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  • Selker, J. S., and D. A. Haith, 1990: Development and testing of single-parameter precipitation distributions. Water Resour. Res.,26, 2733–2740.

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  • Wilks, D. S., 1989: Rainfall intensity, the Weibull distribution, and estimation of daily surface runoff. J. Appl. Meteor.,28, 52–58.

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  • View in gallery

    Epstein’s model for point-to-area rescaling of precipitation probabilities consists of a circular area of averaging, A, and circular precipitation cells, each covering area C. The forecaster’s task is to judge the quotient Q = C/A. The uncertain wetted area is D.

  • View in gallery

    Relation between point probability πO and area probability πA of precipitation occurrence as a function of the cell/area quotient Q, based on Epstein’s model.

  • View in gallery

    Relation between the cell/area quotient Q, to be judged by the forecaster, and the variance reduction factor τ2, for several values of the point probability πO of precipitation occurrence.

  • View in gallery

    Relation between the pattern certainty factor F, to be judged by the forecaster, and the variance reduction factor κ2, for several values of the ratio of the point probability to the area probability πO/πA.

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Point-to-Area Rescaling of Probabilistic Quantitative Precipitation Forecasts

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  • 1 Department of Systems Engineering and Division of Statistics, University of Virginia, Charlottesville, Virginia
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Abstract

A probabilistic quantitative precipitation forecast (PQPF) is prepared judgmentally by a meteorologist based on a guidance PQPF. The predictand of a judgmental PQPF is the spatially averaged precipitation amount. The predictand of a guidance PQPF produced by a statistical model is the point precipitation amount. Therefore, a procedure is needed for point-to-area rescaling of the PQPF. Theoretically based equations for rescaling are presented. The equations incorporate two predictive parameters, which characterize the precipitation field being forecast: the quotient of the area covered by a precipitation cell to the area of averaging (cell/area quotient), and the degree of certainty about the precipitation pattern (pattern certainty factor). Both parameters can be judgmentally quantified by the meteorologist during PQPF preparation. The same parameters can be entered into an inverse procedure for area-to-point rescaling of the judgmental PQPF.

Corresponding author address: Professor Roman Krzysztofowicz, University of Virginia, Thornton Hall, SE, Charlottesville, VA 22903.

Abstract

A probabilistic quantitative precipitation forecast (PQPF) is prepared judgmentally by a meteorologist based on a guidance PQPF. The predictand of a judgmental PQPF is the spatially averaged precipitation amount. The predictand of a guidance PQPF produced by a statistical model is the point precipitation amount. Therefore, a procedure is needed for point-to-area rescaling of the PQPF. Theoretically based equations for rescaling are presented. The equations incorporate two predictive parameters, which characterize the precipitation field being forecast: the quotient of the area covered by a precipitation cell to the area of averaging (cell/area quotient), and the degree of certainty about the precipitation pattern (pattern certainty factor). Both parameters can be judgmentally quantified by the meteorologist during PQPF preparation. The same parameters can be entered into an inverse procedure for area-to-point rescaling of the judgmental PQPF.

Corresponding author address: Professor Roman Krzysztofowicz, University of Virginia, Thornton Hall, SE, Charlottesville, VA 22903.

Introduction

Probabilistic hydrometeorological forecasting is rapidly advancing from theory to application (National Weather Service 1999; Krzysztofowicz 1998). One of the issues in developing an operational procedure for probabilistic quantitative precipitation forecasting (PQPF) is the spatial scale of the predictand. In some forecast systems, the predictand is defined as the point precipitation amount. For example, the guidance PQPF produced via the model output statistics technique is for point amounts (Bermowitz and Zurndorfer 1979; Carter et al. 1989). In other forecast systems, the predictand is defined as the spatially averaged precipitation amount. For example, the guidance QPF prepared by the Hydrometeorological Prediction Center based on numerical model outputs is for spatially averaged amounts (Olson et al. 1995).

The task of a field forecaster is to combine information from various guidance forecasts, numerical models, observations, and local analyses with his knowledge of local hydrometeorological influences into the final forecast (Krzysztofowicz et al. 1993). To aid the forecaster in performing this complex task, every guidance forecast should pertain to the same predictand. Hence there is a need for a rescaling procedure.

This article coalesces several theoretical results into an operational procedure for rescaling a PQPF of the point amount to a PQPF of the spatially averaged amount. (Rescaling in the opposite direction can be done by rearranging scaling equations, each of which is one-to-one.) In contrast to existing methods for rescaling climatic intensity-duration-frequency distributions of rainfall, whose primary purpose is hydrologic simulation studies, a procedure for rescaling the PQPF should satisfy two operational requirements: (i) it should incorporate essential predictive parameters of the current meteorologic situation, as judged by the forecaster, and (ii) it should be implementable in a field office. Meeting these requirements is the aim of the proposed procedure.

The next section defines the predictand and the PQPF. The subsequent three sections present procedures for rescaling elements of the PQPF: probability of precipitation occurrence, conditional distribution of the amount, and expected fractions of the temporal disaggregation. The sixth section formulates a model for judgmental quantification of the variance reduction factor. The last section summarizes the main properties of the overall procedure.

Probabilistic forecast

Predictand

Consider a fixed period beginning at a designated hour of the day and divided into n subperiods of equal or unequal length. Precipitation amounts to be defined are either point amounts or spatially averaged amounts.

