Spatially Averaged versus Point Precipitation in Monongahela Basin: Statistical Distinctions for Forecasting

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1 163 Spatially d versus Point Precipitation in Monongahela Basin: Statistical Distinctions for Forecasting ASHLEY A. SIGREST Department of Systems Engineering, University of Virginia, Charlottesville, Virginia ROMAN KRZYSZTOFOWICZ Department of Systems Engineering and Division of Statistics, University of Virginia, Charlottesville, Virginia (Manuscript received 1 June 1996, in final form 27 April 1998) ABSTRACT The predictand of a probabilistic quantitative precipitation forecast (PQPF) may be either a point precipitation amount or a spatially averaged precipitation (SAP) amount. At the current state of the art, it is the SAP amount (the volume of water accumulated over an area during a period) that is most predictable. This case study compares the climatic PQPFs of the two predictands within a river basin in the Appalachians, then highlights similarities and distinctions of which the forecasters should be aware. Empirical relations reveal whether or not a given statistic of the point precipitation amount is (i) locally invariant, that is, does not vary appreciably within some area so that a single estimate (e.g., a spatial average) can approximate the statistic at every point within the area, and (ii) amenable to averaging, that is, can be averaged over some area to obtain an approximation to the statistic of the SAP amount. The study also illustrates the effect of elevation on the statistics of point precipitation and highlights seasonal differences. The conclusions point to a need for local climatic guidance to help forecasters in calibrating PQPFs. 1. Introduction a. Quantitative precipitation forecasting Precipitation amount has been known as one of the most difficult meteorologic predictands (Georgakakos and Hudlow 1984). But a steady progress in numerical weather prediction and in understanding of precipitation process has changed the outlook on the feasibility of producing quantitative precipitation forecasts (QPFs). The case in point is the QPF guidance provided to field offices of the National Weather Service (NWS) by the Hydrometeorological Prediction Center (HPC), formerly Weather Forecast Branch in the Meteorological Operations Division of the National Meteorological Center (renamed the National Centers for Environmental Prediction). This QPF guidance is prepared judgmentally by the HPC forecasters who apply knowledge, experience, and techniques to observations, analyses, and model outputs (Funk 1991). The guidance is for all 48 conterminous states and has graphical form: it shows isopleths of the spatially averaged precipitation (SAP) amounts that can be interpolated to basin or even subbasin averages (Olson et al. 1995). Verification statis- Corresponding author address: Prof. Roman Krzysztofowicz, University of Virginia, Thornton Hall, SE, Charlottesville, VA 23. tics indicated that the HPC guidance consistently outperforms model outputs upon which it is based. A survey indicated that this is also the most useful guidance to field forecasters (Krzysztofowicz and Drake 1993). Nonetheless, the uncertainty associated with this QPF remains considerable. The task of a field forecaster is to combine information from various guidance products, numerical models, observations, and local analyses with his knowledge of local hydrometeorological influences into the final forecast. To assist the forecaster in performing this complex task and, at the same time, to quantify forecast uncertainty, a probabilistic quantitative precipitation forecast (PQPF) system was developed and tested in the Pittsburgh office of the NWS (Krzysztofowicz et al. 1993). Further development of this system is in progress (Mills and Krzysztofowicz 1998). b. Spatially averaged precipitation The predictand of a PQPF is defined as the SAP amount. There are two coinciding reasons for this. First, the total precipitation amount from a storm is usually far less uncertain than the spatial pattern of precipitation (HPC senior forecasters 1997, personal communication). By averaging the uncertain future precipitation field over an area, the forecaster can encode information 1998 American Meteorological Society

