1. Introduction
One of the most challenging aspects of flood hazard characterization is assessing the spatial extent of flooding. The spatial distribution of flood magnitudes reflects the interplay of rainfall variability in time and space with land surface properties that influence infiltration and runoff production, and with the network structure of the channel system (Blöschl et al. 2007, 2013; Sivapalan et al. 1987; Sivapalan 2003; Laaha et al. 2012; Ayalew et al. 2014a,b; Javier et al. 2010; Merz et al. 2008). Rainfall structure and evolution can vary substantially from storm to storm, resulting in striking contrasts in the spatial distribution of flood magnitudes over the drainage network.
Villarini and Smith (2010) introduced the flood index as the ratio of flood peak discharge at a particular location along a river network to the 10-yr flood discharge at the same location, as a dimensionless representation of flood magnitudes that could be used to examine the spatial distribution of flooding over large regions. Their analyses were based on observations from a network of U.S. Geological Survey (USGS) stream gauging stations. The spatial distribution of flooding was represented by a continuous spatial field of flood index values obtained by inverse-distance-weighted interpolation of discrete point observations. The flood index has been shown to be highly correlated with flood damage for tropical cyclone flooding (Czajkowski et al. 2013, 2017). A disadvantage of this method is that the spatial structure of the river network is not directly taken into consideration.
In this study, we develop tools for examining spatial structure of flooding through integration of the spatial and temporal information from high-resolution observations of rainfall fields into a spatially distributed hydrologic model. We adopt a physics-based representation for runoff generation and routing in the river network, with a particular focus on the interplay of rainfall variability in time and space with network structure of the watershed. The hydrological model CUENCAS (Mantilla and Gupta 2005; Cunha 2012) is chosen based on its successful application for flood characterization in large drainage basins. Its hillslope–link basin decomposition provides an efficient and accurate representation of the river network. In CUENCAS, model formulations are designed to reduce the need for calibration, and parameterization is based on readily available datasets, enabling the model to give realistic simulation of floods continuously over a river network.
We use the flood index, here defined as the ratio of the simulated event flood peak and the historical 10-yr flood magnitude, to represent the flood magnitudes continuously over the drainage network. To summarize the flood characteristics and compare them across events, we develop a statistical framework to characterize the probability distribution of flood index values over the drainage network. These probability distributions are examined for the entire basin and are used to characterize the spatial variation of flood magnitudes for flood events. We examine the scale-dependent distribution of flood magnitudes using multiple representations of the basin scale, including drainage area, stream length, and the Horton stream order. We also develop metrics to characterize the correlation structure of flood index over the network, especially along the main stem. Other approaches for analysis of flood magnitude continuously over the drainage network include those based on topological kriging (see Skøien et al. 2005; Merz et al. 2008).
Regulation by dams affects flood response in portions of the Delaware River basin, as in most major river basins of the United States and around the world. Detailed information on dam operations, however, is rarely available, especially during flood periods. Furthermore, discharge observations from dams are not routinely available. We use a simple low-pass-filter representation of dam impacts on downstream flooding in our CUENCAS model implementation. For the Delaware River basin, this formulation is consistent with previous analyses of the downstream impacts of dams on flood response (Smith et al. 2010). We will show that it also provides an improved representation of flood response for the four flood events that we analyze. This approach is not suited for all applications, especially those that directly address the impacts of operations of specific dams on flood response. It does, however, provide useful insight into the impacts of dams on the spatial extent of flooding over a river network.
We apply the modeling and analysis framework to four storms, which resulted in the largest floods in Delaware River basin (in terms of flood peaks at the basin outlet) since the flood of 1955 caused by Hurricanes Connie and Diane. The four floods reflect the principal flood-generating mechanisms in the eastern United States (Smith et al. 2010): landfalling tropical cyclones (Hurricane Ivan in September 2004 and Hurricane Irene in August 2011), late winter/early spring extratropical systems (April 2005), and warm season convective systems (June 2006). The hydrology and hydrometeorology of the 2004, 2005, and 2006 floods have been summarized in Smith et al. (2010).