Let W denote the total precipitation amount accumulated during the period. Let Wi denote the precipitation amount accumulated during the ith subperiod, i ∈ {1, . . . , n}. Thus
i1520-0450-38-6-786-eq1
Conditional on precipitation occurrence during the period, W > 0, define variate
i1520-0450-38-6-786-eq2
which represents a fraction of the total amount accumulated during subperiod i. Thus
i1520-0450-38-6-786-eq3
The vector of fractions (Θ1, . . . , Θn) defines the temporal disaggregation of the total amount into n subperiods. Because one of the fractions can always be expressed in terms of the remaining fractions through the unit sum constraint, only n − 1 fractions must be forecast. The predictand is the vector (W; Θ1, . . . , Θn).

Forecast

The forecast uncertainty is completely characterized in terms of an n-variate generalized distribution of the vector (W; Θ1, . . . , Θn). The term generalized distribution stems from the fact that each variate is discrete-continuous. Specifically, W = 0 may have probability p (0 < p < 1), while the remaining probability (1 − p) is spread over the unbounded interval (0, ∞) in accordance with some probability density function. Likewise, Θi = 0 and Θi = 1 may have probabilities p0 and p1, respectively (0 ⩽ p0 < 1, 0 ⩽ p1 < 1, 0 < p0 + p1 < 1), while the remaining probability 1 − p0p1 is spread over the open interval (0, 1) in accordance with some probability density function. Because direct assessment of an n-variate generalized distribution by a field forecaster is infeasible, an operational forecast specifies only key elements of the distribution. This forecast consists of two parts (Krzysztofowicz and Sigrest 1997).

The first part is a probabilistic forecast of total amount W. It specifies a probability of precipitation occurrence π during the period,
πPW
and a cumulative distribution function G of amount W, conditional on the hypothesis W > 0; with P standing for probability and ω for a fixed amount,
GωPWωWω
Practically, a continuous distribution is obtained by fitting a parametric model to three conditional exceedance fractiles (x75, x50, x25) of W that are assessed by the forecaster. With p denoting a probability number, 0 < p < 1, the 100p% conditional exceedance fractile of W is an estimate x100p such that P(W > x100p|W > 0) = p.
The second part is a deterministic forecast of temporal disaggregation (Θ1, . . . , Θn). It specifies a vector of expected fractions z = (z1, . . . , zn), conditional on the hypothesis W > 0; with E standing for expectation,
ziEiWin,
where 0 ⩽ zi ⩽ 1 for every subperiod i, and z1 + · · · + zn = 1.

In summary, the PQPF specifies π, G, and z, and these are the elements for which a rescaling procedure is desired.

Examples

In numerical examples presented throughout the paper, the spatially averaged amount is for the Lower Monongahela River basin above Connellsville, which covers 3429 km2 (1325 square miles) in Pennsylvania and Maryland. The point amounts are for three stations within the basin with the following abbreviations and elevations: Connellsville (Cn, 900 ft), Confluence (Cf, 1490 ft), and Sines Deep Creek (Sd, 2040 ft). The spatially averaged amounts were calculated from these and three other stations. The source data were hourly amounts recorded by rain gauges from 1943 to 1993. All examples are for months of March and July, 24-h period beginning at 1200 UTC, and disaggregation into four 6-h subperiods. The sample sizes (equal to the number of complete data records) are 725 for March and 937 for July. The area of spatial averaging is roughly comparable to 5000 km2 initially specified by the National Weather Service as the nominal scale for the PQPF. This is the scale at which the PQPF should be prepared and verified.

Henceforth, variables pertaining to point precipitation will acquire subscript O, and variables pertaining to spatially averaged precipitation will acquire subscript A.

Rescaling probability of precipitation

Area probability

Let πO = P(WO > 0) denote the point probability, that is, the probability of observing measurable precipitation at a fixed point O within the area, and let πA = P(WA > 0) denote the area probability, that is, the probability of observing measurable precipitation at some point within the area. A theoretical relationship between πO and πA was derived by Epstein (1966) under two assumptions: (i) both the area and the precipitation cells are circular, and (ii) the precipitation cells have identical diameter and are distributed at random over space large compared to the area of averaging (see Fig. 1). Then πO is identical at every point and
πA(1 − πO)[1+(1/Q)1/2]2
where Q = C/A is the quotient of the area covered by a precipitation cell, C, to the area of averaging, A; the cell/area quotient for short. One can see that πA > πO whenever πO < 1, and that πA converges to πO as Q increases. These relations are displayed in Fig. 2.

For example, when a small convective storm is forecast for which Q = 0.5 and πO = 0.3, then πA = 0.87. When a large cyclonic system is forecast for which Q = 5 and πO = 0.3, then πA = 0.53. The forecaster’s thought process can thus progress from the type of storm (convective vs stratiform) to the spatial character of precipitation (spotty vs widespread), to the judgment of the cell/area quotient Q. Judging Q is a simple cognitive task, relative to other tasks performed by forecasters. Hence (1) offers an operational method for rescaling point probabilities to area probabilities.

Given πO and πA, the cell/area quotient can be found from (1) as
Qγ1/2γ2
where γ = [ln(1 − πA)/ln(1 − πO)]. Climatic values of πO, πA, and Q are shown in Table 1. At each point, Q is higher in March than it is in July. This implies that, on average, the size of precipitation fields is larger in March than in July, as one would expect. In each month, πO and Q increase with the elevation of a point. This suggests that a spatial nonhomogeneity of precipitation occurrence can be attributed to local orographic effects.

Coverage fraction

The uncertainty about the occurrence of precipitation induces uncertainty about the spatial coverage, which is modeled as follows (see Fig. 1). Let D denote the area covered by precipitation cells within the area of averaging, so that 0 ⩽ DA. Then Φ = D/A denotes the fraction of area that is covered by the precipitation cells, and hence is wetted. Because D is uncertain, Φ is a random variable, constrained by 0 ⩽ Φ ⩽ 1. What one needs for further development is the conditional moments of Φ.

