U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2015–5089

Prepared in cooperation with the Department of the Interior WaterSMART Program

Evaluation of Statistical and Rainfall-Runoff Models

for Predicting Historical Daily Streamflow Time Series

in the Des Moines and Iowa River Watersheds

Evaluation of Statistical and RainfallRunoff Models for Predicting Historical 

Daily Streamflow Time Series in the 

Des Moines and Iowa River Watersheds

By William H. Farmer, Rodney R. Knight, David A. Eash, Kasey J. Hutchinson, 

S. Mike Linhart, Daniel E. Christiansen, Stacey A. Archfield, Thomas M. Over, 

and Julie E. Kiang

Prepared in cooperation with the Department of the Interior WaterSMART Program

Scientific Investigations Report 2015–5089

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

SALLY JEWELL, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2015

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Suggested citation:

Farmer, W.H., Knight, R.R., Eash, D.A., Hutchinson, K.J., Linhart, S.M., Christiansen, D.E., Archfield, S.A., Over, T.M., and 

Kiang, J.E., 2015, Evaluation of statistical and rainfall-runoff models for predicting historical daily streamflow time series 

in the Des Moines and Iowa River watersheds: U.S. Geological Survey Scientific Investigations Report 2015–5089, 34 p., 

http://dx.doi.org/10.3133/sir20155089.

ISSN 2328-0328 (online)
iii

Contents

Abstract...........................................................................................................................................................1

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

Purpose and Scope ..............................................................................................................................2

Study Area and Data Selection ...................................................................................................................3

Methods to Estimate Daily Streamflow......................................................................................................3

Selection of Index Streamgages........................................................................................................3

Drainage-Area Ratio.............................................................................................................................5

Nonlinear Spatial Interpolation Using Flow Duration Curves .......................................................5

Precipitation Runoff Modeling System .............................................................................................5

Methods of Analysis......................................................................................................................................6

Results and Discussion.................................................................................................................................7

Summary and Conclusions.........................................................................................................................12

Acknowledgments.......................................................................................................................................12

References Cited..........................................................................................................................................13

Appendix 1. Stations Used in Analysis .................................................................................................15

Appendix 2. Basin Characteristics Used in Analysis .........................................................................16

Appendix 3. Cross-Validation of Map Correlation...............................................................................18

Appendix 4. Distributions of Each Performance Metric ....................................................................19

Figures

1. Map showing location of the Iowa and Des Moines River Basins, study 

area, and landform regions in Iowa...........................................................................................2

2. Map showing location of 44 streamgages within the Iowa and Des Moines 

River Basins included in this study............................................................................................4

3. Robust rank-based evaluation cloud showing the tradeoff between the 

mean ranking and standard deviation of the ranks of the 32 performance 

metrics for each method of prediction in ungaged basins..................................................11

4. Graph showing for all methods of prediction in ungaged basins, the crosssite mean fraction of the validation record less than a threshold absolute 

percent error, a cumulative distribution of the absolute percent error.............................11

5. Graph showing for all methods of prediction in ungaged basins, the crosssite coefficient of variation of the fraction of the validation record less than 

a threshold absolute percent error..........................................................................................12
iv

Appendix Figures

4–1—4–32. Boxplots showing:

4–1. The distribution of the at-site Nash-Sutcliffe efficiencies of daily streamflow

prediction for each method of prediction in ungaged basins (PUB)................................... 19

4–2. The distribution of the at-site Nash-Sutcliffe efficiencies of the logarithms 

of daily streamflow predictions for each method of prediction in ungaged 

basins (PUB).........................................................................................................................19

4–3. The distribution of the at-site root-mean-square errors of daily streamflow

predictions for each method of prediction in ungaged basins (PUB).............................20

4–4. The distribution of the at-site root-mean-square-normalized errors of 

daily streamflow predictions for each method of prediction in ungaged 

basins (PUB).........................................................................................................................20

4–5. The distribution of at-site average percent errors of daily streamflow 

predictions for each method of prediction in ungaged basins (PUB) ........................21

4–6. The distribution of at-site Pearson correlations between simulated and 

observed daily streamflows for each method of prediction in ungaged 

basins (PUB).........................................................................................................................21

4–7. The distribution of at-site Spearman correlations between simulated

and observed daily streamflows for each method of prediction in ungaged 

basins (PUB) ...............................................................................................................................22

4–8. The distribution of the at-site Nash-Sutcliffe efficiencies of the daily 

storage-yield curve (SYC) for each method of prediction in ungaged 

basins (PUB).........................................................................................................................22

4–9. The distribution of the at-site Nash-Sutcliffe efficiencies of the logarithms 

of the daily storage-yield curve (SYC) for each method of prediction in 

ungaged basins (PUB) ........................................................................................................23

4–10. The distribution of the at-site root-mean-square errors of the daily storageyield curve for each method of prediction in ungaged basins (PUB).............................23

4–11. The distribution of the at-site root-mean-square-normalized errors of 

the daily storage-yield curve for each method of prediction in ungaged 

basins (PUB).........................................................................................................................24

4–12. The distribution of at-site average percent errors of the daily storage-yield

curve for each method of prediction in ungaged basins (PUB).......................................24

4–13. The distribution of at-site Pearson correlations between the simulated 

and observed daily storage-yield curves for each method of prediction 

in ungaged basins (PUB)....................................................................................................25

4–14. The distribution of at-site Spearman correlations between the simulated 

and observed daily storage-yield curves for each method of prediction 

in ungaged basins (PUB)....................................................................................................25

4–15. The distribution of at-site percent errors in the estimated coefficient of 

variation of annual streamflows for each method of prediction in ungaged 

basins (PUB).........................................................................................................................26

4–16. The distribution of at-site percent errors in the estimated coefficient of 

variation of daily streamflows for each method of prediction in ungaged 

basins (PUB).........................................................................................................................26
v

4–17. The distribution of at-site percent errors in the estimated 10th percentile

of the distribution of 7-day average annual-minimum events for each 

method of prediction in ungaged basins (PUB)...................................................................27

4–18. The distribution of at-site percent errors in the estimated 50th percentile 

of the distribution of 7-day average annual-minimum events for each 

method of prediction in ungaged basins (PUB)..............................................................27

4–19. The distribution of at-site percent errors in the estimated 90th percentile

of the distribution of annual-maximum events for each method of prediction 

in ungaged basins (PUB) ............................................................................................................... 28

4–20. The distribution of at-site percent errors in the estimated 90-percentexceedance streamflow for each method of prediction in ungaged 

basins (PUB).........................................................................................................................28

4–21. The distribution of at-site percent errors in the estimated 75-percentexceedance streamflow for each method of prediction in ungaged 

basins (PUB).........................................................................................................................29

4–22. The distribution of at-site percent errors in the estimated 50-percentexceedance streamflow for each method of prediction in ungaged 

basins (PUB).........................................................................................................................29

4–23. The distribution of at-site percent errors in the estimated 25-percentexceedance streamflow for each method of prediction in ungaged 

basins (PUB).........................................................................................................................30

4–24. The distribution of at-site percent errors in the estimated 10-percentexceedance streamflow for each method of prediction in ungaged 

basins (PUB).........................................................................................................................30

4–25. The distribution of at-site percent errors in the estimated mean daily 

streamflow for each method of prediction in ungaged basins (PUB) ........................31

4–26. The distribution of at-site percent errors in the estimated coefficient 

of variation (L-CV) of daily streamflow for each method of prediction 

in ungaged basins (PUB.....................................................................................................31

4–27. The distribution of at-site percent errors in the estimated skewness 

(L-skew) of daily streamflows for each method of prediction in ungaged 

basins (PUB).........................................................................................................................32

4–28. The distribution of at-site percent errors in the estimated kurtosis 

(L-kurtosis) of daily streamflows for each method of prediction in 

ungaged basins (PUB) ........................................................................................................32

4–29. The distribution of at-site percent errors in the estimated lag-1 autocorrelation

of daily streamflows for each method of prediction in ungaged basins (PUB)................ 33

4–30. The distribution of at-site percent errors in the estimated amplitude of 

the sinusoidal seasonal trend of daily streamflows for each method of 

prediction in ungaged basins (PUB).................................................................................33

4–31. The distribution of at-site percent errors in the estimated phase of 

the sinusoidal seasonal trend of daily streamflows for each method of 

prediction in ungaged basins (PUB).................................................................................34

4–32. The distribution of the at-site root-mean-square-normalized errors of 

estimated Fundamental Daily Streamflow Statistics (FDSS) for each 

method of prediction in ungaged basins (PUB)..............................................................34
vi

Tables

1. Regional regressions of 27 flow percentiles along the daily flow duration 

curve for the full set of 44 streamgages in the Des Moines and Iowa Rivers....................6

2. Mean rank for each performance metric and method of prediction in 

ungaged basins (PUB) .................................................................................................................8

3. Standard deviation of the ranks for each performance metric and method 

of prediction in ungaged basins (PUB) ...................................................................................10

Appendix Tables

1. Names, station numbers, fraction of zeros, and period of record for each 

streamgage used for the comparison of methods of prediction in ungaged 

basins in the Des Moines and Iowa River Basins.................................................................15

2. Description of all basin characteristics considered as potential explanatory 

variables in the development of the regressions conducted as part of the 

model comparison in the Des Moines and Iowa River Basins ...........................................16

3. Leave-one-out cross-validated, root-mean-squared error of the fitted 

variograms of intersite hydrograph correlation at 44 streamgages in the 

Des Moines and Iowa River Basins.........................................................................................18

Conversion Factors

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

inch (in.) 25.4 millimeter (mm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

square miles per mile (mi2

/mi) 1.609 square kilometers per kilometer (km2/km)

miles per square mile (mi/mi2

) 0.6214 kilometers per square kilometer (km/km2)

Area

square mile (mi2

) 259.0 hectare (ha)

square mile (mi2

) 2.590 square kilometer (km2) 

Volume

cubic foot (ft3

) 28.32 cubic decimeter (dm3) 

cubic foot (ft3

) 0.02832 cubic meter (m3) 

Flow rate

cubic foot per second (ft3

/s) 0.02832 cubic meter per second (m3/s)

Hydraulic gradient

foot per mile (ft/mi) 0.1894 meter per kilometer (m/km)

Temperature in degrees Celsius (°C) may be converted to degrees Fahrenheit (°F) as follows:

°F = (1.8 × °C) + 32.

Temperature in degrees Fahrenheit (°F) may be converted to degrees Celsius (°C) as follows:

°C = (°F – 32) + / 1.8
vii

Datum

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 

(NAVD 88).

Horizontal coordinate information is referenced to the North American Datum of 1983 (NAD 83).

Elevation, as used in this report, refers to distance above the vertical datum.

Abbreviations and Acronyms

DAR Drainage-Area Ratio, a PUB approach. Often combined with an index 

selection method (NN or MC).

FA Flow-Anywhere, a PUB approach

FDSS Fundamental Daily Streamflow Statistics

L-CV Second L-moment ratio 

L-Skew Third L-moment ratio

L-kurtosis Fourth L-moment ratio

MC Map Correlation, an index selection technique

NN Nearest Neighbor, an index selection technique

NSE Nash-Sutcliffe model efficiency. Also see with an appended L, denoting 

the use of logarithms.

PUB Prediciton in Ungaged Basins

QPPQ Nonlinear Spatial Interpolation using Flow Duration Curves, a PUB approach. 

Often combined with an index selection method (NN or MC).

PRMS Precipitation Runoff Modeling System

RRBE Robust, Rank-Based Evaluation

RMSE Root-Mean-Squared Error

RMSNE Root-Mean-Squared-Normalized Error

WaterSMART Water: Sustain and Manage America’s Resources for Tomorrow

Evaluation of Statistical and Rainfall-Runoff Models 

for Predicting Historical Daily Streamflow Time Series 

in the Des Moines and Iowa River Watersheds

By William H. Farmer, Rodney R. Knight, David A. Eash, Kasey J. Hutchinson, S. Mike Linhart, 

Daniel E. Christiansen, Stacey A. Archfield, Thomas M. Over, and Julie E. Kiang

ability to protect natural resources, design water abstractions 

for drinking or irrigation, develop hydropower resources, 

and protect against hydrologic extremes (Sivapalan, 2003; 

Sivapalan and others, 2003; Vogel, 2011; Hrachowitz and 

others, 2013). Unfortunately, the ability to collect long-range 

streamflow records is constrained by time and resources; as a 

result, the national network of streamgages is both spatially 

and temporally sparse (Kiang and others, 2013). For this reason, the exploration of tools for the prediction or estimation of 

daily streamflow records in unmonitored or ungaged locations 

is essential for an improved hydrologic understanding.

