Subpixel-Scale Rainfall Variability and the Effects on Separation of ...

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Subpixel-Scale Rainfall Variability and the Effects on Separation of Radar and Gauge Rainfall Errors YU ZHANG* National Risk Management Research Laboratory, U.S. EPA, Cincinnati, Ohio

THOMAS ADAMS Ohio River Forecast Center, National Weather Service, Wilmington, Ohio

JAMES V. BONTA North Appalachian Experimental Watershed, USDA–ARS, Coshocton, Ohio (Manuscript received 25 September 2006, in final form 4 April 2007) ABSTRACT This paper presents an extended error variance separation method (EEVS) that allows explicit partitioning of the variance of the errors in gauge- and radar-based representations of areal rainfall. The implementation of EEVS demonstrated in this study combines a kriging scheme for estimating areal rainfall from gauges with a sampling method for determining the correlation between the gauge- and radar-related errors. On the basis of this framework, this study examines scale- and pixel-dependent impacts of subpixelscale rainfall variability on the perceived partitioning of error variance for four conterminous Hydrologic Rainfall Analysis Project (HRAP) pixels in central Ohio with data from Next-Generation Weather Radar (NEXRAD) stage III product and from 11 collocated rain gauges as input. Application of EEVS for 1998–2001 yields proportional contribution of two error terms for July and October for each HRAP pixel and for two fictitious domains containing the gauges (4 and 8 km in size). The results illustrate the importance of considering subpixel variation of spatial correlation and how it varies with the size of domain size, number of gauges, and the subpixel locations of gauges. Further comparisons of error variance separation (EVS) and EEVS across pixels results suggest that accounting for structured variations in the spatial correlation under 8 km might be necessary for more accurate delineation of domain-dependent partitioning of error variance, and especially so for the summer months.

1. Introduction Weather Surveillance Radars (WSRs), by virtue of their extensive spatial and temporal coverage, have long been considered an attractive alternative to rain gauges in supplying precipitation information to a wide range of applications (e.g., Hudlow et al. 1991; Smith et al. 1996). In recent years there have been attempts to integrate radar rainfall data in water resources manage-

* Current affiliation: Office of Hydrologic Development, National Weather Service, Silver Spring, Maryland.

Corresponding author address: Yu Zhang, USEPA National Risk Management Research Laboratory, Cincinnati, OH 45268. E-mail: [email protected]

DOI: 10.1175/2007JHM835.1

ment for relatively small spatial domains [i.e., ⬍100 km2; see, e.g., Vieux et al. (2002)]. The WSR precipitation products being investigated for these purposes are often those provided by the National Weather Service (NWS) for operational streamflow forecasting [e.g., stage III and the Multisensor Precipitation Estimate; Vieux et al. (2002)]. Despite the fact that these radar datasets have been adjusted against rain gauge records (Fulton et al. 1998), the concern about their accuracy remains. Hardegree et al. (2003), for example, found pronounced differences between the stage III data and corresponding rain gauge records in six U.S. Department of Agriculture (USDA) Agricultural Research Service (ARS) stations. In attempting to reconcile the differences between gauge measurements and radar rainfall estimates, we must note that rain gauge data, because of the limited

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dimension of the instrument, may not be representative of areal rainfall for a radar pixel (e.g., Zawadzki 1973; Austin 1987; Krajewski 1987; Ciach and Krajewski 1999). While over an extended period of time the difference may stabilize and therefore be describable by a bias factor, over short time scales (less than a week) the spatial variability may still play an important role. For these time scales, aside from accounting for deterministic errors arising from the limitations of weather radars and processing procedures (e.g., Krajewski et al. 1996; Smith et al. 1996), methods of incorporating radar rainfall (RR) data need to reflect our understanding of small-scale rainfall variability that governs the pointarea difference (e.g., Austin 1987; Krajewski 1987). In response to this need, there have been efforts in recent years to account for the point-area difference through distribution-oriented verification of radar rainfall products (e.g., Habib et al. 2004). Center to such efforts is the decomposition of the variance of radar-gauge (R-G) discrepancies into that of point-area difference and the errors in RR estimates. Ciach and Krajewski (1999) proposed a framework for this purpose. The method, originally termed “error separation method” (ESM; Ciach and Krajewski 1999), was later renamed “error variance separation” (EVS; Ciach et al. 2003). While the original formulation EVS in Ciach and Krajewski (1999) offers a simple way for determining the uncertainties in RR estimates, it is focused exclusively on the differences between rainfall data from a single gauge and the radar rainfall for a radar pixel. Yet, in some of the small-scale applications (such as in urban settings), it is the accuracy of areal rainfall estimates for an area (not necessarily a radar pixel) that interests practitioners. When multiple gauges are present in and around this area, it is desirable to use the areal rainfall on the basis of combined gauge records for estimating the uncertainties in corresponding RR data. Moreover, the number of gauges and the subpixel locations of gauges, which often vary greatly among pixels, may need to be taken into account in translating the partitioning results across areas. This study specifically addresses these needs by extending the EVS approach to allow partitioning of error variance in areal rainfall based on multiple gauge records and RR data. This extended version of EVS (EEVS), is implemented for a small area situated in the northwestern Appalachian plateau, where multiple years of Next-Generation Weather Radar (NEXRAD) stage III data are available and a collocated network of 11 rain gauges (distributed within a radius of approximately 1 km) offers an independent set of precipitation information. This combination offers an ideal opportunity for examining how EEVS can be applied to assess errors in existing

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RR products that have gone through synoptic correction (i.e., based on data from a coarse gauge network spreading over a much larger spatial domain). Following the work by Ciach et al. (2003), the implementation of EEVS in the study considers possible correlation in the radar and gauge-related errors and thereby relaxes the zero-covariance hypothesis (Ciach et al. 2003). In addition, this study takes a step further to address the practical issues involved in applying EEVS, in particular the effects of introducing distancedependent spatial correlation (DDSC) on EEVS results. By comparing the results for July and October of 1998–2001, and those among various spatial domains, this study highlights the dependence of EEVS-based partitioning on scale-dependent rainfall characteristics as well as the number and subpixel locations of collocated gauges. The organization of this paper is of follows. Section 2 offers a brief review of EVS and then presents the framework of EEVS and its implementations. Section 3 describes the study setting and rainfall records. Section 4 presents the results of EEVS applied to July and October of 1998–2001. Section 5 discusses the implications for further application of EEVS. Section 6 presents the summary and conclusions.

2. Error variance separation method and its extension a. Review of error variance separation method The key to the method of Ciach and Krajewski (1999) is the assumption that the errors in RR estimates are uncorrelated with the discrepancy solely because of the difference between “point” rainfall by a single gauge and the actual areal rainfall (P ⫺ A difference). Let Rg(x) designate the rainfall measurement at a gauge whose coordinates are given by x ⫽ (x, y). Let Rr be the RR estimates for the pixel that encompasses x, and Ra the true areal rainfall for that pixel. The assumption can be written as E 关共Rg ⫺ Ra兲共Rr ⫺ Ra兲兴 ⫽ E 关Rg ⫺ Ra兴E 关Rr ⫺ Ra兴.