Let IO denote a Bernoulli variate indicating the occurrence of precipitation at a point and taking on value IO = 0 if WO = 0 or IO = 1 if WO > 0. By employing the total probability law, one can find the probability of precipitation occurrence at a point, conditional on the hypothesis that precipitation occurs within the area: P(WO > 0|WA > 0) = πO/πA. Consequently, the conditional mean and the conditional variance of the indicator are, respectively, E(IO|WA > 0) = πO/πA and var(IO|WA > 0) = (πO/πA)(1 − πO/πA).

When the probability of precipitation occurrence πO is identical at every point within the area, the conditional mean and the conditional variance of the coverage fraction Φ can be expressed in terms of the conditional moments of the indicator IO as follows:
i1520-0450-38-6-786-e2
where τ2 is the variance reduction factor such that 0 ⩽ τ2 ⩽ 1. Equation (2) follows from the fact that E(Φ|WA > 0) = E(IO|WA > 0). Equation (3), which is exact, has been derived in the appendix together with an approximate expression for τ2, which takes the form
i1520-0450-38-6-786-e4
and τ2 = 0 if πBπA. Probability πB lies within the interval πO < πB < 1 and is specified by the formula
πB(1 − πO){1+[(πO/πA)/Q]c/2}2
where c > 1 is a parameter. Its estimate ĉ = 1.7, to be used herein, has been determined empirically from plots of var(Φ|WA > 0) versus A reported by Seo and Smith (1996, Figs. 6 and 7) for precipitation fields observed by two radars over areas ranging from 140 km2 to 100 000 km2. Figure 3 shows the relation between πO, Q, and τ2; the order of calculations is (1), (5), and (4).

Table 1 shows climatic values of τ2 calculated from the climatic values of πO and Q. Also shown are E(Φ|WA > 0) and var(Φ|WA > 0), which are calculated according to (2) and (3). The conditional mean of the coverage fraction Φ is higher in March than in July at each point, as one would expect, and it increases with the elevation of a point in each month, which is the result of increasing πO. Interestingly, the conditional variance of Φ varies little; in fact, it appears to be one of the steadier spatial statistics of precipitation within the Monongahela basin.

In summary, the conditional moments of the coverage fraction Φ are determined solely by the point probability πO and the cell/area quotient Q. This model of coverage differs from the models of Eagleson and Wang (1985) and Seo and Smith (1996). Their models are suited to analyzing observed precipitation fields—situations in which only the conditional probability of point precipitation occurrence is meaningful; this probability is given by the ratio p = πO/πA. Epstein’s (1966) theory could be used to express this conditional probability p as a function of Q. Only then could the behavior of E(Φ|WA > 0) and var(Φ|WA > 0) be compared with the behavior of their counterparts in climatic models. One distinction is conspicuous as A varies from 0 to ∞. When πO and C are held fixed in the present model, the conditional mean of Φ decreases monotonically from 1 to an asymptote at πO. When C is held fixed in the model of Eagleson and Wang (1985, Fig. 5), the conditional mean of Φ decreases from 1 to 0.

Rescaling conditional distribution of amount

Conditional moments

Define the mean and variance of each amount, WO and WA, conditional on the hypothesis that the amount is positive:
i1520-0450-38-6-786-eq8
Next assume that within the area of averaging (i) probability πO is identical at every point, and (ii) the field of point precipitation amounts, conditional on precipitation occurrence at every point, is covariance stationary; this implies, in particular, that each μO and σ2O exists and is identical at every point at which πO > 0.

When πO = 1 at every point within the area of averaging, assumption (ii) is equivalent to a homogeneous precipitation field used in hydrologic rescaling models (e.g., Rodriguez-Iturbe and Mejia 1974; Sivapalan and Blöschl 1998). Mathematical formulations of these models usually suppress the conditioning of amount, WO or WA, on precipitation occurrence, WO > 0 or WA > 0, respectively, and this blurs the distinction between the conditional and unconditional moments of WO or WA. This distinction is essential in forecasting because πO < 1 more often than not, which has a profound impact on scaling of conditional moments (Seo and Smith 1996).

When πO < 1, the wetted area may be less than the averaging area. To account for this possibility, the conditional moments of point amount WO are first rescaled to conditional moments of WD, the spatially averaged amount for the wetted area D. This is accomplished by adapting results from hydrologic rescaling models (e.g., Rodriguez-Iturbe and Mejia 1974; Wood and Hebson 1986; Gupta and Waymire 1990) that are based on assumption (ii) and the hypothesis WO > 0 at every point within area A. In the present case, the hypothesis WA > 0 is equivalent to the hypothesis WD > 0, which in turn is equivalent to the hypothesis WO > 0 at every point within area D. Hence,
i1520-0450-38-6-786-e6
where κ2 is the variance reduction factor such that 0 < κ2 ⩽ 1. In general, κ2 depends upon the spatial correlation structure of the precipitation field, as well as the size and shape of the wetted area. In the present case, however, the wetted area is random and this makes κ2 different from the variance reduction factor found in the hydrologic models. The effect of this randomness will be modeled later.
Mass conservation implies AWA = DWD. Hence WA = ΦWD. Assuming that conditional on event WA > 0, variates Φ and WD are mutually stochastically independent, the conditional moments of WA may be derived via three equations:
i1520-0450-38-6-786-eq9
With all inputs specified by (2)–(3) and (6)–(7), one finds
i1520-0450-38-6-786-e8
Equation (8) establishes a ratio transformation between the conditional mean of the point amount, μO, and the conditional mean of the spatially averaged amount, μA. Equation (9) establishes a linear transformation between the conditional variance of the point amount, σ2O, and the conditional variance of the spatially averaged amount, σ2A.

