Stochastic Empirical Loading and Dilution 

Model (SELDM) Version 1.0.0

By Gregory E. Granato

Chapter 3

Section C, Water Quality,

Book 4, Hydrologic Analysis and Interpretation

Prepared in cooperation with the

U.S. Department of Transportation, Federal Highway Administration, 

Office of Project Development and Environmental Review

Techniques and Methods 4–C3

U.S. Department of the Interior

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

KEN SALAZAR, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2013

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U.S. Government.

Although this information product, for the most part, is in the public domain, it also may contain copyrighted materials 

as noted in the text. Permission to reproduce copyrighted items must be secured from the copyright owner.

Although this software program has been used by the U.S. Geological Survey (USGS), no warranty, expressed or 

implied, is made by the USGS or the U.S. Government as to the accuracy and functioning of the program and related 

program material nor shall the fact of distribution constitute any such warranty, and no responsibility is assumed by 

the USGS in connection therewith.

Suggested citation:

Granato, G.E., 2013, Stochastic empirical loading and dilution model (SELDM) version 1.0.0: U.S. Geological Survey 

Techniques and Methods, book 4, chap. C3, 112 p., CD–ROM. (Also available at http://pubs.usgs.gov/tm/04/c03/.)
iii

Acknowledgments

The author thanks the many people who reviewed this manual, the model, and the associated 

CD–ROM. Susan Jones of the Federal Highway Administration (FHWA), John Risley, Robert 

Runkel, Mary Ashman, and Kevin Breen of the U.S. Geological Survey (USGS), David Graves 

of the New York State Department of Transportation (DOT), William Fletcher of the Oregon 

DOT, and Rachel Herbert of the U.S. Environmental Protection Agency (USEPA) for providing 

thoughtful and thorough technical and editorial reviews of this report, the software, appendixes, 

and the associated CD–ROM. Elizabeth Ahearn and Anthony Gotvald of the USGS and Joseph 

Krolak of the FHWA reviewed the information supporting development and use of the basinlagtime equations and the triangular hydrographs. Richard Vogel of Tufts University provided 

information on use of the probability-plot correlation coefficient method for evaluating the 

distribution of the random numbers generated by SELDM.

The author also thanks the many people who helped ensure that the model and the graphical 

user interface work correctly. William Fletcher of the Oregon DOT, Mark Mattson of the 

Massachusetts Department of Environmental Protection, Andrew McDaniel of the North 

Carolina DOT, and David Graves of the New York DOT provided exceptionally detailed reviews 

for several beta-test versions of the model. Patricia Cazenas, Susan Jones, Cynthia Nurmi, and 

Brian Smith of the FHWA participated in the beta tests. Rachel Herbert, Dino Marshalonis, 

Jesse Pritts, Nelly Smith, and Mark Sievers of the USEPA; Ryan McReynolds of the U.S. Fish 

and Wildlife Service; and Jeffrey Chaplin, Judy Horwatich, and John Risley of the USGS also 

participated in one or more beta tests of the model. Additionally, environmental professionals 

from 14 other State DOTs participated in one or more beta tests: Kristine Benson and Guangyan 

Griffin from Alaska; Dennis Cress, Rik Gay, and Holly Huyck from Colorado; Steven Sisson from 

Delaware; Karen Coffman and Ling Li from Maryland; Henry Barbaro, Ying Jiang, and Alex 

Murray from Massachusetts; John Taylor from Mississippi; James Murphy from Nevada; Paul 

Kieda from New York; Matthew Lauffer from North Carolina; Michele Dolan from Oklahoma; 

Crystal Newcomer from Pennsylvania; Allison Drake and Allison Hamel from Rhode Island; 

Alex Nguyen, Le Nguyen, and Kristianne Sandoval from Washington; and Jackie Fields from 

West Virginia.
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v

Contents

Acknowledgments........................................................................................................................................iii

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

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

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

Data-Quality Objectives.......................................................................................................................2

Theory and Implementation .........................................................................................................................3

Monte Carlo Simulation Methods ......................................................................................................5

Inverse Cumulative Distribution Function Method.................................................................8