The problem of prediction in ungaged basins (PUB) has 

become an international research question (Sivapalan and 

others, 2003; Viglione and others, 2013). In the United States, 

the PUB problem has been of pivotal importance for initiatives 

like WaterSMART (Sustain and Manage America’s Resources 

for Tomorrow; [Salazar, 2010; Alley and others, 2013]) 

and the U.S. Geological Survey’s National Water Census 

(U.S. Geological Survey, 2007; Alley and others, 2013). The 

National Water Census seeks to provide spatially and temporally continuous records of daily streamflow across the United 

States, an endeavor that necessitates an immense amount of 

PUB. The work presented in this report is directly supported 

by the National Water Census and serves as a second case 

study of PUB methods in the United States.

Recently, Farmer and others (2014) completed an 

exploration of PUB methods in the southeastern United 

States. There, 19 variations of PUB methods were compared 

to estimate 30-year records of daily streamflow. Considering 

a wide-range of performance metrics, results indicated that 

several PUB methods performed reasonably well. On average, a transfer and transformation of flow duration curves 

served as the best approach to predicting daily streamflow 

records. The Southeast is a fairly humid region with generally 

homogeneous climate. Trends in precipitation and temperature are largely driven by elevation. The purpose of this 

report is to consider a similar model comparison in another 

hydroclimatic region.

Abstract

Daily records of streamflow are essential to understanding hydrologic systems and managing the interactions between 

human and natural systems. Many watersheds and locations 

lack streamgages to provide accurate and reliable records of 

daily streamflow. In such ungaged watersheds, statistical tools 

and rainfall-runoff models are used to estimate daily streamflow. Previous work compared 19 different techniques for 

predicting daily streamflow records in the southeastern United 

States. Here, five of the better-performing methods are compared in a different hydroclimatic region of the United States, 

in Iowa. The methods fall into three classes: (1) drainage-area 

ratio methods, (2) nonlinear spatial interpolations using flow 

duration curves, and (3) mechanistic rainfall-runoff models. 

The first two classes are each applied with nearest-neighbor 

and map-correlated index streamgages. Using a threefold 

validation and robust rank-based evaluation, the methods are 

assessed for overall goodness of fit of the hydrograph of daily 

streamflow, the ability to reproduce a daily, no-fail storageyield curve, and the ability to reproduce key streamflow statistics. As in the Southeast study, a nonlinear spatial interpolation 

of daily streamflow using flow duration curves is found to be 

a method with the best predictive accuracy. Comparisons with 

previous work in Iowa show that the accuracy of mechanistic 

models with at-site calibration is substantially degraded in the 

ungaged framework.

Introduction

Daily records of streamflow provide foundational 

records of hydrologic processes within gaged watersheds. 

These records are essential to a hydrologic understanding of 

natural systems. Furthermore, these records help in understanding how human systems intersect with and affect natural 

systems. An informed hydrologic understanding affects the 
2 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds

After the Southeast study (Farmer and other, 2014), 

the Des Moines and Iowa Rivers in Iowa were identified as 

another case study. The legacy of model comparison (Linhart 

and others, 2011 and 2013) allows for historical comparison 

as well as the implementation of newer tools. Iowa is substantially smaller than the southeastern region, but presents a 

number of other challenges: a different hydroclimatic regime, 

less variation in elevation, and a variety of different landform 

regions (Prior and others, 2009). The purpose of this Iowa 

model comparison presented in this report is to explore the 

advantages and disadvantages of several PUB methods that 

performed well in the Southeast model comparison.

Purpose and Scope

This report presents a comparison of five different methods for predicting continuous daily records of streamflow in 

the Des Moines and Iowa River Basins (fig. 1). Each prediction method is assessed for its overall accuracy of predicting a hydrograph of daily discharge, its ability to reproduce 

daily, no-fail storage yield curves, and its ability to reproduce 

key observed streamflow statistics. Hydrographs of daily 

discharge are estimated at 44 sites from October 1, 1983, 

through September 30, 2011. The PUB methods are compared 

using Robust Rank-Based Evaluation and the cumulative 

Figure 1. Location of the Iowa and Des Moines River Basins, study area, and landform regions in Iowa.

4

4

o

Iowa

River

Basin

Des Moines

River

Basin

Cedar

River

Basin

Cedar River

Iowa River

Des Moines River

Columbus 

Junction

O

o

 

De s Moines River

Iowa Riv er

Cedar River

Mississi ppi River

Cedar River Basin

Des Moines River Basin

Iowa River Basin

0 25 50 MILES

0 25 50 KILOMETERS

Base from U.S. Geological Survey digital data,

1:2,000,000, 1979

Universal Transverse Mercator projection,

Zone 15

North American Datum of 1983 (NAD 83)

EXPLANATION

Des Moines Lobe

East-Central Iowa Drift Plain

Iowa-Cedar Lowland

Iowan Surface

Missouri River Alluvial Plain

Northwest Iowa Plains

Paleozoic Plateau

Loess Hills Southern Iowa Drift Plain

Mississippi River Alluvial Plain

MINNESOTA

WISCONSIN

ILLINOIS

MISSOURI

NEBRASKA

SOUTH

DAKOTA

96° 95° 94° 93° 92°

91°

43°

42°

41°

Oakville

Columbus

Junction

Keokuk
Methods to Estimate Daily Streamflow 3

distribution of absolute errors. The results provide a compliment to the Southeast model comparison. The purpose of the 

Iowa model comparison presented in this report is to, in a 

different hydroclimatic region, to explore the advantages and 

disadvantages of several PUB methods that performed well in 

the Southeast model comparison.

Study Area and Data Selection

The Iowa and Des Moines Rivers generally flow in a 

southeasterly direction to join the Mississippi River along 

the eastern border of Iowa (fig. 1). (A list of all streamgages 

evaluated in this study is included in Appendix 1) The Iowa 

River Basin drains approximately 12,640 square miles at the 

confluence with the Mississippi River near Oakville, Iowa. 

This includes the Cedar River Basin, which constitutes approximately 62 percent of the Iowa River Basin (Larimer, 1957). 

The Cedar River Basin extends from its headwaters in southern 

Minnesota to its confluence with the Iowa River near Columbus 

Junction in east-central Iowa. The Des Moines River Basin 

drains almost 14,700 square miles at the confluence with the 

Mississippi River near Keokuk, Iowa (Larimer, 1957). Collectively, the Iowa and Des Moines River Basins drain approximately 27,300 square miles at their respective confluences 

with the Mississippi River. The Iowa and Des Moines River 

Basins cross 5 of the 10 distinct landform regions in Iowa (Prior 

and others, 2009), including the Des Moines Lobe, the Iowan 

Surface, the Southern Iowa Drift Plain, and the Iowa-Cedar 

Lowland, with a small portion of the Des Moines River Basin 

draining the Northwest Iowa Plains.

The dominant land use in the Des Moines and Iowa River 

Basins is row-crop agriculture (67 and 73 percent, respectively). Corn and soybean production represents approximately 

60 percent of the row-crop agriculture, with the remaining 

portion distributed across noncropland covers (Falcone, 2011). 

Artificial drainage networks, including subsurface tile and open 

ditches, are extensively used in Iowa to remove excess water 

and support row-crop agriculture. Other land use throughout these basins includes confined and unconfined livestock 

operations for beef and dairy cattle, hogs, sheep, and poultry 

(Iowa Department of Natural Resources, 2006). For the period 

from 1971 to 2000 average annual precipitation was 33 inches 

in the Des Moines Basin and 34.7 inches in the Iowa River 

Basin. Average annual temperature for the same period in the 

Des Moines River Basin was 47.6 °Fahrenheit (F) (minimum 

of 37.2 °F; maximum of 58.73 °F), while the average annual 

temperature in the Iowa River Basin was 47 °F (minimum of 

36.9 °F; maximum of 57.9 °F) (Falcone, 2011).

This study uses the same 44 streamgages in the Iowa and 

Des Moines River Basins that were used in a previous statewide comparison of PUB methods in Iowa (Linhart and others, 

2012). The locations of the 44 streamgages are shown in figure 2. Appendix 2 lists the basin-characteristic values that were 

measured for these 44 streamgages as part of a peak-flow study 

for Iowa (Eash and others, 2013) and were used for the development of 27 regression equations in this study.

Methods to Estimate Daily Streamflow

Three approaches to PUB were used to estimate daily 

streamflow records, including a drainage-area ratio method, 

a nonlinear spatial interpolation using flow duration curves, 

and the Precipitation Runoff Modeling System (PRMS). The 

former two are direct-transfer, statistical models that do not 

explicitly rely on a process-understanding of basin hydrology. 

The PRMS, on the other hand, models the physical processes of 

streamflow generation in gaged basins and transfers that physical understanding to ungaged locations. All methods require 

the selection of an index streamgage from which to transfer the 

physical or statistical understanding to the ungaged location. 

Two techniques for the selection of index streamgages, nearest 

neighbor and map correlation, were used. A description of the 

index selection techniques and PUB approaches are provided in 

the Selection of Index Streamgages section of this report. The 

coupling of an index-selection technique and a PUB approach, 

both of which are required to produce estimates of streamflow, 

is termed a PUB method. Farmer and others (2014) and Farmer 

(2015) implemented these methods in the southeastern United 

States and provide further information on the development and 

application of these methods.

Selection of Index Streamgages

All of the PUB approaches considered here require the 

selection of an index streamgage from which to transfer hydrologic information to an ungaged location. Two techniques for 

the selection of index streamgages, nearest neighbor and map 

correlation, were used for each of the direct-transfer, statistical 

approaches, drainage area ratio-, and nonlinear spatial interpolation using flow duration curves. A modified version of the 

nearest-neighbor selection technique was used with the PRMS. 

The combinations of index selection techniques and PUB 

approaches define five PUB methods and yields five unique 

estimates of daily streamflow at each location.

An index streamgage is any streamgage from which 

hydrologic information is transferred. In the nearest-neighbor 

technique, the optimal index streamgage is the streamgage 

geographically closest to the ungaged location of interest. Map 

correlation (Archfield and Vogel, 2010) uses geostatistical tools 

to identify the streamgage with the greatest estimated correlation with the anticipated daily record at the ungaged location 

and selects it as the optimal index streamgage. As described by 

Archfield and Vogel (2010), map correlation proceeds by fitting 

a regional variogram for each site in the network. Each variogram, which is tied to a specific site, estimates the Pearson correlation coefficient between the streamflow at the linked site and 

the records at any other location. By inputting the coordinates of 

the an ungaged site into each of the site-specific variograms, it 

is possible to determine which network streamgage is expected 

to provide the greatest level of correlation with the ungaged 

site; this site is selected as the optimal index streamgage. 

(Appendix 3 provides the leave-one-out validation root-meansquared errors of map correlation, indicating the applicability 
4 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds

of this index streamgage selection technique in this region. The 

average root-mean-squared error is 0.0274.) For both nearest 

neighbor and map correlation, the location of each stream site, 

gaged or ungaged, is defined by the latitude and longitude of the 

basin outlet. Both nearest neighbor and map correlation were 

coupled with each of the direct-transfer statistical methods. For 

the PRMS, the nearest-neighbor procedure was amended: Only 

streamgages with an at-site calibration of the PRMS producing a Nash-Sutcliffe model efficiency (Nash and Sutcliffe, 

1970) greater than 0.80 were considered as potential index 

streamgages. From this subset of streamgages, the optimal index 

streamgage was selected by spatial proximity.

Unlike the PRMS, which transfers model parameters 

rather than time series, the direct-transfer, statistical PUB 

approaches require observations from an index streamgage for 

each day of prediction. If the most optimal index streamgage, 

based on spatial proximity or estimated correlation, is not 

operational on the target date, then an alternative must be 

selected for that date. This was accomplished by ranking the 

potential index streamgages based on spatial proximity or 

estimated correlation. Farmer and others (2014) provide an 

illustrative example. The date-range gaps are then filled by 

working successively down the ranked population of potential 

index streamgages.

Figure 2. Location of 44 streamgages within the Iowa and Des Moines River Basins included in this study.

o

Des Moines

River Basin

Iowa River Basin

MINNESOTA

WISCONSIN

ILLINOIS

MISSOURI

NEBRASKA

SOUTH

DAKOTA

96°

43°

42°

41°

95° 94° 93° 92°

91°

Base from U.S. Geological Survey digital data,

1:2,000,000, 1979

Universal Transverse Mercator projection,

Zone 15

North American Datum of 1983 (NAD 83)

#

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15

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1

22

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44

42 43 41

40 39

33

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9

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EXPLANATION

Scenario 1 validation site and map number

Scenario 2 validation site and map number

Scenario 3 validation site and map number

0

0

25

25 50 MILES

50 KILOMETERS
Methods to Estimate Daily Streamflow 5

Drainage-Area Ratio

The prediction of streamflow records using the 

drainage-area ratio (DAR) PUB approach has been widely 

practiced in the fields of hydrology and water resources 

engineering (Hirsch, 1979; Asquith and others, 2006). The 

DAR method is predicated on the assumption that streamflow 

per unit area is equivalent across hydrologically similar sites. 