Tropical cyclones represent a particularly important flood hazard for the eastern United States, especially in controlling the upper tail of flood peak distributions (Villarini and Smith 2010). The framework that we develop for examining the spatial distribution of flood magnitudes over a drainage network will be used for examining the distribution of tropical cyclone flood peaks, in both the current and future climates. We will use large samples of tropical cyclone storm sets (Emanuel et al. 2006, 2008; Zhu et al. 2013) to assess flood hazards over the eastern United States.
The paper is organized as follows. In section 2, we describe the hydrological model CUENCAS and introduce the flood index and statistical framework for examining flood magnitudes over the drainage network. In section 3, we describe the study area and the four storm events. In section 4, we present hydrological modeling results, flood index statistics, and rainfall field analysis. In section 5, we summarize major findings.
2. Methodology
a. Hydrologic modeling
In this study we apply a hillslope-based hydrological model, CUENCAS, which has been applied in a wide range of flood hydrology applications (Mantilla and Gupta 2005; Cunha 2012). In CUENCAS, the drainage basin is treated as a collection of hillslopes connected by channel links, which are compartmentalized and derived from a high-resolution digital elevation model (DEM). The river network is represented as a binary tree defined by links and nodes. From the DEM, we extract information about the position and geometry of each link, and the way links are connected. CUENCAS provides a flexible representation of hydrologic modeling components. At the hillslope scale, infiltration is calculated based on the Soil Conservation Service Curve Number (SCS-CN) method (NRCS 1986; Mishra and Singh 1999). The advantages of the SCS-CN approach are its parsimonious parameterization and broad application (Ponce and Hawkins 1996). In CUENCAS, the SCS-CN method is enhanced for continuous hydrologic simulation, taking into account the temporal and spatial variability of the curve number (Michel et al. 2005; Durbude et al. 2011; Cunha 2012).
The Delaware River basin is decomposed into 1891 hillslopes and 1891 links. The area of hillslopes varies from less than 0.1 to 78 km2. The length of links varies from less than 0.1 to 22 km. The model runs in a storm event mode, with the initial soil moisture taken from monthly mean CPC soil moisture data and hourly rainfall taken from NCEP stage IV at 4-km spatial resolution. Model parameters (i.e., 
b. Flood index
The 10-yr flood can be calculated in a variety of ways. The approach we take in this paper is designed to provide a simple implementation and illustration of the modeling framework. For specific applications, more detailed representations of the 10-yr flood may be developed. Flood quantiles in a region can vary with both drainage area and other physical properties of the drainage basin (Ayalew 2015), and multivariate statistical methods have been widely used to develop regional estimation equations for flood quantiles (Stedinger 1983; Stedinger and Tasker 1985, 1986). These approaches can utilize the information in covariates for which systematic datasets are available in the United States (Falcone 2011). In addition to the applications-oriented development of regional regression equations, this line of research provides important insights to dependence of flood magnitudes on drainage basin properties. In some settings the 10-yr flood can be estimated based on computed peak discharges from hydrologic modeling using long rainfall records (Moser et al. 2015). A challenge in implementing such a model-based approach is developing suitable rainfall datasets. The resampling methods introduced by Wright et al. (2014) provide a promising line of research for advancing this approach. Future research will examine the development of improved statistical and model-based methods for computing the 10-yr floods based on resampling methods.
c. Statistics for spatial analysis of the flood index
Previous studies have analyzed the distributions and joint distributions of flood peaks, flood volumes, and flood durations based on samples of observations at stream gauging stations (Krstanovic and Singh 1987; Sackl and Bergmann 1987; Goel et al. 1998; Yue 1999; Villarini and Smith 2010). A disadvantage of analyses based solely on point observations is that the river network structure is not fully taken into consideration. In addition, the analysis is dependent on the specific locations of gauging stations and the number of available stations.
The drainage network can be represented as a tree graph that consists of a set of nodes connected to one another through links such that there are no loops in the graph. The outlet node imposes flow direction. For each node there is a unique path (i.e., sequence of nodes and links) to the outlet. A source is a node, other than the outlet, connected only to one link. A junction is a node connected to more than one link. Exterior links connect sources to junctions. Interior links connect two junctions. The number of links in the drainage network will be denoted by n, and an order is imposed on the stream links. The Horton stream order represents the relative size of a river segment. Links with Horton stream order 1 are stream segments with no tributaries, links with Horton stream order 2 are on streams formed by the confluence of first-order streams, and more generally, links downstream of the confluence of two links of order i have order 
n = number of links in the drainage network;
The statistics introduced above provide useful tools for characterizing the aggregate distribution of flooding over a drainage network and representations of correlation structure of flood magnitudes over the network. Throughout the remainder of the paper, we will suppress dependence on storm j whenever possible. Large samples of flood events can be used to characterize distributional properties of these quantities; in section 5 we point to methods of examining the climatology of tropical cyclone flooding.