共1兲

Assuming E[Ra] ⫽ E[Rg] ⫽ E[Rr], the P ⫺ A difference and the error associated with RR estimates and can be partitioned as follows: Var关Rg ⫺ Rr兴 ⫽ Var关Rg ⫺ Ra兴 ⫹ Var关Ra ⫺ Rr兴.

共2兲

The utility of EVS is primarily that of computing the fractional contribution of the two right-hand side (rhs) terms to the total discrepancy E[(Rg ⫺ Rr)2], that is, (Var[Rg ⫺ Ra]/Var[Rg ⫺ Rr]) and (Var[Rr ⫺ Ra]/ Var[Rg ⫺ Rr]) (denoted in this study by ␣ and ␤, re-

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spectively). To do so, Ciach and Krajewski (1999) assumed that the rain gauge measurement at point x, Rg(x) is an exact estimate of point rainfall. By also assuming second-order stationarity of point rainfall in space, Ciach and Krajewski (1999) derived the covariance between single gauge rainfall Rg and the true areal rainfall Ra, Cov[Rg, Ra], and the variance in the P ⫺ A difference Rg ⫺ Ra, Var[Rg ⫺ Ra]: Cov关Rg, Ra兴 ⫽

␴ 2g 㛳A 㛳

冕

冋

␳共xg, x兲 dx

x∈A

冕 冕冕

Var关Rg ⫺ Ra兴 ⫽ ␴ 2g 1 ⫺ ⫹

共3兲

1 㛳A 㛳2

2 㛳A 㛳

A

␳共xg, x兲 dx

A

␳共x,xⴕ兲 dx dxⴕ

A

册

共4兲

共5兲

where d is distance and d0 is the decorrelation distance. Ciach and Krajewski (1999) treated spatial correlation as a constant within each grid (4 km) on the grounds that the grid size is much smaller than typical decorrelation distances (of the order of 30–70 km), and that correlation may exhibit nugget effect. The nugget effect may arise from instrumentation errors, which tend to blur the dependence of spatial correlation on distance (Ciach et al. 2003; Ciach and Krajewski 2006). This gives the following: Var关Rg ⫺ Ra兴 ⫽ ␴ 2g共1 ⫺ ␳0兲.

共6兲

An estimator of ␳0 can be derived by pooling the records from multiple gauges

␳ˆ 0 ⫽

n␴ˆ 2av

⫺

␴ˆ 2g

共n ⫺ 1兲␴ˆ 2g

A, and their spatial coordinates are given by xi, where i ⫽ 1 . . . n. Let Rg(xi) denote the rainfall recorded at gauge i located at xi, and Rr(Sj) the radar rainfall estimate for pixel Sj. The areal rainfall estimates for A based entirely on rain gauge measurements can be expressed as ˆ 共A 兲 ⫽ G共R 共x 兲, . . . , R 共x 兲兲 R a,g g 1 g n

共8兲

where G(. .) denotes the function that maps records from individual gauges Rg(Xi) to the areal rainfall estiˆ (A). mate R a,g Similarly, the rainfall estimates for A based on radar data can be written as ˆ 共A 兲 ⫽ H共R 共S 兲, . . . , R 共S 兲兲 R a,r r 1 r M

where A is the domain of a pixel, and ␴ 2g ⫽ Var[Rg]. Equation (4) can be substituted into (2) to yield the estimates of ␣ and ␤. Ciach and Krajewski (1999) employed the isotropic, exponentially decaying spatial correlation in the form of

␳共d兲 ⫽ ␳0e⫺dⲐd0

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共7兲

where ␴ˆ g and ␴ˆ av are the standard errors for singlegauge rainfall and the multigauge average, respectively; ␴ˆ g can be obtained by combining the gauge records.

b. Extended EVS When multiple gauges are present within the area of interest, the records from the gauges can be combined to provide error variance decomposition for an arbitrary area. To start let us consider a spatial domain A. Let Ra(A) denote the actual areal rainfall over area A. Suppose a series of n gauges are present in and around

共9兲

where M is the total number of pixels that intersect with A, and H maps rain rates for each radar pixel to the ˆ (A). Again, it is radar-based areal rainfall estimate R a,r ˆ ˆ assumed that E[Ra,g] ⫽ E[Ra,r] ⫽ E[Ra]. To accommodate the situation where RR data for a given area has been corrected with gauge data, the assumption of lack ˆ ⫺ R is reˆ ⫺ R and R of correlation between R a,g a a,r a laxed here to yield ˆ ⫺R ˆ 兴 ⫽ Var关R ˆ ⫺ R 兴 ⫹ Var关R ˆ ⫺R 兴 Var关R a,g a,r a,g a a,r a ˆ ⫺R ,R ˆ ⫺ R 兴. ⫺ 2Cov关R a,g a a,r a

共10兲

The objective is hence to determine the following quantities: ˆ ⫺R 兴 Var关R a,g a ˆ ⫺R ˆ 兴 Var关R

␣⫽

a,g

共11兲

a,r

and ˆ ⫺R ,R ˆ ⫺R 兴 Cov关R a,g a a,r a , ˆ ˆ Var关R ⫺ R 兴

␤⫽1⫺␣⫹2

a,g

共12兲

a,r

that is, the fractional contributions of point-area difference and radar-related errors, respectively. ˆ ], ˆ ⫺R The denominator in (11) and (12), Var[R a,g a,r ˆ at ˆ and R can be estimated empirically by pooling R a,g a,r all time instants within a designated time interval T ⫽ {ti; t ⫽ 1, 2 . . . , K}: ˆ ⫺R ˆ 兴⬇ Var关R a,g a,r

1 K⫺1

K

兺 关Rˆ

a,g共ti 兲

ˆ 共t 兲兴2. ⫺R a,r i

共13兲

i⫽1

To reduce the impacts of bias, that is, the difference ˆ ], on the result of Eq. (4), the ˆ ] and E[R between E[R a,g a,r stage III estimates can be adjusted for each time interval T using a constant factor so that the cumulative rainfall matches that from the gauges. The EEVS framework as given by (11) and (12) al-

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lows incorporation of different methods of estimating areal rainfall. For demonstration purposes, let us consider the situation where areal rainfall is represented by the weighted average of gauge records: ˆ 共A 兲 ⫽ R a,g

兺

兵i,᭙xi∈A其

␻iRg共xi兲.