The Frequency-Factor Method..................................................................................................9

Stochastic Regression Method.................................................................................................9

Correlated Random Numbers ..................................................................................................11

Stormflow.............................................................................................................................................14

Storm-Event Characteristics....................................................................................................15

Prestorm Streamflow Volumes................................................................................................19

Storm-Runoff Volumes ..............................................................................................................22

Storm-Event Hydrographs........................................................................................................30

Dilution Factors ..........................................................................................................................35

Highway and Upstream Stormwater Concentrations and Loads ...............................................35

Sources of Water-Quality Data................................................................................................35

Highway-Runoff-Quality Data.........................................................................................35

Upstream-Water-Quality Data........................................................................................39

Random Water-Quality Modeling............................................................................................42

Dependent Water-Quality Modeling.......................................................................................43

Upstream Water-Quality Transport-Curve Modeling...........................................................47

Runoff Modification by Best Management Practices (BMPs)....................................................49

Volume Reduction......................................................................................................................50

Hydrograph Extension and Concurrent Upstream Flows....................................................54

Water-Quality Treatment ..........................................................................................................55

Downstream Stormwater Concentrations and Loads ..................................................................59

Water-Quality Pairs ...................................................................................................................59

Concentration of Concern ........................................................................................................59

Mixing .........................................................................................................................................61

Lake-Basin Analysis...........................................................................................................................64

Mass Balance Approach..........................................................................................................67

Attenuation Factors...................................................................................................................71

Nutrients.............................................................................................................................71

Sediment ............................................................................................................................72

Trace Elements..................................................................................................................77

Organic Chemicals ...........................................................................................................79

Interpreting the Results of an Analysis...........................................................................................81

Storm-Event Results..................................................................................................................82

Annual Results............................................................................................................................87
vi

Figures

1. Schematic diagram showing the stochastic mass-balance approach for estimating 

stormflow, concentration, and loads of water-quality constituents upstream of 

a highway-runoff outfall, from the highway, and downstream of the outfall .....................4

2. Schematic diagram showing the inverse cumulative distribution function method 

for generating data that fit a given distribution .......................................................................8

3. Schematic diagram showing the normal distribution of data above and below the 

regression lines for A, a one-segment regression model with positive slope, 

B, a one-segment regression model with zero slope, and C, a two-segment 

regression model ........................................................................................................................10

4. Graphs showing the results of a Monte Carlo analysis to demonstrate scatter of 

paired uniform random-number samples around a one-to-one relation for eight 

different values of the rank correlation coefficient (Spearman’s rho) ..............................12

5. Schematic diagram showing the upstream-flow and highway-runoff components 

that must be estimated for a mass-balance analysis of receiving-water quality ............14

6. Map showing the spatial distribution of 2,610 hourly-precipitation data stations 

with respect to the U.S. Environmental Protection Agency rain zones and the 

U.S. Environmental Protection Agency (2003) Level III ecoregions that have been 

discretized to a 15-minute grid in the conterminous United States ...................................16

7. Schematic diagram showing A, the characterization of hourly rainfall data, and 

B, the synoptic storm-event definition. ...................................................................................17

8. Map showing the spatial distribution of 2,783 selected U.S. Geological Survey 

streamgages with respect to U.S. Environmental Protection Agency (2003) Level III 

ecoregions, which have been discretized to a 15-minute grid in the conterminous 

United States ...............................................................................................................................20

9. Schematic diagram showing the potential variability in prestorm flows that may 

occur if the definition of the minimum time between precipitation events is less than 

the duration of stormflow recession for a given basin .........................................................21

Use of the Stochastic Empirical Loading and Dilution Model Interface ............................................90

Installing the Model ............................................................................................................................91

Establishing the FHWA-SELDM Output Directory .........................................................................91

Using the Source Code or Modifying the Model ...........................................................................91

Navigating the Graphical User Interface ........................................................................................92

Model-Output Files .............................................................................................................................95

Output-Documentation File ......................................................................................................96

Precipitation-Event Output File ................................................................................................96

Prestorm-Streamflow Output File ...........................................................................................96

Stormflow Output File ................................................................................................................97