When precipitation is certain to occur at the point, πO = πA = 1 and σ2A = σ2Oκ2; thus (9) is consistent with the hydrologic rescaling models that assume πO = πA = 1. However, when πO < πA < 1, Eqs. (8) and (9) reveal that the ratio of the point probability to the area probability, πO/πA, becomes the scaling factor for the conditional mean and a primary scaling factor for the conditional variance, along the variance reduction factors τ2 and κ2. In addition, μO enters the intercept of (9).

Unlike the case with certain occurrence of point precipitation, which leads to rescaling of the conditional variance via a ratio transformation, the case with uncertain occurrence of point precipitation leads to rescaling of the conditional variance via a linear transformation. This may be rationalized by considering a situation wherein, conditional on the hypothesis that precipitation will occur at the point, WO > 0, the forecaster predicts the point amount WO with certainty, so that P(WO = μO|WO > 0) = 1 and consequently σ2O = 0. Now (9) yields σ2A > 0. In other words, conditional on the hypothesis that precipitation will occur at one point or more within the area, the spatially averaged amount WA is not predicted with certainty. The reason is that the spatial coverage of precipitation remains uncertain, as indicated by πO < 1. Only when this uncertainty vanishes as well, so that πO = 1 and πO/πA = 1, does σ2O = 0 imply σ2A = 0.

Conditional distribution

Because forecasters judge conditional exceedance fractiles, not conditional moments, expressions for rescaling fractiles are desired. Toward this end, (point or spatially averaged) amount W, conditional on precipitation occurrence, W > 0, is modeled in terms of the Weibull family of distributions:
Gωωαβω
where α > 0 is the scale parameter and β > 0 is the shape parameter. The moments of W are
i1520-0450-38-6-786-e11
For any probability p, 0 < p < 1, the 100p% conditional exceedance fractile of W can be found from equation p = 1 − G(ω) and takes the form
ωαp1/β

It is the closed form of (10) and (13) that makes the Weibull model advantageous for operational forecasting. Most importantly, the model usually fits well to empirical distributions of daily amounts (e.g., Hershenhorn and Woolhiser 1987; Wilks 1989; Selker and Haith 1990). However, I do not know if a proof exists that a Weibull distribution of point amount is sufficient for a Weibull distribution of spatially averaged amount. Good fits of the Weibull model to both amounts suggest that at least some approximate sufficiency exists. This observation parallels the assumption of Wood and Hebson (1986) that a gamma distribution of point amount is approximately sufficient for a gamma distribution of the spatially averaged amount.

Conditional distribution parameters

Under the assumption that the conditional distribution of each amount, WO and WA, belongs to the Weibull family, point parameters (αO, βO) are uniquely rescaled to area parameters (αA, βA) via equations derived from (8)–(9) and (11)–(12). These equations are analytic, but not closed form:
i1520-0450-38-6-786-e14

When expressed as functions of all inputs, the equations take the following general form: αA = α(αO, βO, πO/πA, τ2, κ2) and βA = β(βO, πO/πA, τ2, κ2). It is now transparent that three factors determine the scaling:the ratio of the point probability to the area probability, πO/πA; the factor reducing the conditional variance of the point precipitation occurrence, τ2; and the factor reducing the conditional variance of the point amount, κ2. Interestingly, the area scale parameter αA depends upon both point parameters, scale αO and shape βO, whereas the area shape parameter βA depends only upon the point shape parameter βO.

The equations imply that (i) βA = βO if and only if αA = αO(πO/πA), and (ii) βA = βO = β if and only if
i1520-0450-38-6-786-eq10
where γ = Γ(1 + 2/β)/Γ2(1 + 1/β) − 1. If, in addition, πA = πO, then κ2 = 1. Because in reality κ2 < 1, the equalities βA = βO and πA = πO cannot occur simultaneously. Hence, the shape parameters for areal amount and point amount can be identical only when πA > πO, which, according to (1), occurs whenever πO < 1. Last, the equations imply that (iii) αA = αO if and only if Γ(1 + 1/βA)πA = Γ(1 + 1/βO)πO. If, in addition, πA = πO, then βA = βO and κ2 = 1, which is unrealistic. Hence in practice, the scale parameters can be identical only if the shape parameters are not, and vice versa.

Climatic estimates of the distribution parameters are shown in Table 2. In both months and for all points, αA < αO and βA < 1 < βO. The latter two inequalities imply that WA has the mode at zero, whereas WO has the mode at a positive amount (Johnson and Kotz 1970, p. 251). Thus the difference between βA and βO, which may appear small, has a significant impact on the shape of the distribution. In summary, spatial averaging of precipitation has two effects on the Weibull conditional distribution of the amount: (i) both parameters decrease, especially the scale parameter, and (ii) the mode of the amount shifts toward zero.

Table 2 also shows estimates of the variance reduction factor κ2 inferred via (9) from climatic values of (πO, μO, σ2O) and (πA, μA, σ2A). At each point, κ2 is larger in March than in July. This suggests that κ2 decreases as precipitation becomes more scattered (Q decreases) and point amount becomes more uncertain (σ2O increases).

