Accepted Manuscript

Rainfall Organization Control on the Flood Response of mild-slope Basins

Yiwen Mei, Emmanouil N. Anagnostou, Dimitrios Stampoulis, Efthymios I.Nikolopoulos, Marco Borga, Humberto J. Vegara

PII: S0022-1694(13)00910-4DOI: http://dx.doi.org/10.1016/j.jhydrol.2013.12.013Reference: HYDROL 19275

To appear in: Journal of Hydrology

Received Date: 21 May 2013Revised Date: 26 November 2013Accepted Date: 7 December 2013

Please cite this article as: Mei, Y., Anagnostou, E.N., Stampoulis, D., Nikolopoulos, E.I., Borga, M., Vegara, H.J.,Rainfall Organization Control on the Flood Response of mild-slope Basins, Journal of Hydrology (2013), doi: http://dx.doi.org/10.1016/j.jhydrol.2013.12.013

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Rainfall Organization Control on the Flood Response of mild-slope

Basins

Yiwen Mei, Emmanouil N. Anagnostou, Dimitrios Stampoulis

Civil and Environmental Engineering

University of Connecticut, Storrs, CT, USA

Efthymios I. Nikolopoulos and Marco Borga

Department of Land, Environment, Agriculture and Forestry,

University of Padova, Legnaro, PD, Italy

Humberto J. Vegara

School of Civil Engineering and Environmental Sciences

University of Oklahoma, Norman, OK, USA

Submitted to: Journal of Hydrology

Submission date: April 2013

Revised Manuscript Submitted: October 2013

Corresponding Author: Emmanouil N. Anagnostou, CEE, University of Connecticut, Storrs, CT

06269, (860) 486-6806, Email: manos@engr.uconn.edu

i

Abstract

This study uses a long-term (8 years) dataset of radar-rainfall and runoff observations for the Tar River

Basin in North Carolina, to explore the rainfall space-time organization control on the flood response of

mild-slope (max slope < 32 degrees) basins. We employ the concepts of ―spatial moments of catchment

rainfall‖ and ―catchment scale storm velocity‖ to quantify the effect of spatial rainfall variability and

basin geomorphology on flood response. A calibrated distributed hydrologic model is employed to assess

the relevance of these statistics in describing the degree of spatial rainfall organization, which is important

for runoff modeling. Furthermore, the Tar River Basin is divided into four nested sub-basins ranging from

1106 km2 to 5654 km

2, in order to investigate the scale dependence of results. The rainfall spatiotemporal

distribution represented in the analytical framework is shown to describe well the differences in

hydrograph timing (less so in terms of magnitude of the simulated hydrographs) determined from forcing

the hydrologic model with lumped vs. distributed rainfall. Specifically, the first moment exhibits a linear

relationship with the difference in timing between lumped and distributed rainfall forcing. The analysis

shows that the catchment scale storm velocity is scale dependent in terms of variability and rainfall

dependent in terms of its value, assuming typically small values. Accordingly, the error in dispersion of

simulated hydrographs between lumped and distributed rainfall forcing is relatively insensitive to the

catchment scale storm velocity, which is attributed to the spatial variability of routing and hillslope

velocities that is not accounted by the conceptual framework used in this study.

Keywords: Spatiotemporal Variability of Rainfall; Rainfall Moment; Storm Velocity; Hydrologic Model

1

1 Introduction 1

The prediction of catchment flood response is one of the recurrent themes in hydrology (Nicótina, et 2

al., 2008; Smith, et al., 2004; Woods & Sivapalan, 1999). A question asked is that, given certain 3

catchment and precipitation characteristics, what are the dominant processes in flood response? In 4

practical terms, operational flood and flash flood guidance systems are based on continuous hydrological 5

models that provide catchment-scale soil moisture estimates representing the antecedent moisture 6

conditions at spatial scales closer to the need of flash flood forecasting (Carpenter, et al., 1999). Based on 7

these antecedent moisture conditions, it is possible to provide an estimate of rainfall depth for a given 8

duration that can lead to flooding without explicitly modeling the event dynamics (Georgakakos, 2005; 9

Reed, et al., 2004). At the ground of flood/flash flood guidance is the computation of the basin-average 10

rainfall for a given duration that is necessary to cause flooding. 11

The fundamental problem facing flood modeling associated with flash flood guidance (FFG) is that 12

small to medium size basins prone to flash floods are rarely gauged and, therefore, prediction of flash 13

flood response relies on basin-average rainfall information and the parameterized response of the larger 14

scale, parent basins. As several past studies have indicated the basin flood response is sensitive to the 15

spatial rainfall patterns (Dawdy & Bergmann, 1969; Naden, 1992; Viglione, et al., 2010a; Wilson, et al., 16

1979; Wood, et al., 1988; de Lima & Singh, 2002). Therefore, from the perspective of developing 17

effective flood forecasting systems, it is important to develop a consistent framework to quantify the 18

effects of space-time rainfall aggregations on the prediction accuracy of flood response for different basin 19

and storm characteristics (Borga, et al., 2007; Merz & Blöschl, 2003; Norbiato, et al., 2008; Parajka, et 20

al., 2010; Saulnier & Le Lay, 2009; Zoccatelli, et al., 2010). 21

An important feature frequently reported in hydrologic modeling studies is the catchment dampening 22

effect, which is often interpreted as the property of the catchment to filter out specific space-time 23

characteristics of the precipitation input in the flood response (Skøien & Blöschl, 2006; van De Giesen, et 24

al., 2000; van Loon & Keesman, 2000). Therefore, only a portion of rainfall space-time characteristics 25

2

(e.g. spatial concentration in specific catchment regions and storm motion) will emerge to control the 26

hydrograph shape (Skøien, et al., 2003). As shown in a number of works (Smith, et al., 2002; 2005; 27

Woods & Sivapalan, 1999), the river network geometry plays an essential role in the structure of the 28

catchment smoothing properties. Therefore a flow distance coordinate, i.e. the coordinate defined by the 29

distance along the runoff flow path to the basin outlet, has been introduced with the aim of providing 30

information on rainfall spatial organization relative to the basin network structure as represented by the 31

routing time (Borga, et al., 2007; Smith, et al., 2005; Zhang, et al., 2001). Similar works (Smith, et al., 32