Dilution-Factor Output File .......................................................................................................97

Highway-Runoff-Quality Output File .......................................................................................97

Annual Highway-Runoff Output File .......................................................................................98

Upstream Runoff-Quality Output File ......................................................................................98

Downstream Runoff-Quality Output File ................................................................................98

Lake-Analysis Output File .........................................................................................................98

Summary........................................................................................................................................................99

References Cited........................................................................................................................................100

Appendixes 1-4 Accessible from Index Page (linked here)
vii

10. Graphs showing A, the mean, B, standard deviation, and C, coefficient of skew of 

runoff coefficients for 58 highway-runoff monitoring sites and 167 other stormrunoff monitoring sites with 9 or more storm events............................................................27

11. Probability plot showing the upper and lower 95th-percentile confidence limits 

and the mean values of the nonparametric rank correlation coefficients (Spearman’s 

rho) for each of the 43 monitoring sites with 7 or more paired base-flow and runoff 

measurements.............................................................................................................................28

12. Contour plot of calculated Spearman’s rho values showing the relations between 

the impervious fractions of the upstream basin and the highway site..............................31

13. Simplified schematic diagrams showing A, hypothetical triangular hydrographs, 

and B, the hypothetical cumulative upstream-stormflow volume that would coincide 

with unmodified runoff from a highway and with runoff from an extended 

detention structure.....................................................................................................................32

14. Schematic diagram showing time factors for a triangular storm-event hydrograph .....33

15. Graphs showing the basin-lagtime data and regression equations based on the 

basin-lag factor and A, the basin development factor with the primary dataset, or 

B, the total impervious area with the secondary dataset....................................................36

16. Boxplot showing distributions of the minimum, most probable value, and maximum 

for the statistics of the best-fit triangular hydrograph-recession ratio estimated 

from 20 or more storm-event hydrographs from each of the 41 streamgages in the 

multistate dataset.......................................................................................................................38

17. Map showing the spatial distribution of 24,581 water-quality-monitoring stations with 

drainage areas less than 1,050 square miles within U.S. Environmental Protection 

Agency (2003) Level III ecoregions that have been discretized to a 15-minute grid 

in the conterminous United States ..........................................................................................39

18. Process-flow diagram of the steps used for regionalization and site selection, 

data compilation, and data interpretation to define relations between surface-water 

discharge and concentrations of selected water-quality constituents ............................42

19. Graph showing an example of random water-quality data generated by using 

statistics from 27 suspended sediment concentrations measured in highwayrunoff samples collected along State Route 2 in Littleton, Massachusetts .....................44

20. Graph showing an example of dependent water-quality data generation using a 

regression relation for predicting suspended sediment concentrations from total 

suspended solids concentrations measured in 94 highway-runoff samples ...................45

21. Graphs showing examples of A, development, and B, use of a three-segment 

water-quality transport curve for stochastic generation of total hardness concentration data from regional-average daily-flow statistics for ecoregion 67 by use of a 

random-error component based on the median absolute deviation of water-quality 

data in each discharge range...................................................................................................48

22. Scatterplots showing A, inflow and outflow volumes from the San Rafael biofilter, 

B, inflow volumes and the ratios of outflow-to-inflow volumes, and C, the ranks of 

the inflow volumes and the ratios of outflow-to-inflow volumes .......................................52

23. Graphs showing statistics for estimating stochastic flow reductions for the 

San Rafael biofilter.....................................................................................................................53

24. Scatterplots showing A, inflow and outflow concentrations of total phosphorus 

from the Lake Ridge detention pond in milligrams per liter, B, inflow concentrations 

and the ratio of outflow-to-inflow concentrations, and C, the ranks of the inflow 

concentrations and the ratios of outflow-to-inflow concentrations..................................57

25. Graphs showing statistics for estimating stochastic reductions in total phosphorus 

by the Lake Ridge detention pond............................................................................................58
viii

26. Graph showing relations between the basin length, defined as the length of the 

main channel in miles from the outlet to the drainage divide, and the basin 

drainage area in square miles for 845 sites that have drainage areas greater than 

0.1 mi2 in the secondary dataset ..............................................................................................65