3. Study area and flood summaries
a. Study area
The Delaware River basin has a drainage area of 35 066 km2 at Trenton, New Jersey, draining parts of Pennsylvania, New York, New Jersey, and Delaware to Delaware Bay. As shown in Fig. 1 (left), the upper basin is mountainous while the downstream portion of the watershed has lower relief. The basin is mostly forest covered, especially in mountainous areas, with modest urban development in low-elevation areas. There are four major tributaries of the Delaware River, the West Branch Delaware River, the East Branch Delaware River, the Lackawaxen River, and the Lehigh River.
Map of Delaware River basin showing (left) elevation and river network and (right) Horton stream order and location of dams.
Citation: Journal of Hydrometeorology 18, 4; 10.1175/JHM-D-16-0071.1
Like most major rivers in the United States, the Delaware River is regulated by dams and reservoirs (Fig. 1, right). Reservoirs in the upper Delaware River basin are mainly operated for New York City water supply. Dams in the Lackawaxen and Lehigh River basins provide both flood control and water supply functions (Smith et al. 2010). Despite the extensive system of dams in the basin, the effects on flood hydrology are poorly understood (Williams and Wolman 1984; Batalla et al. 2004). Here we consider the effects of dams in the hydrologic model by treating them as low-pass filters. This is done by reducing channel velocity on links located with dams, that is, υ in Eq. (1), to 10% of its original value. This simple representation improves the overall performance of the model with minimal computational effort.
The Delaware River basin is decomposed into 1891 links (Fig. 1, right) for hydrologic modeling. There are 67 USGS stream gauging stations in the study area, providing observations for hydrologic model validation and the computation of the T (yr) flood. The 10-yr flood for the 67 observations exhibits a power-law relationship with the drainage area (with correlation of 0.905), as shown by the log–log plot in Fig. 2. With the estimated prefactor and exponent of the power-law relationship, we can compute the 10-yr flood magnitude for any drainage area. Combining CUENCAS model simulations over a drainage network with computed values of 10-yr flood for any drainage area allows us to compute the flood index continuously over the network on an event basis.
Scaling properties of 10-yr flood peaks for 67 USGS gauging stations in Delaware River basin.
Citation: Journal of Hydrometeorology 18, 4; 10.1175/JHM-D-16-0071.1
b. Storms
In this study, we examine four storm events that occurred 17–19 September 2004, 1–2 April 2005, 25–28 June 2006, and 26–28 August 2011, which resulted in the four largest flood peaks at the basin outlet (Trenton) since the flood in 1955 caused by Hurricanes Connie and Diane (Fig. 3).
Annual peak streamflow at Trenton.
Citation: Journal of Hydrometeorology 18, 4; 10.1175/JHM-D-16-0071.1
Hurricane Ivan (2004) produced storm total rainfall accumulations ranging from 35 to 495 mm in the Delaware River basin as the storm center moved from west to east through the southern portion of the basin. The storm had weakened to a tropical depression before it affected the basin (from 1000 UTC 17 September to 1000 UTC 19 September) and completed extratropical transition by 1800 UTC 18 September (Stewart 2005). Hurricane Irene (2011) passed from south to north along the eastern margin of the basin and produced storm total rainfall accumulations ranging from 91 to 348 mm. The storm was a category 1 hurricane when it reached the Delaware River basin (from 0500 UTC 26 August to 0500 UTC 28 August) and decayed to a tropical storm after it passed the basin (0900 UTC 28 August; Avila and Cangialosi 2012). The storm began to undergo extratropical transition around 1200 UTC 27 August (Grumm 2011). Extratropical transition of Ivan and Irene, as with many tropical cyclone flood events in the northeastern United States, played a central role in heavy rainfall and flooding in Delaware River basin (Smith et al. 2010; Liu and Smith 2016).