2002; 2005) have developed dimensionless normalized indicators based on the flow distance coordinate 33

in terms of either routing time or length of flow path (e.g. normalized flow distance, normalized 34

dispersion) and used them for quantifying the geomorphologic driven spatial rainfall patterns. Zanon et al. 35

(2010) showed that the normalized time distances and normalized time dispersion of a severe storm event 36

at four small size cascade basins (less than 150km2) with complex terrain structure had a value very close 37

to one while similar results were also exhibited in Sangati et al. (2009). Sangati et al. (2009) additionally 38

showed that the normalized time distance and dispersion have a considerable increase for basin sizes 39

exceeding 500 km2. Beside the normalized time distance and dispersion, the concept of rainfall movement 40

has been developed to quantify theoretically and experimentally the combined effect between the rainfall 41

space-time properties and basin geomorphology. Viglione et al. (2010b) applied the analytical framework 42

developed in Viglione et al. (2010a) and concluded that due to the high spatial heterogeneity properties of 43

short-rain event, the roles of movement component of flood inducing complex terrain storms can be 44

important in runoff generation; and that for the more spatially uniform long-lasting rain events, the 45

movement component is weak in influencing the hydrograph. Seo et al. (2012) on the other hand has 46

shown that storm motion may have significant impact even for long-lasting precipitation events. 47

In a recent work, Zoccatelli et al. (2011) derived a term named ―catchment scale storm velocity‖, 48

which accounts for the combined effects of total storm motion and temporal storm variability over 49

catchment. The scale dependency of this concept was further examined by Nikolopoulos et al. (2013) who 50

3

showed strong nonlinearity and low value based on a single flash flood event. These studies on 51

―catchment scale storm velocity‖ were focused on a limited number of short-living flood events occurring 52

in small catchments. For this type of floods and basin sizes, differences between the assumptions in the 53

method and the event characteristics are deemed to be small; therefore, the method may be applied to 54

represent the effect of spatial rainfall aggregation on flood modeling. In this work, the method is applied 55

on a large number of moderate storm intensity, long-lasting flood events occurring over large (greater 56

than 1000km2) and mild-slope (max of slope less than 32° with mean at around 3°) catchments from the 57

Tar River Basin in North Carolina. Our aim is to test the representation of ―spatial moments of catchment 58

rainfall‖ and ―catchment scale storm velocity‖ reported in Zoccatelli et al. (2011) as potential metrics for 59

quantifying the effect of rainfall organization and storm motion on the hydrologic response and to analyze 60

the impact of some of its assumptions (e.g. uniform runoff routing velocity). In section 2 we present the 61

study area and data used in our analysis. Section 3 summarizes the analytical framework of Zoccatelli et 62

al. (2011), while section 4 presents the numerical experiments used to determine the two methods, and 63

evaluates the correlations of those statistics to the hydrologic modeling error metrics. Conclusions and 64

recommendations for further research are provided in section 5. 65

66

2 Study Area and Data 67

2.1 Tar River Basin 68

The target area of this study is the Tar River basin in North Carolina, USA. The Tar River originates 69

in Person County as a freshwater spring and flows 225km southeast to Washington, NC. The main stem 70

of the upper river flows through Louisburg, Rocky Mount, Tarboro and Greenville and provides drinking 71

water for these communities. Its major tributaries are Swift, Fishing and Tranters creeks and Cokey 72

Swamp. The area within this basin is relatively undeveloped. Agriculture accounts for 33.6% of the land 73

use while the rest includes mainly forests (29.6%), open water (19.7%), and wetlands (11.4%). Urban 74

lands and scrub growth accounts for 5.2% of the land usage. Fig.1 shows the location and elevation of the 75

4

Tar River basin. The basin is characterized by mild slopes (Table 1) and elevations ranging between 4m 76

a.s.l. (above sea level) close to the outlet on the southeast to about 225m a.s.l. near the headwaters on the 77

northwest. The study area was divided into four cascade basins namely B1, B2, B3, and B4 (shown in 78

Fig.1). Drainage areas for the four basins are approximately 1106, 2012, 2396, and 5654 km2, 79

respectively. Table 1 summarizes some basic statistics on local slope and flow length distribution, derived 80

from available elevation dataset (10 feets) for each basin. Note that slope values reported corresponds to 81

the steepest downhill descent for each point and flow length is equal to the distance, along the runoff flow 82

path, from a given point to basin outlet. 83

2.2 Rainfall Data 84

Rainfall information was based on the US National Weather Service (NWS) Multisensor Precipitation 85

Estimation (MPE) rainfall product available at 4 km/hourly spatiotemporal resolution. MPE data are 86

derived from a blend of automated and interactive procedures that combine information from satellite, the 87

WSR-88D radar network and rain gauges (Breidenbach & Bradberry, 2001; Fulton, 2002). Hourly MPE 88

data are available for the Southeast River Forecast Center (SERFC) region at HRAP (Hydrologic Rainfall 89

Analysis Project) resolution since January 2002. The nominal size of an HRAP grid cell in the area is 90

about 4 km (Greene & Hudlow, 1982; Reed & Maidment, 1999). A total of eight years of data (2002-91

2009) were analyzed for this work. Among the eight years, 2003 was the wettest, which had 1260 mm of 92

annual basin-average rainfall, while the driest year of the study period was in 2007 with 831 mm of 93

annual basin-average rainfall. Fig.2 shows the eight-year average annual rainfall within the study area. A 94

point to note is that the mean annual rainfall increases by 20% going from west (the headwaters of the 95

basin) to east (basin outlet). 96

2.3 Flood Events 97

The streamflow dataset used in this study consists of hourly streamguage measurements from US 98

Geological Survey (USGS) and span the same 8 years period as the precipitation data. Similarly to 99

precipitation, 2003 had the maximum annual flow for the overall basin as well as the three sub-basins. 100

5

The highest peak flow for B1 was 225m3/s and was observed during 2002. For the other three basins, the 101

highest peak flows occurred during 2006 and were 292m3/s, 402m

3/s and 694m

3/s. A procedure was 102

applied on the runoff time series to extract the flood events used in this study. One of the steps in this 103

procedure involved separation of baseflow from discharge time series that was performed following the 104