Finally, Table 2 contains evidence for an assertion that station Cn is representative of the basin in the sense of satisfying the scaling condition μAπA = μOπO. On the contrary, stations Cf and Sd illustrate cases wherein the scaling condition is violated, albeit not excessively. Inasmuch as μOπO increases with the elevation of a point, as does πO, a spatial nonhomogeneity of the mean amount can be attributed to effects of local orography on precipitation occurrence. Implications are discussed at the end of the paper.

Conditional exceedance fractiles

An equation for rescaling the 100p% conditional exceedance fractile ωO of the point amount WO into the corresponding 100p% conditional exceedance fractile ωA of the areal amount WA can be obtained for any p, 0 < p < 1, as follows. Letting c = (πO/πA)[Γ(1 + 1/βO)/Γ(1 + 1/βA)], Eq. (14) takes the form αA = O. Starting from (13) for ωA yields
i1520-0450-38-6-786-eq11
which, after substituting the expression for c, takes the form
i1520-0450-38-6-786-e16
where
i1520-0450-38-6-786-eq12

The rescaling equation takes the form of a power transformation. When βA = βO, the rescaling equation becomes linear: ωA = (πO/πA)ωO. The earlier analysis implies that the necessary condition for this to occur is πA > πO, assuming κ2 < 1.

The scaling factors are functions of all inputs; specifically, m = m(αO, βO, πO/πA, τ2, κ2) and n = n(βO, πO/πA, τ2, κ2). Thus a formal implementation of (16) requires operational estimates of all inputs on each forecasting occasion. An estimate of πO is given; an estimate of πA can readily be calculated from Epstein’s formula;an estimate of τ2 can likewise be calculated. Estimates of (αO, βO) can be obtained when a guidance forecast is probabilistic but not when a guidance is deterministic (i.e., specifies only one estimate of WO). An estimate of κ2 poses a roadblock because it is not provided by any guidance currently available.

It is worthwhile, therefore, to consider an approximation wherein m and n are held fixed across all forecasting occasions and only πO/πA varies. To test the potential of such an approximation, m = 1 has been fixed, whereas n has been estimated for each station and month. Such a stratification is consistent with the dependence of n on τ2 and κ2, each of which varies with storm type and hence season. Next, the estimate ν of n, together with the climatic value of πO/πA for each station and month, have been inserted into (16) to rescale climatic exceedance fractiles ωO for p = 0.75, 0.50, 0.25. The resultant estimates
ω̂AπOπAωνO
are compared with the climatic exceedance fractiles ωA in Table 3. The maximum error of 10% and the average error of just 3.3% suggest that formula (17) alone may offer an approximation suitable for operational application. (This conclusion notwithstanding, the problem of quantifying κ2 operationally is solved in section 6.)

Rescaling expected fractions

Daily precipitation exhibits the conditional disaggregative invariance—a property demonstrated empirically for point amounts and spatially averaged amounts (Krzysztofowicz and Pomroy 1997). Operational forecasting takes advantage of this property by assuming that on any forecasting occasion, the vector of fractions (Θ1, . . . , Θn) is stochastically independent of the total amount W. This implies, inter alia, that for every i ∈ {1, . . . , n},
i1520-0450-38-6-786-eq13
Therefore, the expected fraction can be expressed as follows:
i1520-0450-38-6-786-eq14
Next assume that within the area of averaging the field of point precipitation amounts in subperiod i is covariance stationary; thus E(WOi) is identical at every point. The immediate implication is that E(WAi) = E(WOi) for each subperiod i = 1, . . . , n. This parallels the equality E(WA) = E(WO) for the period assumed earlier. Consequently,
i1520-0450-38-6-786-e18
The vector of expected fractions (zAl, . . . , zAn) defines the expected temporal disaggregation of spatially averaged amount WA, whereas the vector of expected fractions (zOl, . . . , zOn) defines the expected temporal disaggregation of point amount WO. Equation (18) states that the expected temporal disaggregation is scale invariant.

To illustrate the property, Table 4 compares climatic estimates of expected fractions zOi = EOi|WO > 0) of point amounts with climatic estimates of expected fractions zAi = EAi|WA > 0) of spatially averaged amounts. These estimates have been purposely calculated as sample means of fractions, not as ratios of sample means of amounts, in order to examine the more general hypothesis EAi|WA > 0) = EOi|WO > 0) for i = 1, 2, 3, 4. Data in Table 4 suggest that the hypothesis is plausible and offers an approximation sufficient for the purpose of operational forecasting. Moreover, no elevation effect can be discerned. It appears that, at least within the Monongahela basin, a field of the expected fraction is more homogeneous than either a field of the precipitation probability or a field of the mean amount.

Quantifying variance reduction factor

Theoretical relations

It is difficult to imagine how a forecaster could judge a parameter as abstract as the variance reduction factor κ2. Therefore, a measurement model is needed whereby κ2 is related to another parameter that admits an intuitive interpretation.

A simple model of the precipitation field (e.g., Rodriguez-Iturbe and Mejia 1974; Wood and Hebson 1986) assumes that, conditional on precipitation occurrence at every point within the area of averaging, the correlation coefficient ρ(d) between two point amounts separated by distance d is isotropic (within the field) and exponential (with distance):
ρddλ
Function ρ is the spatial correlogram, and parameter λ > 0 is the spatial correlation length that characterizes the random field.
A relation between λ and κ2 was estimated by Sivapalan and Blöschl (1998) under the assumption that the averaging area A is square. The relation was reported as a plot of κ2 versus dimensionless argument A/λ2. We have fitted a parametric function to this plot, which takes the form
κ2aAλ2b−4
where a > 0 and b > 0 are parameters with estimates â = 0.134 and = 0.484. A simple transformation of (20) can reveal that κ is a logistic function of ln(λ/A).