共14兲

ˆ is the areally weighted A convenient estimate of R a,r average of rainfall at all radar pixels that overlap with A . Denote the spatial domain of radar pixel j as Sj, then n

ˆ ⫺ R 兴 ⫽ Var关R 兴 ⫹ Var关R a,g a a

M

j

r

共15兲

j

j⫽1

where aj is the proportional area of the pixel within domain A , and Rr(Sj) is the RR estimates for pixel j. ˆ (A) ⫽ R (S ) when the area of interest A coHere R a,r r j incides with the spatial domain of a radar pixel j, Sj. With (14), the variance of the difference between the single- or multigauge estimate and the true areal mean rainfall can be easily derived. After some manipulations (see, e.g., Journel and Huijbregts 1978)

n

兺兺

兺 a R 共S 兲

ˆ ⫽ R a,r

n

wiwj Cov关Rg共xi兲, Rg共xj兲兴 ⫺ 2

i⫽1 j⫽1

兺 w Cov关R 共x 兲, R 兴. i

g

i

a

共16兲

i⫽1

Substituting Eq. (3) into (16) yields ˆ ⫺R 兴⫽ Var关R a,g a

␴ 2g 㛳A 㛳

2

冕冕 A

n

␳共x, xⴕ兲 dx dxⴕ ⫹ ␴ 2g

n

兺兺 i⫽1 j⫽1

A

Using climatic correlation ␳ (xi, xj) derived from longˆ ⫺ R ] can be minimized term gauge records, Var[R a,g a following block kriging to yield the best linear unbiˆ . The deriased estimate (BLUE) of areal rainfall, R a,g vation of the associated weighting factors wi can be found in Journel and Huijbregts (1978, p. 307), with the

兺 㛳A 㛳 冕 n

wiwj␳共xi, xj兲 ⫺ 2␴ 2g

i⫽1

wi

␳共xi, x兲 dx.

covariance term being that between a point and the area Cov[Rg(xi), Ra]. Substituting Eq. (17) into (11) yields ␣. To estimate ␤ requires estimating the cross term in (12). Note that covariance can expanded thusly:

ˆ ⫺R ,R ˆ ⫺ R 兴 ⫽ cor关R ˆ ⫺R ,R ˆ ⫺ R 兴Var关R ˆ ⫺ R 兴1Ⲑ2Var关R ˆ ⫺ R 兴1Ⲑ2. Cov关R a,g a a,r a a,g a a,r a a,g a a,r a ˆ ⫺ R ] can be approxiAssuming that Ra in Var[R a,r a ˆ mated by Ra,g, it can be shown that ˆ ⫺R ,R ˆ ⫺ R 兴. ␤ ⬇ 1 ⫺ ␣ ⫹ 2␣1Ⲑ2cor关R a,g a a,r a

共19兲

A resampling method is developed to empirically determine ␤ wherein the correlation term in (19) is assumed to be independent of the location and the numˆ . In this method, ber of gauges used for estimating R a,g the true areal rainfall for domains A and B (Fig. 2a) is approximated by the BLUE based on all n gauges ˆ n n ⫽ 11; weights are obtained by minimizing (17)]. [R a, ˆ k , a similar gauge-based ˆ n and R The coupling of R a a,g ˆ k , allows BLUE except based on a subset of k gauges R a,g estimation of the correlation in (19). Following the sampling method outlined in the appendix, multiple samples of such a subset are generated (with k gauges each, 2 ⱕ k ⱕ 10). For each sample the upper and lower ˆn , R ˆ ⫺R ˆn ] ˆk ⫺ R quartiles of the estimated cor[R a,r a,g a,g a,g are used to delineate the range of estimated ␤ (see the appendix for details).

共17兲

x∈A

共18兲

In this study, first following Ciach and Krajewski (1999), uniform spatial correlation (USC) at subpixel scale is used in deriving the partitioning coefficients. Then this assumption is replaced by DDSC in the form of Eq. (5) on the basis of observations. Climatic spatial correlation structure derived from long-term historical gauge data (15 yr) is used for computing the weights that minimize the variance in (17) (see section 3 for details). The results from the two implementations are compared to assess the impacts of structured variations of rainfall on the perceived contributions from P-A difference and radar-related errors. These results are further compared to those from EVS to highlight the differences. In the implementation of both EVS and EEVS, only the hours with positive rainfall are used for computing ␴g and ␳(xi, xj) to avoid an oversuppression of variance by the presence of dry periods. In EVS computation, each gauge is paired with the collocated Hydrologic Rainfall Analysis Project (HRAP) pixel, and all gauge–pixel pairs are treated equally regardless of the number of gauges that are present within one pixel.

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FIG. 1. NAEW near Coshocton, OH, and four neighboring WSR-88Ds, namely, (i) ILN (Wilmington, OH), (ii) CLE (Cleveland, OH), (iii) RLX (Charleston, WV), and (iv) PBZ (Pittsburgh, PA). Each circle represents the 125 n mi limit within which the reflectivity data from the respective radar were used for creating the NEXRAD stage III precipitation dataset.

3. Study setting and preliminary data analysis The study site is located within the North Appalachian Experimental Watershed (NAEW) near Coshocton, OH (about 40°22⬘N, 82°12⬘W; see Fig. 1). NAEW is managed by the USDA–ARS. The regional climate is continental, with about 950 mm of precipitation per annum. For this study NEXRAD stage III hourly precipitation estimates were acquired from the Ohio River Forecast Center (OHRFC), NWS. The stage III data came on the grids of HRAP [see Fulton (1998) and Reed and Maidmant (1999) for descriptions], with each data point representing areally averaged rainfall over an HRAP grid cell that is approximately 4 km ⫻ 4 km in size (Fulton et al. 1998, p. 5). The stage III dataset for this study has been used in operational river forecast by OHRFC. The stage III dataset was created by mosaicing precipitation estimates from three adjacent Weather Surveillance Radar-1988 Doppler (WSR88D), namely, 1) CLE (located near Cleveland, OH); 2) RLX (Charleston, WV), and 3) PBZ (Pittsburgh, PA; Fig. 1) The radar stations are located about 116, 229, and 135 km from the gauge network, respectively. Hourly precipitation estimates for each radar station were derived from reflectivity measurements and were bias-adjusted using records from gauges underneath the radar umbrella [see the methodology in Seo (1998) and related discussions in Fulton et al. (1998)]. Scattered over the site are 11 Belfort weighing bucket gauges with pairwise distance ranging from

FIG. 2. (top) Locations of rain gauges in the NAEW near Coshocton, OH. Superimposed are the boundaries of NEXRAD HRAP grid (in dotted lines) and two fictitious domains A and B (solid lines). (bottom) A magnified version of the top panel.

250 m to 2 km (see Fig. 2 for illustration and Table 1 for coordinates), each providing break-point records of rainfall accumulation on irregular time intervals (the duration varies between 2 min and 24 h). The elevation of the gauges ranges from 281 m at RG119 to 372 m at RG103 (Table 1). These gauges are distributed among four HRAP pixels, with 6 gauges in pixel 1, 2 in pixels 2 and 3, and 1 in pixel 4 (Table 1 and Fig. 2b). In addition to having been quality assured by the USDA– ARS station, the dataset has gone through a filtering procedure devised to correct the records that are deemed spurious (i.e., displaying large deviation from the rest). In this procedure, the hours with significant amount of intergauge variation in rain [i.e., standard deviation (SD) of hourly rainfall exceeding 2 mm] were first identified using the records of 11 gauges. Then

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TABLE 1. Locations of rain gauges in NAEW.