―Smooth Minima Technique‖ (Gustard, et al., 1992). Specifically, the technique first finds the minima of 105

every 144-hour non-overlapping periods by searching from the beginning to the end of the entire period 106

of record. Next, it searches the time series of the minima for values that are less than 11% of its two outer 107

values; such central values are then defined as turning points. Baseflow hydrograph is then constructed by 108

connecting all of the turning points with straight lines. Flood events were then selected from the time 109

series by identifying the peak flows between any two turning points. For a flood event to be considered 110

for further analysis, the ratio of the direct runoff to baseflow at the time of peak flow should be greater 111

than 4 (this value was empirically set to exclude peaks from moderate to low flood events). As a final 112

step, events were visually inspected to separate multimodal hydrographs into respective single peak 113

hydrographs. Fig.3 illustrates three of the selected flood events for B4 basin after applying the event 114

separation method on the streamflow data. 115

Table 2 summarizes the minimum and maximum range values as well as the mean values of the 116

event-based statistics of the selected flood events. Mean rainfall volume increases with basin scale (from 117

42 mm for B1 to 61 mm for B4), which points to the fact that as the basin scale increases, larger portion 118

of the relatively wet part of the east basin is accounted for. Both the mean cumulative direct flow and 119

peak flow rate increase with basin scale as expected (225 mm and 88 m3/s for B1 to 390 mm and 213 m

3/s 120

for B4). The event runoff coefficients (RC), defined as the ratio between direct flow and total 121

precipitation (Blume, et al., 2007; Merz, et al., 2006; Norbiato, et al., 2009), are also reported. The mean 122

RC is shown to decrease with basin area suggesting that runoff generation in west part of the basin is 123

higher than the east part. 124

125

6

3 Catchment Scale Rainfall Organization 126

3.1 Spatial Moments of Catchment Rainfall 127

We apply the concept of Spatial Moments of Catchment Rainfall reported in Zoccatelli et al. (2011) 128

to examine the space-time precipitation organization. This analytical framework was built upon the work 129

of Woods and Sivapalan (1999), Viglione et al. (2010a) and Smith et al. (2002; 2005). The spatial rainfall 130

moments provide a description for spatial integration of rainfall field p(x,y,t) (L·T-1

) within a basin at a 131

certain time t as a function of the flow length d(x,y) (L) defined as the distance along the flow path, from 132

position x,y to the outlet of the basin. The n-th spatial moment of catchment rainfall pn (Ln+1

·T-1

) is 133

defined by Zoccatelli et al. (2011) as: 134

(1)

where A denotes the catchment area. Note that the zero-th order spatial moment of catchment rainfall p0(t) 135

is the catchment-averaged rainfall rate at time t. Analogously, the gn, n-th moments of flow-length yields 136

a similar form as: 137

(2)

Thus, the catchment-averaged flow distance is the first order moment of flow-length (g1). 138

By taking the ratio between moments of catchment rainfall and flow-length, one can obtain a 139

dimensionless form for the spatial moments of catchment rainfall as the formulations below: 140

(3)

(4)

The first order moment δ1(t) reflects the location of catchment rainfall centroid (i.e., the catchment center 141

of mass) over the basin. Values of δ1(t) around one indicate rainfall is distributed uniformly over the 142

catchment or mostly concentrated over the catchment centroid. Values of δ1(t) less (greater) than one 143

7

indicate that rainfall is distributed close to the outlet (the periphery) of the basin. The second order 144

moment δ2(t) describes the dispersion of the rainfall field relative to its mean position with respect to the 145

spreading of the flow-length. Values of δ2(t) close to one reflect that rainfall distributes uniformly over 146

the catchment. Values of δ2(t) less (greater) than one indicate that rainfall is characterized by a unimodal 147

(multimodal) distribution along the flow-length. 148

The above equations can also be extended to reflect rainfall organization corresponding to the 149

cumulative rainfall over the time period Ts (e.g. storm period). These statistics are termed Pn and Δn 150

(Zoccatelli, et al., 2011) and are defined as follow: 151

(5)

(6)

(7)

Therefore, P0 stands for the average rainfall intensity over the given time period and product between P0 152

and the time period Ts is the cumulative rainfall volume for that period. 153

3.2 Catchment Scale Storm Velocity 154

Since the first order moment of rainfall reflects the location of rainfall mass center, it is then 155

interesting to note that its first order temporal derivative of evolution over time reveals the effect of storm 156

motion. For a simple case of a storm element characterized by constant rainfall intensity moving along the 157

catchment, the definition of the catchment scale storm velocity V(L·T-1

) is as follows: 158

(8)

Storm velocity defined in this way reflects the kinematic movement of rainfall barycenter with respect to 159

basin geomorphology (i.e. movement of rainfall mass center across the flow-length coordinate). 160

8

For a more realistic case of storm motion with changes in mean areal precipitation, Zoccatelli et al. 161

(2011) provided the following definition: 162

(9)

(10)

where w(t) is termed as the rainfall weight. Eq.(9) contains two slope terms; the first slope term is 163

estimated based on the time regression between rainfall-weighted first scaled moments and time, while 164

the second one is based on the regression between rainfall weights and time. The difference between these 165

two terms describes the storm velocity Vs, which reflects the combined effect of storm kinematic motion 166

and dynamics of the catchment mean areal rainfall rate. For temporally variable but spatially uniform 167

rainfall (δ1 = Δ1 = 1) the two slope terms will be equal in value and opposite in sign, which implies a null 168

value for the catchment scale storm velocity. In this case, the storm is effectively stationary. On the other 169

hand, if rainfall is variable in space, but uniform in time (w = 1), Vs takes the same value as V. Note that 170

the sign of the velocity is positive (negative) for the case of upstream (downstream) storm motion. 171

Solving for Eq.(9) requires a numerical evaluation of the linear regression over a specified space and 172

time window. The spatial window is defined by the catchment size, whereas the temporal window is 173

chosen in this study to have two distinct durations: a) a time length equal to the entire storm event 174

duration, and b) a shorter-length moving time window (arbitrarily set to be 6 hours). Vs obtained from 175

time window (a) reflects the overall velocity of the entire event representing the concept of ―catchment 176

scale storm velocity‖, while Vs(t) computed based on time window (b) represents a time-dependent storm 177

velocity. 178

3.3 Events Analysis 179

Rainfall variability is evaluated here by means of the ―Spatial Moments of Catchment Rainfall‖, 180

described in the previous section. A long duration flood event associated with more than 35 hours of 181

rainfall (January 13th to 14

nd, 2005) is selected to illustrate the conceptualization of rainfall spatial 182