(One may note that a square area is assumed to rescale the conditional variance of the amount, and a circular area is assumed to rescale the probability of precipitation occurrence. While this introduces an aesthetically displeasing inconsistency, I do not suppose it will affect the calibration of operational PQPFs in any significant measure.)

Coverage uncertainty

In a forecasting situation with πO < 1, the wetted area D is uncertain, as discussed in sections 3b and 4a. Consequently, the conditioning of κ2 in (20) on the occurrence of precipitation at every point of A is not satisfied. An approximate way of modeling this situation is to replace A with an estimate of D. Inasmuch as D = ΦA, one possible estimate is the mean of D, conditional on the hypothesis that precipitation occurs within the area: E(D|WA > 0) = E(Φ|WA > 0)A. With the conditional mean of coverage fraction Φ specified by (2), one finds E(D|WA > 0) = (πO/πA)A. Using this estimate in lieu of A in (20) yields
i1520-0450-38-6-786-e21
In effect, κ2 is now conditioned on event WA > 0, which is consistent with conditioning the variance of amount WD on event WA > 0 in (7).

Pattern uncertainty

The statistical definition of the correlation coefficient, which serves well for estimating ρ(d) from climatic data, is not meaningful for quantifying ρ(d) by a forecaster. In the context of forecasting, for which subjective probability is the appropriate theory, the correlation coefficient should be interpreted as a measure of uncertainty about the precipitation pattern.

For a realization of the precipitation field, its pattern can be represented in terms of isopleths. If a clairvoyant supplied isopleths (drawn at fixed fractions of the maximum point amount), but without the amounts attached, then only uncertainty about the amounts would remain. The forecaster could quantify this uncertainty by assessing a distribution of the amount WO at a point on any isopleth. A realization of the field could next be obtained by generating a realization of WO from the assessed distribution and scaling the remaining isopleths. Across all probable realizations, the ratio of amounts at two fixed points would be constant. Hence, the amounts would be perfectly positively correlated, ρ(d) = 1. At the other extreme, if the forecaster were totally ignorant about the pattern (to the point of not knowing how a realization of the isopleths can look), then in his judgment any realization of point amounts would be probable. To generate all such realizations, the amounts at any two points would have to be uncorrelated, ρ(d) = 0. In reality, the forecaster is neither clairvoyant nor ignorant. At the minimum, he is familiar with precipitation patterns that are climatically probable. Hence, a climatic estimate of ρ(d) constitutes a reference point (which may vary with storm type, season, and possibly other climatic predictors of pattern uncertainty).

Measurement model

With the exponential correlogram ρ, the limiting cases may be characterized in terms of parameter λ. When the precipitation pattern is certain, we let λ → ∞; hence ρ(d) → 1 and κ2 → 1. When the precipitation pattern is totally uncertain, we let λ → 0; hence ρ(d) → 0 and κ2 → 0. However, λ is not suitable for quantifying and communicating the forecaster’s uncertainty. First, it is not an intuitive measure of uncertainty because it has a unit of distance (e.g., kilometers or miles). Second, it is unbounded from above, which makes it difficult to judge the degree of uncertainty relative to the limiting cases.

A judgmental measure of uncertainty is constructed as follows. Define a pattern certainty factor, F, with range 0 < F < 1, lower bound F = 0 if the pattern is totally uncertain, and upper bound F = 1 if the pattern is certain. The pattern in question is the pattern of point precipitation over the averaging area A. (It is, in fact, the uncertainty about this pattern that is the primary reason for forecasting the spatially averaged amount rather than the point amount.) To connect F with λ, two scaling conditions are imposed: (i) the area of averaging is square (consistent with an earlier assumption), and (ii) the pattern certainty factor equals the correlation coefficient between the amount at the center of the area and the amount at a most distant point of the area. In a square, such maximum distance is half of the diagonal, A/2. The second scaling condition states that F = ρ(A/2). When (19) is substituted for ρ, the following relation results:
i1520-0450-38-6-786-e22
After inserting (22) into (19), one obtains
ρdFd2A
In effect, the correlogram has been reparameterized by supplanting λ with F and A. Finally, (21) and (22) yield
i1520-0450-38-6-786-e24

This relation is plotted in Fig. 4. The plot shows that the variance reduction factor κ2 increases with the pattern certainty factor F, and decreases with the ratio of the point probability to the area probability πO/πA. For instance, the same value κ2 = 0.69 results in two distinct situations: (i) when the forecaster is relatively uncertain about the pattern, F = 0.2, and precipitation is spotty, πO/πA = 0.1, and (ii) when the forecaster is considerably more certain about the pattern, F = 0.6, and precipitation is widespread, πO/πA = 1. Inasmuch as πO/πA depends upon the cell/area quotient Q, the variance reduction factor κ2 is affected by both parameters, Q and F.

Given the climatic estimates of κ2 in Table 2, climatic estimates of λ and F may be inferred via (21) and (24), respectively. Their values are listed in Table 5. The pattern certainty factor F is higher in March than in July at each point, and decreases with the elevation of a point, decisively in March but only somewhat in July.

