Estimating the frequency of extreme rainfall using weather radar and stochastic storm transposition

Journal of Hydrology 488 (2013) 150–165

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Estimating the frequency of extreme rainfall using weather radar and stochastic storm transposition Daniel B. Wright a,⇑, James A. Smith a, Gabriele Villarini b, Mary Lynn Baeck a a b

Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ, USA Department of Civil and Environmental Engineering, University of Iowa, IA, USA

a r t i c l e

i n f o

Article history: Received 1 November 2012 Received in revised form 17 February 2013 Accepted 2 March 2013 Available online 13 March 2013 This manuscript was handled by Konstantine P. Georgakakos, Editor-in-Chief, with the assistance of Marco Borga, Associate Editor Keywords: Extreme events Extreme rainfall Rainfall frequency analysis Flood frequency analysis Radar rainfall

s u m m a r y Spatial and temporal variability in extreme rainfall, and its interactions with land cover and the drainage network, is an important driver of flood response. ‘‘Design storms,’’ which are commonly used for flood risk assessment, however, are assumed to be uniform in space and either uniform or highly idealized in time. The impacts of these and other commonly-made assumptions are rarely considered, and their impacts on flood risk estimates are poorly understood. This study presents an alternate framework for rainfall frequency analysis that couples stochastic storm transposition (SST) with ‘‘storm catalogs’’ developed from a ten-year high-resolution (15-min, 1-km2) radar rainfall dataset for the region surrounding Charlotte, North Carolina, USA. The SST procedure involves spatial and temporal resampling from these storm catalogs to reconstruct the regional climatology of extreme rainfall. SST-based intensity–duration– frequency (IDF) estimates are driven by the spatial and temporal rainfall variability from weather radar observations, are tailored specifically to the chosen watershed, and do not require simplifying assumptions of storm structure. We are able to use the SST procedure to reproduce IDF estimates from conventional methods for four urban watersheds in Charlotte. We demonstrate that extreme rainfall can vary substantially in time and in space, with potentially important flood risk implications that cannot be assessed using conventional techniques. SST coupled with high-resolution radar rainfall fields represents a useful alternative to conventional design storms for flood risk assessment, the full advantages of which can be realized when the concept is extended to flood frequency analysis using a distributed hydrologic model. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Spatial and temporal variability in extreme rainfall (e.g. Ramos et al., 2005; Norbiato et al., 2007), and its interactions with heterogeneous land cover (e.g. Mejia and Moglen, 2010; Wright et al., 2012c) and drainage networks (e.g. Smith et al., 2002; Smith et al., 2005; Morin et al., 2006; Meierdiercks et al., 2010), is a crucial driver of flood response, especially in urban settings where flow path lengths are short, runoff velocities are high, and the spatial distribution of land use and hydrologic infrastructure is highly heterogeneous (e.g. Schilling, 1991; Berne et al., 2004b). Rainfall and flood frequency analyses should be tied directly to the structure and variability of the storm systems that can produce extreme rainfall and flooding in the watershed or hydrologic system of interest. In hydrologic engineering practice, however, a number of simplifying assumptions are made about the structure of extreme rainfall to facilitate frequency analysis, including spatially and oftentimes temporally uniform ‘design ⇑ Corresponding author. Tel.: +1 609 258 4600; fax: +1 609 2582760. E-mail address: [email protected] (D.B. Wright). 0022-1694/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jhydrol.2013.03.003

storms’ (see, for example, Koutsoyiannis, 1994), and area reduction factors (ARFs, see Svensson and Jones, 2010). Some of these assumptions, which were developed at a time when high-resolution measurement of rainfall was not practical, are discussed in Wright et al. (2012). They neglect the variety and complexity of hydrometeorological processes that contribute to the extreme rainfall and flood climatology of a watershed or hydrologic system. Furthermore, they are often invoked without consideration of the substantial uncertainties that underlie them. Conventional flood frequency analysis involves additional assumptions regarding the routing of runoff through the drainage network. These assumptions vary but typically disregard the spatial and temporal variability in runoff processes and how they are impacted by variability in rainfall, land cover, and drainage network properties. The results of flood frequency analyses, therefore, are subject to joint errors arising from multiple assumptions regarding a range of flood-driving mechanisms. The impacts of these simplifying assumptions on subsequent design or risk assessment have seldom been quantified. With the the advent of the National Weather Service (NWS) Next–Generation Radar network (NEXRAD, Heiss et al., 1990),

D.B. Wright et al. / Journal of Hydrology 488 (2013) 150–165

high-resolution rainfall estimation has become routine across most of the United States. Bias-corrected weather radar offers accurate and high resolution estimates of extreme rainfall (e.g. Smith et al., 1996; Fulton et al., 1998; Baeck and Smith, 1998; Vieux and Bedient, 1998; Krajewski and Smith, 2002; Creutin and Borga, 2003; Einfalt et al., 2004; Villarini et al., 2010; Krajewski et al., 2010a; Smith et al., 2012; Wright et al., submitted for publication) and provides opportunities to test conventional methods and to develop more robust techniques for estimating the frequency of extreme rainfall (see, for example, Overeem et al., 2008; Overeem et al., 2009; Overeem et al., 2010). In this paper, we couple ‘‘storm catalogs’’ developed using highresolution radar rainfall estimates with an alternative framework for rainfall frequency analysis known as stochastic storm transposition (SST). SST-based rainfall frequency analysis is a technique for reconstructing the long-term climatology of extreme rainfall of a watershed using ‘‘space-for-time substitution,’’ nested within a probabilistic framework. Inferences derived from any frequency analysis technique are inherently limited by the length of observational records. SST aims to effectively lengthen the period of record by resampling from a catalog of observed storms not only from the location of interest but also from the surrounding region. The challenges of accurately estimating long-return interval rainfall that arise from short observation records are at least partially addressed through the inclusion of additional observations of nearby storms. Descriptions of the general framework and associated challenges are provided in Foufoula-Georgiou (1989) and Fontaine and Potter (1989). The method is employed for rainfall frequency analysis in Wilson and Foufoula-Georgiou (1990), and extended to flood frequency analysis in Franchini et al. (1996). The SST procedure used in this study is similar in formulation but attempts to address some of the limitations of previous studies, principally the lack of high-quality spatially and temporally continuous rainfall fields. This paper presents intensity–duration–frequency relationships (IDF; sometimes referred to as IFD) calculated using an SST-based approach coupled with bias-corrected radar rainfall estimates. SSTIDF curves are compared to curves computed using conventional (rain gage-based) methods. Two sets of IDF estimates are created. One set uses all storm types and the other set uses only nontropical storms, in order to examine the role that tropical and nontropical systems have in shaping the climatology of extreme rainfall and to demonstrate the flexibility of the SST method. High-resolution radar rainfall estimates coupled with SST allow us to examine aspects of the spatial and temporal structure of extreme rainfall that previous SST studies and conventional rainfall frequency analysis techniques have not been able to address. It is noted that the transposition step of SST is similar to that used in the estimation of the Probable Maximum Precipitation (PMP) and Probable Maximum Flood (PMF), which are used for the design of spillways and other flood defenses for high value infrastructure such as large dams and nuclear power plants Hansen (1987). PMP estimation typically combines spatio-temporal estimates of extreme rainfall based on regional observations with a theoretical estimate of the maximum possible atmospheric water vapor supply for the region to produce a hypothetical storm with realistic spatio-temporal properties but higher-thanobserved intensity. This hypothetical storm is then transposed over a watershed or hydrologic system such that precipitation or modeled discharge is maximized. A commonly-cited weakness of PMP/PMF is that despite its name, the method is not probabilistic, complicating its use in risk assessment (see, for example, Alexander, 1963). This criticism notwithstanding, the concept is closely linked to SST through the transposition, and a PMP estimate can be considered as an upper bound on a SST-based frequency analysis.