2 164 WEATHER AND FORECASTING VOLUME 13 about the amount while filtering noise from the unpredictable details of the spatial pattern. In principle, averaging should be performed at a scale at which the forecaster is most certain about details of the spatial pattern. This scale may depend upon the predictability of the pattern on a particular occasion, storm type (stratiform versus convective), orography (planes versus mountains), and possibly other factors. Most significantly, this is the scale at which the PQPF reaches its maximum informativeness, given current state of precipitation predictability. Second, one of the main purposes of the PQPF is to provide input to a hydrologic model that the NWS employs to forecast rivers (Fread et al. 1995). This model is driven by SAP amounts that are defined for basins or subbasins, often called mean areal precipitation units. Thus, a clairvoyant stating the actual SAP amount for each subbasin would provide a forecast that is perfect for the current operational forecasting model even though such a forecast lacks the detail of the actual precipitation field. For this reason, the smallest subbasins define the desired, but not necessarily optimal, scale of spatial averaging for PQPF. c. Calibration of forecaster s judgment A paramount question in training forecasters who prepare PQPFs is the distinction between characteristics of point precipitation and characteristics of SAP. The purpose of this paper is twofold: (i) to illustrate how such characteristics can be obtained from the local climatic guidance (LCG) an interactive software developed as a component of the PQPF system (Krzysztofowicz and Sigrest 1997); and (ii) to highlight major distinctions between point precipitation and SAP by comparing climatic statistics of the two predictands within a river basin in the Appalachians. Similar statistical comparisons could be performed for other forecast areas. The objective of a comparison is to aid forecasters in calibrating (or recalibrating) their judgments about the frequency, timing, and magnitude of the SAP amounts. The ultimate objective is to ensure that operational PQPFs are well calibrated (reliable) and thus can be uniquely interpreted by users. 2. Methodological background a. Predictand The following definition of the predictand applies to SAP amounts or to point precipitation amounts. Let W denote the precipitation amount accumulated during a fixed period (e.g., 24 h). Let W i denote the precipitation amount accumulated during the ith subperiod, i {1,...,n}, where n is the number of subperiods. Thus W ; Wi, i 1,...,n; W W W. 1 n TABLE 1. Duration index* d C 2C 2C 2N 2N 2N 3C 3C 3N 3N 4 Definition of precipitation duration and timing pattern for four subperiods. Timing pattern t Definition in terms of fractions , , 3 3, 4 1, 3 1, 4 2, 4 1, 2, 3 2, 3, 4 1, 2, 4 1, 3, 4 1, 2, 3, *C consecutive subperiods; N nonconsecutive subperiods. Given a precipitation event, W, define for each i a variate i W i /W representing a fraction of the total amount accumulated during subperiod i. Thus i 1, i 1,...,n; 1 n 1. The vector of fractions ( 1,..., n ) defines the temporal disaggregation of the total precipitation amount accumulated during a period into n subperiods. Note that only n 1 fractions must be forecast because one of the fractions can always be expressed in terms of the remaining fractions through the unit sum constraint. The predictand is the vector (W; 1,..., n ). It will be characterized in terms of statistics provided by the LCG, which employs two models. The first model describes the total precipitation amount W and the vector of fractions ( 1,..., n ) independently, and for each provides marginal climatic statistics. This model is relatively simple but incomplete because it does not describe the stochastic dependence between the total amount and its temporal disaggregation. The second model introduces two additional descriptors of precipitation. The timing T is defined by the occurrence or nonoccurrence of precipitation in subperiods. A particular timing pattern is identified by the concatenation of indices of subperiods that are wet. For example, if all precipitation occurs in subperiods 1 and 3, then 1, 3, 1 3 1, and T 13. Another descriptor of precipitation is its duration D, which is defined by the number of wet subperiods. Duration may be refined to indicate also whether the wet subperiods are consecutive (C) or nonconsecutive (N). Table 1 lists all possible realizations of T and D for n 4 subperiods. The model gives the probability of any timing pattern T t and for each pattern provides conditional climatic statistics of the total precipitation