9

moments (the corresponding flood event is shown in Fig.9). Rainfall statistics and spatial moments are 183

presented for the first 35 hours of the rain event, which is the period contributing the majority of the rain 184

accumulation. This event induced flooding in all four sub-basins of the Tar River basin enabling the 185

illustration of the scale dependency of the statistics. The time series of the first and second scaled 186

moments (δ1 and δ2) of the catchment rainfall are reported in Fig.4, along with the basin-averaged rainfall 187

rate (p0). 188

The temporal patterns of the basin-averaged rainfall rate (Fig.4 Panel a) of these cases are quite 189

similar since they are from the same event. All of them begin with very low rain rate followed by a 190

smooth peak after 22:00 of January 13th. Subsequently, a sharp peak with the highest rainfall rate 191

(11.7mm/h for B1, 7.6mm/h for B2, 7.2mm/h for B3 and 7.1mm/h for B4) occurred between 11:00 and 192

17:00 on January 14th. A pause during 17:00 and 22:00 on January 13

th is also seen. The time series of 193

both first and second scaled spatial moment (Fig.4 Panel b and c) exhibit a relatively large variability with 194

δ1 varying between 0.19 and 1.91 in B4 basin. Based on the trend of δ1 it can be seen that the storm 195

centroid was located towards the headwater of the basin (except for B1 sub-basin) within most of the time 196

(from 22:00 of 13th to 14:00 of 14

th). The overall movement of storm center is from the periphery towards 197

the outlet at the phase of the 2nd

rainfall peak (from January 14th 12:00 to 17:00, the last part of the second 198

segment in Fig.4 Panel b and c). This down basin movement is captured by a sequence of snapshots of 199

hourly rainfall maps rendered in Fig.5 between 12:00 and 17:00 of January 14th. The highest value of δ2 200

for B1 in that period is 2.63. The value of δ2 appears to be greater than 1 for several hours in B1, while 201

those in the other basins rarely exceeded 1 (except for δ2 in the 2nd

hour) during most of the hours (i.e. the 202

storm only has one core during almost the entire event in these basins), specifically at the late period of 203

the storm shown in Fig.5. The temporal evolution of δ1 and δ2 for this storm are similar to each other 204

especially for the B2 and B3 basins due to the high shape and geomorphologic (see Table 1) similarity of 205

the two basins. More importantly, values of δ2 generally reflect the trend of δ1, as expected, with small 206

10

values of dispersion when δ1 is either larger or smaller than one and values of dispersion close to one 207

when δ1 is also close to unity. 208

The main direction of the storm centroid during the main part of this event (i.e. after 11:00 UTC) is 209

downstream, which is interpreted by the negative Vs(t) values reported in Fig.6. Results from this figure 210

show that for all basins, magnitude of catchment scale storm velocity for the main event was relatively 211

low (within 0.5 m/s). In addition, although Vs has been accounted for the combined-effect of storm 212

movement and basin-average rainfall rate of change, it still reveals the time evolution of δ1. For example, 213

the time that Vs(t) reaches the maximum value of 1.51m/s for B2, 2.37m/s for B3 and 2.76m/s for B4 at 214

15:00 of January 13th is overlapping with the period that δ1(t) increases rapidly (i.e. between 13:00 and 215

16:00 of January 13th). The overall dependency of Vs(t) to the change rate in δ1(t) is not clear because the 216

regression is performed over δ1(t) weighted by w(t). Hence, during periods with a minor rate of change of 217

w(t), Vs(t) can explain the change of δ1(t) and Eq.(9) becomes 218

(11)

This equation contains only one regression performed over t and δ1(t). 219

It was stated in Eq.(9) that the Vs term can partly reflect the dynamics of rainfall rate. To verify this 220

aspect, Fig.7 is produced by juxtaposing the time series of basin-averaged rainfall rate and the catchment 221

scale storm velocity. Upslope deceleration characterized as positive decreasing values of Vs(t) can be 222

revealed between 22:00 of 13th and 4:00 of 14

th for all basins (start 2 hours earlier for B1). Coincidentally, 223

these time ranges are also the time periods for which rainfall grows from 0 to the first peak rain rate. 224

Conversely, upstream acceleration of the rainfall mass center appears between 7:00 and 9:00 of the 14th 225

with a decreasing of rainfall rate. A clearer example period occurs around the second rainfall peak. These 226

two example periods lead to the observation that the extreme values of Vs(t) are likely to occur at the time 227

that rainfall rate reaches its maxima (peak value) or minima (gap between two rainfall peaks). This 228

observation highlights the dependence between rainfall intensity to the storm velocity, which was also 229

demonstrated in previous studies (Tarolli, et al., 2013). Possible cause of this observational result is that 230

11

the second linear regression term offsets the first term due to the high values that it can take when rainfall 231

rate is changing rapidly. This further affects the magnitude of Vs(t). Consequently, it seems that Vs(t) can 232

represent the change of δ1(t) when the change of basin areal rainfall rain is null or negligibly small. On 233

the other hand Vs(t) would reflect the temporal variability of p0(t) rather than the barycenter movement 234

when p0(t) is significantly fluctuated over time. 235

From Fig.6, it is shown that the range of Vs(t) is increasing with scale, which implies that the 236

variability of Vs(t) could be scale dependent. For better understanding of this aspect we present the mean 237

and variability of the hourly Vs determined based on all storm events in our database, and render box plots 238

(Fig.8) to demonstrate the change in distributions of mean and STD of Vs(t) with respect to basins size. 239

The medians (in terms of absolute values) and length of value ranges of the mean of Vs(t) are increasing 240

with scale in panel a) of Fig.8 which is consistent with other works on this topic (Nikolopoulos, et al., 241

2013; Tarolli, et al., 2013). This implies that mean of event Vs(t) from larger basins are in general larger 242

and have higher degree of variability. Fig.8 panel b) displays another aspect of scale dependency by 243

means of standard deviation of event Vs(t). A positive shift (denoted by the shift of medians) from the 244

smallest to the largest basin can be revealed on the plot in panel b) of Fig.8, which corresponds to our 245

observation regarding the value range of Vs(t) in Fig.6. Hence, values of the event-based hourly Vs from 246

larger basin events are more extreme. In addition, it is interesting to note that the value range for standard 247

deviation of Vs(t) is expanding from the smallest to the largest basin pointing to the fact that the inter-248

event variability of Vs(t) is also scale dependent. 249

250

4 Framework Examination with hydrologic modeling 251

4.1 Hydrologic Model 252

In this section we carried out a series of hydrologic simulations, for all events examined, using as 253

input a) lumped and b) distributed rainfall, in order to quantify the effect of neglecting the rainfall spatial 254

variability on the hydrologic model application. The hourly discharge rate for the selected events from the 255