151

This paper is structured as follows: the study region and four study watersheds are introduced in Section 2; radar and rain gage observations, radar bias-correction techniques, and other hydrometeorological data are discussed in Section 3.1; the SST-based procedure for IDF estimation is presented in Section 3.2; and techniques for examining the spatial and temporal structure of extreme rainfall are discussed in Section 3.3. The accuracy of radar rainfall estimates is demonstrated in Section 4.1; criteria for selecting the domain for SST are shown in Section 4.2; SST-based IDF estimates are presented and compared with conventional estimates in Section 4.3; SST-based analyses of the temporal and spatial structure of extreme rainfall are presented in Sections 4.4 and 4.5, respectively. Section 5 discusses remaining challenges and Section 6 contains a summary and concluding remarks. 2. Study area The study region is centered on the Charlotte, North Carolina metropolitan area. Charlotte is an ideal setting for flood hydrology research due to the quantity and quality of data resources and the variety of flood-producing hydrometeorological processes (see, for example, Smith et al., 2002; Turner-Gillespie et al., 2003; Villarini et al., 2010 and Wright et al., submitted for publication). Two NWSoperated Weather Surveillance Radar 1988 Doppler (WSR-88D) radars, Greer and Columbia (radar codes KGSP and KCAE, respectively) cover the Charlotte area. The siting of the KCAE radar is such that beam blockage over the Charlotte area is a major problem (Villarini and Krajewski, 2010). The KGSP radar has been selected for use in this study due to the lack of beam blockage. Charlotte is located approximately 130 km from the radar (Fig. 1, left panel). Our analyses indicate that range effects on radar rainfall estimation for the Charlotte metropolitan region are minimal (results not shown). The Blue Ridge Mountains to the west affect storm initiation and evolution across the region (Weisman, 1990a; Weisman, 1990b; Murphy and Konrad, 2005) and local topographic relief is minimal. This study focuses on rainfall frequency analyses for four urban watersheds in the Charlotte metropolitan area (Fig. 1), named by their associated US Geological Survey (USGS) stream gages. Most analyses in this paper focus on Little Sugar Creek at Archdale, a 110 km2 watershed that drains much of downtown Charlotte. Rainfall and flooding in subbasins of Little Sugar Creek have been extensively examined in Smith et al. (2002), Turner-Gillespie et al. (2003), Villarini et al. (2009), Villarini et al. (2010), and Wright et al. (submitted for publication). We also examine rainfall frequency relationships for Edwards Branch at Sheffield, a 2.5 km2 subbasin of Little Sugar Creek and two subwatersheds of McAlpine Creek: McAlpine Creek near Sardis (92 km2) and McAlpine Creek below McMullen (240 km2). These four watersheds have been included in this study because they span a range of spatial scales that are relevant for urban flood processes. In addition, McAlpine Creek near Sardis and Little Sugar Creek at Archdale are both approximately 100 km2 but have different shapes (McAlpine is approximately square whereas Little Sugar Creek is elongated) and correspondingly different drainage network configurations. Characteristics for the four study watersheds can be found in Table 1. 3. Data and methods 3.1. Radar rainfall estimation This study uses a ten-year (2001–2010) 15-min, 1 km2 radarestimated rainfall dataset developed using the Hydro-NEXRAD system presented in Wright et al. (submitted for publication). The Hydro-NEXRAD processing system was developed specifically

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D.B. Wright et al. / Journal of Hydrology 488 (2013) 150–165

km

km

Fig. 1. Study region. Left panel: KGSP 200 km radar range umbrella with state boundaries, regional topography and the Charlotte metropolitan boundary. Right panel: Charlotte metropolitan area with the rain gage locations of a subset of the CRN network, the four study watersheds, and their corresponding USGS stream gages and DEMderived drainage networks. The background map shows percent of land cover that is impervious (darker colors indicate greater imperviousness).

to create rainfall estimates for hydrologic applications, and converts three-dimensional polar-coordinate volume scan reflectivity fields from NWS WSR-88D radars into two-dimensional Cartesian surface rainfall fields (Krajewski et al., 2010b). The standard convective rainfall–reflectivity (Z–R) relationship (R = aZb, where a = 0.017, b = 0.714; R is rain rate in mm h1, and Z is the radar reflectivity factor in mm6 m3), a 53 dBZ hail threshold, and several standard quality control algorithms were used (Seo et al., 2011). Our analyses indicate that range effects on radar rainfall estimation for the Charlotte metropolitan region are minimal (results not shown). No range correction algorithms are used in this study. In other settings, range correction based on vertical profile of reflectivity corrections may be necessary (see, for example, Andrieu and Creutin, 1995; Vignal and Krajewski, 2001; Seo et al., 2000; Borga et al., 2002; Berne et al., 2004a; Germann et al., 2006; Bellon et al., 2007, and Krajewski et al., 2011). Radar rainfall estimates from the KGSP radar are bias-corrected using rain gage measurements from the Charlotte Raingage Network (CRN), which consists of 71 rain gages (Fig. 1, right panel) operated jointly by the US Geological Survey (USGS) and Charlotte–Mecklenburg Storm Water Services. Most of these gages have been operational since at least 1995 and are subject to a high degree of quality control. In addition to nearly continuous radar and rain gage data availability for the 2001–2010 period, data are available for the remnants of Hurricane Danny, which caused catastrophic flooding in much of the Charlotte area on 23–24 July 1997. CRN and weather radar and radar rainfall measurements for the July 1997 event are analyzed in Villarini et al. (2010). Two types of radar rainfall biases and corresponding correction procedures are examined in this study: multiplicative mean-field bias which is uniform in space, and conditional bias which is a function of rain rate. Both types of bias correction depend on the aggregation period being considered. Both bias correction procedures are discussed in more detail below. A detailed examination of meanfield and conditional bias for the 2001–2010 radar dataset used in this study can be found in Wright et al. (submitted for publication). Mean-field bias correction removes systematic bias due to variability in Z–R relationships and radar calibration errors (Smith and Krajewski, 1991; Fulton et al., 1998; Seo et al., 1999; Borga et al., 2002; Villarini and Krajewski, 2010). We compute and compare

mean-field multiplicative bias at the daily (12 to 12 UTC) time scale. Analyses presented in Wright et al. (submitted for publication) demonstrate that daily mean-field bias correction is in most cases comparable in accuracy to mean-field bias correction at finer time scales. Both the radar and CRN rain gage rainfall observations are aggregated to daily accumulations prior to bias estimation. The mean-field bias computation takes the form:

P

S Bi ¼ P i

Gij

Si Rij

ð1Þ

where Gij is the rainfall accumulation for gage j on day i, Rij is the daily rainfall accumulation for the co-located radar pixel accumulation on day i and Si is the index of the rain gage stations for which both the rain gage and the radar report positive rainfall accumulations during time period i. Each 15-min radar rainfall field from day i is then multiplied by the bias correction factor Bi. For time periods in which less than five gages reported positive rainfall, a bias value was computed from long-term seasonal rainfall totals (warm season: April–September, or cold season: October–March) excluding tropical storms. The mean-field bias correction procedure is the same as that used in Smith et al. (2012), Wright et al. (2012c), and Wright et al. (2012). Conditional biases can result from radar calibration or from estimation errors associated with hydrometeor type, particularly hail (see Ciach et al., 2007; Villarini and Krajewski, 2009). In this study, conditional bias models were developed using the methodology presented in Villarini and Krajewski (2009), which involves fitting a power law function of the form:

b ¼ aRR ðtÞbðtÞ RðtÞ

ð2Þ

to the result of a locally weighted scatterplot smoothing (LOESS) of the gage-radar pairs, where t is the aggregation period, a and b (t) are empirical parameters, RR(t) is the radar rainfall estimate, and b RðtÞ is the estimate of true rainfall. In this dataset, the parameter a does not appear to be strongly dependent on the temporal scale in question (results not shown), and is taken as the average of power law prefactor for all time scales. The parameter b (t) is modeled by a power law function dependent on the accumulation time t:

