multivariate elasticity of extreme streamflow the united

Multivariate Elasticity of Extreme Streamflow

in the United States

A thesis submitted by

Irina Gumennik

in partial fulfillment of the requirements for the degree of

Master of Science in

Civil and Environmental Engineering

Concentration: Environmental and Water Resources Engineering

Tufts University

August 2015

Advisor: Professor Richard M. Vogel

ii

Abstract

Increasingly, hydrological research reveals that streamflow may be changing in response

to changes in precipitation and land development. The concept of elasticity is used to

investigate the generalized sensitivity of streamflow to changes in multiple basin

characteristics. Multivariate regional regression models are developed for median annual

high and low flows for 18 hydrologic units in the United States. The resulting coefficients

of the multivariate regression models are shown to provide elasticity estimates for basin

characteristics that capture several climatic, morphological, hydrological, and

development influences on watersheds.

Explanatory variables that consistently appear in the regression models reveal patterns

that demonstrate the importance of employing a multivariate sensitivity approach.

Regression estimates of precipitation elasticity based on single variable models are

consistently biased when compared to more accurate multivariate approaches. Overall,

precipitation and groundwater flow exhibit the strongest influence on streamflow with

elasticity values that are an order of magnitude larger than for other explanatory

variables. An increase in land development is associated with an increase in both high

and low flows, but at an order of magnitude smaller compared to precipitation and

groundwater.

iii

Acknowledgments

Professor Richard M. Vogel, Tufts University

Jeffrey McCollum, PhD, FM Global

Annalise Blum, PhD Candidate, Tufts University

iv

Table of Contents

1. Introduction .................................................................................................................... 1

2. Methods .......................................................................................................................... 3

Regression model for streamflow elasticity ................................................................... 3

Study Area ....................................................................................................................... 8

Data Sources ................................................................................................................... 8

Data Preparation ............................................................................................................. 9

Model Selection ............................................................................................................ 10

3. Results ........................................................................................................................... 11

Climate elasticity ........................................................................................................... 12

Land development elasticity ......................................................................................... 12

Hydrological elasticity ................................................................................................... 14

4. Discussion...................................................................................................................... 15

5. Conclusion ..................................................................................................................... 16

Figures ............................................................................................................................... 17

Tables ................................................................................................................................ 20

References ........................................................................................................................ 25

List of Figures

Figure 1. U.S. Geological Survey HUC Regions

Figure 2. Observed Streamflow by HUC Region

Figure 3. Multivariate Elasticities

Figure 4. Multivariate vs Singular Elasticity of Precipitation

List of Tables

Table 1. USGS Regions and Number of Gages in Regional Regression Model

Table 2. GAGESII Basin Characteristics and Elasticity Results

Table 3. Regional Models for Median Annual High Streamflow

Table 4. Regional Models for Median Annual Low Streamflow

Table 5. Regression Diagnostics

Appendices

Appendix A. Physical Interpretation of Regression Model for Streamflow Elasticity

Appendix B. Boxplots of GAGESII Basin Characteristics by Region

Appendix C. Regression Model Diagnostics

1

1. Introduction 1

Effective management of surface water supply and design of flood control systems relies 2

on accurate forecasting of streamflow conditions. Increasingly, hydrological research 3

reveals that streamflow may be changing in response to changes in climate and land 4

development (Beighley and Moglen, 2002; Peel and Bloschl, 2011; Wagener et al., 2010). 5

Analyses related to the non‐stationarity of streamflow can be broadly categorized as 6

either trend detection or trend attribution studies. Trend detection studies focus on 7

identifying variation in streamflow metrics over time (Kundzewicz and Robson, 2004; 8

Vogel et al., 2011), whereas trend attribution studies focus on identifying the underlying 9

cause of the variation (Poff et al., 2006). However, challenges with trend analysis, such as 10

data availability and long term persistence (Cohn and Lins, 2005), limit the ability to 11

extrapolate to future streamflow conditions. Streamflow sensitivity analysis provides an 12

alternative approach to investigate change by quantifying streamflow response to 13

changes in underlying physical processes. 14

Several methods have been employed to investigate the response of streamflow to basin 15

changes. These methods can be utilized to quantify streamflow alteration should basin 16

conditions change from existing conditions. Such streamflow sensitivity studies broadly 17

take the form of either physical hydrological modeling or statistical methods. 18

Hydrological modeling methods include analytical equations and computer simulation of 19

basin runoff. Schaake (1990) transferred the concept of elasticity from economics to 20

hydrology to quantify sensitivity of streamflow to a changing climate using water balance 21

equations. Sankarasubramanian et al. (2001) evaluated the performance of several 22

analytical models and introduced a non‐parametric estimator that performed well in 23

2

comparison to parametric estimators of streamflow elasticity. Other comparative studies 24

of? hydrological models have found that the results are variable among models 25

depending on model assumptions and calibration techniques (Chiew, 2006; Sun et al., 26

2014). 27

Statistical methods utilize observed discharge data to model streamflow on measured 28

basin characteristics. Time series analyses are generally limited to few explanatory 29

variables, such as precipitation, temperature, and population density. Regression 30

methods that substitute space for time to utilize several gaged sites over one time period 31

are ubiquitous in hydrology (Benson, 1967; Reis III, 2007). While these types of statistical 32

studies are most often utilized for prediction in ungauged basins, the results can also be 33

interpreted as sensitivity estimates because the methodology assumes a power law 34

model relationship between streamflow and explanatory variables (Lacombe et al., 2014; 35

Vogel et al., 1999). Steinschneider et al. (2013) recently introduced panel regression, 36

another concept from economics, to hydrological sensitivity studies to simultaneously 37

account for temporal and spatial effects. Other statistical methods have been employed 38

to investigate streamflow sensitivity, such as random forest regression (Eng et al., 2013). 39

The scope of streamflow sensitivity analyses are often focused on assessing the impacts 40

of climate change on undisturbed basins (Arora, 2002; Dooge et al., 1999; Gyawali et al., 41

2015). These studies focus on isolating response of streamflow to changes in 42

precipitation, temperature, and evapotranspiration by limiting the dataset to gages with 43

no flow regulation and minimal to no land development. Studies have increasingly also 44

sought to isolate the effect of urbanization and land development on streamflow using 45

measures of impervious surface, population density, or land cover (Poff et al., 2006; Yang 46

et al., 2011), or to compare the relative impacts of human activities and climate (Ahn and 47

3

Merwade, 2014). Few studies consider the impacts of direct streamflow modification 48

through regulation, channelization, dams, and water withdrawals (Fitzhugh and Vogel, 49