Judgmental quantification

To quantify the pattern certainty factor F on a particular occasion and for a particular area A, the forecaster should ponder the question, “How certain am I about the pattern of point precipitation over the area?” The forecaster’s task is to express his degree of certainty on a scale between F = 0 (totally uncertain) and F = 1 (certain). A reference point Fc, which is the climatic value of F (say, for a day within a season), should be located on the scale to facilitate calibration of judgment. When the forecaster judges the predictability of the precipitation pattern to be higher on this particular day than on an average rainy day of the season, his F should be higher than Fc. When the forecaster judges the predictability of the precipitation pattern to be lower on this particular day than on an average rainy day of the season, his F should be lower than Fc. Otherwise, he should set F equal to Fc. As a surrogate for predictability, the forecaster may judge the difficulty of predicting the pattern on a particular occasion relative to the average difficulty, which he knows from experience.

Judgment of F is also relative to area A. This can be inferred from (22). For a fixed λ, factor F decreases with area A. In other words, the degree of certainty about the pattern of point precipitation increases as the area for which the pattern must be predicted decreases. Assuredly, the easiest isopleth to predict is that for a point. The relativity of F with respect to A is desirable, from a cognitive point of view, because the forecaster’s mind naturally focuses on the area of averaging.

In summary, quantification of F requires a judgment on predictability of the precipitation pattern. Such judgment is a part of the methodology used by forecasters who prepare PQPFs operationally (Krzysztofowicz et al. 1993). Thus the only novelty will be the quantification task.

Conclusions

A procedure has been formulated for rescaling a PQPF of the point amount to a PQPF of the spatially averaged amount, or vice versa. The procedure is intended to meet operational needs for (i) rescaling a guidance PQPF for points to a guidance PQPF for a nominal area, and (ii) rescaling a PQPF prepared by the forecaster for a nominal area to a PQPF for points.

To prevent a degradation of informativeness of the PQPF due to rescaling, theoretically based scaling equations must incorporate predictive parameters, which characterize the precipitation field being forecast, as opposed to climatic parameters, which characterize a sample of precipitation fields observed in the past. Two such predictive parameters are the cell/area quotient, Q, and the pattern certainty factor, F. When both Q and F are quantified by the forecaster, they are sufficient for rescaling all elements of the PQPF. When only Q is quantified, one other parameter must be estimated from climatic data (it may be the variance reduction factor κ2, or the exponent ν in the approximate formula that rescales the conditional exceedance fractiles). The quantification of each predictive parameter appears to lie well within the scope of judgmental tasks routinely performed by forecasters.

The assumption that the uncertainty about the precipitation field being forecast is spatially homogeneous leads to compact scaling equations. The validity of this assumption is obviously limited, as demonstrated via examples. To stay within the limits of validity, at least approximately, the nominal area of averaging should be chosen so that any local hydrometeorological influences, such as elevation effect or lake effect, are uniform throughout the area. To establish the robustness of the rescaling procedure, further tests should be conducted on data from other river basins.

Acknowledgments

I am indebted to John C. Schaake Jr. for discussions in which he pointed out the significance of the scaling problem in operational forecasting, and to Dong-Jun Seo for alerting me to a seminal reference. This article is based upon work supported by the National Oceanic and Atmospheric Administration under Award NA67WD0486, “Probabilistic Hydrometeorological Forecast System.”

REFERENCES

  • Bermowitz, R. J., and E. A. Zurndorfer, 1979: Automated guidance for predicting quantitative precipitation. Mon. Wea. Rev.,107, 122–128.

  • Carter, G. M., J. P. Dallavalle, and H. R. Glahn, 1989: Statistical forecasts based on the National Meteorological Center’s numerical weather prediction system. Wea. Forecasting,4, 401–412.

  • Eagleson, P. S., and Q. Wang, 1985: Moments of catchment storm area. Water Resour. Res.,21, 1185–1194.

  • Epstein, E. S., 1966: Point and area precipitation probabilities. Mon. Wea. Rev.,94, 595–598.

  • Gupta, V. K., and E. Waymire, 1990: Multiscaling properties of spatial rainfall and river flow distributions. J. Geophys. Res.,95 (D3), 1999–2009.

  • Hershenhorn, J., and D. A. Woolhiser, 1987: Disaggregation of daily rainfall. J. Hydrol.,95, 299–322.

  • Johnson, N. L., and S. Kotz, 1970: Distributions in Statistics: Continuous Univariate Distributions-1. Wiley, 300 pp.

  • Krzysztofowicz, R., 1998: Probabilistic hydrometeorological forecasts: Toward a new era in operational forecasting. Bull. Amer. Meteor. Soc.,79, 243–251.

  • ——, and T. A. Pomroy, 1997: Disaggregative invariance of daily precipitation. J. Appl. Meteor.,36, 721–734.

  • ——, and A. A. Sigrest, 1997: Local climatic guidance for probabilistic quantitative precipitation forecasting. Mon. Wea. Rev.,125, 305–316.

  • ——, W. J. Drzal, T. R. Drake, J. C. Weyman, and L. A. Giordano, 1993: Probabilistic quantitative precipitation forecasts for river basins. Wea. Forecasting,8, 424–439.

  • National Weather Service, 1999: The modernized end-to-end forecast process for quantitative precipitation information: Hydrometeorological requirements, scientific issues, and service concepts. NWS, Silver Spring, MD, 188 pp. [Available from National Weather Service, Office of Meteorology, 1325 East–West Highway, Silver Spring, MD 20910.].

  • Olson, D. A., N. W. Junker, and B. Korty, 1995: Evaluation of 33 years of quantitative precipitation forecasting at the NMC. Wea. Forecasting,10, 498–511.

  • Rodriguez-Iturbe, I., and J. M. Mejia, 1974: On the transformation of point rainfall to areal rainfall. Water Resour. Res.,10, 729–735.

