (L/D)-aT-β for (L/D) ? 8, (L/D) ? 0, or T ? 8. The nonnegative constants a and β depend somewhat on the geometry of the region (regular or highly elongated) and differ in the three limiting cases above, but are independent of rainfall advection and can be found easily from the multifractal properties of rainfall. The behavior for (L/D) ? 0 is simply ARF ? 1, whereas the other two limiting cases are nontrivial.
 The ARF depends on T in two ways: through the term T-β and through a, which has different values for T finite and T ? 8. The latter is usually the dominant influence. The effect of T on the ARF may be numerically important. This confirms qualitatively the findings of Bell , Asquith and Famiglietti , and De Michele et al. . A reason why empirical studies like that of NERC  failed to detect significant T dependence is that available space-time rainfall records allow ARF estimation over only a small range of return periods.
 For (L/D) ? 8 or T ? 8, the above scaling relationships have prefactors that depend on the rainfall advection velocity parameter uad and to a lesser extent the shape of the basin. For nonextremely elongated basins this prefactor is of the type (1 + uad2)c, where c is a positive constant with typical values between 0.1 and 0.5. Hence, depending on uad (see section 2.3), the effect of advection on the ARF may be as large as 2. To our knowledge, this is the first time that this effect has been quantified. Of course, advection is implicitly included in empirical estimates of the ARF, but its effect should be added when the ARF is theoretically estimated from nonadvecting rainfall models.
 We have studied the effect of basin shape by considering two limiting cases: basins with nearly circular or square shape and highly elongated basins that can be approximated as line segments. Basin shape affects the exponent β when T ? 8 and the prefactor in the case of advecting rainfall. These effects are generally small (that on β is important for T beyond the typical range of return periods encountered in practice). Also, very highly elongated basins are rare. Hence, for most applications, one may use the results for regularly shaped regions.
 Rainfall has been observed to deviate from perfect multifractality. The main deviation is that local intensity fluctuations are smaller than required for scale invariance. We have modeled this behavior by using “bounded cascades”, in which the fluctuations at smaller scales are progressively reduced in amplitude. As a result of this reduction, the ARF is closer to 1 and the scaling properties mentioned above are lost. In particular, the ARF no longer depends on L and D through the ratio L/D and its contour lines on the (log(L), log(D)) plane are no longer straight, becoming flatter as L and D decrease. A curvature of this type is often noted in empirical ARF charts.
 Another reason for the curvature of empirical contour lines is the bias induced by estimating area rainfalls from point (rain gauge) data. As L approaches the interstation distance, only one station is used to estimate area rainfall and the ARF is consequently calculated as 1. This saturation causes the ARF contours to bend in a way similar to the case of bounded cascades. We have found that both deviations from multifractality and sparse sampling bias affect NERC's  empirical ARF charts.
 Although this study covers a wide range of factors affecting the ARF, some issues remain unexplored. One is the existence and effect of anisotropic scaling of rainfall in space and time [see, e.g., Venugopal et al., 1999]. Qualitatively, anisotropic scaling changes the 45° slope of the contour lines on the (log(L), log(D)) plane. No such tilt was observed in NERC's  ARF results. However, the absence of scaling anisotropy (and other aspects of rainfall and ARF modeling discussed in the paper) should be confirmed through additional rainfall data analysis.
 Sparseness of the NERC  data set did not allow us to adequately investigate the ARF behavior for large D/L ratios. While also this issue could be resolved by using more extensive data sets, the fact that for large D/L the ARF is close to 1 makes its resolution less critical for practical applications.
 This work was supported in part by the Department of Civil and Environmental Engineering of MIT under the Schoettler Fellowship, in part by the National Science Foundation under grant EAR-0228835, and in part by the Alexander S. Onassis Public Benefit Foundation under scholarship F-ZA 054/2004-2005.