3 165 amount W and the vector of fractions ( 1,..., n ). This model is complete because, conditional on timing T t, the total amount and its temporal disaggregation are stochastically independent (Krzysztofowicz and Pomroy 1997). Consequently, the forecaster can decompose the problem into three tasks: (i) forecasting the precipitation timing T; (ii) forecasting the total amount W, conditional on timing T t; and (iii) forecasting the temporal disaggregation ( 1,..., n ), conditional on timing T t. b. Data The source data used in this study consist of hourly precipitation amounts recorded by rain gauges within or near the basin of interest from 1948 to For each hour, the SAP amount is approximated by a weighted sum of raingauge observations. Hourly amounts are recorded by rain gauges with the precision of.1 in. (.254 mm). The same precision is maintained in the calculated SAP amounts, which are rounded to the nearest hundredth of an inch. Consequently, the occurrence of point precipitation is synonymous with at least.1 in. of water at a rain gauge, whereas the occurrence of SAP is synonymous with at least.1 in. of water over the entire basin. For the specified beginning hour and durations of the n subperiods, a realization of the random vector (W 1,...,W n ) is obtained for each station and the basin by summing up the hourly amounts over the appropriate subperiods. In general, the source of data, the definition of areas (as basins, subbasins, or otherwise), and the method of estimating the SAP amounts can be decided by the user because they are external to the statistical models employed by the LCG. c. Case study The case study described throughout this paper is for the Lower Monongahela River basin () above Connellsville, which covers 39 km 2 (1324 mi 2 )in Pennsylvania and Maryland. Hourly basin average precipitation amounts are estimated from observations at six stations with the following weights and elevations: Connellsville (.14; 9 ft), Confluence (.26; 149 ft), Glencoe (.1; 161 ft), Boswell (.15; 183 ft), Sines Deep Creek (.24; 24 ft), and New Germany (.11; 259 ft). Figure 1 shows the locations of the raingauge stations as well as the Theissen polygons that determine the station weights for spatial averaging. This study compares climatic statistics of the SAP over the basin with climatic statistics of the precipitation at each of the six stations. The statistics being compared are for the months of March and July, 24-h period beginning at 12 UTC, and disaggregation into four 6-h subperiods: 12 18, 18, 6, All statistics are calculated using the LCG software (Krzysztofowicz and Sigrest 1997). FIG. 1. Lower Monongahela River basin () raingauge stations with elevations in ft, and Theissen polygons that determine station weights for spatial averaging. 3. Guidance without the timing hypothesis When the forecaster is highly uncertain about the timing of the precipitation, he may request timing-independent guidance. The precipitation amount accumulated during the period is characterized in terms of an exceedance function, which is determined in part by the climatic probability of precipitation occurrence, and 75%, 5%, and 25% exceedance fractiles. These are precipitation amounts that are exceeded 75%, 5%, and 25% of the time, respectively. When preparing a PQPF, the forecaster is asked to assess the values of these three fractiles. The probability of precipitation (PoP) and the exceedance fractiles for the point precipitation at each of the six stations are compared with each other and with the PoP and the exceedance fractiles for the SAP over the basin. a. Probability of precipitation The PoPs in March at the six stations are displayed as empty squares in Fig. 2. The values range from.36 at Connellsville to.5 at Sines Deep Creek. The stations are listed in order of increasing elevation, and one may note a subtle effect of elevation on PoP. The PoP for the basin, which is shown as a filled square, is significantly higher than the probability at any of the stations. Similar results hold in July. At the stations, the PoPs

4 166 WEATHER AND FORECASTING VOLUME 13 FIG. 2. Probabilities of precipitation for the basin and the six stations during the months of March and July. drop between March and July, when they range from.31 at New Germany to.39 at Sines Deep Creek (the empty circles in Fig. 2). If there was a slight elevation effect in March, it no longer exists in July, and the variability between the stations is even smaller in July. As in March, the PoP for the basin (the filled circle) is much higher than at the stations. It is also slightly higher than the PoP for the basin in March (.62 versus.6). By and large, if a forecaster was asked to estimate the PoP at any station for a typical day in March, or in July, he could obtain a reasonable estimate by averaging the climatic probabilities at the other stations. However, the PoP for the basin could not be obtained as a weighted average of the probabilities at the stations. A theoretical relation between point probability, PoP P, and areal probability, PoP A, was derived by Epstein (1966) under two assumptions: (i) both the basin and the precipitation cells are circular, and (ii) precipitation cells have identical diameter and are distributed at random over an area large compared to the basin. Then PoP P is identical at every point and PoP 1 (1 PoP ) [1 (1/Q) ] A P 1/2 2, where Q is the ratio of the area covered by the precipitation cell to the area of the basin. One can see that PoP A PoP P always and that PoP A approaches PoP P as Q increases. For example, when PoP P.3 is associated with a small convective storm, say Q.5, one finds PoP A.87. When PoP P.3 is associated with a large cyclonic system, say Q 5, one finds PoP A.53. Climatic values of Q estimated from probabilities displayed in Fig. 2 range between 5.9 and in March and between 2.58 and 6.9 in July. These ranges imply that, on the average, the scale of precipitation fields is smaller and less affected by elevation in July than it is in March, as one would expect. Thus Epstein s formula captures the most essential determinant of the relation between the point PoP and the areal PoP. Notwithstanding the assumptions, it could be used operationally as an approximation. b. Exceedance fractiles Figure 3 shows the means and 25% exceedance fractiles of the point precipitation amounts (the empty symbols) and the SAP amount (the filled symbols). The 5% and 75% exceedance fractiles are not shown because they are, in most cases,. in. The range of the 25% exceedance fractiles at the stations in March is.6 in., and their average is approximately.15 in. There is even less variability among the stations in the mean precipitation amount, whose average is.11 in. The mean and 25% exceedance fractile at any station could be approximated by averaging the corresponding estimates at the other stations. The 25% exceedance fractile of the SAP in March is.13 in., and the mean is.9 in. These amounts are similar to those at the stations. Thus, averaging the estimates at the stations would result in a good approximation of the 25% exceedance fractile and mean for the basin. (It should be noted that, theoretically, the mean SAP amount is equal to the spatial average of the mean point precipitation amounts.) In July, the 25% exceedance fractiles are lower than in March at all stations except Connellsville; there is still a maximum difference of.6 in. between the stations. The averages of the 25% exceedance fractiles and means are both.12 in. Once again, these averages approximate the estimates at any station. The same is true for estimating the mean SAP amount in July; averaging the mean precipitation amounts at the stations would give a good approximation. On the other hand, averaging the 25% exceedance fractiles at the stations would underestimate the 25% exceedance fractile of the SAP. c. Conditional exceedance fractiles The LCG software allows the forecaster to replace the climatic PoP with the forecast PoP. If the forecaster