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D.B. Wright et al. / Journal of Hydrology 488 (2013) 150–165 Table 1 Land surface characteristics for four urban study watersheds in the Charlotte area. Watershed/USGS gage name

USGS gage ID

Area (km2)

2006 Urban land use (%)

2006 Impervious (%)

Mean flow distance (km)

Max flow distance (km)

Norm. mean flow distance (–)

Little Sugar Cr. at Archdale McAlpine Cr. at Sardis McAlpine Cr. below McMullen Edwards Br. at Sheffield

02146507 02146600 02146750 0214643820

110 92 240 2.5

97 81 84 93

32 20 20 32

13.6 11.3 20.0 1.4

24.5 19.0 37.4 2.6

0.56 0.59 0.53 0.54

bðtÞ ¼ ctd

ð3Þ

where c and d are empirical parameters estimated using nonlinear least squares. Estimates for a, c, and d for the warm (April–September) and cold (October–March) seasons are shown in Table 2. This procedure can remove conditional bias, but only at the expense of other performance criteria such as mean square error (Ciach et al., 2000). Conditional bias correction involves a tradeoff between the minimization of performance criteria such as RMSE and MAE and the accurate representation of high rainfall rates. Since the focus in this study is on rain rate estimation, conditional bias correction has been employed. Further work is needed to assess the utility of conditional bias correction in practice. In this study, we compare IDF estimates developed using SST to conventional rain gage-based estimates from the National Oceanic and Atmospheric Administration (NOAA) Atlas 14 (Bonnin et al., 2004). NOAA Atlas 14 IDF estimates are updated based on recent rain gage observations and for the study region are based on observations from a network of over 200 rain gages in North and South Carolina. The Atlas 14 IDF estimates for Charlotte are among the most accurate conventional IDF estimates available due to the high gage density and relatively long gaging records. The area reduction factors (ARFs) used in this study come from the US Weather Bureau Technical Paper No. 29 (TP-29, US Weather Bureau, 1958), which is probably the most commonly used source of ARFs in the United States and globally. These ARFs vary by storm duration but not by region, return period, or storm type. Analyses using the CRN network, however, suggest that TP-29 ARFs are too high for Charlotte area (Wright et al., 2012; see Asquith and Famiglietti, 2000 for similar conclusions drawn from three rain gage networks in Texas). Further discussion of IDF estimation, Atlas 14, and ARFs can be found in Durrans (2010). The sensitivity of conventional IDF estimates in the Charlotte region to the July 1997 event is evaluated in Weaver (2006). Rainfall associated with tropical cyclones is identified using the Hurricane database from NOAA’s National Hurricane Center (HURDAT; see Jarvinen et al., 1984; Neumann et al., 1993). Any rainfall occurring 12 h before to 12 h after a HURDAT storm track passes within 500 km of Charlotte is classified as tropical in origin (see Hart and Evans, 2001; Kunkel et al., 2011 for similar classification criteria for tropical rainfall). This classification system is used to generate two sets of SST-based IDF estimates, one set using all storm types (tropical and nontropical), and the other set using only nontropical systems. Observations of cloud-to-ground lightning from the National Lightning Detection Network (NLDN) are used in Section 4.2 as a surrogate for convective activity (e.g. Tapia et al., 1998; Ntelekos et al., 2007; Price et al., 2011a; Price et al., 2011b).

3.2. SST procedure The steps used in this study for watershed-specific IDF estimation using bias-corrected radar rainfall fields and SST are as follows:

1. Identify a spatial ‘‘domain’’ that contains the watershed of interest and over which the extreme rainfall climatology is homogeneous. Homogeneity can be assessed using a number of metrics, including storm counts, mean storm depths or intensities, measures of convective activity such as cloud-to-ground lightning observations, or analyses of spatial and temporal rainfall structure. Most of the SST results in this study were generated using a 3600 km2 domain centered on the watershed or point of interest. Some criteria for assessing homogeneity are examined in Section 4.2 and the possibility of relaxing the homogeneity requirement is discussed in Section 5. 2. Identify the largest m storms within the domain at the t-hour time scale. This set of storms is henceforth referred to as a ‘‘storm catalog.’’ Since the goal of the procedure is to estimate rainfall and flood exceedance probabilities for a specific watershed, the m largest storms are selected with respect to shape and orientation of that watershed. For example, the principal axis of Little Sugar Creek at Archdale is orientated roughly north–south and has an area of 110 km2, so the m storms that constitute the t-hour storm catalog are those associated with high t-hour rainfall intensities over an area of 110 km2 with the same shape and orientation as Little Sugar Creek. The m storms are selected from an n-year record, such that an average of k = m/n storms per year are included in the storm catalog. We then assume that the number of annual storm occurrences follows a Poisson distribution with a rate parameter equal to k storms per year. In this study, we chose m = 50 storms over the ten year radar record (k = 5.0 storms per year). When tropical systems were excluded from the SST procedure, k was decreased. The assumption of Poisson-distributed storm occurrences was also used in Wilson and Foufoula-Georgiou (1990), and is valid for the Charlotte area (analyses not shown). We conduct the procedure for t = 1, 3, 6, and 12 h for the four study watersheds and for a single point/radar pixel. In the event that two or more distinct t-hour periods of heavy rainfall occurred within 24 h, only the period with the highest rate is included in the catalog. Since storms are selected with respect to the size and orientation of the watershed of interest, the storms that compose each catalog can vary. For example, the 50 storms that comprise the 1-h catalog for Little Sugar Creek are not the same storms that comprise the 12-h catalog for Little Sugar Creek or the 1-h catalog for Edwards Branch. Once a storm has been selected with respect to the size and shape of a particular watershed, the entire spatial extent of the storm is retained for subsequent analysis. 3. Randomly select a storm from the storm catalog. Since the extreme rainfall climatology over the domain is assumed to be homogeneous, this storm could have occurred with equal likelihood anywhere within the domain. Therefore, the storm is transposed an east–west distance Dx and a north–south distance Dy, where Dx and Dy are drawn from uniform distributions. Since the radar data used in this study have a temporal resolution of 15-min, a 1-h storm, for example, consists of four periods, all transposed by the same Dx and Dy. The motion and evolution of the storm at all periods is thus unaltered during

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Table 2 Parameters for conditional bias correction procedure. The cold season is October– March, the warm season is April–September. Season

a

c

d

Warm season Cold season

0.9950 0.9700

1.0774 1.04045

0.0151 0.0110

motion and orientation are preserved, and unlike conventional assumptions of uniform rainfall, T-year rainfall variability and spatial and temporal structure can be directly examined. If the procedure is coupled with a distributed hydrologic model, the impacts of storm motion and structure can be explicitly accounted for and no design storm assumptions are needed. 3.3. Rainfall structure

transposition and only the location is changed. This step is illustrated schematically at the 1-h scale for the 27 June 2006 storm in Fig. 2. This procedure is repeated for a randomly selected subset of k storms where k is a Poisson-distributed number of storm occurrences with rate parameter k storms per year. This step is analogous to recreating one year of heavy rainfall over the domain. 4. For each of the k transposed storms, compute the t-hour basinaveraged rainfall rate that occurs over the watershed of interest. Once t-hour rain rates have been computed for each the k storms, the maximum computed rain rate is retained. This retained value is analogous to the the annual maximum t-hour rain rate for one year for the watershed. Additional rainfall characteristics can also be examined for the retained storms. 5. Repeat steps 3 and 4 to recreate multiple years of t-hour ‘‘annual’’ rainfall maxima for the watershed of interest. This procedure of repeatedly selecting subsets of data to make inferences is referred to as resampling. In this study, steps 3 and 4 are repeated 1000 times and the ordered ‘‘annual’’ maxima are then used to generate rainfall return period estimates of up to 1000 years. Associated statistics that describe rainfall structure, outlined in the following section, are also computed. Five thousand realizations of 1000-year series are generated to examine the variability of the estimates at each return interval. The procedure described above takes advantage of the spatial and temporal structure of radar rainfall fields to estimate the frequency of extreme rainfall for a particular watershed or hydrologic system without any assumption of spatial and temporal uniformity or of point-to-area transformation using ARFs. The effects of storm