12

four study basins is simulated with the Hydrology Laboratory Research Modeling System (HL-RMS) 256

developed by National Weather Service (Koren, et al., 2004). HL-RMS is a flexible hydrologic modeling 257

system consisting of a well-tested conceptual water balance model currently adopted as the Sacramento 258

Soil Moisture Accounting Model (SAC-SMA) applied on a regular spatial grid and kinematic hillslope 259

and channel routing models. 260

The SAC-SMA is a well-developed and widely used conceptual rainfall-runoff watershed model 261

(refer to Koren et al. (2004) for model structure and other details) with 17 conceptual parameters (16 of 262

which cannot be measured directly). Koren et al. (2000) derived 11 major SAC-SMA parameters based 263

on a set of equations and the State Soil Geographic soil data (STATSGO). The flow routing component in 264

HL-RMS is divided into hillslope routing (overland flow) and channel routing. Values for the parameters 265

controlling overland flow were derived from high-resolution digital elevation model (DEM), land use 266

data, and results reported from initial tests on the model (NWS, et al., 2008). The channel routing 267

parameters were determined based on the measurements of Q and A data at the outlets of B1, B2, B3 and 268

B4 by following the rating curve method used by Koren et al. (2004). 269

Initial estimates of the remaining six model parameters were established based on previous 270

experience on different basins (Pokhrel, et al., 2008; Yilmaz, et al., 2008), and subsequently calibrated 271

using the Differential Evolution Adaptive Metropolis, DREAM (Vrugt, et al., 2009). DREAM employs 272

Markov Chain Monte Carlo sampling, which uses a formal likelihood function to estimate the posterior 273

probability density function of parameters in complex, high-dimensional sampling problems and separates 274

behavioral from non-behavioral solutions using a cutoff threshold that is based on the sampled probability 275

mass (Vrugt, et al., 2008). Hourly streamflow observations from 2004 to 2006 were used as the reference 276

in the calibration process to compute the sum of squared residuals (between observed and simulated 277

streamflow) that was chosen as the optimization criterion. The model validation was based on two 278

separate periods: 2002-2003 and 2007-2009. Fig.9 displays the simulation hydrographs for the selected 279

event discussed in previous section. Apart from the underestimation of peak flow, consistently for all 280

13

basins, the model captures well the time to peak and the general shape of observed hydrographs. 281

Quantitative statistics of the model performance are discussed next. 282

4.2 Evaluation of HL-RMS performance 283

In this section, we evaluate the performance of the calibrated HL-RMS model forced with the 284

distributed MPE radar-rainfall. Four comparative metrics based on simulated and observed runoff data are 285

calculated on event basis and include: the absolute relative error of event runoff volume (εV), absolute 286

relative error of event peak flow rate (εp), normalized root mean square error (εNRMS) and correlation 287

coefficient (ρ), which are determined as following: 288

(12)

(13)

(14)

(15)

where Vo (Qpo) and Vs (Qps) are the observed and simulated event runoff volume (event peak flow rate) 289

and Qo , Qs are the observed and simulated hourly runoff rate for every event; M is the total number of 290

hours for an event. 291

Table 3 lists the mean value and standard deviations of values of the metrics defined above. The mean 292

value of εV and εp for all the basins is around 30%; more than 85% of the events exhibit εV and εp values 293

less than 60%. These error metrics indicate reasonable performance results from the HL-RMS model 294

simulations. The flow volume simulations provided better error statistics than the peak flow rate 295

simulations. Results reported on Table 3 do not reveal any dependency of the error statistics on basin 296

scale, except for the εNRMS metric, which is shown to be inversely proportional to basin scale. Correlation 297

14

between simulated and observed streamflow is high for the majority of the events, with more than 80% of 298

B2 and B3 (and ~80% for B1) cases associated with correlation values above 0.8, while for the entire 299

basin (B4 basin) only 50% of the events gave correlations above 0.8. 300

4.3 Role of the Analytical Framework in Hydrologic Modeling 301

Aiming at understanding the interaction between the catchment morphological properties and rainfall 302

organization, Zoccatelli et al. (2011) (referred as Z2011 hereafter) introduce the concept of spatial 303

moments of catchment rainfall based on the framework proposed in Viglione et al. (2010a). Z2011 304

assumed than rainfall is not stationary and 2) calculated the hillslopes and channel routing time in terms 305

of a joint runoff travel time by considering a spatial constant runoff routing velocity (v). The Z2011 306

method is linked with two quantities of the hydrograph: a) the mean runoff time (i.e., the time of the 307

center of mass of the runoff hydrograph at a catchment outlet), and b) the variance of the timing of runoff 308

(i.e., the temporal dispersion of the runoff hydrograph). The expected value of catchment runoff time is a 309

surrogate for the time to peak; the variance of runoff time is indicative of the magnitude of the peakedness 310

(i.e. shape of the hydrograph). For a given event duration and volume of runoff, a sharply peaked 311

hydrograph will have a relatively low variance compared to a more gradually varying hydrograph 312

(Woods, 1997). 313

According to Z2011, the catchment flood response is conceptualized into two subsequent steps: 1) 314

rain falls on the catchment and is converted to an amount of runoff by the surface; 2) runoff transport 315

from that point to the catchment outlet. Hence, catchment runoff time Tq is treated as the summation of 316

the holding time for the two stages: 317

(16)

where Tr and Tc are the holding time for the first and the second stages respectively. Term Tr is assumed 318

to be equal to the instantaneous time t which is uniformly distributed as in Viglione et al. (2010a) 319

(V2010a hereafter). Therefore, the expectation and variance of catchment runoff time Tq are expressed as: 320

15

(17)

(18)

For E(Tr) and var(Tr), we refer to V2010a. E(Tc), var(Tc) and cov(Tr,Tc) are given in Z2011 as: 321

(19)

(20)