27. Graph showing the downstream flow-accretion factors, which are the ratios of 

downstream flows at the ends of the specified mixing-zone lengths to the flows 

immediately downstream from the highway outfall..............................................................66

28. Process-flow diagram of the steps used by the Stochastic Empirical Loading and 

Dilution Model to estimate lake-basin fluxes, loads, and concentrations........................67

29. Schematic diagram showing examples of lake-basin areas defined by A, a physical 

delineation and B, a hypothetical delineation .......................................................................69

30. Graph showing the fall velocities of spheres with various specific gravities and 

diameters in quiescent water at 20° Celsius..........................................................................74

31. Graph showing the fall velocities of spheres in quiescent water at 0, 10, 20, and 

30° Celsius with specific gravities of 1.5 and 2.5 grams per cubic centimeter and 

different grain-size ranges........................................................................................................76

32. Graph showing the stochastic population of total phosphorus concentrations 

calculated with statistics from highway-runoff data collected on State Route 2 in 

Littleton, Massachusetts ...........................................................................................................83

33. Graph showing the stochastic population of total phosphorus concentrations 

calculated with statistics from highway-runoff data collected on State Route 2 in 

Littleton, Massachusetts ...........................................................................................................84

34. Graph showing the stochastic population of total phosphorus concentrations in 

highway runoff and in discharge from a best management practice................................86

35. Graphs showing the stochastic population of total phosphorus concentrations in 

A, upstream stormflow, B, highway runoff and discharge from best management 

practices, and C, downstream stormflows generated for the I–81 example problem....88

36. Graph showing the stochastic population of annual total-phosphorus loads in 

highway runoff and in discharge from a best management practice................................89

37. Graph showing the stochastic population of annual total-phosphorus loads 

calculated with statistics from highway-runoff data collected on State Route 2 in 

Littleton, Massachusetts ...........................................................................................................90

38. Schematic diagram showing the flowchart for entering input data and for running 

an analysis ...................................................................................................................................93

39. Screen capture showing the Analysis Identification Form in the A, select current 

and B, edit current modes showing the basic components of many SELDM data 

input forms ...................................................................................................................................94

Tables

1. Probability distributions used to model primary environmental variables in the 

Stochastic Empirical Loading and Dilution Model ..................................................................6

2. Summary of synoptic storm-event precipitation statistics for the 15 U.S. Environmental Protection Agency rain zones within the conterminous United States ...............18

3. Medians of selected streamflow statistics for the 84 U.S. Environmental Protection 

Agency Level III nutrient ecoregions in the conterminous United States calculated 

by using daily mean streamflow data from 2,783 selected U.S. Geological Survey 

streamgages for the period 1960–2003 ...................................................................................23
ix

4. Paired water-quality and streamflow measurements made between October 1900 

and September 2004 from data for 24,581 water-quality-monitoring stations 

throughout the conterminous United States..........................................................................41

5. Example for calculating the effective sediment-attenuation factor for a 

hypothetical lake.........................................................................................................................78

6. Example for calculating the effective sediment- and trace-metal-attenuation 

factors for a hypothetical lake..................................................................................................80

7. Output files from the Stochastic Empirical Loading and Dilution Model associated 

with the stream-basin, stream- and lake-basin, and lake-basin analyses .......................95

Conversion Factors, Datum, and Abbreviations

Inch/Pound to SI

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

acre 4,047 square meter (m2

)

acre 0.4047 hectare (ha)

square foot (ft2

) 0.09290 square meter (m2)

square inch (in2

) 6.452 square centimeter (cm2)

square mile (mi2

) 259.0 hectare (ha)

square mile (mi2

) 2.590 square kilometer (km2)

Volume

cubic foot (ft3

) 28.32 liter (L)

cubic foot (ft3

) 0.02832 cubic meter (m3)

Flow rate

foot per year (ft/yr) 0.3048 meter per year (m/yr)

cubic foot per second (ft3

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

cubic foot per second per square 

mile [(ft3

/s)/mi2]

0.01093 cubic meter per second per square 

kilometer [(m3/s)/km2]

inch per hour (in/h) 0.0254 meter per hour (m/h)

Mass

pound, avoirdupois (lb) 0.4536 kilogram (kg)

Density

pound per cubic foot (lb/ft3

) 0.01602 gram per cubic centimeter (g/cm3)

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).
x

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

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

Chemical concentrations in water are given in units of milligrams per liter (mg/L) or micrograms 

per liter (μg/L), which express the mass of solute per unit volume (liter) of water. Milligrams 

per liter are equivalent to “parts per million.” Micrograms per liter are equivalent to “parts per 

billion.”