+30

Outline of SST domain Δx~uniform(-30,30) Transposed Storm t4 t3

+10

t2

Δy~uniform(-30,30)

y distance (km)

+20

0

−10

t1

et ¼ D i

Z A

wti ðxÞdðxÞdx

t3: 1315 UTC

wti ðxÞ ¼ R

t2: 1300 UTC −30

t1: 1245 UTC −30

−20

−10

0

+10

+20

ð4Þ

where d(x) is the flow distance from point x to the basin outlet and wti ðxÞ is the weight function:

27 June 2006 Storm t4: 1330 UTC

−20

Hydrologists and engineers use a variety of methods to temporally disaggregate IDF estimates in order to develop design storms. The most common methods in the United States are SCS 24-h dimensionless design storms and the ‘alternating block ’ method, both of which are described in many introductory textbooks on engineering hydrology (see, for example, McCuen, 2005). While these methods provide more realistic representations of extreme rainfall than assuming a temporally uniform rain rate, they still involve considerable assumptions and do not reflect the temporal variability of observed extreme rainfall. For example, a T-year, thour design storm developed using the alternating block method requires the selection of a time interval Dt such that t = KDt. The peak Dt-hour rain rate for the design storm is taken to be the Tyear, Dt-hour IDF estimate, the second-highest Dt-hour rain rate is taken to be the T-year, 2Dt-hour IDF estimate minus the T-year, Dt-hour IDF estimate, and so on. The resulting hyetograph shape can then be normalized and rescaled so that the total rainfall depth matches that of the T-year, t-hour IDF estimate. Implicit in the alternating block method, therefore, are a temporally unimodal rainfall distribution and a fixed relationship between T-year rainfall rates for all durations between Dt and KDt. These assumptions are difficult to support, especially in areas such as Charlotte where different types of storm systems (for example, tropical cyclones and summertime thunderstorms) tend to produce different extreme rain rates at different time scales. In Section 4.5, we compare the temporal structure of basin-averaged rainfall time series to conventional design storms developed using the alternating block method, paying particular attention to the role of storm type (tropical vs. nontropical) on extreme rain rates at a range of time scales. One useful way of assessing the spatial structure of extreme rainfall at the watershed scale is through the use of rainfallweighted flow distance statistics (see Smith et al., 2002; Smith et al., 2005), which consider how the structure of rainfall relates to the drainage network. The normalized rainfall-weighted flow distance Dti is a measure of the spatial distribution of rainfall of duration t for storm i, relative to the drainage network of a watershed of area A. Dti is computed by dividing the rainfall-weighted e t by the length of the longest flow path in the mean flow distance D i e t is given by: watershed. D i

+30

x distance (km) Fig. 2. Depiction of one possible realization of storm transposition of the 27 June 2006 storm. The dashed gray square shows the outline 3600 km2 SST domain. The black outline shows the basin boundary of Little Sugar Creek at Archdale. Solid gray ellipses show the locations of the 30 mm/h rainfall contours at four 15-min time intervals; dashed gray ellipses show the corresponding transposed contours.

Rti ðxÞ Rt ðyÞdy A i

ð5Þ

Uniform rainfall over the watershed (i.e. wti ðxÞ ¼ jAj1 ) produces e t is the mean flow path length. Storms or time periods in which D i less than the mean flow path length are characterized by rainfall concentrated closer to the basin outlet relative to spatially uniform e t is greater rainfall. Likewise, storms or time periods in which D i than the mean flow path length are characterized by rainfall concentrated farther from the basin outlet relative to spatially uniform rainfall.

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D.B. Wright et al. / Journal of Hydrology 488 (2013) 150–165

Table 3 Error statistics for 1, 3, 6, and 12-h rainfall accumulations for uncorrected radar and bias corrected radar, calculated with respect to co-located CRN rain gage accumulations. Season

Statistic

1h

3h

6h

12 h

Uncorrected

Corrected

Uncorrected

Corrected

Uncorrected

Corrected

Uncorrected

Corrected

Warm

r RMSE (mm) MAE (mm)

0.800 0.731 0.087

0.856 0.606 0.071

0.822 1.468 0.237

0.885 1.161 0.183

0.825 2.288 0.453

0.895 1.765 0.336

0.828 3.468 0.873

0.911 2.530 0.612

Cold

r RMSE (mm) MAE (mm)

0.546 0.821 0.077

0.777 0.436 0.060

0.590 2.012 0.216

0.843 0.906 0.153

0.602 3.514 0.411

0.882 1.348 0.264

0.883 6.171 0.789

0.899 2.020 0.458

The normalized rainfall-weighted flow distance dispersion Sti for storm i at time t is computed by dividing the rainfall-weighted flow distance dispersion e S ti by the dispersion for uniform rainfall. e S ti is given by:

e S ti ¼

Z A

h i2 12 e t dx wti ðxÞ dðxÞ  D i

overland flow paths. Flow paths, especially in urban areas, can differ substantially from those derived from DEMs due to elaboration of the drainage network through storm drains and the straightening of channel meanders (see Smith et al., 2002).

ð6Þ

4. Results 4.1. Radar rainfall estimation

The normalized dispersion takes a value of 1.0 for spatially uniform rainfall, a value less than 1.0 for rainfall with a unimodal spatial peak, and a value greater than 1.0 for multimodel spatial peaks (for example, if a storm is characterized by intense rain cells in both the upper and lower portions of the basin and lighter rainfall in the intervening area). The duration t used for calculating these statistics can range from the native resolution of the radar rainfall fields (Smith et al., 2002) to the total storm duration (Smith et al., 2005). In Section 4.5, rainfall weighted flow distance and dispersion are examined at the 1, 3, 6, and 12-h time scales. For this study, flow distances were estimated from flow paths derived from 30 meter digital elevation models (DEMs) and include both channel and

Mean-field bias correction substantially improves the accuracy of radar rainfall estimates (Table 3). At the hourly scale, the Pearson correlation coefficient (r) improves from 0.80 to 0.86 during the warm season and from 0.55 to 0.78 during the cold season. Root mean-square error (RMSE) improves from 0.73 mm to 0.61 mm during the warm season and from 0.82 mm to 0.44 mm during the cold season; and mean absolute error (MAE) improves from 0.09 mm to 0.07 mm during the warm season and from 0.08 mm to 0.06 mm during the cold season. Results are similar for 3-h, 6-h, and 12-h accumulations. Error statistics and other radar-rainfall estimation properties for the KGSP radar for Hydro-

50

100

3−hour

Gage Rainfall Rate (mm/h)

1−hour 80

40

60

30

40

20

20

10 0

0 0

20

40

60

80

100

25

0

10

20

30

40

50

15 6−hour

12−hour

20 10 15 10 5 5 0

0 0

5

10

15

20

25

0

5

10

15

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Fig. 4. Assessment of rainfall climatology over study region. Mean annual rainfall in mm over 2001–2010 period (top-left). Mean annual number of days over 2001–2010 period with rainfall accumulations greater than 25 mm (top right). Mean number of cloud-to-ground lightning strikes per year per km2 over the 2001–2010 period (bottomleft). Mean rainfall in mm of the 50 highest 12-h storms selected with respect to the size and shape of Little Sugar Creek (bottom-right). Dashed outline shows the square 3600 km2 SST domain used throughout this study.