2011; Homa et al., 2013; Sauer et al., 1983). 50

The approach implemented in physical modeling studies of leaving out one variable at a 51

time to investigate sensitivity of runoff can misrepresent effects due to the questionable 52

assumption of model linearity (Saltelli and Annoni, 2010). Similarly, statistical regression 53

analyses are subject to omitted variable bias that may lead to biased sensitivity estimates. 54

This study aims to quantify sensitivity of streamflow with consideration for multiple 55

variables simultaneously to 1) identify the basin characteristics that exhibit the highest 56

degree of influence on both high and low annual streamflow, and 2) assess the 57

comparative effect of each variable. Regional models are developed using ordinary least 58

squares regression for median annual high and low flows with consideration for multiple 59

variables that represent climatic, morphological, hydrological, land cover, and human 60

influences on a watershed. The concept of elasticity is used to interpret the results in 61

terms of physical impact on streamflow attributable to each explanatory variable. 62

2. Methods 63

Regression model for streamflow elasticity 64

Elasticity is a common concept in the study of economics that provides a measure of how 65

the price of goods changes in response to market changes. More broadly, elasticity is 66

measure of sensitivity to change. Schaake (1990) applied the concept of elasticity to 67

hydrology to quantify the response of streamflow to climate change using derivatives of 68

water balance equations. Schaake defined precipitation elasticity of streamflow as the 69

4

proportional change in streamflow Q divided by the proportional change in precipitation 70

P: 71

/

/ (1) 72

The resulting elasticity values are interpreted to mean that, for example, if εP = 2 for mean 73

annual streamflow, then a 1 percent change in precipitation leads to a 2 percent change 74

in mean annual streamflow. Several studies have applied the concept of elasticity to 75

streamflow since the idea was introduced by Schaake (Chiew, 2006; Sankarasubramanian 76

et al., 2001; Sun et al., 2014; Zhan et al., 2014). While Schaake utilized water balance 77

models to obtain estimates of elasticity, non‐parametric methods have also been 78

introduced to quantify sensitivity (Allaire et al., 2015; Sankarasubramanian et al., 2001). 79

This study assumes a power law model relationship between streamflow and basin 80

characteristics as the basis for obtaining estimates of streamflow elasticity. In its most 81

simple form, the power law model relates a streamflow metric Q to a single basin 82

characteristic X as an explanatory variable with parameters α and β: 83

(2) 84

The power law model is commonly assumed in statistical hydrology studies for the 85

purpose of prediction in ungauged basins with an applied log transformation to estimate 86

the model coefficients using linear regression (Hrachowitz et al., 2013; Reis III, 2007). As 87

indicated by Vogel et al. (1999), the parameter β can be shown to equal the elasticity of 88

streamflow to the explanatory variable X. A derivatives analysis of the power law model 89

is presented here to illustrate this relationship. The derivative of Q with respect to X in 90

Eq. 2 is given by: 91

5

β 1 (3) 92

Eq. (2) and Eq. (3) are inserted into Eq. (1) with the more generic variable X substituted 93

for precipitation, P. Further algebraic transformation using the rule of multiplying 94

exponents shows that the elasticity of streamflow to the predictor variable is equal to its 95

model coefficient: 96

β 1 (4) 97

The resulting elasticity is a constant, equal to the value of the exponent in the power law 98

model. In physical terms, constant elasticity does not mean that the relationship between 99

streamflow and the explanatory variable is linear. Recall a power law model assumption 100

for the underlying relationship between the explanatory variable and stream flow to 101

obtain the constant elasticity result. 102

The same result is obtained for a multivariate power law model: 103

(5) 104

22

1 11 1

1β1

22

22

11 1 1

1β1

22 (6) 105

and so on for any number of variables. 106

Log transformation of the power law model yields a simple linear model that allows for 107

estimation of the parameters using linear regression (Eq. 7). This simple model can be 108

expanded to multiple variables (Eq. 8): 109

ln ln ln ln (7) 110

6

ln ln ln ln ⋯ ln ln (8) 111

The measured values of many explanatory variables are adjusted as X + 1 in practice to 112

allow log transformation of zero values of X for the purpose of linear regression. Such 113

modifications are often required for land cover variables such as percent impervious 114

surface, which frequently contain zero values in the dataset. The assumed underlying 115

model for such variables is therefore actually depicted by Eq. 9: 116

1 (9) 117

The derivatives analysis for Eq. 9 shows that the elasticity of streamflow Q with respect 118

to the variable X does not equal the model parameter β when defined at X: 119

β 1 1 1 (10) 120

β 1 1

1 (11) 121

However, defining elasticity at X + 1 restores the relationship between elasticity and the 122

model parameter β to allow estimate of elasticity from β: 123

β 1 1

1 (12) 124

As shown in more detail in Appendix A, defining elasticity at X + 1 is equivalent to defining 125

elasticity at X for large values of X. For small values of X, the model parameter β must be 126

adjusted by a factor term that results in εP less than β; however, in all instances, the 127

elasticity can be calculated from β. 128

By employing a multivariate linear regression model to obtain the coefficients, hypothesis 129

tests can be utilized to assess the statistical significance of model coefficients, and 130

7

therefore, to assess the statistical significance of elasticity values. Unlike applications of 131

this methodology for prediction in ungaged basins, which focus on obtaining the best 132

estimate of streamflow Q on the left hand side of the equation, this study is equally 133

interested in obtaining the best estimate of the coefficients on the right hand side of the 134

equation. Further, the number and type of explanatory variables are equally important, 135

whereas prediction‐focused models may overlook explanatory variables in favor of model 136

simplicity. The following hypothesis tests and regression diagnostics are selected for the 137

purposes of this study to support selection of explanatory variables and quality of the 138

regression coefficients: 139

1. Normally distributed residuals, assessed using quantile probability plots and 140

probability plot correlation coefficients (Heo et al., 2008). 141

2. Constant variance (i.e. homoscedastic) residuals, visually assessed using plots of 142

log‐transformed residuals versus fitted values, with a mean equal to zero. 143

3. Model coefficients with p‐values significant at the 0.05 level (2‐sided), or in other 144

words, t‐ratios with absolute value greater than 2. 145

4. Variance inflation factors less than approximately 5 to account for correlated 146

explanatory variables (Kroll and Song, 2013). 147

Statistical metrics such as adjusted R2, prediction R2, and Nash‐Sutcliff efficiency, and 148

standard error in both real space and log space are still considered and reported to 149

understand the performance of the model overall and potential for additional 150

explanatory power of omitted variables. 151

8

Study Area 152

Regional regression models for median annual high and low flows were developed for 153

each of the 18 USGS hydrological regions that comprise the continental United States 154