Densities of trace elements in sediment are given in units of micrograms per kilogram (μg/kg).

For water-quality loads, 28.32 liters per second (L/s) = 1 cubic foot per second (ft3/s), and 

10.93 liters per second per square kilometer (L/s/km2) = 1 cubic foot per second per square mile 

(ft3/s/mi2).

Sediment-grain diameters and the diameters of spheres are given in microns.

Scientific notation is used on some graph figures for very large and very small numbers. For 

example, 1 × 10-3 and 1E-3 are equivalent to 0.001.

Gravitational acceleration is given in centimeters per second per second (cm/s2).

Abbreviations

BDF basin development factor

BLF basin lag factor

BMP best management practice

CMRRNG combined multiple recursive random number generator

CDF cumulative distribution function

CMC criteria maximum concentration

COV coefficient of variation

DOT department of transportation

DQO data-quality objectives

EMC event mean concentration

EPM effluent probability method

FAV final acute value

FHWA Federal Highway Administration

GIS geographic information system

GUI graphical user interface

HRDB Highway Runoff Database

IET interevent time
xi

IF impervious fraction

IQR interquartile range

MAD median absolute deviation

MPV most probable value

NPDES National Pollution Discharge Elimination System

NSQD National Stormwater Quality Database

NURP Nationwide Urban Runoff Program

NWISWeb National Water-Quality Information System Web

NWISSC National Water Information System Site Cleaner

NWiz National Water Information System Wizard

PCode U.S. Environmental Protection Agency parameter code

PDF probability-density function

PPCC probability-plot correlation coefficient

RDBP Relational DataBase File Processor

SSC suspended-sediment concentration

SELDM Stochastic Empirical Loading and Dilution Model

STORET USEPA STOrage and RETrieval database

SWQDM Surface-Water-Quality Data Miner

TIA total impervious area

TMDL Total Maximum Daily Load

TSS total suspended solids

USEPA U.S. Environmental Protection Agency

VBA Visual Basic for Applications
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Stochastic Empirical Loading and Dilution Model (SELDM) 

Version 1.0.0

By Gregory E. Granato

Abstract

The Stochastic Empirical Loading and Dilution Model 

(SELDM) is designed to transform complex scientific data 

into meaningful information about the risk of adverse effects 

of runoff on receiving waters, the potential need for mitigation 

measures, and the potential effectiveness of such management 

measures for reducing these risks. The U.S. Geological Survey 

developed SELDM in cooperation with the Federal Highway 

Administration to help develop planning-level estimates of 

event mean concentrations, flows, and loads in stormwater 

from a site of interest and from an upstream basin. Planninglevel estimates are defined as the results of analyses used to 

evaluate alternative management measures; planning-level 

estimates are recognized to include substantial uncertainties 

(commonly orders of magnitude). SELDM uses information 

about a highway site, the associated receiving-water basin, 

precipitation events, stormflow, water quality, and the 

performance of mitigation measures to produce a stochastic 

population of runoff-quality variables. SELDM provides input 

statistics for precipitation, prestorm flow, runoff coefficients, 

and concentrations of selected water-quality constituents from 

National datasets. Input statistics may be selected on the basis 

of the latitude, longitude, and physical characteristics of the 

site of interest and the upstream basin. The user also may 

derive and input statistics for each variable that are specific to 

a given site of interest or a given area.