NEXRAD using several different bias correction methods are examined in greater detail in Wright et al. (submitted for publication). Conditional bias was examined for time scales ranging from 1, 3, 6, and 12-h timescales (Fig. 3). Both the effects of conditional bias and of measurement error are more pronounced at shorter time scales (Fig. 3, see also Villarini and Krajewski, 2009). The radar tends to underestimate heavy rainfall relative to rain gages (see the dashed lines representing the results of a LOESS regression in Fig. 3). Table 2 contains the parameters a, c, and d for the conditional bias power law model for the cold (October–March) and warm (April–September) seasons estimated using nonlinear least squares regression on Eqs. (2) and (3). As mentioned in Section 3.1, conditional bias correction will increase estimation errors such as RMSE and MAE relative to when only mean field bias correction is used (note the increased scatter above the 1:1 line in Fig. 3) but should yield improved estimates of high rain rates. All SSTbased IDF results presented in this study are corrected for both mean field and conditional bias. 4.2. SST domain selection Homogeneity in heavy rainfall over the SST domain was evaluated in several ways. Mean annual rainfall and the number of days with rainfall accumulations greater than 25 mm were computed based on the bias-corrected radar rainfall data for the 2001–2010 period (Fig. 4, top-left and top-right panels, respectively). Range effects on radar rainfall estimates are visible at far ranges northeast

of the city, but are minimal within the square 3600 km2 domain that is used throughout much of this study. There is a maximum in mean annual rainfall and number of days of heavy rainfall in the vicinity of Little Sugar Creek. This maximum is also evident in the mean number of cloud-to-ground lightning strikes per km2 per year, suggesting that the rainfall maximum is a real feature rather than a sampling or radar estimation artifact (Fig. 4, bottom-left panel). The maxima in rainfall and lightning are likely attributable to urban modification of convective rainfall (for further discussion of urban rainfall modification in Charlotte, see Wright et al. (submitted for publication)). The mean storm depth for the 50 largest 12-h storms selected with respect to the size and shape of Little Sugar Creek shows a predominant southwest-northeast pattern consistent with the patterns observable in longer term rainfall averages (Fig. 4, bottomright panel; results are similar for other time scales and basins). This pattern reflects the predominant direction of storm motion toward the northeast along the eastern edge of the Blue Ridge Mountains. The locations of the rainfall centroids of the 50 largest 1, 3, 6, and 12-h storms selected with respect to the size and shape of Little Sugar Creek are shown in Fig. 5. There appears to be a small increasing trend in nontropical storms occurrences from west to east at all time scales. There does not appear to be any trend in the north–south direction for nontropical storm occurrences or in any direction for tropical occurrences. There is no systematic spatial trend visible in rainfall intensity for any storm type at any time

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Fig. 5. Locations of the 50 largest storms selected with respect to the size and shape of Little Sugar Creek at Archdale (black outline) at 1, 3, 6, and 12-h time scales, selected over a square 3600 km2 domain centered on the watershed. Storms of tropical origin are dark gray, storms of nontropical origin are light gray. Size of circle denotes relative magnitude of rainfall intensity. Histograms show the storm counts in the east–west and north–south directions, by storm type (tropical vs. nontropical).

scale. Results are similar for the other study watersheds (not shown). The SST procedure used in this study ignores possible spatial trends by transposing storm locations in both the north–south and east–west directions according to independent uniform distributions. The impact of observed spatial trends on the results is likely small, but could be a topic of future research. Wilson and Foufoula-Georgiou (1990) use nonuniform spatial transposition, but the domain they considered (a nine-state region of the midwestern United States) was much larger and more heterogeneous than in this study. See Section 5 for additional discussion of spatial heterogeneity in storm selection and resampling. 4.3. IDF results NOAA Atlas 14 and SST-based 1, 3, 6, and 12-h IDF estimates generally agree at the point/radar-pixel scale and for Little Sugar Creek at Archdale (Fig. 6). For long return periods (500 and 1000year), the 1-h duration point/pixel-scale SST-based estimates are larger than the Atlas 14 estimates. This probably reflects the limitations of the relatively sparse rain gage networks used to generate Atlas 14 IDF estimates. Infrequent extreme rain rates at short time scales are typically associated with summertime thunderstorms that are small in spatial extent and duration (often on the order of several square kilometers and less than one hour for single-cell thunderstorms), meaning there is a high likelihood that a sparse rain gage network will fail to sample the location of maximum rain rate. Weather radar, due to its continuous high-resolution sampling, is well suited to measuring small but intense storm cells. The SST procedure yields IDF estimates that are greater than the NOAA 90% confidence interval for some durations and some return periods, though the results tend to converge for very long (500–

1000-year) return periods. This tendency is more pronounced in the Little Sugar Creek IDF estimates than the point estimates. The SST results do not vary as the NOAA estimates for return periods between 100 and 1000 years for Little Sugar Creek at Archdale. It is difficult to interpret these differences. Both conventional IDF point estimates and the ARFs used to convert them to areal rainfall estimates are subject to considerable uncertainties (see Wright et al., 2012), while questions remain regarding the use of conditional bias correction for radar rainfall estimation (Wright et al., submitted for publication). We recommend that the accuracy of ARF estimates and the use of conditional bias correction be topics of future research. IDF estimates based only on nontropical storms are developed using SST to examine the sensitivity of extreme rainfall estimates to storm type (Fig. 6, dark gray symbols). The minimal differences at the 1-h scale between SST IDF estimates based on all storm types and based only on nontropical systems is due to the predominance of short-duration, high-intensity, and spatially small thunderstorms in the 1-h storm catalogs. This is true for both the pixeland watershed-scale SST-based IDF estimates. Extreme rain rates for longer time scales are typically associated with tropical storms, due to their large spatial extents. This is reflected in the differences between SST-based IDF estimates generated using all event types and using only nontropical storms at time scales longer than 1 h. It should be noted that common sources of IDF estimates such as Atlas 14 do not discriminate between storm types. The SST-based IDF results in this study tend to underestimate rainfall for short return periods relative to Atlas 14, especially for short durations. This tendency can also be seen in Foufoula-Georgiou (1989), Wilson and Foufoula-Georgiou (1990), and Franchini et al. (1996) and is the result of the relatively high likelihood that for a given synthetic SST ‘‘year’’, all k storms are transposed in such

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a way that little or no rainfall occurs over the watershed of interest, which is clearly unrealistic. This tendency would be more evident for shorter duration storms, which tend to be smaller in spatial extent (see, for example, the severe underestimation using SST in the 1-h point/pixel estimates in Fig. 6) and for smaller watersheds. The severity of this underestimation could be reduced by increasing both the number of storms included in the storm catalog and the magnitude of k used in resampling. Increased resampling of less-intense storms would not affect estimates of long return period rain rates, but would improve estimates for short return periods. SST and Atlas 14-based IDF estimates were computed for three other study watersheds to examine the effects of characteristics such as basin size and shape (Fig. 7). Results for the three additional basins are generally consistent with those from the point/ pixel-scale and for Little Sugar Creek at Archdale. There is a tendency for SST using all storms to produce higher IDF estimates than Atlas 14, though in some cases these estimates are within the 90% confidence levels of the NOAA estimates. T-year rainfall estimates decrease in magnitude with increasing area. Additionally, the spread between the SST results using all storms and using only nontropical storms increases with increasing basin area for all durations except 1 h. The magnitude and variability of SST-derived rainfall rates for 10-year, 100-year, and 1000-year return periods using all storm types and using only nontropical storms were computed for the four study watersheds to examine the interactions of basin size and shape with rainfall structure (Table 4). Variability, represented in Table 4 by the unitless coefficient of variation (CV; the sample standard deviation divided by the sample mean) for the smallest basin (Edwards Branch; 2.5 km2) increases with increasing return period but shows no clear trend with increasing duration. Variabil-

ity in 100-year rain rates for the three larger basins is greater than for 10-year or 1000-year rain rates at most durations. Rain rate variability will generally increase with return period since the overall rainfall magnitudes are greater, but the upper tail of extreme rainfall rates (i.e. 1000-year events) using SST, even with 5000 SST realizations, are dominated by a fairly small number of storms transposed in a generally consistent way, leading to relatively lower variability for 1000-year events. For example, for Little Sugar Creek using SST with all storms, the 1000-year, 3-h rainfall estimates were produced by just four storms and are mainly dominated by the 23–24 July 1997 storm (4738 out of 5000 realizations), while the 5000 realizations of 100-year 3-h rainfall are more evenly distributed between five storms. Therefore, like other frequency analysis methods, upper-tail rainfall estimates using SST are highly sensitive to a small number of the most extreme observed events. SST-based 1-h rain rates do not vary significantly for any basin according to storm type, since short duration, high-intensity rainfall rates tend to be produced by nontropical convective systems which are often small in spatial extent. The differences among IDFs estimated using all storm types and using only nontropical systems increase with increasing duration, since high-intensity, long duration rain rates tend to be produced by tropical cyclones which are large in spatial extent. The magnitude of estimated rain rates in Edwards Creek is somewhat less dependent on storm type than in the larger basins because fractional coverage of rainfall has less impact on rain rates over small areas. Edwards Creek has high variability of 100-year and 1000-year rain rates for all durations compared with the other basins. This is also likely tied to basin size, since the probability of a spatially small storm cell being transposed directly over a small basin is relatively low compared to the probability of the cell being transposed somewhere over a larger basin.