Hydrologic similarity can be defined several ways; here it is 

approximated by the selection of an optimal index streamgage. 

With an index streamgage, the ungaged streamflow on day t

can be approximated as:

ˆQ , ,

A

A u t Q u

g

g t = ⋅ (1)

where Qˆ

u,t is the estimated streamflow at the ungaged location on day t, Qg,t is the corresponding measured streamflow 

at the index streamgage and Au

 and A

g

 are the drainage areas 

associated with the ungaged and gaged locations, respectively. 

The DAR approach is the foundation for the Flow Anywhere 

method presented and assessed by Linhart and others (2012). 

Here, the application of DAR with nearest-neighbor index 

streamgages is noted as DAR-NN, while the application with 

map-correlation index streamgages is noted as DAR-MC.

Nonlinear Spatial Interpolation Using Flow 

Duration Curves

Where the DAR approach seeks to scale the daily streamflow by drainage area, the approach of nonlinear spatial interpolation using flow duration curves can be thought of as a 

statistical scaling of the entire flow duration curve at each site. 

This approach has been widely used (Fennessey, 1994; Hughes 

and Smakhtin, 1996; Smakhtin, 1999; Mohamoud, 2008; 

Archfield and others, 2010; Shu and Ouarda, 2012) and shown 

to perform well across a range of performance metrics (Farmer 

and others, 2014). As noted by Farmer and others (2014), this 

approach has gone by many names; here the nearest-neighbor 

implementation is abbreviated QPPQ-NN, while the mapcorrelation implementation is noted as QPPQ-MC. The abbreviation of QPPQ, as described in the next paragraph, refers to 

the use of an index discharge (Q) to produce an index probability (p), which yields an estimated, ungaged probability (p) and 

produces an estimate of ungaged discharge (Q).

QPPQ predicts ungaged streamflow records by assuming 

that the exceedance probabilities of streamflow on a particular 

day are identical between two hydrologically similar sites:

Pu,t = P(Qu ≥ Qu,t) = P(Qg ≥ Qg,t) = Pg,t (2)

where Qu,t and Qg,t are the ungaged and index streamflows on 

day t, respectively, and Pu,t and Pg,t are the complementary 

cumulative probabilities (or exceedance probabilities) of the 

ungaged and index streamflow. By this method, the streamflow 

at the index streamgage (Qg,t) is converted to a complimentary 

cumulative probability (Pg,t) using an empirical flow duration 

curve. Using the assumption of equation (2), this exceedance probability is then transferred to the ungaged site as an 

estimate of the ungaged complimentary cumulative probability 

(Pˆ

u,t). The estimate exceedance probability is then converted 

into an estimate of ungaged streamflow (Qˆ

u,t) by means of an 

estimated flow-duration curve at the ungaged site.

For this application of QPPQ, as in Farmer and others 

(2014), the estimated flow duration curve at the ungaged site 

was obtained by regional regressions of 27 key exceedance 

probabilities. The exceedance probabilities considered were 

0.02, 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 25, 30, 40, 50, 60, 70, 

75, 80, 90, 95, 98, 99, 99.5, 99.8, 99.9, 99.95, and 99.98 percent. Defining the entire study area as a single region, regional 

regressions for each exceedance probability were developed 

in the same way as Farmer and others (2014): The streamflow 

quantiles associated with each exceedance probability at each 

gaged location were identified empirically using the Blom 

plotting position (Stedinger and others, 1983). Several basin 

characteristics (Appendix 2) were considered as explanatory variables in a log-linear regression framework. The best 

regional regressions for each quantile are summarized in 

table 1. Using these regressions, the 27 quantiles composing 

the flow-duration curve were estimated at an ungaged site. 

In the prediction of streamflow values, log-linear interpolation was used for estimated exceedance probabilities falling between any two of the 27 exceedance probabilities. 

For extrapolations beyond the 27 exceedance probabilities, 

streamflows with complimentary cumulative probabilities 

less than 0.02 percent and more than 99.98 percent, the QPPQ 

approach defaulted to the DAR approach. Additional details 

on the development of this regression approach are provided 

in Farmer and others (2014).

Precipitation Runoff Modeling System

Unlike the statistical approaches, DAR and QPPQ, the 

Precipitation-Runoff Modeling System (PRMS; Leavesley 

and others, 1983; Markstrom and others, 2008; 2014) provides 

a physically-based, mechanistic model of streamflow. As 

described in Farmer and others (2014), the PRMS is a modular, deterministic, distributed-parameter watershed model that 

was designed to simulate, as driven by climatic inputs, landsurface hydrologic processes and water budgets at various 

temporal scales. The PRMS models aggregate watersheds by 

representing individual stream segments and unique, homogenous hydrologic response units. The PRMS was calibrated for 

gaged watersheds using the Shuffled Complex Evolution algorithm (Duan and others, 1992, 1993, and 1994), which allowed 

adjustment of the initial calibration parameters derived 

from basin characteristics to account for the idiosyncrasies 

of specific hydrologic response units. In ungaged basins, 

the adjustments from initial parameters, as opposed to the 
6 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds

parameter values themselves, were transferred from the index 

streamgage and used to adjust the initial parameters in the 

ungaged watershed. Farmer and others (2014) provide a more 

detailed description of this ungaged application of the PRMS; 

their approach was followed in this report. This approach is 

only a first-order approach to the application of the PRMS in 

ungaged watersheds and differs significantly from the at-site 

calibrated application of Linhart and others (2013).

Methods of Analysis

The performance of the PUB methods was assessed using 

robust rank-based evaluation (RRBE), a technique presented 

in Farmer and others (2014) and refined in Farmer (2015). 

RRBE blends several measures of performance to provide 

a broad assessment of the relative accuracy of streamflow 

predictions. This model comparison of PUB methods in Iowa 

considered the same goodness-of-fit metrics used by Farmer 

and others (2014). Additionally, the cumulative distributions of daily percent errors were used to further distinguish 

between PUB methods. All analyses were conducted using 

a threefold validation: two-thirds of the sites were used to 

calibrate the PUB methods used to predict streamflow at the 

remaining, validation one-third of the sites. In this manner, 

the neglected one-third of the network is modeled without any 

effect from the calibration, just as if it were truly ungaged. The 

calibration and validation sets were iterated so that an ungaged 

estimate was created for each site. The definitions of scenarios 

are provided in figure 2, with particular streamgages noted in 

Appendix 1.

Thirty-two performance metrics were used for the RRBE. 

The overall goodness of fit was assessed with seven metrics, 

including the Nash-Sutcliffe of the untransformed (NSE) 

and natural-logarithmically transformed (NSEL) streamflow 

predictions, the root-mean-square error in streamflow predictions (RMSE), the root-mean-square-normalized error in 

streamflow predictions (RMSNE), the average percent error 

in streamflow estimates, the Pearson correlation coefficient 

between observed and predicted streamflow and the Spearman 

correlation coefficient between observed and predicted streamflow. The daily, no-fail storage-yield curve indicates how 

much storage would be required, given historical properties 

Table 1. Regional regressions of 27 flow percentiles along the daily flow duration curve for the full set of 44 streamgages in the 

Des Moines and Iowa Rivers.

[NA, not applicable. Shaded rows indicate left-censored regression. Basin characteristics considered as potential explanatory variables are listed in Appendix 2]

Flow

percentile

(percent)

Equation

Number of

censored

streamgages

Standard error

of prediction

(percent)

Standard error of 

estiamte divied by 

root-mean-squared error 

(percent)

0.02 10–2.201 DRNAREA0.760 JULYAVEPRE6.520 CCM–0.507 0 19.04 18.23

0.05 10–2.114 DRNAREA0.770 JULYAVEPRE6.431 (DESMOIN+1)–0.064 0 13.11 12.57

0.1 10–3.049 DRNAREA0.763 JULYAVEPRE5.827 CLAY0.835 0 14.12 13.52

0.2 10–3.097 DRNAREA0.798 JULYAVEPRE5.511 CLAY0.875 0 12.63 12.10

0.5 10–2.704 DRNAREA0.842 JULYAVEPRE4.316 CLAY0.950 0 11.40 10.92

1 10–0.805 DRNAREA0.882 100.368(JULYAVEPRE) 100.014(CLAY) 0 11.78 11.28

2 10–0.642 DRNAREA0.926 100.289(JULYAVEPRE) 100.011(CLAY) 0 11.91 11.40

5 10–0.028 DRNAREA0.967 100.117(JUNEAVPRE) 100.013(SOILD) 0 9.16 8.79

10 10–0.462 DRNAREA1.019 SOILC–0.053 PRC81.103 0 8.65 8.29

20 10–0.481 DRNAREA1.020 10–0.003(SOILC) 100.110(PRC8) 0 8.57 8.21

25 10–0.654 DRNAREA1.017 10–0.004(SOILC) 100.131(PRC8) 0 8.64 8.27

30 10–0.857 DRNAREA1.022 10–0.004(SOILC) 100.155(PRC8) 0 8.86 8.48

40 10–1.221 DRNAREA1.028 10–0.004(SOILC) 100.199(PRC8) 0 11.40 10.90

50 10–1.281 DRNAREA1.027 100.018(HYSEP) 10–0.320(CCM) 0 14.60 14.00

60 10–1.684 DRNAREA1.047 100.023(HYSEP) 10–0.421(CCM) 0 18.90 18.10

70 10–2.379 DRNAREA1.089 100.028(HYSEP) 10–0.003(DESMOIN) 0 21.96 21.04

75 10–2.678 DRNAREA1.126 100.030(HYSEP) 10–0.004(DESMOIN) 0 25.15 24.09

80 100.534 DRNAREA1.125 10–3.148(STREAM_VAR) (SOILD+1)–0.202 0 29.52 28.23

90 10–3.636 DRNAREA1.281 103.445(BFI) 10–0.005(DESMOIN) 0 43.41 41.47

95 10–6.699 DRNAREA1.244 100.086(TAU) 100.512(JUNEAVPRE) 0 59.44 56.68

98 10–6.963 DRNAREA1.301 100.095(TAU) 100.443(JUNEAVPRE) 2 NA 66.11

99 10–5.038 DRNAREA1.320 100.093(TAU) 100.049(SOILA) 2 NA 78.82

99.5 10–5.214 DRNAREA1.301 100.096(TAU) (SOILA+1)0.423 3 NA 86.40

99.8 10–11.562 DRNAREA1.306 TAU6.161 (SOILA+1)0.472 3 NA 90.53

99.9 10–12.636 DRNAREA1.301 TAU6.852 (SOILA+1)0.552 6 NA 109.86

99.95 100.368 DRNAREA1.377 10–6.138(STREAM_VAR) 6 NA 116.00

99.98 100.374 DRNAREA1.403 10–6.380(STREAM_VAR) 8 NA 129.70
Results and Discussion 7

of streamflow, to produce a constant yield from the stream. 

As an indication of streamflow signature, the reproducibility 

of the daily, no-fail storage-yield curve was assessed by applying the same seven metrics to predictions of the daily, no-fail 

storage. Farmer and others (2014) detail how the daily, no-fail 

storage-yield curve was estimated. In addition to overall 

goodness of fit and streamflow signature, the reproducibility 

of several streamflow statistics was assessed by observing the ratio of estimated to observed streamflow statistics; 

these statistics included the coefficient of variation of annual 

streamflow, the coefficient of variation of daily streamflow, 

the 10th and 50th percentiles of the empirical distribution of 

7-day-average annual minimum flows, the 90th percentile 

of the empirical distribution of annual maximum flows, the 

10th, 25th, 50th, 75th and 90th percentiles of the distribution 

of daily streamflow (note that annual minima and maxima 

were assessed using empirical distributions; no parametric 

fitting was applied). This analysis also considered the reproducibility of the seven fundamental daily streamflow statistics, 

consisting of the mean streamflow, coefficient of variation 

as an L-moment ratio (L-CV), skewness (L-skew), kurtosis 

(L-kurtosis) and autoregressive lag-one autocorrelation coefficient of daily streamflow, and the amplitude and the phase 

of the sinusoidal seasonal signal (Archfield and others, 2013). 

To strengthen the weight of the fundamental daily streamflow 

statistics, a RMSNE was calculated across all seven statistics. 