5 167 FIG. 3. Means and 25% exceedance fractiles of the precipitation amount for the basin and the six stations during the months of (a) March and (b) July. is certain that measurable precipitation will occur, he should assign a PoP close to 1. Figure 4 shows the 75%, 5%, and 25% exceedance fractiles conditional on PoP 1. In March, each of the conditional exceedance fractiles of the point precipitation is similar at all stations. The 5% credible interval, defined as the difference between the 25% and 75% exceedance fractiles, characterizes the uncertainty about the precipitation amount. This interval is almost constant across stations. There is no elevation effect, and averaging each fractile over the stations would yield a good estimate of the conditional distribution of the point precipitation at any station. The conditional distribution of the SAP amount is distinct: all three exceedance fractiles are lower and the 5% credible interval is narrower than the corresponding estimates at the stations. The credible intervals indicate that the uncertainty about the SAP amount is smaller than the uncertainty about the point precipitation amount. This statistical fact confirms the meteorologic knowledge that forecasting SAP amount should be easier than forecasting point amounts. In July, the conditional exceedance fractiles at the stations are higher than they are in March, especially the 25% and 5% fractiles. Therefore, when precipitation occurs, higher amounts are more likely in July than in March. On the other hand, Fig. 3 shows that on an average day, the unconditional 25% exceedance fractile is higher in March, not July. This is because the PoPs are slightly higher in March than in July. The interpretation of this relation is intuitive: it precipitates more often in March, but when it does rain in July, it pours. The 5% credible intervals are wider in July than in March, indicating higher uncertainty about the amount per rainy day in July. Again, the exceedance fractiles at the six stations are similar, suggesting the same conditional distribution of the point precipitation amount. And again, the conditional distribution of the SAP amount is distinct: the exceedance fractiles are lower and the 5% credible interval is narrower than the corresponding estimates at the stations. These results are consistent with a general relationship between a point precipitation amount and the SAP amount documented in various forms by earlier studies (e.g., Huff and Neill 1957; Rodriguez-Iturbe and Mejia 1974; Flitcroft et al. 1989; Kunkel et al. 1993; Sivapalan and Blöschl 1998). In general, a given point precipitation amount must be reduced to obtain the expected SAP amount. The reduction factor depends upon the size of the area of averaging and the precipitation pattern

6 168 WEATHER AND FORECASTING VOLUME 13 FIG. 4. Exceedance fractiles of the precipitation amount conditional on PoP 1 for the basin and the six stations during the months of (a) March and (b) July. (which in climatic studies is captured by the location of the rain gauge relative to the area contour, the observed precipitation amount, the season, or the storm type). The absence of the effect of elevation on a distribution of the point precipitation amount, conditional on precipitation occurrence (PoP 1), has been noted in many earlier studies. For example, Loukas and Quick (1996) analyzed hourly precipitation recorded by rain gauges at elevations ranging from to 2799 ft (853 m) in southwestern British Columbia. They found that the precipitation amount per storm increases with elevation up till about 984 ft (3 m), and then levels off and oscillates in the remaining elevation range, ft. This elevation range is almost the same as that of the basin. A theoretical relation between the 1p% conditional exceedance fractile of the point precipitation amount, P, and the 1p% conditional exceedance fractile of the SAP amount, A, for any probability value p, can be derived under two assumptions: (i) both the SAP amount and the point precipitation amount have Weibull distributions, and (ii) the precipitation field is covariance stationary. The relation takes the form A m( P ) n. Both parameters m and n depend, in general, upon the spatial correlation of point precipitation amounts, size and shape of the basin, distribution parameters of the point amount, and the ratio of the point probability to the areal probability of precipitation occurrence, PoP P / PoP A. In fact, m is proportional to this ratio and thus an approximate relation is A (PoP P /PoP A ) P. This relation encapsulates a key property: the SAP amount cannot be close to the point precipitation amount unless the precipitation is widespread so that PoP P PoP A. As an example, the approximation is applied to data from Figs. 2 and 4 for the basin (as A) and Connellsville (as P), for the month of March, and the three conditional exceedance fractiles, as follows: 25%:. (.36/.6).37., 5%:.1 (.36/.6).19.11, 75%:.4 (.36/.6).9.5. The estimates on the right happened to be very close to the true conditional exceedance fractiles of the SAP