(Figure 1; Table 1). While one model for the entire region of the United States may be 155

developed, this study elected to subdivide the national dataset into regional samples to 156

identify patterns of which explanatory variables most frequently appear in regression 157

models and consider a possible range of model coefficients for the same explanatory 158

variable. 159

Data Sources 160

Readily available datasets from the U.S. Geological Survey (USGS) for streamflow and 161

basin attributes were utilized for this study. Annual peak flow data was obtained from the 162

USGS peak flow database (USGS, 2015a). Annual minimum flow data was extracted from 163

the USGS daily stream flow database (USGS, 2015b). 164

The USGS dataset entitled “Geospatial Attributes of Gages for Evaluating Streamflow”, 165

version II, (GAGESII) provides several hundred watershed characteristics corresponding 166

to 9,322 stream gages maintained by the USGS within the United States (Falcone, 2011). 167

The basin attributes compiled into the dataset include both environmental characteristics 168

(e.g. historical precipitation, temperature, geology, soils, and topography) and 169

anthropogenic influences (e.g. land use, impervious cover, road density, and presence of 170

dams). Table 2 lists the basin characteristics from the GAGESII dataset that were 171

considered for this study. Appendix B provides boxplots showing the range of values for 172

each variable by HUC region. These characteristics were selected to account for a wide 173

range of factors, including climate (e.g. precipitation, temperature), morphology (e.g. 174

9

basin shape, topography), hydrology (e.g. runoff, infiltration, base flow), land cover (e.g. 175

developed, forest, agriculture), and human influences (e.g. dams, roads). 176

Some of the characteristics capture multiple types of factors that affect streamflow. For 177

example, the “HIRES_LENTIC_PCT” term includes both natural hydrological storage in the 178

form of lakes and man‐made reservoir storage. Note also that mean annual precipitation 179

was used in this study. This metric represents a general climate characteristic and not an 180

extreme precipitation condition that may be more appropriate for investigation of 181

precipitation elasticity of extreme flows (i.e. droughts and floods). 182

Data Preparation 183

The median annual high flow and low flow were calculated for each gage included in the 184

GAGESII database within the 18 hydrological regions comprising the continental United 185

States. The annual high flows were obtained from the USGS peak flow dataset, which 186

directly provides annual maximum instantaneous peak flows. The median annual high 187

flow was utilized as a non‐parametric estimator of the 2‐year high flow. The median value 188

of the annual peak instantaneous flow for each water year from 1991 through 2010 (i.e., 189

October 1, 1990 through September 30, 2010) was obtained for each gage. These twenty 190

years were selected to generally encompass the time period captured by the basin 191

characteristics for land use, water withdrawals, and other factors. While changes in 192

climate and land use may have occurred in some basins during that time period, a 193

sufficient record length is required to estimate extreme flows. Only gages with 10 or more 194

years of data within the time period of interest were included in the analysis. 195

Similarly, annual minimum daily flows were extracted from the USGS daily streamflow 196

data for each water year from 1991 through 2010. The median annual low flow was 197

10

utilized as a non‐parametric estimator of the 2‐year low flow. Only gages with 10 or more 198

years of data between water years 1991 through 2010 were selected for analysis. Further, 199

only sites with positive values for the two year minimum flow were included in the 200

analysis. 201

Table 1 presents the resulting sample size for each hydrological region. Figure 2 shows 202

the range of observed values for median annual high and low flows by HUC region. The 203

calculated values of the median annual high and low flow and all measurements of basin 204

characteristics were log transformed for multivariate linear regression. Most explanatory 205

variables were adjusted by adding 1 prior to the log transformation, as indicated in Table 206

2. All data preparation was completed in the R statistical computing software with basic 207

functionality and the following packages: leaps, car, reshape2, plyr, and ggplot2 (R Core 208

Team, 2014). 209

Model Selection 210

Selection of the explanatory variables for each regional regression model relied primarily 211

on a forward stepwise evaluation of model performance based on a leave‐one‐out cross 212

validation (LOOCV). To account for differences in streamflow due to varying drainage area 213

size, drainage area was automatically included as the first variable of each model to 214

initiate a forward stepwise selection. Figure 2 shows the range of drainage area sizes for 215

sample gages by HUC region. 216

The successive explanatory variable to be added to the model was selected as the variable 217

that resulted in the model with the smallest mean‐square error (MSE). However, rather 218

than calculating the MSE based on the entire sample, the errors were obtained from a 219

LOOCV of the model that included that variable. At each step of the model selection 220

11

process, the variables were ranked in terms of their MSE, nash‐sutcliffe efficiency (NSE) 221

and F statistic. Generally, the variable that resulted in the lowest MSE (and consequently 222

the lowest NSE) with a significant F value was chosen for the model. 223

The LOOCV for each variable was automated in R; however, selection of the variable to 224

include in the model involved review at each step. In some instances, a variable with a 225

slightly higher MSE was selected based on a review of which variables appeared most 226

frequently in a best subsets analysis of the entire sample. Further, given that there is no 227

comparable hypothesis test for the difference between NSE values and F statistics as 228

there is for the difference in model errors (which is measured by the F statistic), a variable 229

that has a higher MSE, NSE, or F statistics at the second, third, or fourth decimal place 230

may be statistically equivalent in its performance as the variable with the lowest values. 231

In other words, consideration for overall performance of the explanatory variable in 232

models of different sizes as shown in best subsets analysis or a variable that results in 233

better behaved residuals and t‐values is preferable to fully automated selection. 234

3. Results 235

The regression model developed for each HUC region is presented in Table 3 for median 236

annual high flows and Table 4 for median annual low flows. Regression diagnostics, 237

including residual plots, are provided in Appendix B. Overall, the regression model 238

approach produced better results for high flows than for low flows. Approximately half of 239

the regression models for high flows exhibited well‐behaved residuals; whereas all of the 240

low flow regression models exhibit non‐normality in the residuals with an over‐prediction 241

bias in the fitted values. 242

12

Figure 3 also presents a visual summary of elasticity values for the explanatory variables 243

that appeared in at least four regression models for median annual high flows and at least 244

four regression models for median annual low flow. 245

Climate elasticity 246

Precipitation appeared in the majority of the regression models for both the median 247

annual high and low flows with an average elasticity value of ε = 2.2 for high flows and ε 248