Despite the strong correlation between some performance 

metrics, like the RMSE and NSE, these metrics were included 

because they are commonly considered and quantify extremely 

important aspects of the model performance. As reported by 

Farmer and others (2014), this did not hamper the analysis, but 

alternative collections of metrics can be similarly compared.

The performance metrics were computed only on complete water years between October 1, 1982 and September 30, 

2011. Zero-flows were included with a censoring value. Most 

of the statistics used complete water years, but the two percentiles of the distribution of the 7-day-average annual minimum 

flow were computed with complete climatic years. Climatic 

years are defined such that they begin on April 1 and end on 

March 31. (The year between April 1, 1983 and March 31, 

1984, is referred to as the 1984 climatic year.) Because the 

percent errors and logarithmic performance metrics cannot 

be applied to zero-valued flows, the zero value was censored. 

The smallest non-zero flow published by the U.S. Geological 

Survey (USGS) is 0.01 cubic foot per second (ft3

/s). For this 

reason, any predictions on the inclusive range between 0.005 

and 0.01 ft3

/s were corrected to 0.01 ft3/s. Predictions less 

than 0.005 ft3

/s were set to a censored value of 0.001 ft3/s. 

Observations of 0 ft3

/s were censored to 0.001 ft3/s. While 

the choice of a censoring value may affect the performance 

metrics, only 0.07 percent of all the days at all the sites had 

observations with a streamflow value of zero. This fraction 

was considered to be small enough so as not to significantly 

affect the analysis.

RRBE combines the relative rankings of the PUB 

methods according to each performance metric. For a particular performance metric at a particular site, the five PUB 

methods were ranked, with best method receiving the lowest 

rank. Ties were ranked identically with the lowest numerical 

ranking available. This process was repeated for each site. The 

ranks for a particular PUB method were summarized across 

all sites, taking the average ranking and the standard deviation of the ranks. This ranking procedure was then conducted 

for each additional performance metric. The result was an 

average ranking and standard deviation of the ranks of each 

PUB method for each performance metric. Each PUB method 

can then be represented as a cloud of points in Cartesian space 

with the cross-site average ranking for each metric on one 

axis and the cross-site standard deviation of the ranks for each 

metric on the other axis. For a particular PUB method, this 

cloud of performance metrics can be summarized by averaging 

to obtain the centroid of the cloud. An indication of the spread 

of this cloud, for a particular PUB method, can be obtained by 

wrapping a variability ellipse around the centroid. The axes of 

the ellipse are defined by the cross-metric standard deviation 

among the cross-site averages and the cross-site standard deviations. Farmer (2015) provides examples of this analysis. The 

most optimal PUB methods are those with a centroid closest to 

the origin, representing the best average cross-site ranking and 

the least cross-site variability. Furthermore, the ideal method 

should have a smaller variability ellipse, indicating only a low 

level of variability across performance metrics.

The cumulative distribution of absolute percent errors 

in daily streamflow gives an indication of the degree of error 

in streamflow predictions from a particular PUB method. At 

a particular site, the absolute percent error of each day in the 

validation record was calculated (the validation record refers 

to complete water years for which observations and predictions are available). The distribution of errors is represented as 

the fraction of the validation record below a certain threshold. 

The cumulative distribution is obtained by increasing the 

threshold. The best PUB methods would show a steep rise 

in the percentage of record below an error threshold for low 

threshold values. This analysis was conducted for all sites in 

the study and the mean percent of the record below a threshold 

was documented. Because of the use of error thresholds, this 

technique is referred to herein as a threshold analysis.

Results and Discussion

The mean of the ranks of each PUB method according to each performance metric are shown in table 2. (The 

distributions of each performance metric are provided in 

Appendix 4.) Except for some metrics where the mapcorrelation (MC) application of QPPQ is marginally better 

when compared to its nearest-neighbor (NN) complement, 

the nearest-neighbor applications have a better average 

ranking. QPPQ-MC performed better than QPPQ-NN in 
8 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds Table 2. Mean rank for each performance metric and method of prediction in ungaged basins (PUB). [At each site, the PUB methods were ranked according to a single metric, with the best method receiving the lowest rank. These ranks were then averaged across sites for each PUB method. The process was repeated for each of the 32 metrics and 5 PUB methods. The PUB methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC) and nonlinear spatial interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC)] Class Metric PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC Overall goodness of fit Nash-Sutcliffe efficiency of streamflows 2.64 2.64 2.70 3.02 3.23 Nash-Sutcliffe efficiency of the logarithms of streamflow 4.98 1.95 2.09 2.66 2.55 Root-mean-squared error of streamflow 2.64 2.64 2.70 3.02 3.23 Root-mean-square-normalized error of streamflow 4.91 2.18 1.80 2.82 2.52 Average percent error of streamflow 4.64 2.59 2.05 2.84 2.11 Pearson correlation between observed and simulated streamflow 2.93 2.55 2.73 2.89 3.14 Spearman correlation between observed and simulated streamflow 5.00 1.50 2.23 2.27 3.16 Average 3.96 2.29 2.33 2.79 2.85 Standard deviation 1.16 0.44 0.38 0.26 0.45 Storage-yield curve (SYC) goodness of fit Nash-Sutcliffe efficiency of required storage 4.41 2.27 2.39 2.68 2.48 Nash-Sutcliffe efficiency of the logarithms of required storage 4.57 2.34 2.20 2.86 2.18 Root-mean-squared error of required storage 4.41 2.27 2.39 2.68 2.48 Root-mean-square-normalized error of required storage 4.73 2.48 2.11 2.84 2.07 Average percent error of required storage 4.70 2.45 2.07 2.89 2.11 Pearson correlation between observed and simulated required storage 4.32 2.09 2.34 2.50 2.98 Spearman correlation between observed and simulated required storage 1.55 1.00 1.05 1.05 1.09 Average 4.10 2.13 2.08 2.50 2.20 Standard deviation 1.14 0.51 0.47 0.66 0.58 Percent error in streamflow statistics Root-mean-square-normalized error of fundamental daily streamflow statistics 4.09 2.57 2.07 3.25 2.25 Coefficient of variation of annual streamflow 3.86 2.50 2.39 2.61 2.86 Coefficient of variation of daily streamflow 3.48 2.82 2.52 3.05 2.36 10th percentile, 7-day average annual-minimum streamflow 3.86 2.82 2.09 3.30 2.11 50th percentile, 7-day average annual-minimum streamflow 4.43 2.75 2.02 2.98 2.02 90th percentile, annual-maximum streamflow 2.89 3.14 2.25 3.64 2.27 90-percent-exceedance streamflow 3.89 2.84 2.39 2.93 2.14 75-percent-exceedance streamflow 3.82 2.59 2.36 3.14 2.30 50-percent-exceedance streamflow 3.68 2.27 2.80 2.64 2.80 25-percent-exceedance streamflow 3.32 2.23 2.98 2.64 3.00 10-percent-exceedance streamflow 3.64 2.43 2.68 2.77 2.68 Mean streamflow 2.66 2.32 3.18 2.77 3.30 L-CV of daily streamflow 3.70 2.86 2.34 3.09 2.23 L-skewness of daily streamflow 4.16 2.45 2.16 3.25 2.20 L-kurtosis of daily streamflow 3.82 2.34 2.36 3.18 2.52 Lag-1 autocorrelation of daily streamflow 1.93 2.93 2.93 3.23 3.20

Amplitude of seasonal trend in daily streamflow 3.07 2.75 2.43 3.41 2.57

Phase of seasonal trend in daily streamflow 4.57 1.84 2.64 2.34 2.84

Average 3.60 2.58 2.48 3.01 2.54

Standard deviation 0.65 0.32 0.33 0.33 0.39

All metrics Average 3.79 2.42 2.36 2.85 2.53

Standard deviation 0.89 0.42 0.40 0.45 0.48
Results and Discussion 9

logarithmic NSE of the daily, no-fail storage-yield curve and 

with regard to the coefficient of variation of streamflows and 

low-flow metrics including the 10th and 25th percentiles of 

daily streamflow. Previously, Farmer and others (2014) found 

a much larger difference between the two methods for the 

selection of index streamgages, but, for this study, the slight 

advantage of the NN method is consistent with their findings. 

For the majority of the performance metrics, QPPQ-NN is 

ranked more favorably than DAR-NN. However, DAR-NN 

yielded a better average ranking with the NSE, NSEL, RMSE 

and correlations of streamflow. Additionally, DAR-NN more 

accurately estimated the higher percentiles of daily streamflow, while QPPQ-NN was better at reproducing the variability of streamflow and the low-flow behavior.

The PRMS shows a consistently poor ranking across 

almost all metrics, indicating that the process-based approach 

does not perform as well in this region. This finding is again 

consistent with the findings of Farmer and others (2014). The 

PRMS does provide a strong ranking with respect to the NSE 

and RMSE of streamflows. Since the PRMS was calibrated 

at reference sites using the NSE, this strong ranking is not 

surprising. More interesting is the superiority of the PRMS in 

reproducing the lag-one correlation of streamflow. This provides evidence that the mechanistic, process-based structure of 

the PRMS is accurately reproducing the at-site lag behavior. 

The transfer-based methods may not be reproducing the at-site 

lag behavior as well because they rely on the lag structure of 

an index streamgage alone.

The standard deviations of the ranks of each PUB method 

according to each performance metric (table 3) indicate the 

consistency of the performance for each PUB metric. Across 

all the metrics, the PRMS and DAR-MC show the most variability. For some metrics, like RMSNE and average percent 

error of streamflow, the PRMS is the least variable. However, 

the QPPQ methods show the least variability when averaged 

across all metrics, and, coupled with the average performance 

presented earlier, QPPQ-NN shows the most consistent (leastvariable) accuracy.

The information in tables 2 and 3 is summarized in an 

RRBE cloud with variability ellipses in figure 3. QPPQ-NN 

demonstrates the best and most consistent performance out of 

these five methods. The point representing QPPQ-NN is on 

the optimal edge of the RRBE cloud, having the best average 

ranking and the smallest average standard deviation. DAR-NN 

has slightly more variability with an increased average rank. 

QPPQ-NN also has the smallest variability ellipse. QPPQ-MC 

shows a more flattened ellipse, demonstrating that QPPQ-MC 

has a more consistent amount of variability with a greater variability in the mean across metrics. The ellipses demonstrate the 

substantial overlap between the methods. The substantial overlap between QPPQ-NN and QPPQ-MC shows that the QPPQ 

method is not overly sensitive to the index-selection algorithm. 

The smaller overlap between DAR-NN and DAR-MC shows 

that DAR is more sensitive to the index-selection algorithm. 

Finally, DAR-NN and QPPQ-NN, the two best methods, show 

a high degree of overlap. While QPPQ-NN appears to be better 

overall, DAR-NN may be more appropriate in some situations. 

The transfer-based methods are strongest and have similarlysized variability ellipses, but the PRMS is set back much further 

and has a large variability ellipse.

The threshold analysis in figure 4 again shows that the 

transfer-based methods are superior to the ungaged mechanistic 

model. The curves represent the mean, across sites, percent of 

the validation record that has an absolute percent error below 

the error threshold. For all error thresholds, the transfer-based 

methods have a greater portion of the validation record below 

the threshold. The curves of the transfer-based methods rise 

more quickly and begin to level out at a higher proportion than 

the PRMS. For the transfer-based methods, nearly 50 percent of 

the validation record is less than the 25 percent error threshold, 

compared to only 26 percent of the validation record for the 

PRMS. At an error threshold of 100 percent of the observations, 

more than 90 percent of the validation record is less than the 

threshold for transfer-based methods; only 75 percent for the 

PRMS. The transfer-based methods are clustered quite closely, 

but QPPQ-NN is on the leading edge. For the validation record, 

90 percent of the QPPQ-NN predictions, on average, are less 

than the 80 percent threshold. As these are just averages, figure 5 

characterizes the variability of each PUB method at each threshold by displaying the coefficient of variation of the percentage 

of the validation record less than the error threshold across sites. 

The variability decreases for higher thresholds and the PRMS 

shows a greater variability than the transfer-based methods.