The April 2005 flood event was associated with an extratropical cyclone. From 1200 UTC 2 April to 0600 UTC 3 April, the storm moved from northern Virginia to the northwestern corner of the Delaware River basin, with minimum sea level pressure decreasing from 995.5 to 981 hPa. Heavy rainfall was associated with a rapidly deepening cyclone, frontogenesis, and strong flow of moist air at 850 hPa (Smith et al. 2010).
The June 2006 flood was caused by heavy rainfall from warm season convective systems. The synoptic setting was characterized by a stationary midtropospheric trough over the Ohio Valley. The storm environment was characterized by a deep layer of humidity and large values of precipitable water. There was strong moisture transport in the lower atmosphere and heavy rainfall along the western margin of the Appalachians (Smith et al. 2010).
c. Rainfall distribution
The southwest–northeast axis of the Appalachians shapes the rainfall distribution for all four storms (Fig. 4). For the 2004, 2005, and 2006 events, rainfall maxima were distributed along or to the west of the axis, while for the 2011 storm, extreme rainfall was located to the east of the axis. Although it is difficult to separate the contribution of orographic precipitation from other rainfall mechanisms, these patterns demonstrate the influence of complex terrain in modifying precipitation from the principal weather systems that produce major flooding in the basin. Other notable examples of weather systems that produce extreme rainfall along the eastern margin of the Appalachians include Hurricane Camille (1969) and Tropical Storm Allison (2001) (Javier et al. 2010). The connection between extreme rainfall and orographic precipitation mechanisms has been summarized in Smith (2006).
Storm total rainfall accumulations with 20-mm interval contour for the (top left) 2004 storm (from 1000 UTC 17 Sep to 1000 UTC 19 Sep), (top right) 2005 storm (from 2300 UTC 31 Mar to 2300 UTC 2 Apr), (bottom left) 2006 storm (from 0900 UTC 25 Jun to 1900 UTC 28 Jun), and (bottom right) 2011 storm (from 0500 UTC 26 Aug to 0500 UTC 28 Aug).
Citation: Journal of Hydrometeorology 18, 4; 10.1175/JHM-D-16-0071.1
The spatial distribution of rainfall is strongly dependent on storm tracks (Smith 2006). The 2004 storm (Hurricane Ivan), for example, moved from west to east along the southern margin of the basin, producing strong upslope flow along the western slopes of the Appalachians. The 2011 storm (Hurricane Irene), however, moved from south to north along the eastern margin of the basin, producing strong upslope flow along the eastern margin of the Appalachians, resulting in heavy precipitation on the east side [see Lin et al. (1999) for similar features in Taiwan].
Time series of basin-averaged rain rate illustrate the intensity and duration of the four storms (Fig. 5). The 2004 and 2011 storms, both tropical cyclones, have rainfall concentrated in a 20-h window, with the peak basin-averaged rain rate of approximately 11–12 mm h−1. The 2005 storm, an extratropical cyclone, had a longer duration (approximately 30 h), with a lower peak rain rate of approximately 9 mm h−1. Rainfall from the 2006 warm season convective systems extended over 3 days, while the highest basin-averaged rain rate is similar to that of the 2005 storm. We will explore the linkage between flood distribution and the intensity and duration of the storms.
Basin-averaged rainfall series for the (top left) 2004 storm (from 1000 UTC 17 Sep to 1000 UTC 19 Sep), (top right) 2005 storm (from 2300 UTC 31 Mar to 2300 UTC 2 Apr), (bottom left) 2006 storm (from 0900 UTC 25 Jun to 1900 UTC 28 Jun), and (bottom right) 2011 storm (from 0500 UTC 26 Aug to 0500 UTC 28 Aug).
Citation: Journal of Hydrometeorology 18, 4; 10.1175/JHM-D-16-0071.1
4. Flood index modeling results
The hydrologic model simulations are compared with discharge observations (typically at 15-min time resolution) from 67 USGS stream gauging stations across the basin. Simulated hydrographs for the four storms generally compare well with observed hydrographs in terms of timing and peak magnitudes (see Fig. 6 for examples). The correlation coefficient between simulated and observed peaks is greater than 0.92 for all four storms (Fig. 7). Representation of dams, as described in the previous section, improves the performance of the model, increasing the correlation coefficient above 0.94 for all events. Sources of uncertainty in the model intercomparisons include errors in radar rainfall estimates, antecedent soil moisture, hydrologic model parameterization, and errors in discharge measurements (Javier et al. 2010; Villarini and Krajewski 2010; Mandapaka and Germann 2013; Domeneghetti et al. 2012; Le Coz 2012). As described in previous studies (Ogden et al. 2000; Westrick and Mass 2001; Vieux and Bedient 2004; Smith et al. 2007), errors in the temporal and spatial distribution of rainfall are the leading determinants in hydrologic simulation uncertainties. Therefore, the most important improvement for hydrologic simulations is to have the best possible rainfall fields in time and space across different scales, as we will address in future work.