(21)

where v is the runoff routing velocity in m/s. Based on the components of E(Tq) and var(Tq) we note that 322

timing of hydrograph is not controlled by storm movement while both the rainfall spatial dispersion and 323

storm motion contributes to the hydrograph shape and to further extend the peak runoff. 324

Two sensitivity tests are conducted in the next two sub-sections to verify the controls on hydrograph 325

timing and magnitude by neglecting the rainfall spatial variability in modeling the hydrologic response of 326

basins. Two event-based difference statistics are derived for relating to Δ1, Δ2, and Vs: a) the difference in 327

timing of the mean values (dE) and b) the difference in dispersion around the mean values (dvar) between 328

distributed and basin-averaged rainfall forced hydrologic modeling. Before describing the statistics two 329

concepts are introduced first. The first concept is the hydrograph centroid (V2010a). The term ―centroid‖ 330

here refers to the expected value of time with respect to the hourly flow rate, i.e. statistically, the first 331

order origin moment of variable t weighted by hourly flow rate: 332

(22)

where Tf is the period of runoff event. Another concept is the dispersion of hydrograph around the 333

centroid; it is determined as the variance of time related to the flow rate. Thus, it is the second order 334

center moment of variable t weighted by hourly flow rate: 335

16

(23)

4.4 Control on Timing of Hydrograph 336

As previously mentioned, Δ1 reflects the spatial organization of catchment rainfall mass center with 337

respect to the flow length coordinate. For the cases with Δ1 larger than one (i.e. rainfall concentrated 338

towards basin’s periphery), a detention of the hydrograph centroid relative to the lumped (spatially 339

uniform) rainfall is expected. For events with Δ1 smaller than one (i.e. rainfall concentrated over basin’s 340

outlet), an advance in hydrograph timing is anticipated compared to hydrographs from the lumped rain. 341

Hence, the difference in timing of the mean value, or timing error, which is the time difference between 342

the centroid of the two types of hydrographs attained from the distributed and lumped rainfall forcing is 343

applied here to show its relation to Δ1: 344

(24)

Where Tq,d and Tq,l correspond to runoff time according to distributed and lumped rainfall forcing, 345

respectively. Applying the spatial moments of catchment rainfall and the associated assumptions we 346

derived dE as function of Δ1 (see Appendix for derivations): 347

(25)

A positive/negative value of dE implies a positive/negative shift in time of the distributed hydrograph with 348

respect to the one produced by using uniform precipitation. Namely, a positive correlation between Δ1 and 349

dE is anticipated. Additionally, the value of the slope and intercept are expected to be the same in number, 350

but different in sign. 351

The relationship between Δ1 and dE derived from simulated hydrographs according to Eq.(25) is 352

reported in Fig.10, depicting a linear trend especially for B2 and B3 basins. This suggests that Δ1 is able 353

to describe the control of rainfall spatial organization on the timing error of flood response for the basins 354

examined. Note from Fig.10 that for all basins, a value of Δ1 around 1 corresponds to a turning point 355

17

below (above) which the timing error becomes negative (positive) as indicated by the analytical 356

relationship of Eq.(25). Furthermore, a line was fitted (based on least-squares) to demonstrate the degree 357

of correlation between dE and Δ1 and investigate also the agreement between fitted coefficients (slope and 358

intercept) as indicated by Eq.(25). Results are summarized in Table 4. It is noted that the absolute values 359

of slope and intercept parameters are increasing with basin scale, which implies an increase in mean flow-360

length g1 values with basin areas referred as the Hack’s Law (Rigon, et al., 1996), considering that runoff 361

routing velocity v is similar in magnitude. The r2 values are greater than 0.8 for B2 and B3 indicating a 362

clear linear trend while drops to ~0.6-0.65 for B1 and B4. From a statistical perspective, presence of few 363

outliers (shown in Fig.10) may be the possible cause for the significant difference in correlation between 364

B2, B3 and B1, B4. Examination of the regression coefficients (Table 4) showed that slope and intercept 365

have almost equal and opposite in sign values, which further corroborates the validity of Eq.(25). 366

According to these results we may conclude that overall the difference in timing between distributed and 367

lumped rainfall hydrographs is well described by Eq.(25). 368

4.5 Control on the Shape of Hydrograph 369

Analogously, we tested the sensitivity of statistic Δ2 of every event to the difference in hydrograph 370

dispersion from distributed and basin-average rainfall forcing. It is noted that with the rainfall excess 371

volume remaining unchanged the effect of decreasing the variance of runoff time is to increase the flood 372

peak (Gaál, et al., 2012). However, since Eq.(18) involves the covariance between Tr and Tc, the 373

catchment storm velocity should have an effect in shaping the hydrograph. Consequently, we may write 374

dvar as: 375

(26)

Apply the spatial moments of catchment rainfall as well as the catchment scale storm velocity to develop 376

dvar (see Appendix): 377

(27)

18

Eq.(27) indicates that the event-based dvar is manipulated by both Δ2 and the product of Vs and square of 378

event duration |Ts|2. A positive (negative) value of dvar would imply a decrease (increase) of temporal 379

dispersion of the hydrograph generated by lumped rainfall with respect to the one produced by the 380

spatially distributed precipitation. 381

Events characterized by a near stationary storm (Vs very close to 0), dvar is the marginal distribution 382

of Δ2 based on Eq.(27): 383

(28)

Observations from a number of case studies have shown that flash floods are frequently characterized by 384

near stationary storm movement (Smith, et al., 2000; Sturdevant-Rees, et al., 2001). From this sense, a 385

relatively strong linearity between dvar and Δ2 may be seen for the flash flood generation rainfall events. 386

For events with extremely high rainfall spatial concentration (Δ2 fairly close to zero) for every time step 387

(rarely occur), dvar is the marginal distribution of Vs times |Ts|2: 388

(29)

In this case, the degree of dispersion is a function of Vs|Ts|2. Furthermore, if Vs is evaluated based on a 389

fixed length time window, dvar is proportional to Vs (|Ts| is constant). 390

Table 4 lists the regression coefficients between dvar, Δ2 and VsTs2. It is seen that the regression 391

intercept and the first slope coefficient are close to each other (with the exception of the B1 basin) while 392

the second slope coefficient is close to zero. Since Eq.(27) is unable to capture the tendency of the bulk 393