The transfer-based methods are able to reproduce daily 

streamflow records more accurately than the ungaged application of the PRMS, a mechanistic model. This result agrees 

with the findings of Farmer and others (2014) in the southeastern United States. Linhart and others (2013), in contrast, 

showed that a fully-calibrated, at-site application of processbased models, a technique that could not be applied at ungaged 

locations, out-performed regionally-calibrated, transfer-based 

methods in the same region as this study. Linhart and others 

(2013) considered both QPPQ and the Flow-Anywhere (FA) 

method, a calibrated version of the DAR method, but did not 

consider a validation framework. Linhart and others (2012) 

demonstrated that their at-site calibration of PRMS showed a 

greater NSE than the transfer-based methods. From Linhart 

and others (2013), PRMS had a median NSE of 0.66, while FA 

and QPPQ had medians of 0.37 and 0.55, respectively. Though 

a larger and different set of sites was used in this report, the 

PRMS had a median of 0.69, while DAR-NN and QPPQ-NN 

yielded NSE values of 0.65 and 0.67, respectively. The relative 

comparison is the same, but the results in table 2 show that the 

consideration of additional metrics, an analysis which Linhart 

and others (2013) call for, does not favor the performance 

of the PRMS. This suggests that the model structure of the 

PRMS is sufficient for reproducing fairly accurate, independent, day-to-day predictions, but, when run in an ungaged 
10 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds Table 3. Standard deviation of the ranks for each performance metric and method of prediction in ungaged basins (PUB). [At each site, the PUB methods were ranked according to a single metric, with the best method receiving the lowest rank. These standard deviations were taken across 44 sites for each PUB method. The process was repeated for each of the 32 metrics and 5 PUB methods. The PUB methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC) and nonlinear spatial interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC)] Class Metric PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC Overall goodness of fit Nash-Sutcliffe efficiency of streamflows 1.66 1.24 1.32 1.39 1.29 Nash-Sutcliffe efficiency of the logarithms of streamflow 0.15 1.14 0.88 1.27 1.00 Root-mean-squared error of streamflow 1.66 1.24 1.32 1.39 1.29 Root-mean-square-normalized error of streamflow 0.60 0.99 1.07 1.17 1.00 Average percent error of streamflow 0.94 1.19 1.01 1.35 1.04 Pearson correlation between observed and simulated streamflow 1.70 1.32 1.23 1.45 1.27 Spearman correlation between observed and simulated streamflow 0.00 0.85 0.89 1.11 0.91 Average 0.96 1.14 1.10 1.30 1.11 Standard deviation 0.73 0.16 0.19 0.13 0.16 Storage-yield curve (SYC) goodness of fit Nash-Sutcliffe efficiency of required storage 1.19 1.21 1.10 1.39 1.19 Nash-Sutcliffe efficiency of the logarithms of required storage 0.76 1.22 1.07 1.39 1.13 Root-mean-squared error of required storage 1.19 1.21 1.10 1.39 1.19 Root-mean-square-normalized error of required storage 0.54 1.28 0.99 1.31 1.11 Average percent error of required storage 0.59 1.27 1.04 1.28 1.13 Pearson correlation between observed and simulated required storage 1.27 1.20 1.16 1.27 1.19 Spearman correlation between observed and simulated required storage 1.32 0.00 0.30 0.30 0.42 Average 0.98 1.05 0.97 1.19 1.05 Standard deviation 0.34 0.47 0.30 0.40 0.28 Percent error in streamflow statistics Root-mean-square-normalized error of fundamental daily streamflow statistics 1.44 1.25 0.93 1.38 1.12 Coefficient of variation of annual streamflow 1.59 1.27 1.32 1.33 1.19 Coefficient of variation of daily streamflow 1.55 1.45 1.09 1.63 1.10 10th percentile, 7-day average annual-minimum streamflow 1.46 1.28 0.96 1.44 1.10 50th percentile, 7-day average annual-minimum streamflow 1.13 1.26 1.13 1.28 0.93 90th percentile, annual-maximum streamflow 1.56 1.19 1.20 1.30 1.32 90-percent-exceedance streamflow 1.51 1.36 1.26 1.40 1.07 75-percent-exceedance streamflow 1.48 1.39 1.26 1.39 1.09 50-percent-exceedance streamflow 1.68 1.30 1.11 1.46 1.15 25-percent-exceedance streamflow 1.60 1.27 1.27 1.40 1.28 10-percent-exceedance streamflow 1.64 1.30 1.25 1.43 1.23 Mean streamflow 1.45 1.25 1.37 1.46 1.34 L-CV of daily streamflow 1.55 1.32 1.12 1.60 1.08 L-skewness of daily streamflow 1.22 1.25 1.10 1.30 1.21 L-kurtosis of daily streamflow 1.35 1.36 1.14 1.51 1.19 Lag-1 autocorrelation of daily streamflow 1.35 1.30 1.30 1.33 1.21

Amplitude of seasonal trend in daily streamflow 1.70 1.48 1.00 1.42 1.13

Phase of seasonal trend in daily streamflow 1.19 0.94 1.08 1.18 1.20

Average 1.47 1.29 1.16 1.40 1.16

Standard deviation 0.17 0.11 0.13 0.11 0.10

All metrics Average 1.25 1.21 1.11 1.33 1.13

Standard deviation 0.45 0.25 0.20 0.22 0.17
Results and Discussion 11

comparisons of NSE and NSEL, among many metrics, also 

agree with the results of Farmer and others (2014). Though 

Farmer and others (2014) considered a wider range of transferbased methods, QPPQ-NN out-performed the other methods, 

on average.

While QPPQ-NN was shown to be the best PUB method 

considered on average, it still has some limitations. Linhart 

and others (2013) noted that the relative performance of PUB 

methods may vary across flow regimes, from high flows to low 

flows. Farmer and others (2014) make a similar observation, 

looking across performance metrics. QPPQ-NN may be the best 

PUB method on average, but may not be the best for particular 

applications. For example, the PRMS is able to more accurately 

reproduce the lag behavior, even if QPPQ-NN is superior in 

many other ways. It is therefore useful to consider the intended 

application before selecting a PUB method. In addition, the 

three-step nature of transfer methods (index selection, regression parameterization and transfer between index and ungaged 

location) like QPPQ opens several ingresses for model uncertainty (Linhart and others, 2012; Farmer and Vogel, 2013). With 

these multiple uncertainties it is difficult to state the confidence 

in any one prediction or identify the areas for improvement. A 

better understanding of the error structure of ungaged predictions may improve the application of PUB methods.

Average standard deviation of ranks

0.0

0.5

1.0

1.5

2.0

Average mean ranking

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

PRMS

DAR-NN

QPPQ-NN

DAR-MC

QPPQ-MC

EXPLANATION

Figure 3. Robust rank-based evaluation cloud showing the 

tradeoff between the mean ranking and standard deviation of the 

ranks of the 32 performance metrics for each method of prediction 

in ungaged basins. See text for a description of each performance 

metric. Horizontal axis shows the mean average rank, while the 

vertical axis gives the average standard deviation of the ranks. 

Optimal methods would display minimal spread and a low meanaverage ranking. Variability ellipses indicate the spread across the 

32 metrics for each prediction method. The PUB methods include 

the Precipitation Runoff Modeling System (PRMS), drainage-area 

ratio with the nearest-neighboring index streamgage (DAR-NN) or 

the map-correlated index streamgage (DAR-MC), and nonlinear 

spatial interpolation using flow duration curves with the same index 

selection techniques (QPPQ-NN and QPPQ-MC)]

Figure 4. For all methods of prediction in ungaged basins, the 

cross-site mean fraction of the validation record less than a 

threshold absolute percent error, a cumulative distribution of the 

absolute percent error, is presented here. The horizontal axis 

indicates the absolute percent error threshold and the vertical axis 

indicated the percent of predictions less than the corresponding 

threshold. The PUB methods include the Precipitation Runoff 

Modeling System (PRMS), drainage-area ratio with the nearestneighboring index streamgage (DAR-NN) or the map-correlated 

index streamgage (DAR-MC), and nonlinear spatial interpolation 

using flow duration curves with the same index selection 

techniques (QPPQ-NN and QPPQ-MC). Mean percent of the streamflow record

0

10

20

40

30

50

60

70

80

90

100

Absolute percent error

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

PRMS

DAR-NN

QPPQ-NN

DAR-MC

QPPQ-MC

EXPLANATION

mode, the PRMS is unable to reproduce the statistical and 

distributional properties of the historical record. Linhart and 

others (2013) noted that the PRMS was limited by the region 

of calibration and Farmer and others (2014) postulated that a 

more robust method of ungaged calibration could improve the 

performance of the PRMS.

Of the transfer-based methods, QPPQ-NN produces a 

predicted record that most accurately reproduces the statistical 

and distributional properties of historical record on average. 

Linhart and others (2012) also showed that, in Iowa, QPPQ 

performed better than the FA method, a method quite similar to the DAR method. When applied by Linhart and others 

(2012) in a study that did not consider the PRMS, FA had a 

median NSE of 0.66 and QPPQ had a median of 0.73. As is 

affirmed by the findings here, the comparison NSELs was 

much tighter: Linhart and others (2012) showed FA to have 

a median NSEL of 0.76 (DAR-NN had a median of 0.86 in 

this report) and QPPQ had a median of 0.80 (QPPQ-NN had 

a median of 0.87 in this report). The results of Linhart and 

others (2013), when the process-based models are left aside, 

also show that QPPQ is preferred over FA. These relative 
12 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds

Summary and Conclusions

Records of daily streamflow are essential to water 

resources management and planning as well as hydrologic 

understanding. Many regions of the world, even in the United 

States, remain sparsely gaged. The U.S. Geological Survey’s 

National Water Census has sought to fill those gaps using 

various tools for prediction in ungaged basins. Building on 

previous research in Iowa and the southeastern United States, 

this report documents the comparison of several methods 

for prediction of ungaged basins, including nearest-neighbor 

and map-correlation applications of drainage area ratios and 

nonlinear spatial interpolation using flow duration curves, 

along with an ungaged application of the Precipitation-Runoff 

Modeling System. In contrast to the southeastern United 

States, the Des Moines and Iowa Rivers in Iowa are characterized by a different hydroclimatic regime, less variation in 

elevation, and a variety of different landform regions.

Using five methods for prediction in ungaged basins, 

streamflow records were reproduced in a threefold validation across 44 watersheds in the Des Moines and Iowa River 

Basins. The drainage area ratio technique requires no calibration, but is similar to other “Flow Anywhere” techniques. 

Nonlinear spatial interpolation using flow duration curves 

was applied using regional-regression estimated flow duration curves at ungaged locations. Each of these statistical 

techniques was implemented with a nearest-neighbor index 

streamgage based on spatial proximity and a map-correlated 

index streamgage based on estimated hydrograph correlation. The Precipitation-Runoff Modeling System was run in 

an ungaged mode to produce a mechanistic, process-based 

estimate of ungaged streamflow.

The methods for prediction in ungaged basins were 

analyzed using the Robust Rank-Based Evaluation and an 

assessment of the cumulative distribution of the daily percent 

errors. Averaging across all performance metrics, the nearestneighbor implementation of the nonlinear spatial interpolation 

using flow duration curves showed the best ability to reproduce the statistical and distributional properties of the historical record. This agrees with previous work in the region and 

similar studies in other regions. The transfer-based methods 

were shown to be generally superior to the ungaged application of the process-based Precipitation-Runoff Modeling 

System. Considering a restricted set of performance metrics in 

the Cedar River Basin, calibrated iterations of the PRMS have 

previously been shown to out-perform transfer-based methods. 

The broader set of performance metric shown in this report, 

however, highlights the weaknesses of an ungaged calibration 

of the Precipitation Runoff Modeling System.

Additional research may improve the performance of 

prediction in ungaged basins. Prediction of historical records 

can continue to be improved by considering additional study 

regions in which to assess prediction in ungaged basins. In 

particular, an arid or a snow-dominated region may provide 

additional information not gathered from this analysis or 

previous work done in the southeastern United States. Furthermore, understanding the uncertainty of streamflow predictions could further refine prediction in ungaged basins and 

inform the use of predictions in water management decisions 

and planning. Such research might show that predictions for 

specific flow regimes may be improved using specific prediction methods. Continuing to explore regional studies of 

prediction in ungaged basins will surely improve national and 

international implementations and lead to more informed and 

responsible use and management of water resources.

Acknowledgments

This report was produced as a part of the USGS National 

Water Census and also supports the goals of the National 

Streamflow Information Program.

We offer our special thanks to Sara Levin and 

Parker Norton (USGS) who provided exceptional reviews of 

the original manuscript and vastly improved the final product.

0

0

10

50

40

30

20

10

20 30 40 50 60 70 80 90 100 110 120 130 140 150

Absolute percent error

Coefficient of variation of the percent

of the streamflow record

PRMS

EXPLANATION

DAR-NN

QPPQ-NN

DAR-MC

QPPQ-MC

Figure 5. For all methods of prediction in ungaged basins, the

cross-site coefficient of variation of the fraction of the validation 

record less than a threshold absolute percent error is presented here. 