= 2.9 for low flows. Precipitation elasticity was also calculated using regression models for 249

each HUC region with precipitation as the only variable in addition to drainage area. As 250

shown in Figure 4, the multivariate precipitation elasticity has a lower average value and 251

lower variability compared to precipitation elasticity obtained from the simple variable 252

model with only precipitation added to drainage area. 253

Temperature appeared less frequently in the regression models than precipitation with a 254

relatively wide range of coefficient values. The temperature elasticity has a negative value 255

in all models, suggesting that in general, increasing temperature decreases flow. This 256

result may reflect the association between higher temperature and increased 257

evapotranspiration. 258

Land development elasticity 259

Explanatory variables that represent measures of land development do not display as 260

definitive a pattern among the regression models as climate and hydrology variables. 261

Population density occurs most frequently, appearing in four of the 19 regressions for 262

high flows with an elasticity value of approximately ε = 0.12. The one low flow model that 263

includes population density also has a similar value of ε = 0.16. Variables such as 264

impervious surface and developed land cover appear in few regression models. When 265

13

considering all of the developed land cover variables (i.e. percent developed watershed 266

and percent developed within 100 and 800 meters of the main stem and all riparian 267

areas), the models suggest that developed land cover increases both high and low flows 268

with elasticity values between ε = 0.22 and ε = 0.75. These values are consistently higher 269

than the elasticity for population density, which is a variable often used as a proxy for 270

developed land cover in hydrological regression modeling. 271

In addition to indirect human influence throughout the watershed, most of the gages in 272

the nationwide dataset occur in watersheds with some degree of direct flow regulation. 273

The influence of dams in particular was considered using variables to account for the 274

number of dams and the amount of associated storage. The number of dams exhibits a 275

strong correlation with the drainage area size but did not appear as a significant 276

contributor to regression models in terms of decreasing error, with one questionable 277

model appearance for high flows and one for low flows. Reservoir storage directly 278

attributable to dams appeared in only three high flow models and none of the low flow 279

models. However, the storage term combining all surface water storage in the form of 280

lakes, ponds, and reservoirs appeared in 12 of the high flow models with an average 281

elasticity value of ε = ‐0.40 and in 3 of the low flow models also with a negative and 282

comparable average elasticity value of ε = ‐0.67. 283

Conversely, the influence of non‐developed forested area on both high and low flows is 284

inconclusive. Percent forest cover at various scales did not exhibit a consistent pattern 285

among regression models, with both negative and positive values ranging from ‐0.20 to ‐286

0.90 for negative coefficients and from 0.19 to 1.33 for positive values. 287

14

Hydrological elasticity 288

Conveyance of water into rivers as runoff, baseflow, and directly through lower order 289

streams strongly exhibited influence on high and low flows. Runoff appeared in 290

approximately half of the models, with an elasticity of ε = 0.72 on average for high flows 291

compared to an elasticity of almost double ε = 1.2 for low flows. Baseflow, as measured 292

by the baseflow index, exhibits elasticity values of comparable magnitude to precipitation 293

but with negative values for high flows and positive values for low flows. The elasticity 294

values indicate that baseflow decreases the median annual high flow on average 1.4 295

percent but increases the median annual low flow on average 2.5 percent for each 296

percent increase in baseflow. In other words, the effect of groundwater storage is to 297

decrease the overall variability in flows within rivers. 298

The elasticity of permeability, which provides a measure of infiltration potential, indicates 299

that increasing permeability reduces high flows (ε = ‐0.361) and increases low flows (ε = 300

0.737). This result is also consistent with general hydrological understanding that the 301

ability of the water to enter into ground storage will reduce flood flows and increase low 302

flows through an increase in baseflow. 303

In addition, greater density of lower order streams draining into a given point increases 304

high flows with an elasticity value for watershed stream density of ε = 2.3 and 305

questionable negative elasticity for low flows. Measures of basin morphology are 306

inconclusive with few variables appearing in the regression models. 307

15

4. Discussion 308

The overall patterns in the sensitivity of stream flow to multiple variables that emerge 309

from the regional regression models are consistent with general hydrological principles. 310

Increases in precipitation, runoff, and stream density increase flows; storage in the form 311

of groundwater, surface water, and man‐made reservoirs reduce high flows and increase 312

low flows. Consequently, despite questionable residuals, the patterns emerging from the 313

39 regression models allow for an examination of multivariate elasticity of streamflow. 314

Precipitation and groundwater storage comparably exhibit the strongest influence on 315

streamflow based on the order of magnitude of their associated elasticity values. The 316

elasticities of variables associated with land development indicate a positive relationship 317

between flow and land development but at an order of magnitude less impact than 318

precipitation and groundwater influence. 319

The results also suggest that the measures of land cover used in this study may not 320

adequately capture the underlying physical processes that are believed to occur as 321

development increases, namely increase in runoff and decrease in surface permeability. 322

Confounding factors associated with impervious surface, such as channelization altering 323

the topographical based model of watershed definition, may obscure the relationship 324

with streamflow. The explanatory variables from GAGESII that provide more direct 325

estimates of both runoff and permeability show the strong influence of these factors on 326

streamflow in accordance with general understanding of the hydrological process. 327

Finally, the addition of explanatory variables to a base model of drainage area and 328

precipitation reduces the value and variability of the elasticity of precipitation on average. 329

16

Consequently, changes in streamflow based solely on changes in precipitation may be 330

over‐predicted if other changes in the watershed are not also considered. 331

5. Conclusion 332

This study demonstrates the importance of considering multiple variables simultaneously 333

for attributing changes in streamflow. Consideration of multiple variables allow for 334

capturing the interaction of the variables to avoid over‐estimation of the impact from any 335

one variable. 336

17

Figures

Figure 1. U.S. Geological Survey HUC Regions

(Reproduced from Seaber et al., 1987)

18

Figure 2. Observed Streamflow by HUC Region for

Median Annual High Flows (left) and Median Annual Low Flows (right)

19

Figure 3. Multivariate Elasticities for

Median Annual High Flow (left) and (b) Median Annual Low Flow (right)

(see Table 2 for detailed descriptions of the variables)

Figure 4. Multivariate vs Singular Elasticity of Precipitation

Median Annual High Flow (left) and (b) Median Annual Low Flow (right)

20

Tables

Table 1. USGS Regions and Number of Gages in Regional Regression Model

Region Region Name High Flow Sample Size

Low Flow Sample Size

1 New England 225 213

2 Mid‐Atlantic 638 613

3 South Atlantic‐Gulf 730 715

4 Great Lakes 303 295

5 Ohio 486 436

6 Tennessee 88 83

7 Upper Mississippi 402 376

8 Lower Mississippi 98 86

9 Souris‐Red‐Rainy 62 54

10L Lower Missouri 299 266

10U Upper Missouri 278 234

11 Arkansas‐White‐Red 345 280

12 Texas‐Gulf 275 196

13 Rio Grande 83 65

14 Upper Colorado 212 207

15 Lower Colorado 133 88

16 Great Basin 172 167

17 Pacific Northwest 523 529

18 California 390 359

21

Table 2. GAGESII Basin Characteristics and Elasticity Results

Variable Name Description Units Source

Q2max Elasticity

Avg (#HUCS)