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Rain rates are higher for McAlpine Creek at Sardis than for Little Sugar Creek at Archdale. It is unclear if this is primarily due to its slightly smaller (90 km2 compared with 110 km2) or because its shape is less elongated (see Fig. 1). The relationship between basin size and SST-estimated rain rate is not clear; rain rates generated using both tropical and nontropical storms are actually slightly higher at the 12-h timescale for the larger McAlpine below McCullen (240 km2) than for Little Sugar Creek at Archdale. More work is necessary to understand the relationships between basin size, shape, and orientation, and how these characteristics relate to storm motion and structure at a variety of time scales. SST-based approaches provide a framework for doing this. We also examine the sensitivity of SST-based IDF estimates to domain size. In addition to IDFs for Little Sugar Creek at Archdale generated using a square 3600 km2 domain, results using square domains of 1600 and 6400 km2 were examined. For durations longer than 1 h, the SST method using all storm types is somewhat sensitive to the size of the domain, although there is no apparent systematic structure to this sensitivity according to duration or return period (Fig. 8, top panel). For long return period 1-h estimates, the method appears insensitive to domain size, but for shorter return periods there is some sensitivity. The SST results based on nontropical systems are similar to those based on all storm types for 1-h durations but are much less sensitive to domain size for longer durations (Fig. 8, bottom panel). The results highlight the

importance that relatively rare tropical systems (ten or fewer of the fifty storms for all storm catalogs for Little Sugar Creek are tropical in origin) play in rainfall frequency analysis. Conventional frequency analyses would likely also be sensitive to storm type. More work is necessary to understand the impacts of domain size on SST, but the lack of systematic variation in Fig. 8 suggests that the results presented throughout this study are not strongly coupled to the choice of domain area. 4.4. Temporal structure We examine the temporal structure of extreme rainfall using SST and compare it with conventional design storms for Little Sugar Creek at Archdale (Fig. 9). 100-year, 1, 3, 6, and 12-h design storms were developed using the alternating block method and NOAA Atlas 14 100-year IDF estimates. Time series of 100-year basin-averaged rainfall for the same durations were obtained from 5000 SST realizations for Little Sugar Creek at Archdale using all storm types and again using nontropical storms. The median and the 0.10 and 0.90 quantiles of the basin-averaged rainfall were computed from the 5000 SST realizations. The SST-based 100-year rainfall time series show significant heterogeneity for all durations. The conventional design storms are bounded by the 0.10 and 0.90 SST quantiles, but the SST results show substantial variability in 100-year rainfall including multiple

Edwards Br. 7.1 (0.016) 9.5 (0.036) 14.0 (0.090) McAlpine below McCull. 6.5 (0.015) 7.8 (0.011) 8.3 (0.020)

McAlpine below McCull. 14.5 (0.028) 25.6 (0.024) 27.7 (0.013)

McAlpine below McCull. 10.0 (0.043) 17.9 (0.049) 20.7 (0.014)

6 h rain rate (mm/h), all storms L. Sug. Crk. at Arch. McAlpine at Sardis 14.2 (0.024) 15.6 (0.028) 25.4 (0.043) 27.3 (0.029) 29.2 (0.017) 30.7 (0.023)

12 h rain rate (mm/h), all storms L. Sug. Crk. at Arch. McAlpine at Sardis 9.1 (0.026) 9.8 (0.055) 15.4 (0.023) 18.5 (0.062) 17.8 (0.039) 22.5 (0.030)

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Edwards Br. 9.5 (0.032) 17.6 (0.061) 25.9 (0.092)

12 h rain rate (mm/h), nontrop. L. Sug. Crk. at Arch. McAlpine at Sardis 6.2 (0.011) 6.8 (0.014) 7.6 (0.018) 8.0 (0.011) 8.5 (0.025) 8.9 (0.041)

Edwards Br. 12.8 (0.020) 19.1 (0.046) 28.0 (0.070) McAlpine below McCull. 11.8 (0.012) 14.6 (0.021) 16.3 (0.018)

McAlpine below McCull. 23.8 (0.022) 35.9 (0.045) 42.0 (0.019) 3 h rain rate (mm/h), all storms L. Sug. Crk. at Arch. McAlpine at Sardis 22.5 (0.020) 25.8 (0.022) 32.9 (0.037) 39.8 (0.057) 42.4 (0.064) 50.7 (0.049)

Edwards Br. 14.9 (0.022) 22.5 (0.068) 34.1 (0.115)

6 h rain rate (mm/h), nontrop. L. Sug. Crk. at Arch. McAlpine at Sardis 11.4 (0.017) 12.3 (0.018) 15.4 (0.032) 16.0 (0.025) 17.9 (0.028) 18.2 (0.028)

Edwards Br. 20.8 (0.037) 36.6 (0.041) 49.8 (0.083) McAlpine below McCull. 19.2 (0.019) 29.1 (0.039) 33.2 (0.019) 3 h rain rate (mm/h), nontrop. L. Sug. Crk. at Arch. McAlpine at Sardis 18.4 (0.024) 20.6 (0.028) 29.6 (0.045) 32.2 (0.027) 35.9 (0.031) 36.5 (0.024)

60.0 (0.041) 82.2 (0.035) 106.0 (0.099)

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Table 4 Mean rainfall rate statistics for four basins for 10, 100, and 1000-year events at the 1, 3, 6, and 12-h scales for SST performed using all storm types and using only nontropical storms. Values inside parentheses are dimensionless coefficients of variation (the standard deviation divided by the mean).