The horizontal axis indicates the absolute percent error threshold 

and the vertical axis indicates the cross-site coefficient of variation 

of the fraction of the validation record less than the corresponding 

threshold. The PUB methods include the Precipitation Runoff 

Modeling System (PRMS), drainage-area ratio with the nearestneighboring index streamgage (DAR-NN) or the map-correlated index 

streamgage (DAR-MC), and nonlinear spatial interpolation using flow 

duration curves with the same index selection techniques (QPPQ-NN 

and QPPQ-MC).
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Appendix 1. Stations Used in Analysis 15

Appendix 1. Stations Used in Analysis

Appendix 1. Names, station numbers, fraction of zeros, and period of record for each streamgage used for the comparison of methods 

of prediction in ungaged basins in the Des Moines and Iowa River Basins.

[Water year is defined as a continuous period from October 1 through Sptember 30]

Station

number

Station name

(index number on figure 2)

Validation

set

Latitude

(decimal

degrees)

Longitude

(decimal

degrees)

Drainage

area

(square

miles)

Mean

streamflow

(cubic feet

per second)

Zero-flow

days

(Percent of 

record)

Number of

complete

water years

First

water

year

Last

water

year

05449500 Iowa River near Rowan (1) 3 42.76 –93.62 429.0 339.5 0.00 29 1982 2011

05451210 South Fork Iowa River NE of New Providence (2) 1 42.32 –93.15 224.0 195.1 0.00 15 1996 2011

05451500 Iowa River at Marshalltown (3) 2 42.07 –92.91 1,532.0 1,286.6 0.00 29 1982 2011

05451700 Timber Creek near Marshalltown (4) 3 42.01 –92.85 118.0 110.3 0.00 29 1982 2011

05451900 Richland Creek near Haven (5) 1 41.90 –92.47 56.1 49.9 0.02 29 1982 2011

05452000 Salt Creek near Elberon (6) 3 41.96 –92.31 201.0 175.1 0.00 29 1982 2011

05452200 Walnut Creek near Hartwick (7) 1 41.84 –92.39 70.9 61.7 0.00 29 1982 2011

05453000 Big Bear Creek at Ladora (8) 2 41.75 –92.18 189.0 165.5 0.00 29 1982 2011

05453100 Iowa River at Marengo (9) 1 41.81 –92.06 2,794.0 2,424.1 0.00 29 1982 2011

05454000 Rapid Creek near Iowa City (10) 1 41.70 –91.49 25.3 20.9 2.62 29 1982 2011

05454220 Clear Creek near Oxford (11) 3 41.72 –91.74 58.4 51.3 0.00 17 1994 2011

05454300 Clear Creek near Coralville (12) 1 41.68 –91.60 98.1 86.1 0.00 29 1982 2011

05455100 Old Mans Creek near Iowa City (13) 1 41.61 –91.62 201.0 167.2 0.00 27 1984 2011

05455500 English River at Kalona (14) 3 41.47 –91.71 574.0 484.8 0.00 29 1982 2011

05457000 Cedar River near Austin, Minnesota (15) 1 43.64 –92.97 399.0 347.3 0.00 29 1982 2011

05457700 Cedar River at Charles City (16) 3 43.06 –92.67 1,054.0 884.2 0.00 25 1982 2011

05458000 Little Cedar River near Ionia (17) 2 43.03 –92.50 306.0 229.7 0.00 29 1982 2011

05458500 Cedar River at Janesville (18) 3 42.65 –92.47 1,661.0 1,411.7 0.00 29 1982 2011

05458900 West Fork Cedar River at Finchford (19) 2 42.63 –92.54 846.0 760.2 0.00 29 1982 2011

05459500 Winnebago River at Mason City (20) 2 43.16 –93.19 526.0 403.6 0.00 29 1982 2011

05462000 Shell Rock River at Shell Rock (21) 3 42.71 –92.58 1,746.0 1,370.0 0.00 29 1982 2011

05463000 Beaver Creek at New Hartford (22) 3 42.57 –92.62 347.0 318.3 0.00 29 1982 2011

05463500 Black Hawk Creek at Hudson (23) 2 42.41 –92.46 303.0 277.2 0.00 23 1982 2011

05464000 Cedar River at Waterloo (24) 3 42.50 –92.33 5,146.0 4,458.3 0.00 29 1982 2011

05464220 Wolf Creek near Dysart (25) 2 42.25 –92.30 299.0 275.2 0.00 12 1996 2011

05464500 Cedar River at Cedar Rapids (26) 1 41.97 –91.67 6,510.0 5,696.8 0.00 29 1982 2011

05465000 Cedar River near Conesville (27) 2 41.41 –91.29 7,787.0 6,890.1 0.00 29 1982 2011

05476750 Des Moines River at Humboldt (28) 1 42.72 –94.22 2,256.0 1,415.8 0.00 29 1982 2011

05479000 East Fork Des Moines River at Dakota City (29) 1 42.72 –94.19 1,308.0 902.8 0.00 29 1982 2011

05480500 Des Moines River at Fort Dodge (30) 2 42.51 –94.20 4,190.0 2,765.5 0.00 29 1982 2011

05481000 Boone River near Webster City (31) 3 42.43 –93.81 844.0 688.5 0.00 29 1982 2011

05481300 Des Moines River near Stratford (32) 2 42.25 –94.00 5,452.0 3,745.5 0.00 29 1982 2011

05481950 Beaver Creek near Grimes (33) 3 41.69 –93.74 358.0 265.3 0.00 29 1982 2011

05482300 North Raccoon River near Sac City (34) 2 42.35 –94.99 700.0 510.3 0.00 29 1982 2011

05482500 North Raccoon River near Jefferson (35) 2 41.99 –94.38 1,619.0 1,143.0 0.00 29 1982 2011

05483450 Middle Raccoon River near Bayard (36) 2 41.78 –94.49 375.0 269.3 0.00 29 1982 2011

05484000 South Raccoon River at Redfield (37) 3 41.59 –94.15 994.0 679.3 0.00 29 1982 2011

05484500 Raccoon River at Van Meter (38) 3 41.53 –93.95 3,441.0 2,406.5 0.00 29 1982 2011

05486000 North River near Norwalk (39) 1 41.46 –93.65 349.0 239.4 0.03 29 1982 2011

05486490 Middle River near Indianola (40) 3 41.42 –93.59 489.0 350.3 0.00 29 1982 2011

05487470 South River near Ackworth (41) 2 41.34 –93.49 460.0 306.2 0.00 29 1982 2011

05487980 White Breast Creek near Dallas (42) 1 41.25 –93.29 333.0 263.1 0.00 29 1982 2011

05488200 English Creek near Knoxville (43) 1 41.30 –93.05 90.1 71.8 0.33 26 1985 2011

05489000 Cedar Creek near Bussey (44) 2 41.22 –92.91 374.0 286.8 0.02 29 1982 2011
16 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds

Appendix 2. Basin Characteristics Used in Analysis

Appendix 2. Description of all basin characteristics considered as potential explanatory variables in the development of the regressions

conducted as part of the model comparison in the Des Moines and Iowa River Basins.—Continued

[USGS, U.S. Geological Survey; DEM, digital elevation model; WBD, watershed boundary dataset; m, meters; 24K, 1:24,000-scale; NHD, national hydrography 

dataset; GIS, geographic information system; NRSC, Natural Resource Conservation Service; SSURGO, Soil Survey Geographic database; IDNR, Iowa Department 

of Natural Resources; PRISM, parameter-elevation regressions on independent slopes model]

Morphometric characteristics Source

DRNAREA—Drainage area (square miles) USGS DEM (10 m), WBD (24k)

BASINPERIM—Basin perimeter (miles) USGS DEM (10 m), WBD (24k)

BASLEN—Basin length (miles) USGS DEM (150 m), WBD (24k)

BSLDEM10M—Average basin slope computed from 10 meter DEM (percent) USGS DEM (10 m)

RELIEF—Basin relief computed as maximum elevation minus minimum elevation (feet) USGS DEM (10 m)

RELRELF—Relative relief computed as RELIEF divided by BASINPERIM (feet per mile) USGS DEM (10 m), WBD (24k)

BSHAPE—Shape factor measure of basin shape computed as BASLEN squared divided by DRNAREA (dimensionless) USGS DEM (10 m), WBD (24k)

ELONRATIO—Elongation ratio measure of basin shape (dimensionless) (Eash, 2001) USGS DEM (10 m), WBD (24k)

ROTUND—Rotundity of basin measure of basin shape (dimensionless) (Eash, 2001) USGS DEM (10 m), WBD (24k)

COMPRAT—Compactness ratio measure of basin shape (dimensionless) (Eash, 2001) USGS DEM (10 m), WBD (24k)

LENGTH—Main-channel length as measured from basin outlet to basin divide (miles) USGS DEM NHD (24k)

MCSR—Main-channel sinuosity ratio computed as LENGTH divided by BASLEN (dimensionless) USGS DEM (10 m), WBD, NHD (24k)

STRMTOT—Total length of mapped streams in basin (miles) USGS DEM NHD (24k)

STRDEN—Stream density computed as STRMTOT divided by DRNAREA (miles per square mile) USGS DEM (10 m), WBD, NHD (24k)

SLENRAT—Slenderness ratio computed as LENGTH squared divided by DRNAREA (dimensionless) USGS DEM (10 m), WBD, NHD (24k)

CCM—Constant of channel maintenance computed as DRNAREA divided by STRMTOT (square miles per mile) USGS DEM (10 m), WBD, NHD (24k)

CSL1085LFP—Stream slope computed as the change in elevation between points 10 and 85 percent of length along the 

longest flow path determined by a GIS divided by length between the points (feet per mile)

USGS DEM (10 m), NHD (24k)

CSL100—Stream slope computed as entire LENGTH (feet per mile) USGS DEM (10 m), NHD (24k)

MCSP—Main-channel slope proportion computed as LENGTH divided by the square root of CSL1085LFP (dimensionless) USGS DEM (10 m), NHD (24k)

RUGGED—Ruggedness number computed as STRMTOT multiplied by RELIEF and divided by DRNAREA (feet per mile) USGS DEM (10 m), WBD, NHD (24k)

SLOPERAT—Slope ratio computed as CSL1085LFP divided by BSLDEM10M (dimensionless) USGS DEM (10 m), NHD (24k)

FOSTREAM—Number of first-order streams within basin using the Strahler stream ordering method (dimensionless) USGS DEM NHD (24k)

DRNFREQ—Drainage frequency computed as FOSTREAM divided by DRNAREA (number of first-order streams 

per square mile) 

USGS DEM (10 m), WBD, NHD (24k)

RSD—Relative stream density computed as FOSTREAM multiplied by DRNAREA and divided by STRMTOT squared

(dimensionless) 

USGS DEM (10 m), WBD, NHD (24k)

SLOP30—Percent area with slopes greater than 30 percent USGS DEM (10 m)

NFSL30—Percent area with slopes greater than 30 percent facing north USGS DEM (10 m)

Hydrologic characteristics Source

BFI—Base-flow index is the mean ratio of base flow to annual streamflow (dimensionless) (Wahl and Wahl, 1988) USGS kriged BFI grid

HYSEP—Hydrograph separation and analysis is the median percentage of baseflow to annual streamflow (percent) 

(Sloto and Crouse, 1996)

USGS kriged HYSEP grid

TAU_ANN—Annual base-flow recession time constant computes the rate of baseflow recession between storms (days) 

(Eng and Milly, 2007)

USGS kriged TAU_ANN grid

STREAM_VAR—Streamflow-variability index is a measure of the steepness of the slope of a duration curve (dimensionless) 

(Koltun and Whitehead, 2002)

USGS kriged STREAM_VAR grid

Pedologic/geologic/land-use characteristics Source

SOILASSURGO—Percent area underlain by hydrologic soil type A (percent area) NRCS SSURGO Web Soil Survey

SOILBSSURGO—Percent area underlain by hydrologic soil type B (percent area) NRCS SSURGO Web Soil Survey

SOILCSSURGO—Percent area underlain by hydrologic soil type C (percent area) NRCS SSURGO Web Soil Survey

SOILDSSURGO—Percent area underlain by hydrologic soil type D (percent area) NRCS SSURGO Web Soil Survey

SAND—Percent volume of sand content of soil (percent volume) NRCS SSURGO Web Soil Survey

CLAY—Percent volume of clay content of soil (percent volume) NRCS SSURGO Web Soil Survey

KSATSSUR—Average saturated hydraulic conductivity of soil (micrometers per second) NRCS SSURGO Web Soil Survey

DESMOIN—Percent area of basin within Des Moines Lobe landform region (percent area) Iowa Geological & Water Survey, 

IDNR grid

ROWCROP—Percent area of cultivated crops (percent area), see Web page: http://www.mrlc.gov/index.php, 

and Homer and others (2004) 