SELDM is a stochastic model because it uses Monte 

Carlo methods to produce the random combinations of input 

variable values needed to generate the stochastic population 

of values for each component variable. SELDM calculates 

the dilution of runoff in the receiving waters and the resulting 

downstream event mean concentrations and annual average 

lake concentrations. Results are ranked, and plotting positions 

are calculated, to indicate the level of risk of adverse effects 

caused by runoff concentrations, flows, and loads on receiving 

waters by storm and by year. Unlike deterministic hydrologic 

models, SELDM is not calibrated by changing values of 

input variables to match a historical record of values. Instead, 

input values for SELDM are based on site characteristics and 

representative statistics for each hydrologic variable. Thus, 

SELDM is an empirical model based on data and statistics 

rather than theoretical physiochemical equations.

SELDM is a lumped parameter model because the 

highway site, the upstream basin, and the lake basin each 

are represented as a single homogeneous unit. Each of these 

source areas is represented by average basin properties, and 

results from SELDM are calculated as point estimates for 

the site of interest. Use of the lumped parameter approach 

facilitates rapid specification of model parameters to develop 

planning-level estimates with available data. The approach 

allows for parsimony in the required inputs to and outputs 

from the model and flexibility in the use of the model. For 

example, SELDM can be used to model runoff from various 

land covers or land uses by using the highway-site definition 

as long as representative water quality and impervious-fraction 

data are available.

Introduction

Water-resource managers are concerned about the 

frequencies, magnitudes, and durations of concentrations and 

loads (the products of measured stormflow and concentration) 

that may have an adverse effect on the quality of receiving 

waters (Driscoll and others, 1979, 1989; Athayde and others, 

1983; Di Toro, 1984; U.S. Environmental Protection Agency, 

1996b, 2002b, 2007a; Smith and others, 2001; Borsuk and 

others, 2002; Bonta and Cleland, 2003; Gibbons, 2003; 

Novotny, 2004; Elshorbagy and others, 2007; Brouwer 

and De Blois, 2008; Langseth and Brown, 2011). These 

decisionmakers commonly use specified estimates of 

streamflow and upstream constituent concentrations to 

estimate allowable concentrations and flows for discharges 

to receiving waters (U.S. Environmental Protection Agency, 

1986b, 2002b). Evaluating the potential effects of stormwater, 

however, poses many unique challenges (Athayde and 

others, 1983; Di Toro, 1984). Intermittent and highly variable 

concentrations, flows, and loads complicate the monitoring, 

characterization, and evaluation of potential effects of runoff 

on receiving waters. For example, the U.S. Environmental 

Protection Agency’s (USEPA) Nationwide Urban Runoff 

Program (NURP) evaluated the effects of short-term exposures 

that would result from intermittent stormwater runoff and 

estimated that acute concentrations in runoff would be about 
2 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

twice those for continuous discharges during steady low-flow 

conditions (Athayde and others, 1983). The NURP used event 

mean concentrations (EMCs) of constituents in runoff and 

receiving waters to evaluate the potential effects of runoff. 

EMCs are operationally defined as the total constituent load 

from a storm event divided by the total volume of runoff from 

the storm. EMCs are commonly estimated from data collected 

with flow-proportional water-quality-sampling methods. 

Planning-level estimates of EMCs in runoff and receiving 

waters at monitored sites can be used to evaluate the potential 

for adverse effects from highway and urban runoff in receiving 

waters at unmonitored sites (Athayde and others, 1983; 

Di Toro, 1984; Driscoll, Shelley and others, 1989; Driscoll 

and others, 1990b; Marsalek, 1991). 

The Federal Highway Administration (FHWA) developed 

a highway-runoff model that used analytical approximations 

to estimate the potential effects of runoff on receiving 

waters. Publication of the 1990 FHWA runoff-quality model 

with data from 24 highway-runoff monitoring sites was the 

culmination of the FHWA runoff-quality research conducted 

during the 1970s and 1980s (Driscoll and others, 1990a, b). 

The 1990 FHWA runoff-quality model was based on this older 

runoff-quality data and the assumption that concentrations 

of constituents in receiving waters were equal to 0. By the 

mid-1990s, however, it was recognized that the existing 

data and modeling methods were reaching obsolescence 

(Bank and others, 1996). Changes in highway construction 

and maintenance (such as the use of pulverized rubber tires 

in pavement mixtures) and automobile technology (such as 

the disappearance of leaded fuel, continuing improvements 

in catalytic converters, and a trend from asbestos to 

organometallic brake pads) may affect the quality of highway 

runoff. Changes in atmospheric deposition and other ambient 

sources of pollution from surrounding land uses also could 

affect the quality of highway runoff and receiving waters. 