Gauge ID RG100 RG103 RG107 RY101 RY102 RY103 RG108 RG109 RG113 RG115 RG119

Latitude 40°22⬘19⬙ 40°22⬘15⬙ 40°21⬘59⬙ 40°22⬘28⬙ 40°22⬘15⬙ 40°22⬘25⬙ 40°21⬘54⬙ 40°22⬘04⬙ 40°21⬘42⬙ 40°21⬘32⬙ 40°21⬘30⬙

N N N N N N N N N N N

Longitude 81°47⬘48⬙ 81°47⬘59⬙ 81°47⬘44⬙ 81°47⬘39⬙ 81°47⬘38⬙ 81°47⬘20⬙ 81°47⬘11⬙ 81°46⬘54⬙ 81°47⬘55⬙ 81°47⬘37⬙ 81°46⬘53⬙

W W W W W W W W W W W

Elevation (m)

Pixel ID

354 373 352 361 363 343 315 334 353 349 281

1 1 1 1 1 1 2 2 3 3 4

among these periods, the gauge records whose departure from the multigauge median exceeds 2.5 times the SD were designated as anomalies. These records were subsequently replaced by the mean of the rest of gauge record for that hour. One hour of anomalous data in July and six hours in October were corrected through this procedure. Note that none of these gauges has been used by NWS in deriving radar rainfall products, though a nearby cooperative (COOP) gauge (COOP ID: 331905; 40°22⬘N, 81°47⬘W) is within the NWS gauge network and has provided limited number of hourly observations (days with record between 1998 and 2001: 24 in July and 14 in October). Despite the gauge-based adjustment (as described in Seo 1998), discrepancies between stage III and ground gauge records for this specific location, as will be demonstrated in later parts of this section, remain considerable. To illustrate the dependence of results on temporal resolution and to facilitate comparisons with Ciach and Krajewski (1999), in this study, gauge data were interpolated into hourly scale first and then both gauge and stage III data were aggregated into 2-, 3-, 4-, 6-, 12-, 24-, and 48-h increments. To fully utilize the gauge records, two fictitious domains, labeled A and B, were created from the HRAP pixels, each covering the entire gauge network (Fig. 2a). Domain A is composed of four quarter-pixels surrounding the intersection of the four HRAP pixels, and domain B encompasses four entire HRAP pixels (Fig. 2a). In addition, comparisons of monthly precipitation across the gauges for July and October from 1990 to 2004 reveal no obvious dependence on elevation (Fig. 3), therefore justifying the assumption of secondorder stationarity among the gauge records and the use of a single ␴ 2g. Over central Ohio, convective storms are responsible for a significant fraction of summertime precipitation, whereas stratiform systems are more prevalent during

FIG. 3. Box plots of monthly rainfall accumulations (between 1990 and 2004) at 11 gauges for (a) July and (b) October. The solid line represents the elevation of each gauge.

the fall. To examine the impacts of seasonally dependent rainfall structure on EEVS, the records for July and October of 1998–2001 were used separately for deriving two sets of results (one for July and one for October). These records were first used to compile a set of events, where each event is defined as a period with continuously positive hourly rainfall (in a multigauge average sense). A total of 54 events was thus identified for July and 37 for October (Table 2). The contrasting nature of precipitation systems is aptly reflected in the difference in average event duration (3 h for July and 7 h for October, Table 2) and maximum hourly rainfall (49 mm in July versus 20 mm in October; Table 2). It is also evident in the quantiles of hourly rainfall accumulation (relative abundance of light rain; Fig. 4), and in the difference in the estimated variance of gauge-based rainfall ␴ˆ 2g (consistently higher values for July; Fig. 5). TABLE 2. Statistics of rainfall events for July and October of 1998–2001.

Month

Monthly total (mm)

Number of events

Mean duration (h)

Maximum duration (h)

Mean hourly rain (mm)

7 10

434 277

54 37

3 7

9 21

49 20

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FIG. 4. Quantile–quantile plot of mean hourly rainfall accumulation by all gauges for July and October 1998–2001.

The gauge- and radar-based hourly precipitation at the gauge locations are compared in Fig. 6, and in Tables 3 and 4. The cumulative rainfall from the NEXRAD stage III dataset (averaged over the 4 HRAP pixels; Fig. 6) is comparable to the corresponding gauge record (averaged over 11 gauges) for July (Table 3), but is considerably less than the latter for October (Table 4). This underestimation of rainfall accumulation by radar for October is related to both underestimation of total hours of positive rain during October (135 h by radar versus 232 h by gauges; Table 4) and that of maximum rainfall intensity (10 mm h⫺1 by radar and 20 mm h⫺1 by gauges; Table 4). Prior to applying EEVS, the assumption of constant subpixel spatial rainfall correlation that Ciach and Krajewski (1999) employed is examined on the basis of gauge records. The bases of this assumption are, namely, a) the presence of the nugget effect and b) that the scale of interest (⬍4 km) is an order of magnitude lower than the decorrelation distance d0 in Eq. (5). It follows from these two assumptions that, at subpixel scale (⬍4 km), spatial correlation approaches, and therefore can be approximated by, the value of the nugget ␳0. In this study, the nugget effect assumption is assessed by examining the distance dependence in the correlation coefficients among pairs of gauges, with the tacit assumption that correlation structure is time invariant for each period. The first step of this analysis entails computing pairwise correlation for July and October on the basis of 15

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FIG. 5. Estimate of variance of gauge-based rainfall (␴ˆ 2g), vs temporal resolution, for July and October of 1998–2001. The values were computed by combining the records from all 11 gauges (only the periods with at least one positive record were used).

yr of gauge records between 1990 and 2004. Again, only periods with positive rainfall in at least one of the 11 gauges were considered. The pairwise correlation is then regressed against the corresponding betweengauge distance following the exponential model specified in (5) to derive the estimate of the nugget ␳˜ 0 and decorrelation distance d˜ 0 (the symbol ⬃ denotes the estimates from regression). The dependence of pairwise correlations ␳(xi, xj) on separation distance for July and October are illustrated in Figs. 7a,b. The distance dependence in the correlation coefficients is evident for both periods at both time resolutions. The results of regression indicate a significant reduction in variance by considering distance dependence (R2 ⫽ 0.81 for both July and October at hourly increments; Table 5). In addition, the estimate ␳˜ obtained from regression is nearly 1.00 for almost all time resolutions (Table 5). These observations, collectively, suggest a highly structured spatial correlation and lack of significant nugget effect at the scale of interest (⬎300 m) for this particular setting. The regression-based estimates of d˜ 0 and correlation at 4 km ␳˜ (4 km) are illustrated in Figs. 8a,b along with the respective error ranges (i.e., ⫹/⫺ standard error). The values of d˜ 0 and ␳˜ at 4 km are comparable for July and October at hourly resolution. The contrast between the two periods becomes increasingly pronounced with decreasing temporal resolution (Figs. 8a,b), as progressive temporal aggregation induces a stronger homog-

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FIG. 6. Scatterplots of hourly rainfall accumulation based on NEXRAD stage III data (from all four HRAP pixels depicted in Fig. 2) vs those at corresponding individual gauges, for (a) July 1998–2001 and (b) October 1998–2001.

enizing effect for October. Between the two months, the error ranges of d˜ 0 are appreciably higher for October, as a result of higher values of d˜ 0. Further comparisons of regression-based ␳˜ at 4 km and uniform ␳ˆ 0 via Eq. (7) show the former being appreciably higher at fine temporal resolutions (i.e., ⬍4 h; Fig. 8b). In the subsequent section, the actual impacts of incorporating the DDSC on EEVS results will be addressed.