Observed and simulated hydrographs at four sites on Delaware main stem for the 2004 event (location of four sites is shown in Fig. 1).
Citation: Journal of Hydrometeorology 18, 4; 10.1175/JHM-D-16-0071.1
Scatterplot of simulated and observed peaks at 67 USGS stream gauging stations for the (from top to bottom) 2004, 2005, 2006, and 2011 flood events.
Citation: Journal of Hydrometeorology 18, 4; 10.1175/JHM-D-16-0071.1
Despite having similar peak discharge values at the basin outlet at Trenton (Fig. 3), the distributions of flood magnitudes for the four flood events, as represented by the flood index in Fig. 8, vary markedly over the Delaware River basin. The 2005 event produced the largest flood peak at Trenton, 6853 m3 s−1, but had the most uniform distribution of flood index over the drainage basin. The 2004 and 2011 events produced smaller flood peaks at Trenton, 5522 and 5692 m3 s−1, respectively, but had flood index values that were more than twice as large as the peak flood index values for the 2005 event.
Map of flood index for the (top left) 2004, (top right) 2005, (bottom left) 2006, and (bottom right) 2011 flood events.
Citation: Journal of Hydrometeorology 18, 4; 10.1175/JHM-D-16-0071.1
Figure 9 shows the box plots characterizing the central tendency (median), spread, and extremes of flood index values for the four storms. These results point to the relatively compact distribution of flood index values for the April 2005 storm and the relatively large number of outlier values for the September 2004 storm. Despite large variations in the mean flood index (Table 1), the coefficient of variation (standard deviation divided by the mean, Table 2) of flood index is relatively stable across storms, taking values of 0.45, 0.38, 0.47, and 0.45 for the 2004, 2005, 2006, and 2011 simulations with dams and values of 0.40, 0.36, 0.46, and 0.40 for the simulations without dams.
Box plot of flood index for the four events.
Citation: Journal of Hydrometeorology 18, 4; 10.1175/JHM-D-16-0071.1
Mean of flood index for simulations with dam (WD) and no dam (ND).
Coefficient of variation for simulations with dam (WD) and no dam (ND).
Regulation effects on annual flood peaks have been studied in various watersheds (Asquith 2001; Peters and Prowse 2001). Here we highlight in Fig. 9 the regulation effects on spatial distribution of event flood peaks over the Delaware River basin. For each of the storms, dams systematically reduce median values as well as the 0.25 and 0.75 quantiles of the flood index. Overall, there is only a modest change in the interquartile range, representing the spread of flood index values, but there is a pronounced decrease in extreme values of flood index for the 2011 flood, reflecting the occurrence of the most extreme rainfall in the Upper Delaware River basin, which includes watersheds controlled by water supply dams for New York City.
The contrasting distribution of flood index values over the Delaware River drainage network is clearly linked to the contrasting properties of the rainfall distribution of the four storms (Fig. 4). Large values of flood index are associated with areas of anomalously high storm accumulations, especially on small streams. For the 2004, 2006, and 2011 cases, we see high flood index values lying along a northeast–southwest axis; the location of the axis varies by storm, with high values for the 2004 and 2006 events concentrated in the western portion of the watershed and for the 2011 event in the eastern portion of the watershed. The spatial distribution of storm total accumulation, however, does not provide a comprehensive picture of flood magnitudes over the basin. For the 2004 event, rainfall accumulations exceeding 140 mm produced flood index values above 2 (similar to 2011 event), while similar storm total accumulations are associated with flood index values close to 1 for the 2006 storm. Rainfall–flood peak correlation will be discussed in more detail later in this section.
The fractional area function 
Distribution of flood index over the entire river network in Delaware River basin where (bottom) gives a detailed look into tails in (top).