7 169 FIG. 5. Fractions statistics for the basin and the six stations during the months of (a) March and (b) July. amount. This is not always the case; for instance, data for Boswell yield the following estimates:.27,.15,.7. Still, the approximation could be used operationally to obtain an initial estimate of the SAP amount from a given estimate of the point amount. This initial estimate can be adjusted next to account for other factors mentioned earlier. d. Fractions statistics A forecaster using the LCG may request fractions statistics, which provide guidance concerning the temporal disaggregation of the total precipitation amount, given that precipitation occurs. Included in the statistics are the mean and the 75% and 25% exceedance fractiles of each fraction. Three-quarters of the time, the actual fraction (percentage) of the total precipitation observed in a particular subperiod is greater than the 75% exceedance fractile, and one-quarter of the time it is greater than the 25% exceedance fractile. As before, the 5% credible interval between the 25% and 75% exceedance fractiles conveys the uncertainty about the fraction. The wider the interval, the greater the uncertainty. Figure 5 displays timing-independent fractions statistics for each of the six stations and the basin in March and July. The 75% exceedance fractiles are. in all cases. In March, the means are similar at all six stations, and the variability between the stations is low. There is slightly greater variability in the 25% exceedance fractiles than in the means. The 5% credible intervals are wider in the first and the fourth subperiods than they are in the second and third at all six stations. For the most part, averaging over the stations would provide a fairly good estimate of the temporal disaggregation at any particular station. The timing-independent fractions statistics of the SAP over the basin in March look very much like the fractions statistics at the stations. The weighted averages of the mean and the exceedance fractiles at the stations were computed with the same station weights that had been used to calculate the SAP amount. These averages are compared with the fractions statistics for the basin in Table 2. In March, the mean for the basin is exactly the same as the weighted average of the means at the stations in the first and third subperiods, and there is only a 2% difference in the second and fourth subperiods. The differences between the basin and the weighted average estimates are slightly greater in the 25% ex-

8 17 WEATHER AND FORECASTING VOLUME 13 TABLE 2. Comparison of the fractions statistics for the basin and the weighted averages of the fractions statistics at the six stations. Timing pattern Month Basin or average 1 75% Mean 25% 2 75% Mean 25% 3 75% Mean 25% 4 75% Mean 25% Timing independent T 12 T 23 T 34 T 123 T 234 T 1234 March July March July March July March July March July March July March July ceedance fractiles, with a maximum difference of 9% in the fourth subperiod and a minimum difference of 2% in the third subperiod. On the whole, the timingindependent fractions statistics of the SAP could reasonably be approximated by spatially averaging the fractions statistics at the stations. The timing-independent fractions statistics in July are somewhat different from those in March. There is more variability between the stations in the 25% exceedance fractiles, especially in the first and fourth subperiods. Whereas in March the highest uncertainties are about the first and fourth fractions, in July the greatest uncertainty is about the second fraction. Generally but not always, a fairly good estimate of the timing-independent fractions statistics at one station could be obtained by averaging the statistics at the other stations. As before, the fractions statistics for the basin in July are compared with the weighted averages of the fractions statistics at the stations in Table 2. The basin statistics are very close to the weighted average statistics in the third and fourth subperiods. In the second subperiod there is 5% difference in the means and 4% difference in the 25% exceedance fractiles. The maximum difference is observed in the first subperiod, where the 25% exceedance fractile for the basin is 1% less than the weighted average. Nevertheless, in July as in March, the timing-independent fractions statistics of the SAP could be approximated by spatially averaging the fractions statistics at the stations. 4. Hypothesis of timing and duration In most situations, the forecaster is able to form a judgment about the timing of the precipitation. The LCG displays the timing tree (based on Table 1) to assist the forecaster in hypothesizing a timing pattern. The timing tree shows the unconditional probabilities of all possible timing patterns, the unconditional probabilities of all durations, and the probabilities of the timing patterns conditional on the duration. The forecaster may directly select a timing pattern based on the unconditional probabilities, or he may perform sequential inference by first hypothesizing the duration of the precipitation and then selecting a timing pattern based on the conditional probabilities. a. Probabilities of duration The unconditional probabilities of precipitation occurring in one, two consecutive, three consecutive, and four subperiods over the basin and at the six stations are displayed in Fig. 6. In both March and July, the overall trend for the stations and the basin is that the longer the duration, the less frequent its occurrence.