Δ2, Vs|Ts|2 and dvar in an efficient way (refer to the r

2 values in Table 4), we present a box plot (Fig.11) of 394

residuals between the model-calibrated dvar with respect to dvar attained from its regression function. As it 395

demonstrated in Fig.11, the residuals are distributed within a considerable range of values ([-448 402] for 396

B1, [-498 606] for B2, [-646 395] for B3 and [-1563 2079] for B4) with the medians having a non-397

negligible displacement from zero (especially for the B4 case). This implies a considerable large degree 398

of disagreement between the model-calculated dvar and the dvar derived from analytical framework. 399

19

Possible reason lies in the differences on assumption of routing kinematics. The Z2011 framework 400

lumped the hillslope routing and channel routing as runoff routing with constant routing velocity 401

spatiotemporally while deriving the relative parameters. These assumptions are acceptable for small scale 402

basins and flood events with bank-full condition (e.g. flash floods) (Rodríguez-Iturbe & Valdés, 1979; 403

Saco & Kumar, 2002). On the other hand, the HL-RMS differentiates hillslope from channel routing and 404

assigns spatiotemporal variable as the channel routing velocity. Namely, the analytical framework in 405

Z2011 and the hydrologic model in this study share different assumptions on the routing kinematics. 406

From the model, the standard deviations of channel routing velocity range between 0.03 and 0.06 m/s, 407

which indicates that the assumption of constant routing velocity does not hold for the hydrologic model. 408

Consequently, differences in routing schemes are the main cause of low efficiency in the regression of 409

Eq.(27). Moreover, due to the square operator of v in Eq.(27) the disagreement between the framework 410

and hydrologic model may be amplified. Nevertheless, as shown above the timing error of the hydrograph 411

is not as susceptible as the shape error to the manners of routing kinematics. 412

413

5 Conclusions 414

In this study we applied the concept of ―spatial moments of catchment rainfall‖ (i.e. first and second 415

order rainfall moment and catchment scale storm velocity presented in Z2011) to investigate the role of 416

rainfall spatial organization on flood response of mild-slope basins in the south Atlantic region of US. We 417

have concluded that values of the time-integrate second scale moment generally reflect the trend of the 418

time-integrate first scale moment with values of δ2 close to zero when δ1 is far from one, and values of δ2 419

around one when δ1 is close to unity. Neither of the Vs values determined based on the entire storm 420

duration or computed with a fixed-length (6 hour time) moving time-window were significantly different 421

than zero. With considerably large space-time rainfall variability, Vs(t) was found to be rainfall intensity 422

dependent, and its mean and variability was basin-scale dependent. 423

20

Another essential aspect of this study was to test the framework sensitivity with a distributed 424

hydrologic model. We used two error metrics determined based on model simulations forced with 425

distributed and basin-averaged rainfall to evaluate the framework statistics in representing the errors in 426

flood response. Results from both empirical equations relating modeling error metrics to the framework 427

statistics showed reasonable fitting. It was concluded that catchment response is relatively sensitive to the 428

spatial heterogeneity of rainfall quantified on the basis of Δ1, Δ2, and Vs especially for the dependency 429

between timing difference dE and Δ1. Specifically, strong linear dependency was exhibited between dE and 430

Δ1 even though they were carried out under routing kinematics with different properties. However, the 431

correlation was weak in terms of difference in peakedness of the hydrograph, due to again the differing 432

assumptions on routing between the conceptual framework and the distributed hydrologic model, i.e. 433

separated hillslope and channel vs. lumped routing processes as well as constant vs. spatiotemporal 434

variability v for the framework and model, respectively. It is concluded that different processes in routing 435

kinematics disrupt the shape of hydrograph more profoundly than the timing of it. 436

Overall, this study has demonstrated that the application of the concept of spatial moments of 437

catchment rainfall in the case of mild-slope terrain basins and moderate flood regimes can be used to 438

understand the effect of rainfall space-time organization on the timing of hydrologic response, but does 439

not provide a strong dependence on the variance of runoff time series. Further research should consider 440

performing hydrologic modeling on smaller scale basins with more uniform channel routing velocities 441

and where the hillslope routing process is not significant, and therefore, it would be reasonable to verify 442

the role of Δ2 and Vs on the dispersion of hydrograph. In addition, future studies could focus on complex-443

terrain flash-flood inducing storm events that are characterized with stationary storm velocity to evaluate 444

the rigorousness of the marginal relationship of dvar to Δ2. 445

446

Acknowledgment: This work was supported by a NASA Precipitation Measurement Mission award 447

(NNX07AE31G). 448

21

Appendix 449

Difference between centroids of hydrographs attained from distributed and lumped rain forcing in this 450

paper is defined as: 451

Since Tr is a temporal variable which is assumed uniformly distributed over Ts, E(Tr) from both models 452

are the same (refer to V2010a). Also note that Δ1 is equal to one for lumped rain. Therefore: 453

Difference in hydrograph peakedness related to the centroid is determined as: 454

Similar to the expectation, variance of Tr for both rainfall forcing is the same (see V2010a for details). Δ2 455

is one and Vs, or cov(Tr,l,Tc,l), is zero for lumped rain. So: 456

(A.1)

By applying the assumption that the instantaneous time t is uniformly distributed over Ts (V2010a), 457

Eq.(A.1) becomes Eq.(27): 458

Since 459

460

22

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2007 event in Western Slovenia. J. Hydrol., 394(1-2), 182-197. 599

doi:10.1016/j.jhydrol.2010.08.020 600

Zhang, Y., Smith, J. A., & Baeck, M. L. (2001). The hydrology and hydrometeorology of extreme floods 601

in the Great Plains of Eastern Nebraska. Adv. Water Resour., 24(9-10), 1037-1049. 602

doi:10.1016/S0309-1708(01)00037-9 603

26

Zoccatelli, D., Borga, M., Viglione, A., Chirico, G. B., & Blöschl, G. (2011). Spatial moments of 604

catchment rainfall: rainfall spatial organisation, basin morphology, and flood response. Hydrol. 605

Earth Syst. Sci., 8(3), 5811-5847. doi:10.5194/hessd-8-5811-2011 606

Zoccatelli, D., Borga, M., Zannon, F., Antonescu, B., & Stancalie, G. (2010). Which rainfall spatial 607

information for flash flood response modelling? A numerical investigation based on data from the 608