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  • Seo, D.-J., and J. A. Smith, 1996: Characterization of the climatological variability of mean areal rainfall through fractional coverage. Water Resour. Res.,32, 2087–2095.

  • Sivapalan, M., and G. Blöschl, 1998: Transformation of point rainfall to areal rainfall: Intensity-duration-frequency curves. J. Hydrol.,204, 150–167.

  • Wilks, D. S., 1989: Rainfall intensity, the Weibull distribution, and estimation of daily surface runoff. J. Appl. Meteor.,28, 52–58.

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APPENDIX

Variance of Coverage Fraction

Conditional variance

Recall that E(Φ|WA > 0) = πO/πA and denote s = E2|WA > 0). Next rearrange the expression defining the conditional variance as follows:
i1520-0450-38-6-786-ea1
where
i1520-0450-38-6-786-ea2

Bounds

The unknown quantity is s. Its bounds are
i1520-0450-38-6-786-ea3
The lower bound is implied by the nonnegativity of variance, var(Φ|WA > 0) ≥ 0. To find the upper bound, suppose that at all points within area A, indicators IO are equal either to zero or to one; in other words, the correlation between indicators IO at any two points is one. In such a case, either Φ = 1 or Φ = 0. Consequently Φ = IO and var(Φ|WA > 0) = var(IO|WA > 0) = πO/πA − (πO/πA)2, which implies E2|WA > 0) = πO/πA. The bounds of s imply that 0 ⩽ τ2 ⩽ 1.

Approximation

Toward finding s, recall that Q = C/A and observe that Φ = D/A = Q(D/C). Hence,
i1520-0450-38-6-786-eq15
Analogously, Φ2 = ΦQ(D/C) and
i1520-0450-38-6-786-eq16
which forms a basis for a two-step approximation to s. In the first step, we let
i1520-0450-38-6-786-ea4
where CB = C/(πO/πA), QB = CB/A = Q/(πO/πA), and πB is the area probability calculated from the point probability πO and the cell/area quotient QB. When (A4) is inserted into (A2), one obtains (4).
The approximation in the first line of (A4) consists of replacing the expectation of a product of variates with a product of the expectations of individual variates. This leads to a certain linearization of the expectation of Φ2 with respect to Q. In effect, E2|WA > 0) is overvalued for small Q and undervalued for large Q. This effect is compensated for in the second step, whereby QB is redefined as
QBQπOπAc
with parameter c > 1 to be estimated empirically. When (A5) is inserted into (1), one obtains (5).

Finally, one must verify that the approximation to s given by (A4)–(A5) lies within the bounds specified by (A3). This is the case provided πO < πB < πA. The left inequality is assured by (5). The right inequality is satisfied if Q < [Q/(πO/πA)]c. When c = 1, this holds always because πO/πA < 1. When c > 1 and Q ≥ 1, this holds always. When c > 1 and Q < 1, the inequality is violated for Q sufficiently close to 0 (and πA sufficiently close to 1). But the convergence of Q to zero also implies that precipitation becomes spotty and hence the indicators IO at any two points become uncorrelated. In effect, var(Φ|WA > 0) converges to 0, while s converges to its lower bound. This suggests that whenever πBπA, approximately τ2 = 0.

Fig. 1.
Fig. 1.

Epstein’s model for point-to-area rescaling of precipitation probabilities consists of a circular area of averaging, A, and circular precipitation cells, each covering area C. The forecaster’s task is to judge the quotient Q = C/A. The uncertain wetted area is D.

Citation: Journal of Applied Meteorology 38, 6; 10.1175/1520-0450(1999)038<0786:PTAROP>2.0.CO;2

Fig. 2.
Fig. 2.

Relation between point probability πO and area probability πA of precipitation occurrence as a function of the cell/area quotient Q, based on Epstein’s model.

Citation: Journal of Applied Meteorology 38, 6; 10.1175/1520-0450(1999)038<0786:PTAROP>2.0.CO;2

Fig. 3.
Fig. 3.

Relation between the cell/area quotient Q, to be judged by the forecaster, and the variance reduction factor τ2, for several values of the point probability πO of precipitation occurrence.

Citation: Journal of Applied Meteorology 38, 6; 10.1175/1520-0450(1999)038<0786:PTAROP>2.0.CO;2

Fig. 4.
Fig. 4.

Relation between the pattern certainty factor F, to be judged by the forecaster, and the variance reduction factor κ2, for several values of the ratio of the point probability to the area probability πO/πA.

Citation: Journal of Applied Meteorology 38, 6; 10.1175/1520-0450(1999)038<0786:PTAROP>2.0.CO;2

Table 1.

Point probability and area probability of precipitation occurrence, the inferred cell/area quotient and variance reduction factor, and the calculated conditional mean and variance of the coverage fraction; Monongahela basin, climatic data, 1943–93.

Table 1.
Table 2.

Parameters of the Weibull distributions of point and areal amounts, calculated moments, and the inferred variance reduction factor; Monongahela basin, climatic data, 1943–93. Note: μO and σO are in in./24 h.

Table 2.
Table 3.

Approximate rescaling of conditional exceedance fractiles from the point amounts to the spatially averaged amount; Monongahela basin, climatic data, 1943–93. Note: Amounts are in in./24 h.

Table 3.
Table 4.

Expected fractions defining the expected temporal disaggregation of the point amounts and the spatially averaged amount;Monongahela basin, climatic data, 1943–93.

Table 4.
Table 5.

Variance reduction factor, inferred spatial correlation length, and inferred pattern certainty factor; Monongahela basin, climatic data, 1943–93. Note: A = 3429 km2; λ is in km.

Table 5.
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