9 171 FIG. 6. Probabilities of durations for the basin and the six stations during the months of (a) March and (b) July. Among the stations, there is a noticeable elevation effect on the duration: the probability of a short duration (one subperiod) decreases with elevation, whereas the probabilities of longer durations (two, three, and four subperiods) tend to increase with elevation. The effect of season is also quite consistent for the basin and the stations. The probabilities of shorter durations (one and two subperiods) are higher in July than in March. The probabilities of longer durations (three and four subperiods) are higher in March than in July. The most pronounced effect of spatial averaging of precipitation over the basin is a decreased frequency of events covering one subperiod and an increased frequency of events covering four subperiods. As a result, the probability of duration of the SAP cannot be approximated by a weighted average of the probabilities of duration at the stations. Doing so would significantly overestimate the probability of all SAP occurring in a single subperiod and would underestimate the probability of the SAP lasting four subperiods. b. Probabilities of timing pattern The probabilities of the timing patterns at the stations and over the basin are shown in Fig. 7. The timing patterns with nonconsecutive subperiods are not included. As in the case of duration, there does seem to be an elevation effect. The probabilities of timing patterns with duration of one subperiod tend to decrease with elevation. The probabilities of timing patterns with durations of two, three, and especially four subperiods generally increase with elevation. The distribution of timing pattern decisively varies with season. At stations, the most likely timing patterns are T 1, 4 in March, but T 1, 2 in July; the least likely timing patterns are T 2, 3, 23 in March, but T 123, 234, 1234 in July. The distinction between the distribution of timing pattern over the basin and the distributions at the stations is even more pronounced here than in the case of duration. Spatial averaging of precipitation alters the frequency of timing patterns. Relative to the stations, timing patterns with short durations are less likely over the basin, while timing patterns with longer durations are more likely over the basin. The largest differences occur in the frequencies of timing patterns T 1, 4, 1234 in both March and July, and also T 3, 123, 234 in July. In summary, when estimating the probability of duration or timing pattern at any one of the stations, the

10 172 WEATHER AND FORECASTING VOLUME 13 FIG. 7. Probabilities of timing patterns for the basin and the six stations during the months of (a) March and (b) July. forecaster could first average the probabilities at the other stations and then make an adjustment to account for the elevation effect. The spatial averaging of probabilities at the stations would not be suitable, however, for estimating the probability of duration or timing pattern of the SAP over the basin. 5. Guidance with the timing hypothesis a. Total precipitation amount Having hypothesized a timing pattern, the forecaster may request a timing-dependent exceedance function of the precipitation amount. This function shows the probabilities of various precipitation amounts being exceeded, given that measurable precipitation occurs during the subperiods specified by the timing pattern. As in the timing-independent case, three important precipitation amounts are the 75%, 5%, and 25% exceedance fractiles. The exceedance fractiles conditional on the timing T 34 in March and July are shown in Fig. 8. These are the precipitation amounts that are exceeded 75%, 5%, and 25% of the time, given that precipitation occurs in subperiods 3 and 4, but not in subperiods 1 and 2. In March, there is little variation among the stations in the 75% exceedance fractile, slightly more variation in the 5% exceedance fractile, and the most variation is in the 25% exceedance fractile. Nonetheless, the distributions of the total precipitation amount given the timing T 34 are similar at all stations. There is no obvious elevation effect. In July, the magnitudes of the three exceedance fractiles are higher than they are in March. Also, the 5% credible intervals are wider, indicating a larger uncertainty about the precipitation amount in July. The same seasonal differences are observed for the basin: the timing-dependent exceedance fractiles are higher and the 5% credible interval is slightly wider in July than in March. In comparison with the point precipitation amount, the SAP amount has a conditional distribution in which all three exceedance fractiles are lower and the 5% credible interval is narrower. Thus the uncertainty about the SAP amount is smaller than the uncertainty about the point precipitation amount. The above observations apply to all timing patterns in March and July. The exceedance fractiles for the timing T 123 are shown in Fig. 9 to confirm this. In