Citation: Journal of Hydrometeorology 18, 4; 10.1175/JHM-D-16-0071.1
The upper tail of flood index values exhibits sharp contrasts between the four events (Fig. 10, bottom). In particular, the 2005 event stands out for the relatively low values of flood index in the upper tail: 10% of the network has flood index values greater than a value slightly larger than 1. For the 2004, 2006, and 2011 events, the 10% exceedance values cluster around 2, that is, approximately 10% of the drainage network experienced flood peaks that were more than twice the 10-yr flood. For the 1% exceedance value, the 2006 flood is somewhat smaller than the two tropical cyclone floods in 2004 and 2011. Thus, the largest flood index values are associated with the two tropical cyclone floods.
The distribution of flood peaks over the drainage network varies systematically with Horton stream order (see Mantilla et al. 2006; Mandapaka et al. 2009; Menabde et al. 2001; Gupta and Mesa 2014; Gupta et al. 2015). The distribution of flood index values shows a systematic increase in central tendency for Horton stream orders increasing from 1 to 3 (Fig. 11). This feature is characteristic of each storm and for both simulations with and without dams. Results for dam and no-dam simulations diverge significantly with the transition from stream order 3 to 4. For the no-dam simulations, the quantiles of flood index values generally increase for three of the four events; there is a slight decrease in the median and upper quantiles for the 2011 flood. For the simulations with dams, there is a marked decrease in flood index, for all quantiles, as we move from order 3 to 4 channel segments. The difference at the particular stream order follows from the fact that most river segments with Horton stream order 4 are downstream of dams (see Fig. 1, right). These results point to a systematic dependence of flood magnitudes on stream order and a systematic scale-based dependence of flood magnitude on reservoir impacts.
Distribution of flood index by Horton stream order for the four events.
Citation: Journal of Hydrometeorology 18, 4; 10.1175/JHM-D-16-0071.1
Flood magnitudes do not vary continuously over the drainage network; as we go from headwater to basin outlet, we see abrupt transitions of flood index with values gradually changing between the transitions. In Fig. 12, we show the downstream variation of flood magnitude along the main stem of the Delaware River, from its origin in the West Branch Delaware River to its outlet in Trenton, for the two tropical cyclone floods in September 2004 and September 2011. Anomalously large rainfall accumulations for both events were located in the headwaters of the Delaware River. For the 2011 event, a sharp spike in headwater flood index values is followed by a decay in flood index until an abrupt drop occurring at the West Branch Delaware River dam location (noted by “Dam”). For the 2004 event, maximum flood index values are concentrated upstream of the West Branch Delaware River dam.
(a) Downstream variation of flood index along the main stem of Delaware River, from its origin in the West Branch Delaware River to its outlet in Trenton. (b) The main stem of Delaware River. (c) Change of drainage area along the main stem of Delaware River.
Citation: Journal of Hydrometeorology 18, 4; 10.1175/JHM-D-16-0071.1
The largest increases in flood index for both events occur at the confluence of the West Branch and East Branch Delaware River (noted by “A”). Abrupt transitions occur at major tributary junctions, notably the junction of the Lackawaxen River (in the middle basin, Lackawaxen) and Lehigh River (in the lower basin, “B”). For the 2004 flood, there is an abrupt increase in flood index at the confluence of the Lackawaxen, reflecting the anomalously large rainfall in the subbasin (see Fig. 4) and an abrupt, but smaller, decrease in flood index occurs at the confluence of the Lehigh River (reflecting the relatively small rainfall accumulations in the subbasin). For the 2011 event, the change in flood index values at the Lackawaxen and Lehigh confluences are smaller.
As suggested by the preceding analyses, flood index values are correlated along the river network. Flood index values for the four storms remain stable on the Delaware River main stem (from Lackawaxen to Trenton; see Figs. 8 and 12a), where we use Pearson correlation to examine downstream influence, with Lackawaxen as the starting point (Fig. 13). As expected, the correlation decreases with separation distance for all four storms. For the 2006 and 2004 floods (with flood index on main stream varying from 2 to 1.3), the correlation over a distance of 50 km downstream is as high as 0.94. For the 2011 and 2005 events (with flood index varying from 1.3 to 0.8), the correlation over a distance of 50 km drops to 0.65. For these events, larger flood index values are associated with stronger downstream correlation. The change of flood peak along the stream is a combined effect of attenuation and aggregation of flows. While the flood peak attenuation is complicated, the change caused by confluence of streams is smaller with larger floods (higher flood index). This feature is linked to the higher downstream correlation with larger values of flood index. This spatial correlation of the flood index, however, is also highly influenced by the spatial correlation of the rainfall fields.