ˆ ⫺R 兴⫽ Var关R a,g a

1 ⫺ ␳0 2 ␴ g. n

共20兲

Equation (20) differs from (6) only by a factor of n. When DDSC as given in Eq. (5) is considered, the first, second, and third terms of Eq. (17) become

␴ 2g 㛳A 㛳

2

冕冕 l

⫺l

l

⫺l

共l ⫺ |u|兲共l ⫺ |␷|兲␳0e⫺共u

2⫹␷2 1Ⲑ2Ⲑd0

兲

du d␷, 共21兲

4. Application of EEVS and observations For EEVS with constant spatial correlation assumed, the weighting factors in Eq. (17) obtained via the kriging approach would be uniform (i.e., w ⫽ 1/n). Equation (17) is therefore reduced to TABLE 3. Summary of rainfall by gauges and stage III for July 1998–2001. Accumulation (mm)

Hours of rain

n

␴ 2g

Gauges

Radar

Gauges

Radar

Gauges

Radar

Jul 1998 Jul 1999 Jul 2000 Jul 2001 Net

28 164 112 130 434

55 125 149 103 432

38 36 43 37 154

31 46 54 35 166

5 31 21 49 49

9 13 28 30 30

i

j 0

⫺|xi⫺xj|Ⲑd0

共22兲

,

i⫽1 j⫽1

and TABLE 4. Summary of rainfall by gauges and stage III for October 1998–2001.

Maximum rain rates (mm h⫺1)

Month

n

兺 兺ww␳ e

Accumulation (mm) Month Oct Oct Oct Oct Net

1998 1999 2000 2001

Hours of rain

Maximum rain rates (mm h⫺1)

Gauges

Radar

Gauges

Radar

Gauges

Radar

100 24 63 90 277

74 13 53 73 213

53 46 49 84 232

34 19 35 47 135

20 5 9 6 20

10 4 7 9 10

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FIG. 7. Pairwise correlation between each pair of gauges vs separation distance at hourly and daily resolutions, for (a) July 1990–2004 and (b) October 1990–2004. Superimposed are the fitted model (solid lines) and the error ranges (dotted lines) obtained via regression following the exponential model specified in Eq. (5).

⫺

2␴ 2g 㛳A 㛳

兺w冕 n

i

i⫽1

␳0e⫺|x⫺xi|Ⲑd0 dx,

共23兲

x∈A

⌬

⌬

respectively, where u ⫽ x ⫺ x⬘ and ␷ ⫽ y ⫺ y⬘ and l is the size of the pixel. The three terms are numerically evaluated at each temporal resolution, wherein ␳0 and d0 are substituted by the corresponding regression-based estimates ␳˜ 0 and d˜ 0 (see Table 5 for values). The estimate of ␣ with the USC assumption (␣ˆ u) and that by considering DDSC (␣ˆ d) are shown in Fig. 9, where the results are contrasted between domains A (Figs. 9a,c) and B (Figs. 9b,d; see illustration of domains A and B in Fig. 2a), and between July (Figs. 9a,b) and October (Figs. 9c,d). Also shown for domain A (Fig. 2a) are estimates from EVS (Figs. 9a,c). For domain A (4 km; Fig. 2a) EEVS-based a estimates with both USC and DDSC are consistently lower than those from EVS (Figs. 9a,c). The former never exceed 5% irrespective of the month, whereas the latter are somewhere between 10% and 15% at an hourly time scale for July. Between the EEVS results with USC and DDSC, for July the latter values are much higher than the former ones (Fig. 9a), whereas for October the differences are rather minor (Fig. 9c). The impacts of incorporating DDSC are more pronounced for domain B (8 km; Fig. 2a), where ␣ˆ d are significantly higher (close to or above 10% for July and above 5% for October at the hourly resolution; Figs. 9b,d) than ␣ˆ u (remain identical to

those at 4 km). Between the two months, July is associated with comparatively higher ␣ˆ d values and a wider error range (Fig. 9b). The EEVS-based estimates of the contribution from radar-related error ␤ were computed using the correlation between the error terms in (19). The latter was estimated via the sampling method described earlier and in the appendix. The values of median and two quartiles of the correlation are given in Table 6, and ␤ˆ d (estimate of ␤ considering DDSC) for domains A and B (Fig. 2a) are shown in Fig. 10, with the range delineated by the upper and lower quartiles of the ␤ˆ d. The correlation values are conspicuously negative for July, and are slightly less so for October (Table 6). The range of ␤ˆ d is wider for July (Figs. 10a,b) than for October (Figs. 10c,d). The impacts of accounting for DDSC are further ilTABLE 5. Spatial correlation parameters from regression. d˜ 0 (h)

␳˜ 0

␳˜ 4km

R2

␦t (h)

Jul

Oct

Jul

Oct

Jul

Oct

Jul

Oct

1 2 3 4 6 12 24 48

24 27 27 27 28 29 43 58

32 55 58 73 83 101 117 293

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.81 0.80 0.81 0.77 0.75 0.76 0.72 0.72

0.81 0.75 0.73 0.75 0.71 0.69 0.70 0.51

0.97 0.97 0.98 0.98 0.98 0.98 0.98 0.98

0.94 0.96 0.97 0.98 0.98 0.99 1.00 1.00

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effect is particularly notable for July, where ␣ estimates for pixel 1 at hourly resolution increases from nearly 0% to above 20% with the introduction of DDSC (in comparison to about 10% based on EVS; Figs. 11a,b). The ␣ estimates for pixel 1 rise to such an extent that they become comparable with those for pixel 4 (Figs. 11b,d). In contrast to pixels 1 and 4, for domain A (Fig. 2a) the estimates of ␣ experience only limited increase for both July and October (Figs. 11a–d).