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rainfall peaks at 3, 6, and 12-h time scales and a tendency for longer periods of high rain rates than the design storms. The peak rain rates for the median SST-based time series are lower than the peak design storm rainfall. The peak rain rates associated with the upper limits of the SST-based time series, however, are higher than thepeak rain rates from the design storms. Also, the upper limits of peak rain rates from the SST based on nontropical storms are higher than either the design storms or the SST based on all storm types, despite having lower median 100-year peak rain rates. As storm duration increases, SST predicts that rainfall is more broadly distributed in time than the design storm, and that 100year rainfall at time scales longer than 1-h tends to be multimodal. This suggests that several assumptions used to construct the design storms, namely the unimodal temporal structure and the assumed relationship between T-year rainfall rates of across a range of durations, should be reassessed. The impact of this variability in temporal structure on flood risk is unclear. It is possible that for watersheds in which flood response is closely tied to peak rain rates, rather than rainfall volumes, the upper bound of 100year nontropical rain rates shown in Fig. 9 represents a significant flood hazard, despite the fact that time-averaged 3, 6, and 12-h 100-year rain rates associated with nontropical systems are much lower that those of tropical systems. Likewise, the multimodal temporal structure suggested by the SST statistics may have important flood risk implications, which can be evaluated when rainfall SST is coupled with a distributed hydrologic model. 4.5. Spatial structure The spatial structure of storm total rainfall for Little Sugar Creek at Archdale was evaluated using the normalized rainfall-weighted flow distance and dispersion (Figs. 10 and 11, respectively). Variability in normalized rainfall-weighted flow distance and dispersion tends to decrease with increasing rainfall duration, but not necessarily with return period. The boxplots of rainfall-weighted flow distance center around the mean normalized flow distance of 0.56 for Little Sugar Creek at Archdale, the variability does not appear to depend systematically on return period or storm type (except for 1-h durations, which show a monotonic decrease in rainfall-weighted flow distance with return period, and little sensitivity to storm type since tropical storms do not tend to produce extreme 1-h rain rates). For example, variability in 1000-year 3-h nontropical storms is low compared to 1000-year 12-h nontropical storms, but the opposite is true when all storm types are used. It is likely that these features are the result of relatively few large storms (particularly the 23–24 July 1997 event) dominating the 1000-year rainfall estimates and so their particular storm structures dominate the SST statistics. The variability in normalized rainfall-weighted dispersion follows the same general patterns as the variability of flow distance. Median normalized dispersion tends to increase from less than 1.0 to near 1.0 with increasing duration, although the median dispersion is less than 1.0 for all return periods and durations, suggesting that extreme rainfall in Little Sugar Creek at Archdale tends to deviate from uniformity and be characterized by one spatial rain peak. This result is consistent with the modeling results of Morin et al. (2006), which show that high discharge results from a single spatial peak situated relatively near the basin outlet. We also examined the magnitude and variability of SST-derived normalized rainfall-weighted flow distances for 10-year, 100-year, and 1000-year events for all four study watersheds (Table 5). For watersheds at all durations and all return periods, the median flow distances calculated from 5000 realizations using the SST procedure are close to the mean watershed flow path length found in Table 1. There is a tendency, however, for higher return periods to have slightly shorter flow distances than shorter return periods,

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indicating that extreme storms tend to be located closer to the basin outlet (also consistent with the results of Morin et al. (2006)). For example, the median 6-h normalized flow distance for McAlpine below McCullen reduces from 0.55 for 10-year events to 0.53 for 1000-year events. The variability in flow distance for 1000-year events tends to be higher if only nontropical events

are considered, since nontropical convective systems tend to be smaller in spatial extent and so the spatial distribution of rainfall relative to the drainage network becomes more important. The dependence of flow distance variability is less pronounced for the small Edwards Branch, since storms will tend to cover the entire basin.

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Return Period Fig. 10. Boxplots of normalized rainfall-weighted flow distance for 10, 100, and 1000-year storms generated using SST for Little Sugar Creek at Archdale. Statistics are based on 5000 realizations of 1000-year records using a 3600 km2 domain. Boxes denote the lower and upper quartiles and whiskers indicate the extent of the 1.5 interquartile range.

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The analyses presented for flow distance and dispersion for these basins suggest that there can be significant spatial variability in rainfall structure even for 1000-year storms, and therefore standard assumptions of spatially uniform extreme rainfall should be reexamined. The flood frequency implications of the variability in rainfall structure can be examined by coupling rainfall SST with a distributed hydrologic model. It is impossible to assess the importance of this variability using conventional rainfall or flood frequency analysis techniques.

5. Discussion The SST procedure presented in this study addresses some of the shortcomings of previous attempts at SST-based rainfall fre-

quency analysis. Several important considerations and potential future directions for the SST procedure are discussed in this section. Selection of the SST domain size and shape involves tradeoffs: larger domains will tend to enclose greater numbers of extreme storm events, but these storms may be the products of rainfall climatologies that are distinct from that of the watershed of interest. In addition, when radar rainfall fields are used, range effects can impose limits on the feasible size of the domain. The decision to use a square 3600 km2 domain throughout this study reflects these tradeoffs, while analyses show no systematic variation for domains between 1600 and 6400 km2. Storms are assumed to have an equal probability of occurring anywhere within the SST domain, and thus are transposed from their ‘‘true’’ starting positions north–south and east–west according to uniform distributions. We examine

Edwards Br. 0.56 (0.19) 0.55 (0.19) 0.53 (0.21) McAlpine below McCull. 0.53 (0.31) 0.53 (0.28) 0.51 (0.28) 12 h norm. weighted flow dist., nontrop. L. Sug. Crk. at Arch. McAlpine at Sardis 0.55 (0.31) 0.60 (0.24) 0.55 (0.29) 0.59 (0.24) 0.53 (0.30) 0.58 (0.27) McAlpine below McCull. 0.56 (0.30) 0.55 (0.31) 0.52 (0.21) 12 h norm. weighted flow dist., all storms L. Sug. Crk. at Arch. McAlpine at Sardis 0.56 (0.24) 0.61 (0.19) 0.56 (0.24) 0.60 (0.20) 0.56 (0.23) 0.59 (0.19) Return period 10 y 100 y 1000 y

Edwards Br. 0.56 (0.18) 0.56 (0.16) 0.55 (0.16)

Edwards Br. 0.56 (0.20) 0.55 (0.21) 0.53 (0.22) McAlpine below McCull. 0.54 (0.33) 0.52 (0.32) 0.52 (0.22) 6 h norm. weighted flow dist., nontrop. L. Sug. Crk. at Arch. McAlpine at Sardis 0.55 (0.40) 0.60 (0.29) 0.55 (0.35) 0.59 (0.26) 0.53 (0.27) 0.59 (0.23) McAlpine below McCull. 0.55 (0.35) 0.53 (0.30) 0.53 (0.21) 6 h norm. weighted flow dist., all storms L. Sug. Crk. at Arch. McAlpine at Sardis 0.56 (0.34) 0.60 (0.25) 0.56 (0.33) 0.59 (0.22) 0.55 (0.22) 0.59 (0.17) Return period 10 y 100 y 1000 y

Edwards Br. 0.55 (0.21) 0.55 (0.21) 0.55 (0.20)

Edwards Br. 0.55 (0.27) 0.55 (0.22) 0.53 (0.22) McAlpine below McCull. 0.54 (0.44) 0.54 (0.39) 0.53 (0.26) 3 h norm. weighted flow dist., nontrop. L. Sug. Crk. at Arch. McAlpine at Sardis 0.56 (0.48) 0.60 (0.34) 0.56 (0.40) 0.58 (0.28) 0.55 (0.26) 0.59 (0.16) McAlpine below McCull. 0.54 (0.42) 0.52 (0.47) 0.54 (0.26) 3 h norm. weighted flow dist., all storms L. Sug. Crk. at Arch. McAlpine at Sardis 0.55 (0.41) 0.60 (0.30) 0.55 (0.38) 0.59 (0.32) 0.57 (0.33) 0.59 (0.23) Return period 10 y 100 y 1000 y

Edwards Br. 0.56 (0.24) 0.55 (0.20) 0.55 (0.20)

0.55 (0.31) 0.54 (0.26) 0.53 (0.25)

Edwards Br. McAlpine below McCull.

0.54 (0.49) 0.53 (0.41) 0.52 (0.26) 0.59 (0.37) 0.58 (0.26) 0.58 (0.16)

McAlpine at Sardis L. Sug. Crk. at Arch.

0.56 (0.48) 0.55 (0.43) 0.55 (0.29) 0.55 (0.28) 0.54 (0.24) 0.54 (0.24) 0.59 (0.34) 0.59 (0.25) 0.58 (0.17) 0.55 (0.45) 0.55 (0.40) 0.55 (0.29) 10 y 100 y 1000 y

0.54 (0.46) 0.53 (0.36) 0.52 (0.26)

1 h norm. weighted flow dist., nontrop.

Edwards Br. McAlpine below McCull. McAlpine at Sardis

1 h norm. weighted flow dist., all storms

L. Sug. Crk. at Arch. Return period

Table 5 Normalized rainfall-weighted flow distance for four basins for 10, 100, and 1000-year events at the 1 h, 3 h, 6 h, and 12 h scales for SST performed using all storm types and using only nontropical storms. Values inside parentheses are dimensionless coefficients of variation (the standard deviation divided by the mean).