11 173 FIG. 8. Timing-dependent exceedance fractiles conditional on the timing T 34 for the basin and the six stations during the months of (a) March and (b) July. conclusion, the distributions of total precipitation amount for any given timing pattern are generally similar at all six stations. If a forecaster took the average of the exceedance fractiles at the stations, he would obtain a reasonable estimate of the fractile at any particular station; however, he would significantly overestimate each fractile of the SAP amount as well as the uncertainty about the SAP amount. b. Temporal disaggregation When a forecaster hypothesizes a timing pattern, he implies that precipitation will occur during certain subperiods, but not during others. Therefore, fractions are positive only in those subperiods which the forecaster believes will receive precipitation; the remaining fractions are zero. Figure 1 displays fractions statistics for the timing T 34. There is some variability in the fractions statistics at the stations given the timing T 34 in March, and slightly less variability in July. For the most part, averaging across the stations would provide a reasonable estimate of the fractions statistics conditional on the timing T 34 at any one of the stations. The temporal disaggregation for the basin is different from that for the stations in March, but it is not so different in July. The fractions statistics for the basin are compared with the weighted averages of the statistics at the stations in Table 2. In both months, a statistic for the basin in the third subperiod is lower than the weighted average of the same statistic at the stations; the relation is reversed in the fourth subperiod. In March, there is a difference of 8% between the means and a maximum difference of 15% between the fractiles. In July, the difference between the means is only 5%, and the maximum difference between the fractiles is only 7%. Thus, averaging the fractions statistics at the stations in July would result in a good estimate for the basin, but averaging in March would overestimate the basin fraction statistics in the third subperiod and underestimate in the fourth. The fractions statistics conditional on the timing T 123 are shown in Fig. 11. There is much variability among the stations. In the first subperiod, there is a 27% difference between the 25% exceedance fractiles at Connellsville and Glencoe in March, and also between Connellsville and Boswell in July. On the average, the fractions statistics at the stations differ from one another by 14% in March and by 19% in July. There is no consistent relation among the stations. In some cases, averaging

12 174 WEATHER AND FORECASTING VOLUME 13 FIG. 9. Timing-dependent exceedance fractiles conditional on the timing T 123 for the basin and the six stations during the months of (a) March and (b) July. the statistics at all stations would give a reasonable estimate of the fractions statistics at one particular station, but in other cases it would not. On the other hand, the fractions statistics for the basin, especially the mean, could be estimated by taking a weighted average of the statistics at the stations (see Table 2). The largest difference between the basin and the weighted average is in the 75% exceedance fractile for subperiod two; the difference is 1% in March and 8% in July. In summary, the timing-dependent fractions statistics for all possible timing patterns show some or much variability among the stations. There is no consistent seasonal or elevation effect. For the most part, the fractions statistics for the basin resemble those at the stations. Usually, the difference between the basin and the stations is no greater than the differences among the stations. Therefore, in most cases a weighted average of the timing-dependent fractions statistics at the stations adequately approximates the timing-dependent fractions statistics for the basin. 6. Conclusions An operational PQPF consists of several elements from which a probability distribution of the predictand can be constructed. The LCG software provides the forecaster with a climatic PQPF that has the same format as the operational forecast but is estimated from precipitation records. This case study has compared the climatic PQPFs of two predictands in the Lower Monongahela River basin: (i) point precipitation observed at raingauge stations within or near the basin, and (ii) SAP estimated over the basin. The objective was to highlight similarities and distinctions of which the forecasters should be aware. In addition, the study has illustrated seasonal differences by comparing statistics for the months of March and July and has examined the effect of elevation on the statistics of point precipitation. The conclusions are compiled in four lists. Some conclusions are theoretically justifiable; their applicability is as general as the validity of the underlying theories. Other conclusions are empirically based; their applicability should be limited to the case study area unless they are corroborated elsewhere. a. Elevation-dependent statistics The range of elevation between the lowest rain gauge (Connellsville, 9 ft) and the highest rain gauge (New Germany, 259 ft) is merely 169 ft (515 m). This may

13 175 FIG. 1. Timing-dependent fractions statistics conditional on the timing T 34 for the basin and the six stations during the months of (a) March and (b) July. explain the weak effect of elevation on the point precipitation statistics. The only trend, which consistently prevails in March and July, is in the probabilities of duration (Fig. 6): the probability of short duration precipitation (D 1) decreases with elevation, whereas the probability of longer duration precipitation (D 2C, 3C, 4) increases with elevation. b. Locally invariant statistics A locally invariant statistic is a statistic of the point precipitation amount that does not vary appreciably within some area so that a single estimate (such as a spatial average) can approximate the statistic at every point within the area. The following PQPF elements exhibit this property: 1) PoP (Fig. 2); 2) mean and fractiles of the total amount (Fig. 3); 3) fractiles of the total amount, conditional on occurrence of precipitation at every point (Fig. 4); 4) means and fractiles of fractions, though not always (Fig. 5); 5) fractiles of the total amount, conditional on the same timing pattern at every point (Figs. 8, 9); 6) means and fractiles of fractions, conditional on the same timing pattern at every point, though not always (Figs. 1, 11). Properties 1, 2, 3, and 5 are typically assumed, and have been empirically demonstrated, for models of precipitation over homogeneous areas, free of elevation and other local effects. c. Amenable-to-averaging statistics An amenable-to-averaging statistic is a statistic of the point precipitation amount that can be averaged over some area to obtain an approximation to the statistic of the SAP amount. The following PQPF elements exhibit this property: 1) mean of the total amount (Fig. 3); 2) means and fractiles of fractions, though not always (Fig. 5; Table 2); 3) means and fractiles of fractions, conditional on a timing pattern, though not always (Figs. 1, 11; Table 2). The first property has a theoretical justification in the calculus of probability. The other two properties are empirical observations. As can be seen in Figs. 5, 1, and 11, in several cases the spatial averages of fractions statistics for point precipitation consistently overestimate or underestimate the corresponding fractions statistics for SAP. Still, in most cases the spatial average provides a reasonable approximation (Table 2). Cer-