2001 National Landcover Database grid
Appendix 2. Basin Characteristics Used in Analysis 17

Appendix 2. Description of all basin characteristics considered as potential explanatory variables in the development of the regressions 

conducted as part of the model comparison in the Des Moines and Iowa River Basins.—Continued

[USGS, U.S. Geological Survey; DEM, digital elevation model; WBD, watershed boundary dataset; m, meters; 24K, 1:24,000-scale; NHD, national hydrography 

dataset; GIS, geographic information system; NRSC, Natural Resource Conservation Service; SSURGO, Soil Survey Geographic database; IDNR, Iowa Department 

of Natural Resources; PRISM, parameter-elevation regressions on independent slopes model]

Climatic characteristics Source

PRECIP—Mean annual precipitation 1971–2000 (inches), see Web page: http://www.prism.oregonstate.edu/ PRISM Climate Group

I24H2Y—Maximum 24-hour precipitation that occurs on average once in 2 years Midwest Climate Center Bulletin 71

I24H5Y—Maximum 24-hour precipitation that occurs on average once in 5 years Midwest Climate Center Bulletin 71

I24H10Y—Maximum 24-hour precipitation that occurs on average once in 10 years Midwest Climate Center Bulletin 71

I24H25Y—Maximum 24-hour precipitation that occurs on average once in 25 years Midwest Climate Center Bulletin 71

I24H50Y—Maximum 24-hour precipitation that occurs on average once in 50 years Midwest Climate Center Bulletin 71

I24h100Y—Maximum 24-hour precipitation that occurs on average once in 100 years Midwest Climate Center Bulletin 71

PRC1—Mean January precipitation 1971–2000 (inches) PRISM Climate Group

FEBAVPRE—Mean February precipitation 1971–2000 (inches) PRISM Climate Group

MARAVPRE—Mean March precipitation 1971–2000 (inches) PRISM Climate Group

PRC4—Mean April precipitation 1971–2000 (inches) PRISM Climate Group

MAYAVEPRE—Mean May precipitation 1971–2000 (inches) PRISM Climate Group

JUNEAVPRE—Mean June precipitation 1971–2000 (inches) PRISM Climate Group

JULYAVPRE—Mean July precipitation 1971–2000 (inches) PRISM Climate Group

PRC8—Mean August precipitation 1971–2000 (inches) PRISM Climate Group

SEPAVPRE—Mean September precipitation 1971–2000 (inches) PRISM Climate Group

OCTAVPRE—Mean October precipitation 1971–2000 (inches) PRISM Climate Group

NOVAVPRE—Mean November precipitation 1971–2000 (inches) PRISM Climate Group

DECAVPRE—Mean December precipitation 1971–2000 (inches) PRISM Climate Group
18 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds

Appendix 3. Cross-Validation of Map Correlation

Appendix 3. Leave-one-out cross-validated, root-mean-squared error of the fitted variograms

of intersite hydrograph correlation at 44 streamgages in the Des Moines and Iowa River Basins.

Station number Index number

on figure 2

Root-meansquared error

05449500 1 0.024

05451210 2 0.022

05451500 3 0.023

05451700 4 0.028

05451900 5 0.024

05452000 6 0.031

05452200 7 0.026

05453000 8 0.027

05453100 9 0.031

05454000 10 0.023

05454220 11 0.026

05454300 12 0.025

05455100 13 0.027

05455500 14 0.027

05457000 15 0.033

05457700 16 0.034

05458000 17 0.029

05458500 18 0.033

05458900 19 0.021

05459500 20 0.028

05462000 21 0.033

05463000 22 0.023

05463500 23 0.032

05464000 24 0.030

05464220 25 0.034

05464500 26 0.036

05465000 27 0.035

05476750 28 0.030

05479000 29 0.031

05480500 30 0.032

05481000 31 0.022

05481300 32 0.030

05481950 33 0.024

05482300 34 0.025

05482500 35 0.024

05483450 36 0.023

05484000 37 0.023

05484500 38 0.025

05486000 39 0.026

05486490 40 0.026

05487470 41 0.025

05487980 42 0.023

05488200 43 0.022

05489000 44 0.027
Appendix 4. Distributions of Each Performance Metric 19

Appendix 4. Distributions of Each Performance Metric Efficiency

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Method of prediction in ungaged basins

PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC

Efficiency

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Method of prediction in ungaged basins

PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC

Figure 4–1. The distribution of the at-site Nash-Sutcliffe efficiencies of daily streamflow prediction for each method of prediction in ungaged 

basins (PUB) is considered here. The horizontal axis indicates each PUB method. The Nash-Sutcliffe efficiency is along the vertical axis. 

Nash-Sutcliffe efficiency ranges from one to negative infinity; a value of one indicates a perfect fit, while a value of zero indicates that a mean 

value would have produced the same level of accuracy. The dark line indicates the median of the distribution, the box outlines the 25th and 

75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times the interquartile range away from the nearest 

quartile. The PUB methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index 

streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using flow duration curves 

with the same index selection techniques (QPPQ-NN and QPPQ-MC).

Figure 4–2. The distribution of the at-site Nash-Sutcliffe efficiencies of the logarithms of daily streamflow predictions for each method of 

prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The Nash-Sutcliffe efficiency of 

the logarithms is along the vertical axis. Nash-Sutcliffe efficiency ranges from one to negative infinity; a value of one indicates a perfect fit, 

while a value of zero indicates that a mean value would have produced the same level of accuracy. The dark line indicates the median of the 

distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times the 

interquartile range away from the nearest quartile. The PUB methods include the Precipitation Runoff Modeling System (PRMS), drainagearea ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear spatial 

interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC).
20 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds Root-mean-square error

30

50

100

500

1,000

5,000

6,500

Method of prediction in ungaged basins

PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC

Figure 4–3. The distribution of the at-site root-mean-square errors of daily streamflow predictions for each method of prediction 

in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis indicates the rootmean-square error in units of streamflow. Lower values represent less error, on average. The dark line indicates the median of the 

distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times 

the interquartile range away from the nearest quartile. The PUB methods include the Precipitation Runoff Modeling System (PRMS), 

drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), and 

nonlinear spatial interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC). Root-mean-square-normalized error

0.2

0.5

1

5

10

50

100

500

1,000

1,500

Method of prediction in ungaged basins

PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC

Figure 4–4. The distribution of the at-site root-mean-square-normalized errors of daily streamflow predictions for each method of 

prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis indicates 

the root-mean-square-normalized error. Lower values represent less error, on average. The dark line indicates the median of the 

distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times 

the interquartile range away from the nearest quartile. The PUB methods include the Precipitation Runoff Modeling System (PRMS), 

drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), and 

nonlinear spatial interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC).
Appendix 4. Distributions of Each Performance Metric 21

Average percent error

−50

0

50

100

150

200

Method of prediction in ungaged basins

PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC

(8) (2) (2) (3) (2)

Figure 4–5. The distribution of at-site average percent errors of daily streamflow predictions for each method of prediction in ungaged

basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the average percent bias. 

Unbiased methods display a median near zero and minimum variability of at-site bias. The dark line indicates the median of the distribution, 

the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times the interquartile 

range away from the nearest quartile. Numbers at the top of the graph indicate outliers beyond the upper limit of the vertical axis. The PUB 

methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage 

(DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using flow duration curves with the same 

index selection techniques (QPPQ-NN and QPPQ-MC). Pearson correlation

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Method of prediction in ungaged basins

PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC

Figure 4–6. The distribution of at-site Pearson correlations between simulated and observed daily streamflows for each method of 

prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the 

Pearson correlation, which ranges from negative to positive one. A perfect correspondence would exhibit a Pearson correlation of 

one. The dark line indicates the median of the distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to 

the data point a distance not more than 1.5 times the interquartile range away from the nearest quartile. The PUB methods include 

the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or 

the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using flow duration curves with the same index 

selection techniques (QPPQ-NN and QPPQ-MC).
22 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds Spearman correlation

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Method of prediction in ungaged basins

PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC

Efficiency

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Method of prediction in ungaged basins

PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC

Figure 4–7. The distribution of at-site Spearman correlations between simulated and observed daily streamflows for each method of 

prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the 

Spearman correlation, which ranges fromnegative to positive one. A perfect correspondence between ranked streamflow would exhibit 

a Spearman correlation of one. The dark line indicates the median of the distribution, the box outlines the 25th and 75th percentiles, and 

the whiskers extend to the data point a distance not more than 1.5 times the interquartile range away from the nearest quartile. The PUB 

methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage 

(DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using flow duration curves with the 

same index selection techniques (QPPQ-NN and QPPQ-MC).

Figure 4–8. The distribution of the at-site Nash-Sutcliffe efficiencies of the daily storage-yield curve (SYC) for each method of prediction in 

ungaged basins (PUB) is considered here. See text for a description of the methodology used to predict the SYC. The horizontal axis indicates 

each PUB method. The Nash-Sutcliffe efficiency is along the vertical axis. Nash-Sutcliffe efficiency ranges from one to negative infinity; a value 

of one indicates a perfect fit, while a value of zero indicates that a mean value would have produced the same level of accuracy. The dark line 

indicates the median of the distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not 

more than 1.5 times the interquartile range away from the nearest quartile. The PUB methods include the Precipitation Runoff Modeling System 

(PRMS), drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), 

and nonlinear spatial interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC).
Appendix 4. Distributions of Each Performance Metric 23

Efficiency

−1.1

−1.0

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Method of prediction in ungaged basins

PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC

Root-mean-square error

10

50

100

500

550

Method of prediction in ungaged basins

PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC

Figure 4–9. The distribution of the at-site Nash-Sutcliffe efficiencies of the logarithms of the daily storage-yield curve (SYC) for each 

method of prediction in ungaged basins (PUB) is considered here. See text for a description of the methodology used to predict the SYC. 

The horizontal axis indicates each PUB method. The Nash-Sutcliffe efficiency of the logarithms is along the vertical axis. Nash-Sutcliffe 

efficiency ranges from one to negative infinity; a value of one indicates a perfect fit, while a value of zero indicates that a mean value 

would have produced the same level of accuracy. The dark line indicates the median of the distribution, the box outlines the 25th and 

75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times the interquartile range away from the 

nearest quartile. The PUB methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearestneighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using 

flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC).

Figure 4–10. The distribution of the at-site root-mean-square errors of the daily storage-yield curve for each method of prediction in 

ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis indicates the root-meansquare error in units of storage fraction (time, storage volume divided by mean streamflow). Lower values represent less error, on 

average. The dark line indicates the median of the distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend 

to the data point a distance not more than 1.5 times the interquartile range away from the nearest quartile. The PUB methods include 

the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or 

the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using flow duration curves with the same index 

selection techniques (QPPQ-NN and QPPQ-MC).
24 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds Root-mean-square-normalized error

0.01

0.05

0.1

0.5

1

5

10

50

100

300

Method of prediction in ungaged basins

PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC

Average percent error

−100

−50

0

50

100

150

200

Method of prediction in ungaged basins

PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC

(4) (2) (2) (4) (2)

Figure 4–11. The distribution of the at-site root-mean-square-normalized errors of the daily storage-yield curve for each method of 

prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis indicates 

the root-mean-square-normalized error. Lower values represent less error, on average. The dark line indicates the median of the 

distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times 

the interquartile range away from the nearest quartile. The PUB methods include the Precipitation Runoff Modeling System (PRMS), 

drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), and 

nonlinear spatial interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC).

Figure 4–12. The distribution of at-site average percent errors of the daily storage-yield curve for each method of prediction in 

ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the average percent 

bias. Unbiased methods display a median near zero and minimum variability of at-site bias. The dark line indicates the median of the 

distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 

times the interquartile range away from the nearest quartile. Numbers at the top of the graph indicate outliers beyond the upper limit 

of the vertical axis. The PUB methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearestneighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using 

flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC).
Appendix 4. Distributions of Each Performance Metric 25

Pearson correlation

0.90

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1.00

Method of prediction in ungaged basins

PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC

Spearman correlation

0.995

0.996

0.997

0.998

0.999

1.000

Method of prediction in ungaged basins

PRMS DAR-NN QPPQ-NN DAR-MC QPPQ-MC

Figure 4–13. The distribution of at-site Pearson correlations between the simulated and observed daily storage-yield curves for 

each method of prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical 

axis shows the Pearson correlation, which ranges from negative to positive one. A perfect correspondence would exhibit a Pearson 

correlation of one. The dark line indicates the median of the distribution, the box outlines the 25th and 75th percentiles, and the 

whiskers extend to the data point a distance not more than 1.5 times the interquartile range away from the nearest quartile. The PUB 

methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage 

(DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using flow duration curves with the 

same index selection techniques (QPPQ-NN and QPPQ-MC).