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the validity of this assumption in several ways, including inspection of the spatial distribution of rainfall depths, of cloud-toground lightning strikes, and of storm initiation locations. An alternative, used previously in Wilson and Foufoula-Georgiou (1990), is to relax the assumption of spatially uniform storm occurrences. If storm initiation location depends on topographic, urban, or other features, it should be straightforward to alter the resampling procedure to reflect this nonuniformity in initiation. Heterogeneity in storm intensity due to urban rainfall modification or spatial variation in the supply of water vapor would be more difficult to characterize and account for in an SST framework. One challenge in applying the SST method to larger watersheds is the limited range of weather radar. In this study, the 3600 km2 domain was completely enclosed by the KGSP radar domain and range effects on radar rainfall estimation were found to be minimal. The effects of range on radar estimation are apparent in the Northeast corner of the upper panels of Fig. 4. In cases where the storm selection domain must be large with respect to the radar domain, range effects can become a major challenge. In addition, the finite area of the radar umbrella imposes a limit on the size and position of the domain, irrespective of range effects. As with other radar rainfall applications, blockage due to mountainous terrain or poor radar siting represent major challenges. Some radar products such as Stage IV (from the National Center for Environmental Prediction, see http://www.emc.ncep.noaa.gov/mmb/ylin/pcpanl/ stage4) and the upcoming NMQ Q2 (from the NOAA National Severe Storms Laboratory, see http://www.nssl.noaa.gov/projects/ q2/q2.php) merge radar rainfall estimates from multiple radars as well as from gages. The accuracy of these merged estimates should be carefully examined, however, before they are used for flood risk assessment (see, for example, Wright et al., submitted for publication). One final limitation of the SST method employed in this study is the existence of an upper bound in which basin-averaged rainfall is maximized by a particular transposition of the largest storm in the storm catalog. This upper bound is likely undesirable in the SST framework, due to the small samples of storms used and the uncertainties of estimating extreme rainfall. One could develop a stochastic component in the form of a multiplicative ‘‘intensity factor,’’ perhaps based on statistical modeling of the uncertainties in either mean-field or conditional bias estimates. In this study, two sets of SST-based IDF relationships were developed, one set including all storm types and the other set including only nontropical storms. This stratification is done for several reasons. First, we are able to assess the sensitivity of the method to the inclusion of small numbers of truly extreme events (in this case, several tropical storms, particularly 23–24 July 1997). Second, the changing frequency and intensity of both tropical cyclones and extratropical systems under global climate change is an important topic of research in the weather and climate extremes community, but few tools have emerged that are useful for practicing engineers. The SST procedure presented in this study could be one such tool because it allows us to vary the frequency of storm occurrence k and evaluate the impacts on the frequency and intensity of extreme rainfall from different storm types. Finally, the stratification demonstrates that the SST method is flexible enough to be used to test a variety of hypotheses. For example, instead of stratifying by storm type, IDFs could be developed based on month or season to examine reservoir operations for flood control.

6. Summary and conclusions Spatial and temporal variability in extreme rainfall, and its interactions with heterogeneous land cover and drainage networks, is an important driver of flood response, especially in urban

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settings where flow path lengths are short, runoff velocities are high, and the spatial distribution of land use and hydrologic infrastructure is very heterogeneous. Estimates of flood risk, therefore, should ideally consider this variability. Conventional flood risk assessment, however, involves a number of assumptions which tend to neglect rainfall variability. Central to these assumptions is the notion of hypothetical design storms that are spatially uniform and temporally are either uniform or highly idealized. Rainfall rates and depths for design storms come from intensity– duration–frequency (IDF) relationships developed from rain gage networks that are often characterized by low spatial density and short observational periods. Point-to-area transformation of design rainfall is achieved through area reduction factors (ARFs). Techniques for computing ARFs vary widely, but they tend to ignore local rainfall climatology, return period, and storm type, and use an averaging process to reduce considerable variability in rainfall to a single ARF estimate that is usually only a function of watershed area. Uncertainties in IDFs, ARFs, design storm methods, and other assumptions are rarely considered when flood risk assessment is performed. The impacts of these assumptions on flood risk estimates are poorly understood. This study presents an alternative framework for rainfall frequency analysis that couples high-resolution radar rainfall estimates using with a method known as stochastic storm transposition (SST) to reconstruct the climatology of extreme rainfall for the watershed or hydrologic system of interest. SST attempts to effectively lengthen the period of record used for frequency analysis by ‘‘space-for-time’’ substitution using not only rainfall observations from the location of interest but also observations from surrounding locations. Several previous studies have examined SST for rainfall and flood frequency analysis, but were limited by the lack of long-term records of accurate, high-resolution rainfall fields from a wide variety of storm systems. In this study, we use an SST procedure to resample from ‘‘storm catalogs’’ developed from a ten-year high-resolution (15-min, 1-km2) radar rainfall dataset created using the Hydro-NEXRAD processing system to produce basin-specific IDF relationships for four urban watersheds in Charlotte, North Carolina, USA. We are able to use SST to reproduce conventional IDF estimates while avoiding the simplifying assumptions of standard methods. The SST-based IDF estimates are driven by the structure and variability of real observed storms and do not require any assumptions regarding rainfall structure such as spatial and/or temporal uniformity or pointto-area transformation using ARFs. It also allows IDF estimates to be uniquely tailored to the watershed of hydrologic system of interest, and is flexible enough to allow the estimation of IDFs for specific seasons or storm types. This study uses the spatially and temporally rich representations of extreme precipitation made possible by weather radar to examine the variability of long-return period rainfall. We demonstrate that extreme rainfall can vary substantially in space and time, as well as by storm type. The results cast doubt on common assumptions about rainfall structure that are used in flood risk estimation. The full advantages of SST-based techniques can be realized when the concept is extended to flood frequency analysis. Transposed radar rainfall fields can be coupled with a distributed hydrologic model to develop unique and independent flood frequency estimates at any point along a drainage network. Peak discharge return periods can be estimated everywhere in the watershed independently of rainfall return period, and the impact of antecedent conditions on flood response can be assessed (see Bradley et al., 1994). This stands in contrast to conventional flood frequency methods, which typically presume a 1:1 relationship between the return period of rainfall and of discharge and between peak discharge return period along the entirety of a drainage network. The complexity of interactions between extreme rainfall,

distributed land use, and complex drainage networks, especially in urban areas, suggests that these conventional practices may not adequately determine flood risk (see Viglione and Bloeschl, 2009; Viglione et al., 2009). The coupling of SST methods coupled with high-resolution radar rainfall estimates represents a powerful alternative for flood frequency analysis, and will be the topic of a future study. We close by reiterating a very important and often misunderstood aspect of extreme event analysis, made previously by Wilson and Foufoula-Georgiou (1990). Critics wrongly question the validity of extrapolating extreme rainfall estimates out to T years based on short rainfall records, where T can be 1000 years or more, due to nonstationarity in the climate system or incomplete representation of climatic variability in observational records. While climate nonstationarity and variability are important factors in flood risk, rainfall and flood frequency analyses, SST-based or otherwise, do not attempt to represent the range of events that could occur over T years. Rather, these analyses (including nonstationary methods) provide an estimate of the rainfall or flood magnitude associated with an annual probability of exceedance of 1/T, subject to the limitations of the observation record. The only reasonable way to evaluate the efficacy of any new frequency analysis method is to judge its strengths and weaknesses relative to other available techniques. Given that the results of weather radar and SST-based frequency analyses presented in this paper agree well with conventional methods, and that the method has several previously discussed advantages over conventional techniques, most importantly its potential for flood risk estimation, we suggest that it is deserving of attention from the hydrologic engineering community.

Acknowledgments This work was partially funded by the Willis Research Network, the NOAA Cooperative Institute for Climate Sciences (Grant NOAA CICS NA08OAR4320752), and the National Science Foundation (Grant CBET-1058027). We would also thank Steven J. Wright, University of Michigan and the two anonymous reviewers for their helpful comments.

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