14 176 WEATHER AND FORECASTING VOLUME 13 FIG. 11. Timing-dependent fractions statistics conditional on the timing T 123 for the basin and the six stations during the months of (a) March and (b) July. tainly the errors of this approximation will be overshadowed by errors of real-time forecasts of fractions. d. Statistics of SAP versus statistics of point precipitation When assessing the elements of a PQPF for the SAP amount based on guidance products or observations for point precipitation amounts, the forecaster should account for the following consistent distinctions. 1) The PoP for SAP is higher than the PoP for any point within the area of averaging (Fig. 2). 2) When the PoP for SAP is less than one, a fractile of the SAP amount may be larger or smaller than the corresponding fractile of the point amount anywhere within the averaging area (Fig. 3). 3) When the PoP for SAP and for every point within the averaging area is one, each fractile of the SAP amount is smaller than the corresponding fractile of the point amount anywhere within the averaging area, and the credible interval of the SAP amount is narrower than the credible interval of the point amount (Fig. 4). 4) Probability of all 24-h precipitation falling in a single subperiod (D 1) is lower for SAP than for point precipitation. Probability of precipitation lasting four consecutive subperiods (D 4) is higher for SAP than for point precipitation (Figs. 6, 7). 5) Conditional on a timing pattern, each fractile of the SAP amount is smaller than the corresponding fractile of the point amount anywhere within the averaging area, and the credible interval of the SAP amount is narrower than the credible interval of the point amount (Figs. 8, 9). Distinctions 1, 3, 4, and 5 are consistent (general) in the sense that the order between a SAP statistic and a point precipitation statistic can be justified theoretically. e. Disaggregation approach One empirical result stands out: the temporal disaggregation of the total precipitation amount (unconditional or conditional on a timing pattern) is locally invariant and amenable to spatial averaging. This property, together with the property of conditional disaggregative invariance of the SAP and point precipitation (Krzysztofowicz and Pomroy 1997), make the disaggregation approach to PQPF especially advantageous.

15 177 First, the problem can be decomposed into three tasks: (i) forecasting the precipitation timing T; (ii) forecasting the total amount W, conditional on timing T t; and (iii) forecasting the temporal disaggregation ( 1,..., n ), conditional on timing T t. Tasks (ii) and (iii) can be performed independently of one another, and this reduces the complexity of judgments required on the part of forecasters. Second, the distinction between the SAP and point precipitation (i) must be accounted for when forecasting the precipitation timing T and the total amount W, but (ii) need not be considered when forecasting the temporal disaggregation ( 1,..., n ). Property (ii) further reduces the complexity of judgments required on the part of forecasters. f. Training of forecasters Training of forecasters preparing PQPFs should include relations between climatic statistics of SAP amounts and the corresponding statistics of point precipitation amounts. Learning these relations is best accomplished in two stages. First, the forecaster should become familiar with the order between the statistics. For example, is the PoP for SAP lower or higher than the PoP for any point? The empirical relations documented herein and identified as general (theoretically justified) provide the answers. Second, the forecaster should become familiar with statistics for his local area. This is the reason for not emphasizing herein generalized relations that are the staple of models. Such relations are usually too complex and too general to be of utility to the local forecaster. For instance, the scaling factor for a conditional exceedance fractile from point precipitation to SAP may depend not only upon the area of averaging, but also upon orography, storm type, circulation pattern, and seasonal and diurnal effects. Only by studying local statistics can the forecaster acquire expertise that ought to be his forte: the knowledge of local hydrometeorological influences. This article has shown what statistics are relevant to PQPF, what questions should guide the inferences, and how the LCG system could be used to accomplish this. Acknowledgments. The hourly precipitation data were obtained through kind cooperation of John Vogel and Michael Yekta, Hydrometeorological Branch, Office of Hydrology, National Weather Service. This article was completed while Roman Krzysztofowicz was on assignment with the National Weather Service, Eastern Region, under an Intergovernmental Personnel Act agreement. Research leading to this article was supported by the National Weather Service, under the project Development of a Prototype Probabilistic Forecasting System. Leadership of Gary Carter in promoting this project and fostering collaborative research environment within the Eastern Region is gratefully acknowledged. 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