Spatial correlations of flood index (FI) downstream along Delaware main stem between Lackawaxen and Trenton.
Citation: Journal of Hydrometeorology 18, 4; 10.1175/JHM-D-16-0071.1
The confluence covariance statistics also provide information about the storm-to-storm variation of flood peak distributions. In Table 3, we present the confluence correlation statistics (covariance divided by variance) for streams of order 1, 2, and 3 (sample size is too small to develop useful statistics for higher stream orders). The smallest correlation, 0.23, is for third-order streams with Hurricane Irene (2011) for simulations with dams. The correlation increases to 0.60 for third-order streams in Irene simulations without dams. The highest correlations for third-order streams are for the April 2005 extratropical flood event, 0.79 with dams and 0.82 with no dams. For the June 2006 convective event, correlation increases for third-order streams from 0.54 without dams to 0.62 with dams. The June 2006 event has the largest correlations for first-order streams, 0.71 for simulations with and without dams. The smallest correlations for first-order streams are for the 2011 flood, with values of 0.49 without dams and 0.56 with dams. There is not a systematic dependence of confluence correlation on stream order. Confluence correlation statistics provide a useful tool for assessing the effects of rainfall variability in time and space for a particular event (as discussed below) on the spatial structure of flood magnitudes. They also provide insight to the role of dams on the stream order dependence of flood magnitudes.
Confluence correlation for simulations with dam (WD) and no dam (ND).
As noted above, the distribution of flood index is significantly linked to the distribution of rainfall. In Figs. 14 and 15, we summarize the distribution of rainfall over the Delaware River basin based on storm total rainfall, maximum hourly rain rate, and maximum 24-h rainfall rate (i.e., hourly rain rate for each grid in a 24-h period centered at the time of the maxima of basin-averaged rainfall rate). These rainfall distributions are compared to the distribution of flood index (Figs. 9, 10) to explore the role of rainfall properties in determining flood magnitudes. Although maps of storm total accumulations (Fig. 4) provide useful insights to the spatial distribution of flood index (Fig. 8), the distribution of storm total accumulations does not provide a comprehensive characterization of flood potential over the drainage network. The 2005 storm has the most extreme storm total accumulations and the 2004 storm has the least extreme, which is opposite of the distribution of flood index. In Figs. 9 and 10, the 2005 flood stands out much weaker than the other three, and the 2004 flood is the most extreme among the four. The distributions of peak hourly and 24-h rainfall rates clearly distinguish the 2005 storm from the others as the least extreme and 2004 storm as the most extreme in terms of central tendency and extremes.
Distribution of storm total rainfall accumulations for each grid (metric 1; same values as shown in Figs. 4–7), max hourly rain rate for each grid during the entire storm (metric 2), and hourly rain rate for each grid in a 24-h period centered at the time of the maxima of basin-averaged rainfall rate (metric 3). Closer looks into tails are shown in panels on the right.
Citation: Journal of Hydrometeorology 18, 4; 10.1175/JHM-D-16-0071.1
Box plot of storm total rainfall accumulations for each grid (metric 1; same values as shown in Fig. 4), max hourly rain rate for each grid during the entire storm (metric 2), and hourly rain rate for each grid in a 24-h period centered at the time of the maxima of basin-averaged rainfall rate (metric 3).
Citation: Journal of Hydrometeorology 18, 4; 10.1175/JHM-D-16-0071.1
The distribution of hourly rainfall rates, especially in their upper tail, highlight contrasts in the upper tail of flood index distributions, not just between the 2005 flood and the other three events, but also among the 2006, 2004, and 2011 events. The relative high frequency of extreme hourly rainfall rates for the 2004 storm drives the relatively high fraction of flood index values exceeding 3, which indicates that a key issue for assessing spatial structure of extreme flooding using radar rainfall fields and hydrologic models is determining the accuracy of rainfall estimates for the most extreme rainfall rates.