Q2minElasticity

Avg (#HUCS)

DRAIN_SQKM Watershed drainage area sq km USGS NAWQA 0.721 (19) 1.05 (19)

PPT_MED_CM Mean annual precipitation, represented by median of mean annual precipitation from 1991‐2010 reported in GAGESII

cm/year 4‐km PRISM data 2.18 (13) 2.86 (9)

TMP_MED_C (+1) Mean annual air temperature, represented by median of mean annual temperature from 1991‐2010 reported in

GAGESII

degrees C 4‐km PRISM data ‐1.09 (3) ‐1.74 (6)

BAS_COMPACTNESS Watershed compactness ratio (area/perimeter^2*100)

unitless USGS NWIS and NAWQA

‐‐ ‐‐

ELEV_MEAN_M_BASIN_30M Mean watershed elevation meters 30m NHDPlus ‐‐ 0.616 (2)

ELEV_SITE_M_30M (+1) Elevation at gage location meters 30m NHDPlus ‐‐ ‐1.75 (1)

SLOPE_PCT_30M (+1) Mean watershed slope percent 30m NHDPlus 0.293 (2) 0.753 (2)

RUNAVE7100 Estimated watershed annual runoff, mean for the period 1971‐2000 (estimation method integrates climate, land use,

water use, regulation, etc)

mm/year GAGESII, using approach in (Krug

et al., 1989)

0.717 (8) 1.23 (9)

PERMAVE Average permeability inches/hour USGS NAWQA, STATSGO and Wolock (1997)

‐0.361 (4) 0.737 (5)

BFI_AVE Base Flow Index, a ratio of base flow to total streamflow

percent Wolock (2003) ‐1.41 (13) 2.52 (16)

CONTACT Subsurface flow contact time days Wolock and others, 1989; Wolock, 1997

‐0.035 (2) ‐0.310 (1)

STREAMS_KM_SQ_KM (+1) Stream density km of streams per watershed sq km

NHD 100k streams 2.32 (4) ‐1.58 (1)

HIRES_LENTIC_PCT (+1) Percent of watershed surface area covered by "Lakes/Ponds" + "Reservoirs"

percent NHD Hi‐Resolution ‐0.404 (12) ‐0.673 (3)

HIRES_LENTIC_DENS (+1) Density (#/sq km) of Lakes/Ponds + Reservoir

number/sq km NHD Hi‐Resolution ‐‐ ‐1.92 (1)

NDAMS_2009 (+1) Number of dams in watershed number of dams 2009 National Inventory of Dams

(NID)

‐0.416 (1) 0.431 (1)

STOR_NID_2009 (+1) Dam storage in watershed

megaliters total storage/sq km

2009 NID ‐0.138 (3) ‐‐

FRESHW_WITHDRAWAL Freshwater withdrawal from 1995 to 2000 county‐level estimates

megaliters/year/sq km

USGS ‐‐ ‐‐

PCT_IRRIG_AG (+1) Percent of watershed in irrigated agriculture

Percent USGS 2002 250‐m MODIS data

‐0.383 (1) 0.457 (3)

PDEN_2000_BLOCK (+1) Population density in the watershed (2000)

persons/sq km 2000 Census block data regridded to

100m

0.121 (4) 0.160 (1)

ROADS_KM_SQ_KM (+1) Road density km of roads/ watershed sq km

Census 2000 TIGER roads

0.741 (1) 1.30 (3)

RD_STR_INTERS (+1) Number of road/stream intersections, per km of total basin stream length

number of intersections/km of

stream length

2000 TIGER roads and NHD 100k

streams

0.404 (1) ‐‐

CANALS_PCT (+1) Percent of stream kilometers coded as "Canal", "Ditch", or "Pipeline" in NHDPlus

Percent NHDPlus ‐‐ ‐‐

IMPNLCD06 (+1) Watershed percent impervious surface (~ 2006)

percent 30‐m resolution 2006 NLCD data

0.644 (3) 0.323 (2)

DEVNLCD06 (+1) Watershed percent "developed" (urban) (~ 2006)

percent 2006 NLCD ‐‐ 0.320 (1)

RIP800_DEV (+1) Riparian 800m buffer percent "developed" (urban)

percent 2006 NLCD 0.425 (1) ‐‐

RIP100_DEV (+1) Riparian 100m buffer percent "developed" (urban)

percent 2006 NLCD ‐‐ 0.753 (1)

MAINS800_DEV (+1) Mainstem 800m buffer percent "developed" (urban)

percent 2006 NLCD 0.304 (2) 0.709 (2)

MAINS100_DEV (+1) Mainstem 100m buffer percent "developed" (urban)

percent 2006 NLCD 0.219 (1) ‐‐

FORESTNLCD06 (+1) Watershed percent "forest" percent 2006 NLCD ‐0.548 (1) ‐‐

RIP800_FOREST (+1) Riparian 800m buffer percent "forest" percent 2006 NLCD ‐‐ 1.33 (1)

RIP100_FOREST (+1) Riparian 100m buffer percent "forest” percent 2006 NLCD ‐‐ ‐0.899 (1)

MAINS800_FOREST (+1) Mainstem 800m buffer percent "forest" percent 2006 NLCD 0.185 (1) ‐0.256 (2)

MAINS100_FOREST (+1) Mainstem 100m buffer percent "forest” percent 2006 NLCD ‐‐ 0.435 (1)

PLANTNLCD06 (+1) Watershed percent “planted/cultivated” (agriculture) (~2006)

percent 2006 NLCD ‐0.199 (3) 0.511 (3)

22

Table 3. Regional Models for Median Annual High Streamflow

Q = ea(1st Predictor)b(2nd Predictor)c(3rd Predictor)d(4th Predictor)e(5th Predictor)f(6th Predictor)g(7th Predictor)h(8th Predictor)i

Region a b c d e f g h i

1 INTERCEPT 5.37 (3.10)

DRAIN_SQKM 0.819 (48.5)

HIRES_LENTIC_PCT ‐0.454 (‐9.66)

BFI_AVE ‐2.36 (‐8.17)

RUNAVE7100 1.19 (6.07)

PLANTNLCD06 ‐0.115 (‐3.65)