Figure 4–14. The distribution of at-site Spearman correlations between the simulated and observed daily storage-yield curves for each

method of prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows 

the Spearman correlation, which ranges from negative to positive one. A perfect correspondence between ranked values would exhibit 

a Spearman correlation of one. The dark line indicates the median of the distribution, the box outlines the 25th and 75th percentiles, and 

the whiskers extend to the data point a distance not more than 1.5 times the interquartile range away from the nearest quartile. The PUB 

methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage 

(DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using flow duration curves with the same 

index selection techniques (QPPQ-NN and QPPQ-MC). NOTE: The boxes are compressed because most values are one.
26 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds Percent error

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0

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Method of prediction in ungaged basins

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Percent error

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Method of prediction in ungaged basins

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Figure 4–15. The distribution of at-site percent errors in the estimated coefficient of variation of annual streamflows for each method 

of prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the 

percent error. Unbiased methods display a median near zero and minimum variability of at-site bias. The dark line indicates the median 

of the distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 

1.5 times the interquartile range away from the nearest quartile. The PUB methods include the Precipitation Runoff Modeling System 

(PRMS), drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage 

(DAR-MC), and nonlinear spatial interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and 

QPPQ-MC).

Figure 4–16. The distribution of at-site percent errors in the estimated coefficient of variation of daily streamflows for each method of

prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the percent 

error. Unbiased methods display a median near zero and minimum variability of at-site bias. The dark line indicates the median of the 

distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times the 

interquartile range away from the nearest quartile. The PUB methods include the Precipitation Runoff Modeling System (PRMS), drainagearea ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear 

spatial interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC).
Appendix 4. Distributions of Each Performance Metric 27

Figure 4–17. The distribution of at-site percent errors in the estimated 10th percentile of the distribution of 7-day average annualminimum events for each method of prediction in ungaged basins (PUB) is considered here. This event is related to the 10-year, 7-day 

average annual event. The horizontal axis indicates each PUB method. The vertical axis shows the percent error. Unbiased methods 

display a median near zero and minimum variability of at-site bias. The dark line indicates the median of the distribution, the box outlines 

the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times the interquartile range 

away from the nearest quartile. Numbers at the top of the graph indicate outliers beyond the upper limit of the vertical axis. The PUB 

methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage 

(DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using flow duration curves with the 

same index selection techniques (QPPQ-NN and QPPQ-MC).

Figure 4–18. The distribution of at-site percent errors in the estimated 50th percentile of the distribution of 7-day average annualminimum events for each method of prediction in ungaged basins (PUB) is considered here. This event is related to the 2-year, 7-day 

annual-minimum event. The horizontal axis indicates each PUB method. The vertical axis shows the percent error. Unbiased methods 

display a median near zero and minimum variability of at-site bias. The dark line indicates the median of the distribution, the box outlines 

the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times the interquartile range 

away from the nearest quartile. Numbers at the top of the graph indicate outliers beyond the upper limit of the vertical axis. The PUB 

methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage 

(DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using flow duration curves with the 

same index selection techniques (QPPQ-NN and QPPQ-MC). Percent error

−50

−40

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0

10

20

30

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Method of prediction in ungaged basins

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Percent error

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300

Method of prediction in ungaged basins

PRMS DAR−NN QPPQ−NN DAR−MC QPPQ−MC

(4) (1)
28 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds

Figure 4–19. The distribution of at-site percent errors in the estimated 90th percentile of the distribution of annual-maximum events 

for each method of prediction in ungaged basins (PUB) is considered here. This event is related to the 10-year annual-maximum 

event. The horizontal axis indicates each PUB method. The vertical axis shows the percent error. Unbiased methods display a median 

near zero and minimum variability of at-site bias. The dark line indicates the median of the distribution, the box outlines the 25th and 

75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times the interquartile range away from the 

nearest quartile. Numbers at the top of the graph indicate outliers beyond the upper limit of the vertical axis. The PUB methods include 

the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or 

the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using flow duration curves with the same index 

selection techniques (QPPQ-NN and QPPQ-MC).

Figure 4–20. The distribution of at-site percent errors in the estimated 90-percent-exceedance streamflow for each method of 

prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the 

percent error. Unbiased methods display a median near zero and minimum variability of at-site bias. The dark line indicates the median 

of the distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 

times the interquartile range away from the nearest quartile. Numbers at the top of the graph indicate outliers beyond the upper limit 

of the vertical axis. The PUB methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearestneighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using 

flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC). Percent error

−100

−50

0

50

100

150

200

Method of prediction in ungaged basins

PRMS DAR−NN QPPQ−NN DAR−MC QPPQ−MC

(1) (1)

Percent error

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0

50

100

150

200

250

300

Method of prediction in ungaged basins

PRMS DAR−NN QPPQ−NN DAR−MC QPPQ−MC

(5) (1)
Appendix 4. Distributions of Each Performance Metric 29

Figure 4–21. The distribution of at-site percent errors in the estimated 75-percent-exceedance streamflow for each method of prediction

in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the percent error. 

Unbiased methods display a median near zero and minimum variability of at-site bias. The dark line indicates the median of the distribution, 

the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times the interquartile 

range away from the nearest quartile. Numbers at the top of the graph indicate outliers beyond the upper limit of the vertical axis. The PUB 

methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage 

(DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using flow duration curves with the same 

index selection techniques (QPPQ-NN and QPPQ-MC).

Figure 4–22. The distribution of at-site percent errors in the estimated 50-percent-exceedance streamflow for each method of prediction

in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the percent error. 

Unbiased methods display a median near zero and minimum variability of at-site bias. The dark line indicates the median of the distribution, 

the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times the interquartile 

range away from the nearest quartile. The PUB methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio 

with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear spatial 

interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC). Percent error

−100

−50

0

50

100

150

200

Method of prediction in ungaged basins

PRMS DAR−NN QPPQ−NN DAR−MC QPPQ−MC

(5)

Percent error

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50

75

100

125

Method of prediction in ungaged basins

PRMS DAR−NN QPPQ−NN DAR−MC QPPQ−MC
30 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds

Figure 4–23. The distribution of at-site percent errors in the estimated 25-percent-exceedance streamflow for each method of prediction

in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the percent error. 

Unbiased methods display a median near zero and minimum variability of at-site bias. The dark line indicates the median of the distribution, 

the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times the interquartile 

range away from the nearest quartile. Numbers at the top of the graph indicate outliers beyond the upper limit of the vertical axis. The PUB 

methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage 

(DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using flow duration curves with the same 

index selection techniques (QPPQ-NN and QPPQ-MC).

Figure 4–24. The distribution of at-site percent errors in the estimated 10-percent-exceedance streamflow for each method of prediction

in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the percent error. 

Unbiased methods display a median near zero and minimum variability of at-site bias. The dark line indicates the median of the distribution, 

the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times the interquartile 

range away from the nearest quartile. Numbers at the top of the graph indicate outliers beyond the upper limit of the vertical axis. The PUB 

methods include the Precipitation Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage 

(DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear spatial interpolation using flow duration curves with the same 

index selection techniques (QPPQ-NN and QPPQ-MC). Percent error

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Method of prediction in ungaged basins

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(2)

Percent error

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Method of prediction in ungaged basins

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(1)
Appendix 4. Distributions of Each Performance Metric 31

Figure 4–25. The distribution of at-site percent errors in the estimated mean daily streamflow for each method of prediction in 

ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the percent 

error. Unbiased methods display a median near zero and minimum variability of at-site bias. The dark line indicates the median of the 

distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times 

the interquartile range away from the nearest quartile. The PUB methods include the Precipitation Runoff Modeling System (PRMS), 

drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), and 

nonlinear spatial interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC).

Figure 4–26. The distribution of at-site percent errors in the estimated coefficient of variation (L-CV) of daily streamflow for each method

of prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the 

percent error. Unbiased methods display a median near zero and minimum variability of at-site bias. The dark line indicates the median 

of the distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 

times the interquartile range away from the nearest quartile. The PUB methods include the Precipitation Runoff Modeling System (PRMS), 

drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), and 

nonlinear spatial interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC). Percent error

−25

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−15

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20

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Method of prediction in ungaged basins

PRMS DAR−NN QPPQ−NN DAR−MC QPPQ−MC

Percent error

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Method of prediction in ungaged basins

PRMS DAR−NN QPPQ−NN DAR−MC QPPQ−MC
32 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds

Figure 4–27. The distribution of at-site percent errors in the estimated skewness (L-skew) of daily streamflows for each method of

prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the percent 

error. Unbiased methods display a median near zero and minimum variability of at-site bias. The dark line indicates the median of the 

distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times the 

interquartile range away from the nearest quartile. The PUB methods include the Precipitation Runoff Modeling System (PRMS), drainagearea ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear 

spatial interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC).

Figure 4–28. The distribution of at-site percent errors in the estimated kurtosis (L-kurtosis) of daily streamflows for each method of

prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the percent 

error. Unbiased methods display a median near zero and minimum variability of at-site bias. The dark line indicates the median of the 

distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 times the 

interquartile range away from the nearest quartile. The PUB methods include the Precipitation Runoff Modeling System (PRMS), drainagearea ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), and nonlinear 

spatial interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC). Percent error

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0

10

20

30

Method of prediction in ungaged basins

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Percent error

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Method of prediction in ungaged basins

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Appendix 4. Distributions of Each Performance Metric 33

Figure 4–29. The distribution of at-site percent errors in the estimated lag-1 autocorrelation of daily streamflows for each method of

prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The vertical axis shows the 

percent error. Unbiased methods display a median near zero and minimum variability of at-site bias. The dark line indicates the median 

of the distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more than 1.5 

times the interquartile range away from the nearest quartile. The PUB methods include the Precipitation Runoff Modeling System (PRMS), 

drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), and 

nonlinear spatial interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC).

Figure 4–30. The distribution of at-site percent errors in the estimated amplitude of the sinusoidal seasonal trend of daily streamflows 

for each method of prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The 

vertical axis shows the percent error. Unbiased methods display a median near zero and minimum variability of at-site bias. The dark 

line indicates the median of the distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point 

a distance not more than 1.5 times the interquartile range away from the nearest quartile. The PUB methods include the Precipitation 

Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated 

index streamgage (DAR-MC), and nonlinear spatial interpolation using flow duration curves with the same index selection techniques 

(QPPQ-NN and QPPQ-MC). Percent error

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Method of prediction in ungaged basins

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Method of prediction in ungaged basins

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34 Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds

Figure 4–31. The distribution of at-site percent errors in the estimated phase of the sinusoidal seasonal trend of daily streamflows 

for each method of prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The 

vertical axis shows the percent error. Unbiased methods display a median near zero and minimum variability of at-site bias. The dark 

line indicates the median of the distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point 

a distance not more than 1.5 times the interquartile range away from the nearest quartile. The PUB methods include the Precipitation 

Runoff Modeling System (PRMS), drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated 

index streamgage (DAR-MC), and nonlinear spatial interpolation using flow duration curves with the same index selection techniques 

(QPPQ-NN and QPPQ-MC).

Figure 4–32. The distribution of the at-site root-mean-square-normalized errors of estimated Fundamental Daily Streamflow Statistics

(FDSS) for each method of prediction in ungaged basins (PUB) is considered here. The horizontal axis indicates each PUB method. The 

vertical axis indicates the root-mean-square-normalized error. Lower values represent less error, on average. The dark line indicates the 

median of the distribution, the box outlines the 25th and 75th percentiles, and the whiskers extend to the data point a distance not more 

than 1.5 times the interquartile range away from the nearest quartile. The PUB methods include the Precipitation Runoff Modeling System 

(PRMS), drainage-area ratio with the nearest-neighboring index streamgage (DAR-NN) or the map-correlated index streamgage (DAR-MC), 

and nonlinear spatial interpolation using flow duration curves with the same index selection techniques (QPPQ-NN and QPPQ-MC). Percent error

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Method of prediction in ungaged basins

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Root-mean-square-normalized error 0.05

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Method of prediction in ungaged basins

PRMS DAR−NN QPPQ−NN DAR−MC QPPQ−MC
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This publication is available online at:

http://dx.doi.org/10.3133/sir20155089
Farmer and others—Evaluation of Statistical and Rainfall-Runoff Models in the Des Moines and Iowa River Watersheds—Scientific Investigations Report 2015–5089

ISSN 2328-0328 (online)

http://dx.doi.org/10.3133/sir20155089