As a result of the implementation of Total Maximum Daily 

Load (TMDL) regulations, decisionmakers have become 

increasingly aware of the importance of considering the 

quality of upstream receiving waters in estimating the 

potential effects of runoff from highways and other land uses. 

Furthermore, awareness has been increasing that statistical 

approaches and Monte Carlo methods are needed to address 

the complexities that affect the probabilities of adverse effects 

from runoff because scientists, engineers, and decisionmakers 

now recognize the stochastic nature of stormflow variables, 

which are partly predictable and partly random. Thus, a model 

that could comprehensively incorporate new data and methods 

was needed.

The SELDM model was developed by the U.S. 

Geological Survey (USGS) in cooperation with the FHWA 

to supersede use of the 1990 FHWA runoff-quality model 

to indicate the risk for stormwater concentrations, flows, 

and loads to be above user-selected water-quality goals. 

The SELDM model was developed and tested during the 

2009–12 period. SELDM is designed to be a tool that can be 

used to transform disparate and complex scientific data into 

meaningful information about the risk for adverse effects of 

runoff on receiving waters, the potential need for mitigation 

measures, and the potential effectiveness of such measures 

for reducing these risks. SELDM was designed to help 

inform water-management decisions for streams and lakes 

receiving highway runoff. Currently (2012), SELDM includes 

precipitation, streamflow, and water-quality data that are 

geographically referenced for sites in the conterminous 

United States. However, SELDM can be used for analysis 

of runoff quality in other areas by setting up geographically 

referenced datasets or by entering user-defined statistics for a 

site of interest.

Purpose and Scope

This report is a user’s manual for the SELDM model. It 

provides information about the theory and implementation of 

the model including the Monte Carlo methods, the methods 

for defining hydrologic variables, numerical methods, and 

governing equations. It provides information for deriving 

model inputs and interpreting model outputs. It also provides 

a detailed discussion of the graphical user interface and the 

format of output files. Four appendixes provide modeling 

information. Appendix 1 describes numerical methods for 

Monte Carlo modeling. Appendix 2 describes specification of 

basin properties needed to characterize the highway site and 

the upstream basin. Appendix 3 provides an illustration of the 

database design. Appendix 4 provides step-by-step use of the 

program’s graphical user interface. 

SELDM was developed as a Microsoft Access® database 

software application that uses a simple graphical user interface 

to facilitate the storage, handling, and use of hydrologic 

datasets. The program is implemented within the database by 

using the Visual Basic for Applications® (VBA) programming 

language. Program source code for the analytical techniques is 

provided in SELDM and in electronic text files accompanying 

this report. Program source code that is specific to Microsoft 

Access®, the graphical user interface, and dataset handling 

is provided in the database. An installation package with a 

run-time version of the software is available with this report 

for potential users who do not have a compatible copy of 

Microsoft Access®. Administrative rights are needed to 

install this version of SELDM. The user needs full control 

(permission to read, write, and modify) of an output directory 

named “FHWA-SELDM” on the root drive of the computer 

to run the model. This directory must be distinct from the 

program-file directories used to install SELDM and the other 

programs developed to facilitate analysis of hydrologic and 

water-quality data (Granato, 2006, 2009, 2010; Granato and 

Cazenas, 2009; Granato and others, 2009).

Data-Quality Objectives

Data-quality objectives (DQOs) are criteria that are 

meant to ensure that data and interpretations are useful for 
Theory and Implementation 3

the intended purpose (U.S. Environmental Protection Agency, 

1986a, 1994, 1996a; Granato and others, 2003). DQOs are 

used to define the information and data necessary to develop 

credible estimates and make defensible decisions for managing 

environmental resources. The DQO process is designed to 

help evaluate the costs of data acquisition in relation to the 

consequences of a decision error caused by inadequate input 

data (U.S. Environmental Protection Agency, 1994, 1996a; 

Granato and others, 2003). SELDM is designed to facilitate an 

iterative DQO approach that is consistent with environmental 

risk-management methods used by the FHWA and the USEPA 

(Sevin, 1987; Cazenas and others, 1996; Federal Highway 

Administration, 1998; U.S. Environmental Protection 

Agency, 1996a). 