‐‐ ‐‐ ‐‐

2 INTERCEPT 10.3 (20.4)

DRAIN_SQKM 0.761 (62.5)

BFI_AVE ‐1.53 (‐11.7)

PERMAVE ‐0.432 (‐11.1)


‐‐ ‐‐ ‐‐ ‐‐

3 ‐‐ DRAIN_SQKM 0.667 (63.0)

PERMAVE ‐0.281 (‐6.12)

RUNAVE7100 0.542 (8.29)

BFI_AVE ‐0.884 (‐11.5)


IMPNLCD06 0.232 (10.5)

SLOPE_PCT_30M 0.206 (5.67)

PPT_MED_CM 0.843 8.59

4 INTERCEPT ‐5.24 (‐5.94)

DRAIN_SQKM 0.842 (52.4)

STREAMS_KM_SQ_KM 2.47 (11.6)

PERMAVE ‐0.391 (‐9.98)

PLANTNLCD06 ‐0.226 (‐12.0)


PPT_MED_CM 1.68 (8.74)

RD_STR_INTERS 0.404 (3.26)

‐‐

5 INTERCEPT ‐7.13 (‐6.83)

DRAIN_SQKM 0.702 (65.2)

PPT_MED_CM 2.83 (14.3)


BFI_AVE ‐0.612 (‐8.67)

PDEN_2000_BLOCK 0.096 (6.12)

‐‐ ‐‐ ‐‐

6 INTERCEPT 6.67 (3.73)

DRAIN_SQKM 0.736 (28.3)

BFI_AVE ‐1.94 (‐9.02)

PPT_MED_CM 1.88 (8.45)

TMP_MED_C ‐1.64 (‐3.69)

‐‐ ‐‐ ‐‐ ‐‐

7 INTERCEPT 4.52 (7.81)

DRAIN_SQKM 0.714 (44.8)

BFI_AVE ‐0.761 (‐12.8)


PLANTNLCD06 ‐0.256 (‐10.7)


RUNAVE7100 0.400 (4.97)

‐‐ ‐‐

8 INTERCEPT 6.14 (3.42)

DRAIN_SQKM 0.524 (11.9)

CONTACT ‐0.265 (‐5.50)

BFI_AVE ‐1.29 (‐4.75


RUNAVE7100 0.878 (3.85)

PDEN_2000_BLOCK 0.142 (2.60)

‐‐ ‐‐

9 INTERCEPT ‐17.0 (‐9.42)

DRAIN_SQKM 0.773 (16.4)


PPT_MED_CM 4.34 (9.48)


PCT_IRRIG_AG ‐0.383 (‐3.11)

MAINS800_DEV 0.327 (2.56)

‐‐ ‐‐

10L INTERCEPT ‐9.21 (‐11.6)

DRAIN_SQKM 0.568 (23.5)

RUNAVE7100 0.694 (15.5)

PPT_MED_CM 1.95 (9.52)

MAINS800_DEV 0.280 (5.77)

CONTACT 0.196 (5.88)


‐‐ ‐‐

10U INTERCEPT ‐3.71 (‐3.25)

DRAIN_SQKM 0.759 (26.3)

RUNAVE7100 0.403 (6.39)

PPT_MED_CM 1.48 (5.90)

BFI_AVE ‐0.761 (‐0.637)


TMP_MED_C ‐0.772 (‐5.17)

IMPNLCD06 0.679 (3.34)

‐‐

11 INTERCEPT ‐‐

DRAIN_SQKM 0.704 (28.6)

RUNAVE7100 0.977 (30.6)

STOR_NID_2009 ‐0.168 (‐6.74)

BFI_AVE ‐0.363 (‐5.768)

PDEN_2000_BLOCK 0.111 (3.21)

‐‐ ‐‐ ‐‐

12 INTERCEPT ‐7.70 (‐9.02)

DRAIN_SQKM 0.527 (17.4)

PPT_MED_CM 2.45 (14.4)

ROADS_KM_SQ_KM 0.741 (6.40)

SLOPE_PCT_30M 0.379 (6.00)

STOR_NID_2009 ‐0.147 (‐6.87)

HIRES_LENTIC_DENS 0.670 (5.89)

PERMAVE ‐0.339 (‐4.45)

13 INTERCEPT ‐4.89 (‐2.64)

DRAIN_SQKM 0.684 (13.4)

PPT_MED_CM 1.60 (4.00)

IMPNLCD06 1.02 (5.33)

‐‐ ‐‐ ‐‐ ‐‐ ‐‐

14 INTERCEPT ‐8.03 (‐10.9)

DRAIN_SQKM 0.785 (33.2)

PPT_MED_CM 2.84 (15.2)

FORESTNLCD06 ‐0.548 (‐5.95)

PDEN_2000_BLOCK 0.133 (2.78)

‐‐ ‐‐ ‐‐ ‐‐

15 INTERCEPT 3.66 (5.48)

DRAIN_SQKM 0.719 (10.7)

RUNAVE7100 0.655 (7.94)

BFI_AVE ‐0.884 (‐5.30)

NDAMS_2009 ‐0.416 (‐4.30)

‐‐ ‐‐ ‐‐ ‐‐

16 INTERCEPT 13.7 (3.30)

DRAIN_SQKM 0.792 (19.9)

PPT_MED_CM 2.50 (10.7)

BFI_AVE ‐5.08 (‐5.55)

STOR_NID_2009 ‐0.098 (‐2.75)

MAINS100_DEV 0.219 (3.49)

TMP_MED_C ‐0.851 (‐2.743)

17 INTERCEPT ‐2.61 (‐3.20)

DRAIN_SQKM 0.917 (62.4)

PPT_MED_CM 1.91 (29.6)


BFI_AVE ‐1.18 (‐7.28)

MAINS800_FOREST 0.185 (6.29)

‐‐ ‐‐ ‐‐

18 INTERCEPT ‐3.29 (‐4.52)

DRAIN_SQKM 0.711 (26.4)

PPT_MED_CM 2.02 (18.6)

RIP800_DEV 0.425 (7.71)


BFI_AVE ‐0.712 (‐4.142)

‐‐ ‐‐ ‐‐

*Table shows coefficient with t‐ratio in parentheses for each variable obtained from the OLS regression equation. Models with best residuals indicated in gray.