Carpathian range, Romania. J. Hydrol., 394(1-2), 148-161. doi:10.1016/j.jhydrol.2010.07.019 609

610

611

27

List of Tables 612

Table 1. Summary of flow-length and slope statistics ................................................................................ 29 613

Table 2. Summary of flood event statistics for the four basins ................................................................... 30 614

Table 3. HL-RMS runoff simulation error statistics ................................................................................... 31 615

Table 4. Parameters for Eq.(25) and (27) .................................................................................................... 32 616

617

28

List of Figures 618

Fig.1. Location and elevation map of the Tar-River basin ......................................................................... 33 619

Fig.2. Spatial distribution of mean annual precipitation (computed over the period 2002 to 2009) .......... 34 620

Fig.3. Illustration of sample flood events derived from the time-series of runoff data (December 2004 to 621

January 2005) .............................................................................................................................................. 35 622

Fig.4. Illustration of rainfall analysis for January 13th to 14

th, 2005: a) basin-averaged rainfall rate; b) 1

st 623

order moment of rainfall and c) 2nd

order moment of rainfall..................................................................... 36 624

Fig.5. Hourly snapshots of rainfall spatial organization from 11:00 to 16:00 on January 14th 2005 (rain 625

intensity in mm/h) ....................................................................................................................................... 37 626

Fig.6. Illustration of the temporal evolution of catchment scale storm velocity Vs(t) for January 13th to 627

14th, 2005..................................................................................................................................................... 38 628

Fig.7. Illustration of basin-averaged rainfall and storm velocity time series for January 13th to 14

th, 200539 629

Fig.8. Boxplot of Vs(t) variance for the four basins and for all storms examined. ...................................... 40 630

Fig.9. Illustration of HL-RMS runoff simulation versus observed runoff for a sample flood event at the 631

four basins ................................................................................................................................................... 41 632

Fig.10. Time-integrated first order moment of rainfall versus timing error for the four basins ................. 42 633

Fig.11. Box plot of residuals between model-calibrated dvar to predicted dvar for the four basins .............. 43 634

635

29

Table 1. Summary of flow-length and slope statistics 636

Basin ID B1 B2 B3 B4

Basin Area (km2) 1106 2012 2396 5654

Flow-length

Statistics

(km)

mean 56 110 111 142

std 26 49 56 66

max 108 199 214 287

Slope

Statistics

(°)

mean 3.2 2.9 2.7 2.3

std 2.2 2.1 2.0 1.9

max 32 32 32 32

637

30

Table 2. Summary of flood event statistics for the four basins 638

Basin ID B1 B2 B3 B4

Num. of event 44 42 40 38

Rainfall

Volume

(mm)

range [12.8, 124.5] [13.4, 123.8] [16.5, 128.7] [18.8, 221.7]

mean 42.1 49.6 54.5 61.0

|Ts|

(h)

range [12, 124] [11, 219] [12, 219] [28, 377]

mean 41 71 81 132

Direct flow

(mm)

range [3.7, 65.9] [4.4, 66.6] [3.9, 58.1] [3.8, 87.0]

mean 16.2 14.7 17.1 18.8

Tq

(h)

range [81, 558] [123, 551] [140, 573] [202, 1030]

mean 225 270 280 390

Peak flow

rate (m3/s)

range [32.6, 225.4] [50.1, 291.7] [53.0, 402.1] [103.9, 693.8]

mean 88.3 102.0 123.7 213.1

Baseflow

(mm)

range [0.1, 14.3] [0.5, 10.5] [0.6, 12. 9] [1.4, 23.8]

mean 3.9 4.8 4.4 9.5

RC

(%)

range [10.0, 80.5] [9.3, 71.4] [7.8, 79.9] [9.0, 73.1]

mean 39.8 30.7 31.5 29.9

639

31

Table 3. HL-RMS runoff simulation error statistics 640

B1 B2 B3 B4

εp

(%)

mean 27.13 31.86 31.64 28.16

STD 6.59 22.61 20.43 15.89

εV

(%)

mean 25.96 28.56 27.55 28.01

STD 5.01 13.12 13.74 9.83

εRMS mean 0.67 0.62 0.56 0.48

STD 0.28 0.44 0.31 0.25

ρ mean 0.82 0.81 0.86 0.75

STD 0.09 0.1 0.05 0.06

641

32

Table 4. Parameters for Eq.(25) and (27) 642

Basin ID B1 B2 B3 B4

Straight

Line

Intercept -31 -57 -61 -99

Slope 31 57 61 97

r2 0.58 0.84 0.82 0.65

Straight

Surface

Intercept -164 -640 -316 -1971

Slope1 202 638 326 1865

Slope2 0.09 0.02 0.03 0.02

r2 0.16 0.16 0.19 0.16

643

33

644

645

Fig.1. Location and elevation map of the Tar-River basin 646

34

647

Fig.2. Spatial distribution of mean annual precipitation (computed over the period 2002 to 2009) 648

35

649

Fig.3. Illustration of sample flood events derived from the time-series of runoff data 650

(December 2004 to January 2005) 651

36

652

Fig.4. Illustration of rainfall analysis for January 13th to 14

th, 2005: a) basin-averaged rainfall rate; b) 1

st 653

order moment of rainfall and c) 2nd

order moment of rainfall 654

37

11:00 12:00

13:00 14:00

15:00 16:00

Fig.5. Hourly snapshots of rainfall spatial organization from 11:00 to 16:00 on January 14th 2005 (rain 655

intensity in mm/h) 656

38

657

Fig.6. Illustration of the temporal evolution of catchment scale storm velocity Vs(t) for January 13th to 658

14th, 2005 659

39

660

Fig.7. Illustration of basin-averaged rainfall and storm velocity time series for January 13th to 14

th, 2005 661

40

662 Fig.8. Boxplot of Vs(t) variance for the four basins and for all storms examined. 663

41

664

Fig.9. Illustration of HL-RMS runoff simulation versus observed runoff for a sample flood event at the 665

four basins 666

42

667

Fig.10. Time-integrated first order moment of rainfall versus timing error for the four basins 668

43

669

Fig.11. Box plot of residuals between model-calibrated dvar to predicted dvar for the four basins 670

44

Highlights 688

Spatio‐temporal rainfall variability effects on flood modeling 689

Rainfall moments and storm velocity conceptual frameworks 690

Experimental framework to assess the conceptual frameworks’ predictability 691

692