5. Discussion a. EEVS framework and factors determining its implementation

FIG. 8. Estimates of (a) decorrelation distance (d0) and (b) spatial correlation (␳) at 4 km vs temporal resolution for July and October 1990–2004. The solid lines depict quantities computed following the exponential formula [Eq. (5)]. The dotted lines in (b) represent the uniform spatial correlation derived via Eq. (7).

lustrated by the comparison of EEVS results for HRAP pixels 1 and 4 and those for domain A (with 6, 1, and 11 collocated gauges, respectively; Figs. 11a–d; see illustration of domain A in Fig. 2a). With USC, ␣ˆ u decreases with increasing number of embedded gauges (n), a feature reflecting the inverse relation between ␣ˆ d and n as described in (20) (Figs. 11a,c). The introduction of DDSC considerably enhances the ␣ estimates for the two HRAP pixels (Figs. 11b,d). While the ␣ estimates from EEVS with USC assumption are almost uniformly lower than those from EVS for pixels 1 and 4 (Figs. 11a,c), the corresponding ones from EEVS with DDSC considerably exceed the EVS results (Figs. 11b,d). This

This study presented a framework (EEVS) that allows incorporation of records from multiple gauges in partitioning the error variance in the gauge- and radarbased areal rainfall for an arbitrary spatial domain. The framework was made sufficiently general to accommodate arbitrary spatial domain and arbitrary methods of estimating areal rainfall from gauge or radar records, although in the demonstration only a block kriging scheme of estimating areal rainfall was deployed. The considerations of implementing and applying EEVS centered on the small-scale rainfall variability. Ciach and Krajewski (1999) employed the exponential model [Eq. (5)], and considered spatial correlation to be nearly constant at the scale of their study (2–7 km) by invoking the assumption of nugget effect. For the study setting (central Ohio), this study shows a conspicuous distance dependence of ␳ between 300 m and 2 km, and a close proximity of ␳˜ 0 to unit value. These results suggest that the nugget effect is inconspicuous over the scale considered here and for this particular climate setting, and points to a possible need of considering subpixel-scale variations in spatial correlation in the implementation of EEVS. Subsequent application of EEVS helped illuminate how small-scale rainfall variability, together with the number and subpixel location of gauges, determines the outcome of EEVS. The comparisons of results from EEVS with USC and DDSC illustrated how the importance of spatial correlation may vary depending on domain size and the subpixel distribution of gauges. With gauge records confined to a small domain (1 km in radius), replacing USC with DDSC yielded rather minor changes to the EEVS results for domain A (Fig. 2a) but much more significant ones for domain B. This contrast is reflective of the amplification of the uncertainty in the gaugebased areal rainfall representation by domain size. The importance of gauge locations was further examined through comparisons between HRAP pixels 1 (6 gauges) with 4 (1 gauge). Though 6 gauges are present

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FIG. 9. Contributions of gauge-related errors (i.e., ␣) computed using EVS and EEVS with uniform and distancedependent spatial correlation (USC and DDSC), for (a) domain A and July, (b) domain B and July, (c) domain A and October, and (d) domain B and October. Shaded areas represent error ranges (i.e., confidence intervals) of correlation distance d˜ 0 and nugget ␳˜ 0 obtained via regression (values shown in Fig. 8). Domains A and B are delineated in Fig. 2a.

in pixel 1, all of them are confined to a small area over its southwestern corner (Fig. 2b). This configuration imposes a potential limitation on the accuracy of gaugebased spatial rainfall estimates. It is therefore unsurprising that factoring in DDSC tended to diminish the difference in ␣ estimates for the two pixels (see Fig. 11). The effects of gauge locations were also evident when comparing the results for HRAP pixels 1 and 4 with those for domain A (Fig. 2a). The presence of a larger number of gauges that are relatively evenly distributed in domain A (Fig. 2a) amounted to much more suppressed ␣ values despite the introduction of DDSC, suggesting that uneven spatial distribution of gauges, as in the case of pixels 1 and 4, may accentuate the importance of accounting for DDSC. This study further illustrated the effects of considering the number and the subpixel location of gauges on

perceived partitioning through comparisons of EEVS and EVS results. EVS-based estimates of partitioning coefficients, which were intended for representing multipixel means, were found to differ in varying degrees from corresponding EEVS results (both with USC and DDSC). The differences are more pronounced for July. For domain A (Fig. 2a), EEVS-based ␣ estimates for July are consistently lower than those from EVS (Fig. 9), again because of the presence of 11 gauges and the fact that the gauges are spreading relatively evenly within the domain (see previous paragraph). For pixel 1 (with 6 gauges), EEVS considering DDSC produced ␣ values that were close to 30% for July, compared to less than 10% from EVS (see related results in Fig. 11). These differences highlighted a potential need for accounting for the interpixel variations in the position of gauges in determining such partitioning coefficients.

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TABLE 6. Correlation between radar and gauge-related errors. Domain A (4 km)

Domain B (8 km)

Month

␦t

25%

50%

75%

25%

50%

75%

Jul Jul Jul Jul Jul Jul Jul Jul Jul Oct Oct Oct Oct Oct Oct Oct Oct Oct

1 2 3 4 6 8 12 24 48 1 2 3 4 6 8 12 24 48

⫺0.35 ⫺0.37 ⫺0.33 ⫺0.37 ⫺0.30 ⫺0.49 ⫺0.32 ⫺0.46 ⫺0.44 ⫺0.12 ⫺0.08 ⫺0.14 ⫺0.23 ⫺0.25 ⫺0.32 ⫺0.33 ⫺0.31 ⫺0.10

⫺0.19 ⫺0.23 ⫺0.19 ⫺0.25 ⫺0.14 ⫺0.37 ⫺0.15 ⫺0.31 ⫺0.28 ⫺0.01 ⫺0.03 ⫺0.04 ⫺0.02 0.01 ⫺0.08 ⫺0.10 ⫺0.15 0.13

0.03 0.01 0.06 0.05 0.12 ⫺0.06 0.10 0.05 0.11 0.08 0.06 0.09 0.22 0.24 0.19 0.17 0.18 0.39

⫺0.34 ⫺0.32 ⫺0.28 ⫺0.34 ⫺0.28 ⫺0.46 ⫺0.29 ⫺0.42 ⫺0.41 ⫺0.10 ⫺0.08 ⫺0.15 ⫺0.28 ⫺0.31 ⫺0.33 ⫺0.32 ⫺0.25 ⫺0.23

⫺0.21 ⫺0.20 ⫺0.16 ⫺0.21 ⫺0.10 ⫺0.34 ⫺0.15 ⫺0.28 ⫺0.29 ⫺0.02 ⫺0.01 ⫺0.04 ⫺0.08 ⫺0.14 ⫺0.16 ⫺0.09 ⫺0.04 ⫺0.06

⫺0.01 ⫺0.00 0.04 ⫺0.02 0.16 ⫺0.11 0.07 0.04 0.00 0.07 0.08 0.06 0.16 0.13 0.18 0.20 0.26 0.17