23

Table 4. Regional Models for Median Annual Low Streamflow

Q = ea(1st Predictor)b(2nd Predictor)c(3rd Predictor)d(4th Predictor)e(5th Predictor)f(6th Predictor)g(7th Predictor)h(8th Predictor)i

Region a b c d e f g h i

1 INTERCEPT ‐30.8 (‐7.52)

DRAIN_SQKM 1.12 (32.8)

ELEV_MEAN_M_BASIN_30M 0.996 (8.93)

MAINS800_FOREST ‐0.962 (‐6.21)

RUNAVE7100 2.03 (5.08)

BFI_AVE 3.02 (3.95)

PERMAVE 0.494 (3.27)

‐‐ ‐‐

2 INTERCEPT ‐31.9 (‐17.1)


BFI_AVE 2.72 (12.4)

RUNAVE7100 1.41 (7.94)

IMPNLCD06 0.358 (7.58)

ELEV_MEAN_M_BASIN_30M 0.236 (4.96)

PPT_MED_CM 1.60 (3.96)

‐‐ ‐‐

3 INTERCEPT ‐27.5 (‐13.1)


BFI_AVE 2.25 (11.7)

SLOPE_PCT_30M 0.594 (9.60)


PPT_MED_CM 2.68 (6.66)

‐‐ ‐‐ ‐‐

4 INTERCEPT ‐15.5 (‐8.96)


BFI_AVE 1.41 (6.04)

RUNAVE7100 1.39 (7.55)

IMPNLCD06 0.287 (5.12)

CONTACT ‐0.310 (‐5.09)

STREAMS_KM_SQ_KM ‐1.58 (‐3.47)

‐‐ ‐‐

5 INTERCEPT ‐16.9 (‐22.0)


BFI_AVE 2.74 (12.4)


HIRES_LENTIC_PCT 0.551 (5.08)

PERMAVE 0.438 (4.09)

‐‐ ‐‐ ‐‐

6 INTERCEPT ‐22.1 (‐9.20)


BFI_AVE 3.11 (7.42)

PERMAVE ‐0.764 (‐3.12)

PPT_MED_CM 2.28 (4.67)

HIRES_LENTIC_DENS ‐1.92 (‐4.46)

RIP100_FOREST ‐0.899 (‐3.46)

‐‐ ‐‐

7 INTERCEPT ‐18.8 (‐15.0)


BFI_AVE 1.39 (7.70)

RUNAVE7100 1.34 (8.42)


SLOPE_PCT_30M 0.911 (9.71)

PERMAVE 0.486 (4.79)

PCT_IRRIG_AG 0.303 (3.29)

TMP_MED_C ‐0.993 (‐2.93)

8 INTERCEPT ‐64.5 (‐7.74)


PPT_MED_CM 9.64 (6.36)

PERMAVE 3.03 (6.54)

PLANTNLCD06 0.767 (4.14)

BFI_AVE 2.46 (2.92)

‐‐ ‐‐ ‐‐

9 INTERCEPT ‐13.3 (‐8.56)


RIP800_FOREST 1.33 (8.36)

‐‐ ‐‐ ‐‐ ‐‐ ‐‐ ‐‐

10L ‐‐ DRAIN_SQKM 0.989 (17.4)

RUANAVE7100 0.536 (7.03)

BFI_AVE 2.27 (8.01)

ELEV_SITE_M_30M ‐1.75 (‐11.7)

TMP_MED_C ‐2.29 (‐12.1)

PDEN_2000_BLOCK 0.160 (2.53)

‐‐ ‐‐

10U INTERCEPT ‐22.4 (‐10.7)

DRAIN_SQKM 0.918 (14.9)

RUNAVE7100 0.589 (4.84)

MAINS100_FOREST 0.435 (4.00)

BFI_AVE 1.08 (3.88)

PPT_MED_CM 2.44 (4.78)


‐‐ ‐‐

11 INTERCEPT ‐13.8 (‐8.67)


RUNAVE7100 0.848 (10.6)

BFI_AVE 1.52 (5.34)

RIP100_DEV 0.753 (4.74)

TMP_MED_C ‐0.864 (‐3.01)

‐‐ ‐‐ ‐‐

12 INTERCEPT ‐22.4 (‐13.9)


RUNAVE7100 1.75 (12.9)

BFI_AVE 1.08 (3.17)

MAINS800_DEV 0.828 (5.33)

PLANTNLCD06 ‐0.314 (‐2.32)

‐‐ ‐‐ ‐‐

13 INTERCEPT 2.73 (3.45)

DRAIN_SQKM 0.381 (3.07)

TMP_MED_C ‐2.27 (‐5.42)

PLANTNCLD06 1.08 (2.66)

MAINS800_DEV 0.589 (2.12)

‐‐ ‐‐ ‐‐ ‐‐

14 INTERCEPT ‐22.3 (‐4.92)


TMP_MED_C ‐1.09 (‐5.70)

PPT_MED_COM 1.82 (5.22)


BFI_AVE 2.93 (3.09)

‐‐ ‐‐ ‐‐

15 INTERCEPT ‐7.83 (‐6.38)

DRAIN_SQKM 0.758 (5.94)

RUNAVE7100 1.14 (5.22)


‐‐ ‐‐ ‐‐ ‐‐ ‐‐

16 INTERCEPT ‐29.3 (‐3.96)

DRAIN_SQKM 0.640 (10.5)

PPT_MED_CM 1.60 (3.72)

TMP_MED_C ‐2.92 (‐5.07)

BFI_AVE 6.11 (3.71)

HIRES_LENTIC_PCT 0.451 (3.06)

‐‐ ‐‐ ‐‐

17 INTERCEPT ‐32.8 (‐19.3)


PPT_MED_CM 2.01 (16.4)

BFI_AVE 4.33 (12.9)

MAINS800_FOREST 0.450 (7.83)

DEVNLCD06 0.320 (5.89)

‐‐ ‐‐ ‐‐

18 INTERCEPT ‐18.2 (‐19.9)

DRAIN_SQKM 0.725 (12.5)

PPT_MED_CM 1.69 (9.10)

BFI_AVE 1.94 (8.47)

NDAMS_2009 0.431 (4.89)

‐‐ ‐‐ ‐‐ ‐‐

*Table shows coefficient with t‐ratio in parentheses for each variable obtain from the OLS regression equation.