5. Summary and conclusions
In this study, we develop a framework to characterize the distribution of flood magnitudes over large river networks. Flood magnitudes are characterized by the flood index, which is defined as the ratio of the peak magnitude of a flood event to the historical 10-yr flood magnitude. Flood peaks are computed continuously over the drainage network using a distributed hydrologic model, CUENCAS, with high-resolution radar rainfall fields as the principal forcing. The magnitudes of 10-yr flood are computed as power-law functions of drainage area, with empirical coefficients of the power-law relationship developed from regional flood peak data. Summary statistics for characterizing probability distribution and spatial correlation of flood magnitudes over the drainage network for a flood event are developed based on the dimensionless flood index.
This framework is applied to four storm events in the Delaware River basin in the northeastern United States. The four storms, which occurred 17–19 September 2004, 1–2 April 2005, 25–28 June 2006, and 26–28 August 2011, resulted in the four largest flood peaks at the Delaware basin outlet since the flood of record in 1955 caused by Hurricanes Connie and Diane. The four floods reflect the principal flood-generating mechanisms in the eastern United States: landfalling tropical cyclones (September 2004 caused by Hurricane Ivan and August 2011 caused by Hurricane Irene), late winter/early spring extratropical systems (April 2005), and warm season convective systems (June 2006).
Despite having similar peak discharge values at the basin outlet, the distributions of flood magnitudes over the drainage network vary markedly for the September 2004, April 2005, June 2006, and September 2011 flood events. The distributions of flood magnitudes also vary markedly in the upper tail, with the April 2005 event having the most tightly concentrated distribution of flood magnitudes and “lightest” upper tail.
The distribution of flood index values shows a systematic increase in central tendency for Horton stream orders increasing from 1 to 3. This feature is characteristic of each storm and for simulations with and without dams. Results for dam and no-dam simulations diverge significantly with the transition from stream order 3 to 4. These results point to a systematic dependence of flood magnitudes on stream order and a systematic scale-based dependence of flood magnitude on reservoir impacts. Dams systematically reduce flood magnitudes over the drainage network. Representation of dams provides the capability of capturing the local impact of dams for storm events, such as the September 2011 tropical cyclone event, which concentrate extreme rainfall in portions of the watershed controlled by dams.
The spatial correlation of flood index along the main stem reaches 0.94 for a downstream distance of 50 km for the 2004 and 2006 floods (flood index varying from 1.3 to 2 on the main stem) and is only 0.65 for the 2005 and 2011 floods (flood index varying from 0.8 to 1.3 on the main stem). For the sample of flood events, higher values of flood index are linked to stronger downstream correlation.
Confluence covariance statistics provide useful information about the storm-to-storm variation of flood peak distributions over the drainage network. Results for the sample of four storms show sharp contrasts in confluence correlation from storm to storm and with stream order. These statistics provide important tools for assessing the role of rainfall variability on flood response and in assessing the dependence of dams on stream order.
In terms of storm total rainfall accumulations as well as the flood peak at the basin outlet, the 2005 extratropical event is the most extreme, while the 2004 and 2011 tropical cyclone events are the least extreme. The 2005 storm, however, produces lower flood magnitudes than the other three over much of the drainage network, while the 2004 and 2011 events produce the most extreme values of flood index. This difference is associated with contrasting spatial structure of rainfall and contrasting rainfall-rate distributions within the storms. Maximum hourly rain-rate distributions over the watershed distinguish the 2005 from the 2004 and 2011 events.
The flood index framework introduced in this study can be utilized to characterize the spatial distribution of floods, especially those caused by landfalling tropical cyclones. Future work will apply this framework to large numbers of tropical cyclone rainfall events to assess and characterize the associated long-term flood hazards. For both historic analyses and future projections, the most important improvement for hydrologic simulations is to have the best possible rainfall fields in time and space across different scales. This goes along with our current research on a simple, physics-based rainfall model that can be coupled with a large number of synthetic tropical cyclones (Emanuel et al. 2006, 2008; Zhu et al. 2013). This study fits into our long-term goal of developing a methodology to quantify the risk of inland flooding associated with landfalling tropical cyclones.
Acknowledgments
This material is based upon work supported by the National Science Foundation under Grant EAR-1520683. We would also like to thank Luciana K. Cunha and Mary Lynn Baeck for their generous help with the hydrological modeling.
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