The effects of spatial rainfall variability were aptly reflected in the July–October contrasts in the magnitude of changes in EEVS results in response to the incorporation of DDSC (Figs. 9 and 11). The pronounced spatial variability of summertime rainfall, as evidenced by the relatively short decorrelation distance d˜ 0 for July (24 km at hourly scale), translated into greater enhancements of ␣ estimates following the introduction of DDSC (see Figs. 9 and 11). By contrast, for October the differences amongst EEVS with USC and DDSC, and those from EVS, were suppressed by lower degree of spatial rainfall variability (d˜ 0 ⫽ 32 km for October at hourly scale). In demonstrating the application of EEVS, an approach was shown that allows relaxation of the zerocovariance assumption and incorporation of the correlation between radar- and gauge-related errors in areal rainfall estimates. The approach is similar to Ciach et al. (2003) in that gauge-based areal rainfall estimates were used as the surrogate for the true rainfall amounts. The correlation thus estimated for two fictitious pixels (i.e., domains A and B, Table 6; illustration of domains in Fig. 2a) were mostly negative for both July and October (Table 6). The presence of negative correlation appeared consistent with Ciach et al. (2003) where the author also noted such negative covariance for a large number of pixel–gauge pairs.

b. Utility, caveats, and further investigation The potential utilities of the methodology are twofold. First, in small-scale applications of even gauge-

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corrected radar rainfall estimates such as the stage III products, it remains desirable to account for subdomain rainfall variations in determining their errors. EEVS can provide a measure of importance of subdomain spatial rainfall variability for a specific spatial domain by utilizing areal rainfall estimates based on multigauge records. The error variance partitioning thus derived can be used for examining the uncertainty in runoff prediction using radar rainfall datasets only. More importantly, as shown in this study, the perceived errors in the gauge-based areal rainfall estimates are closely dependent on the number of gauges and their subpixel locations. While EVS may offer an averaged partitioning of R ⫺ G difference for a radar pixel by pooling multiple gauge–pixel pairs, the use of such results is limited by the interpixel variations in the two aforementioned factors. The presence of these variations, especially in situations where the decorrelation distance is relatively short, may require partitioning of the error variance for an individual pixel. EEVS serves precisely this purpose. In the EEVS implementation demonstrated in this study, it was the spatial correlation, rather than the proportional contribution itself, that was extrapolated to various pixels. This therefore allows the estimated partitions to vary in such a way that reflects the pixel-dependent uncertainty in rainfall estimation. Despite these potential utilities, there are inherent caveats of the methodology that need to be noted in practical applications. First, in using gauge-based areal rainfall as true rainfall for estimating the correlation between the radar- and gauge-related errors in Eq. (12), it is necessary to assume that such correlation is invariant amongst radar pixels and relatively independent of the number and locations of the gauges. However, as Fig. 10 illustrates, the empirical estimates in reality may vary not only with the number and the location of the gauges, but also with rainfall characteristics. The applicability of this approach is furthermore constrained by (a) the accuracy in the gauge-based areal rainfall estimates and (b) the length of records. As shown in this study, the EEVS-based ␣ estimates are quite low for domain A (4 km; Fig. 2a), suggesting high accuracy in the areal rainfall estimates based on 11 ˆ gauges. This implies that approximating Ra by R a,g in ˆ Var[Ra,r ⫺ Ra] [Eq. (19) might be appropriate (see section 2b)]. Yet, for domains with a fewer number of gauges per unit area, such an approximation may introduce considerable uncertainties as the departure of gauge-based estimates from actual values may be significant and particularly so in the summer months (as suggested by the results for domain B in Figs. 9b,d; see Fig. 2a for the illustration of domain B). On the other

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FIG. 10. Contributions of radar-related errors (i.e., ␤) estimated via EEVS with distance-dependent spatial correlation (DDSC), for (a) domain A and July, (b) domain B and July, (c) domain A and October, and (d) domain B and October. The method of computing the bounds can be found in the appendix.

hand, the limited duration of the observation (4 months) means fewer time intervals, and therefore greater uncertainties in the correlation estimates at coarse temporal resolutions (e.g., ⬎12 h). The uncertainties imposed by these limitations were mitigated to a limited extent by using the median estimates for multiple subsamples. However, further investigations, including both theoretical and empirical ones, will still be required to shed light on whether and how consistent positive or negative correlation would arise from differing gauge-based adjustment procedures (such as the one proposed by Seo 1998) and the dependence of this correlation on rainfall characteristics and the gauges used. Another issue relates to the use of gauge data as the representation of true point rainfall, which is out of

necessity and warrants further study when more precise instrumentation records become available (such as those from collocated gauges). The specific implementations demonstrated in this study also followed Ciach and Krajewski (1999) in using the exponential model of spatial correlation [Eq. (5)], and thereby inherited the isotropy and stationarity assumptions intrinsic in its formulation. As this study shows, the scale dependence of the results is also intrinsically linked to the climate and the prevalent precipitation regimes. This means that caution is necessary when attempting to down- or upscale or extrapolate the results toward different spatial and temporal settings. When it comes to temporal extrapolation, for example, the problem of excessively long dry

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FIG. 11. Contributions of gauge-related errors (i.e., ␣) computed using EEVS with uniform and distancedependent spatial correlation (USC and DDSC) for HRAP pixels 1 (6 gauges) and 4 (1 gauge), and for domain A. The results are shown for (a) USC and July, (b) DDSC and July, (c) USC and October, and (d) DDSC and October. Superimposed are the results of EVS.

periods can possibly be accommodated by eliminating the gaps and joining consecutive events, though any gain from such an attempt would come at the expense of losing physical insights into the scale dependence of the results. The uncertainty in the outcome of EEVS due to the intermittency of rainfall may also be addressed in an empirical way following the resampling method proposed by Steiner et al. (2003). Furthermore, the applicability of EEVS is hinged on that of gaugebased areal rainfall estimates. When gauges are located relatively far away from each other with no coherent intergauge cross correlation, interpolation methods that factor in external knowledge concerning rainfall structure, such as Thiessen polygon, may be used in lieu of kriging to reflect this knowledge. Furthermore, in situations where systematic, persistent under- or overestimation is present, it would be best to use EEVS in conjunction with conventional analysis to identify er-

rors from systematic under- or overestimation. There is no substitute for understanding the deterministic causes of systematic errors in correcting such errors. In this study, for October 2001 the number of rainy hours was severely underestimated by NEXRAD stage III data, confirming the problem reported by Hardegree et al. (2003). A likely source of this underestimation is the presence a truncation error in the WSR-88D’s Radar Product Generator (RPG), which was found to cause underestimation of rainfall amounts that can be severe during light rain (Fulton et al. 2003). Correction of this type of errors is expected to enhance the estimated contribution of P ⫺ A difference, while the confirmation of this awaits further investigations.