24

Table 5. Regression Diagnostics

Two‐Year Instantaneous Peak Flow Two‐Year Minimum Daily Flow

Region Adj‐R2 Pred‐R2 NSE Max VIF SE Log SE Real Adj‐R2 Pred‐R NSE Max VIF SElog SEreal

1 0.933 0.930 0.930 1.30 0.378 0.653 0.896 0.889 0.886 3.30 0.751 0.811

2 0.881 0.879 0.879 1.19 0.550 0.706 0.838 0.835 0.834 1.95 0.882 0.899

3 0.901 0.900 0.898 4.12 0.480 0.682 0.762 0.759 0.757 1.33 0.762 0.813

4 0.920 0.916 0.914 1.69 0.384 0.654 0.869 0.864 0.861 2.61 0.778 0.826

5 0.904 0.902 0.901 1.18 0.399 0.657 0.858 0.855 0.853 1.84 0.943 0.952

6 0.913 0.907 0.903 2.27 0.341 0.645 0.889 0.873 0.863 3.09 0.626 0.750

7 0.864 0.861 0.859 1.83 0.474 0.680 0.898 0.895 0.892 3.27 0.861 0.887

8 0.684 0.652 0.629 1.28 0.474 0.684 0.680 0.642 0.620 1.46 1.38 1.69

9 0.880 0.856 0.841 2.16 0.327 0.644 0.713 0.701 0.689 1.01 1.43 1.78

10L 0.838 0.832 0.829 2.77 0.638 0.747 1.45 1.77

10U 0.772 0.765 0.759 2.48 0.763 0.818 0.689 0.673 0.665 3.02 1.35 1.56

11 0.763 0.757 0.754 1.37 0.879 0.898 0.568 0.559 0.551 1.78 1.66 2.47

12 0.651 0.638 0.629 2.25 0.609 0.734 0.672 0.660 0.651 2.45 1.69 2.64

13 0.713 0.697 0.686 1.92 0.794 0.844 0.621 0.585 0.557 3.06 1.30 1.51

14 0.864 0.853 0.856 1.89 0.565 0.714 0.786 0.773 2.33 0.971 0.985

15 0.589 0.562 0.549 2.28 1.03 1.05 0.428 0.400 0.378 1.04 2.02 5.10

16 0.837 0.830 0.823 2.63 0.726 0.798 0.570 0.544 0.530 2.06 1.36 1.58

17 0.989 0.895 0.894 2.43 0.556 0.709 0.791 0.786 0.784 2.35 1.08 1.09

18 0.719 0.713 0.709 2.25 0.885 0.903 0.715 0.709 0.706 2.25 1.38 1.60

Notes:

1. Statistics given in italic, such as Adj‐R2, are reported for the model fitted with intercept.

2. NSE calculated as

1 1

1 ∑

11∑

3. SE = Average Standard Error of Prediction, where SElog is the standard error of the model residuals and SEreal is the standard error transformed to real space using the

following formula where p is the number of sites in the region and n is the number of sites in the region (i.e. sample size).

exp 1 1

25

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Appendix A Physical Interpretation of Regression Model

for Streamflow Elasticity

of 3

Recall from the discussion of the regression model for elasticity, the coefficients provide an estimate of

the elasticity, which can be interpreted as the percent change in streamflow (%∆Q) for a 1 percent

change in the explanatory variable (1%∆X). However, this interpretation is only directly valid when the

model relationship between Q and X is represented as:

(A.1)

Where the elasticity can subsequently be represented as:

/

/ (for X > 0) (A.2)

However, in altering the model to account for explanatory variables with zero values by adding 1 to each

observation for the purpose of regression, the expression for elasticity becomes:

/

/ (A.3)

Further analysis is therefore necessary to be able to interpret the elasticity as the %∆Q for a 1%∆X.

Defining dX as a 1 percent change in X to correspond to the original interpretation of elasticity, Eq. A.3

can be expressed in terms of the observed value of X (X0, i.e the value of X before the change occurs):

/

. / . (A.4)

The corresponding fractional change in Q is represented by the numerator, such that the resulting

fractional change in Q corresponding to a 1 percent change in X is then expressed as:

0.01 0

0 1 (A.5)

Consequently, for large enough values of X0,

0.01 0

0 1 0.01 0

00.01 (A.6)

And therefore, the corresponding percent change in Q for a 1 percent change in X is equivalent to β1,

same as for the model relationship expressed in Eqs. A.1 and A.2:

%∆Qfor1%∆X ∗ 100 0.01 ∗ 100 (A.7)

For small values of X (0 < X < 1), the coefficient β1 must be adjusted using the initial value of X to

determine percent change in Q that would result from that point:

0.01 0

0 1 (A.8)

Ultimately, the value of X increases up to a point where Eq. A.6 begins to apply.

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The model relationship using X+1 shown by Eq. A.3 also allows for definition at X = 0, which is necessary

as many of the observed values for variables of interest such as percent impervious cover and percent

forest land have zero values. However, a 1 percent change in X cannot be defined at X0 = 0. Instead, the

percent change in Q corresponding to a chosen dX, can be obtained:

0 1 0 1 (A.9)

Where dX is expressed as a fraction or decimal.

The change in Q then becomes dependent on how far X changes from zero, with a larger increase in Q

for a larger increase in X and similarly for decreasing values of X.

A few examples with real numbers are helpful to illustrate. Consider that for relatively small incremental

changes in the explanatory variable Eq. (A.3) can also be expressed in terms of the values of X and Q at

present and future time periods:

/

/

/

/ (A.10)

The values for X0, X1, and Q0 can be set, while Q1 becomes the value of interest that results from the

corresponding change from X0 to X1.

Consider the following scenarios based on a 1 percent change in impervious area for a watershed of

interest where the β coefficient corresponding to X is equal to 0.01:

# X0 (sq km)

dX (sq km) X1 (sq km) Q0 (cfs) Q1 (cfs) %∆Q (%) 0.01 0

0 1 ∗ 100 (%)

1 0 0.01 0.01 100 100.01 0.01 ‐‐

2 0 5 5 100 105 5 ‐‐

3 0.5 0.005 0.505 100 100.003 0.003 0.003

4 5 0.05 5.05 100 100.008 0.008 0.008

5 100 1 101 100 100.01 0.01 0.01

Inserting these values into Eq. A.10 for each scenario:

Case #1:

/

. /0.01 0.01 0.01 100 100 100.01 (A.11)

Case #2:

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/

/0.01 0.01 5 100 100 105 (A.12)

Case #3:

/

. / .0.01 0.01

.

.100 100 100.003 (A.13)

Case #4:

/

. /0.01 0.01

.100 100 100.008 (A.14)

Case #5:

/

/0.01 0.01 100 100 100.01 (A.15)

Appendix B Boxplots of GAGESII Basin Characteristics

by Region

Appendix C Regression Model Diagnostics

Region 1Median Annual High Flow

Median Annual Low Flow

(a) Best Subsets (b) Simulated vs Observed (c) Residual Normal Quantile Plot

Region 10LMedian Annual High Flow



Region 10UMedian Annual High Flow



multivariate elasticity of extreme streamflow the united

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