6. Summary EEVS, an extension of the error variance separation method of Ciach and Krajewski (1999) was developed

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to account for the R ⫺ G discrepancies in areal rainfall. The EEVS approach was demonstrated through its application to the gauge- and radar-based areal rainfall estimates at Coshocton, Ohio. The principal findings are the following: 1) Factoring distance-dependent spatial correlation may increase in the proportional contribution of gauge-related errors, especially for the summer months when rainfall is highly variable in space. 2) Contribution from gauge-related errors varies considerably across domains depending on (a) scale of domain, (b) the number, and (c) the subpixel locations of the gauges. This suggests a need for estimating the contributions for each pixel individually. 3) Seasonal contrasts in rainfall characteristics lead to differing impacts of introducing distance-dependence in the EEVS results for July and October.

APPENDIX Sampling Methodology The sampling approach is applied to the fictitious domains A and B (Fig. 2a) so that all 11 gauges can be used (n ⫽ 11). The first step of the approach entails randomly picking a set of unique integers {k} (2 ⱕ k ⱕ n ⫺ 1). For each k, a sample of k gauges are randomly selected from the n gauges. In the second step, only these k gauges are assumed to be present within the domain. For this subset of k gauges, the weights in Eq. ˆ ⫺ R ] are recomputed for (17) that minimize Var[R a,g a ˆ each time resolution. The areal rainfall R a,g are then computed based on the computed weights for domains ˆ A and B (Fig. 2a). An approximation of cor[R a,g ⫺ ˆ ⫺ R ] follows: Ra, R a,r a ˆ ⫺R ,R ˆ ⫺ R 兴 ⬇ cor关R ˆk ⫺R ˆn ,R ˆ ⫺R ˆn 兴 cor关R a,g a a,r a a,g a,g a,r a,g 共A1兲 ˆ k and R ˆ k are BLUE based on k and n gauges, where R a,g a,g respectively. The rhs can be estimated by pooling the ˆ ⬎ areal rainfall estimates at all time instants where R a,g 0. In this study, 5 ks were randomly chosen. For each k, m unique combinations of gauges were determined, where m ⫽ min(Ckn, 100) (i.e., only 100 samples are taken when the number of possible combinations exceed 100). The method yields 兺5j⫽1mj subsets of gauges. ˆ k for June and Each subset provides time series of R a,g k ˆ October. Each of the time series of Ra,g is subsequently coupled with the corresponding one based on all 11

VOLUME 8

ˆ n to produce an estimate of cor[R ˆ gauges R a,g a,g ⫺ Ra, ˆ Ra,r ⫺ Ra]. The upper and lower quartiles of such estimates from the 兺5j⫽1mj samples were then coupled with the upper and lower bounds of ␣˜ (determined by those of d˜ 0 and ␳˜ 0) to generate the bounds of ␤. First let us denote the quartiles of the estimated ˆ ˆ ˆ 25 and ␥ˆ 75. Then let us cor[R a,g ⫺ Ra, Ra,r ⫺ Ra] by ␥ denote the upper and lower bounds of estimated ␣ by ␣ˆ max and ␣ˆ min, and those for ␤ by ␤ˆ max and ␤ˆ min. It follows that 1Ⲑ2 ␤ˆ max ⫽ 1 ⫺ ␣ˆ min ⫹ 2␣ˆ min ␥ˆ 75,

共A2兲

1Ⲑ2 ␤ˆ min ⫽ 1 ⫺ ␣ˆ max ⫹ 2␣ˆ max ␥ˆ 25.

共A3兲

The results are displayed in Fig. 10. REFERENCES Austin, P. M., 1987: Relation between measured radar reflectivity and surface rainfall. Mon. Wea. Rev., 115, 1053–1070. Ciach, G. J., and W. F. Krajewski, 1999: On the estimation of radar rainfall error variance. Adv. Water Resour., 22, 585– 595. ——, and ——, 2006: Analysis and modeling of spatial correlation structure in small-scale rainfall in central Oklahoma. Adv. Water Resour., 29, 1450–1463. ——, E. Habib, and W. F. Krajewski, 2003: Zero-covariance hypothesis in the error variance separation method of radar rainfall verification. Adv. Water Resour., 26, 573–580. Fulton, R. A., 1998: WSR-88D Polar-to-HRAP Mapping. Hydrologic Research Laboratory, Office of Hydrology, National Weather Service Tech. Memo., 34 pp. ——, J. P. Breidenbach, D. J. Seo, D. A. Miller, and T. O’Bannon, 1998: The WSR-88D rainfall algorithm. Wea. Forecasting, 13, 377–395. ——, F. Ding, and D. A. Miller, 2003: Truncation errors in historical WSR-88D rainfall products. Preprints, 31st Int. Conf. on Radar Meteorology, Seattle, WA, Amer. Meteor. Soc., CD-ROM, P2B.7. Habib, E., G. J. Ciach, and W. F. Krajewski, 2004: A method for filtering out raingauge representativeness errors from the verification distributions of radar and raingauge rainfall. Adv. Water Resour., 27, 967–980. Hardegree, S., and Coauthors, 2003: Multi-watershed evaluation of WSR-88D (NEXRAD) radar-precipitation products. Proc. First Interagency Conf. on Research in the Watersheds, Benson, AZ, U.S. Bureau of Land Management and Cosponsors, 486–492. Hudlow, M. D., J. A. Smith, M. L. Walton, and R. C. Shedd, 1991: Hydrological applications of weather radar. NEXRAD: New Era in Hydrometeorology in the USA, I. D. Cluckie and C. G. Collier, Eds., Ellis Horwood, 602–612. Journel, A. G., and C. J. Huijbregts, 1978: Mining Geostatistics. The Blackburn Press, 600 pp. Krajewski, W. F., 1987: Co-kriging of radar-rainfall and rain gage data. J. Geophys. Res., 92, 9571–9580. ——, E. N. Anagnostou, and G. J. Ciach, 1996: Effects of the radar observation process on inferred rainfall statistics. J. Geophys. Res., 101, 26 493–26 502.

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Reed, S. M., and D. R. Maidmant, 1999: Coordinate transformations for using NEXRAD data in GIS-based hydrologic modeling. J. Hydrol. Eng., 4, 174–182. Seo, D. J., 1998: Real-time estimation of rainfall fields using radar rainfall and rain gage data. J. Hydrol., 208, 37–52. Smith, J. A., D. J. Seo, M. L. Baeck, and M. D. Hudlow, 1996: An intercomparison study of NEXRAD precipitation estimates. Water Resour. Res., 32, 2035–2045. Steiner, M., T. L. Bell, Y. Zhang, and E. F. Wood, 2003: Com-

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parison of two methods for estimating the sampling-related uncertainty of satellite rainfall averages based on a large radar data set. J. Climate, 16, 3759–3778. Vieux, B. E., S. Vallabhaneni, S. Moisio, and S. Donovan, 2002: An intercomparison study of NEXRAD facility performance for use in sewer system modeling. Proc. Ninth Int. Conf. on Urban Drainage, Portland, OR, ASCE, 247. Zawadzki, I., 1973: Errors and fluctuations of raingauge estimates of areal rainfall. J. Hydrol., 18, 243–255.