The FHWA has established a system of water-qualityassessment and action plans represented by a decision tree 

that includes different levels of interpretive analysis to 

determine the potential environmental effects of highway 

runoff (Sevin, 1987; Cazenas and others, 1996; Federal 

Highway Administration, 1998). In the FHWA process, the 

state department of transportation (DOT) conducts an initial 

assessment to estimate the probability that the highway 

configuration being considered will produce unacceptable 

environmental effects. If the probable risk of an adverse 

effect is unacceptable to decisionmakers, the assessment is 

refined with more detailed data and analysis. The process is 

concluded if a low probability of unacceptable environmental 

effects can be demonstrated. The decision rule for DQOs in 

this process is dependent on the sensitivity of the receiving 

waters, the presence of water supplies in the watershed, 

uncertainties in available data, and limitations of the analysis 

(Patricia Cazenas, U.S. Department of Transportation, Federal 

Highway Administration, oral commun., 2005). The DOTs, 

however, commonly plan mitigation strategies to minimize 

the potential for adverse effects from highway runoff, even if 

criteria excursions are improbable (Patricia Cazenas, Federal 

Highway Administration, written commun., 2006). Waterquality excursions are defined herein as concentrations, flows, 

and loads in effluent or receiving waters that may cause or 

contribute to unacceptable environmental effects in receiving 

waters (U.S. Environmental Protection Agency, 1986a, 1994).

SELDM is designed to rapidly generate planning level 

estimates with available information and data and to refine 

such estimates if necessary. Planning-level estimates are 

defined as the results of analyses used to evaluate alternative 

management measures; planning-level estimates are 

recognized to include substantial uncertainties (commonly 

orders of magnitude) in all aspects of the decision process 

(Barnwell and Krenkel, 1982; Marsalek and Ng, 1989; 

Marsalek, 1991). To support a step-by-step refinement process, 

SELDM is designed to facilitate initial estimates based on 

available regional statistics determined by the location of the 

site of interest, to help refine statistics by selecting data from 

nearby hydrologically similar sites, and to accept user-defined 

statistics. User-defined statistics may be calculated from 

available data or data from monitoring studies done at the site 

of interest as conditions warrant. Considerable uncertainty 

may remain, however, even if site-specific data are collected 

(Winter, 1981; Granato and others, 2003; Harmel and others, 

2006; Smith and Granato, 2010; Granato, 2010, appendix 1).

Theory and Implementation

SELDM uses Monte Carlo methods to generate a 

stochastic population of the concentrations, flows, and loads 

needed to implement a mass-balance model for a receiving 

stream and (or) lake. SELDM also has a stochastic best 

management practice (BMP) module to assess the potential 

benefits of implementing stormwater controls at a site of 

interest. Monte Carlo methods are used because it is the 

combination of different distributions of precipitation, 

prestorm flows, runoff coefficients, and water-quality 

concentrations that determines the potential risk of waterquality excursions. Excursions are commonly associated with 

constituent concentrations that exceed a maximum allowable 

value, but for properties like pH, concentrations of dissolved 

oxygen, and streamflow, an excursion occurs if the values 

are below desirable limits. Deterministic methods are not 

able to characterize the interaction of different distributions 

for hydrologic parameters and BMP-performance measures. 

Unlike deterministic hydrologic models, SELDM is not 

calibrated by changing values of input variables to match 

a historical record of values. Instead, input variables for 

SELDM are based on site characteristics and representative 

statistics for each hydrologic variable. Each of these variables 

may be characterized by different probability distributions. 

Monte Carlo methods are needed because theoretical 

solutions depend too heavily on assumptions about the 

resultant distributions of concentrations, flows, and loads. 

The output results from SELDM, however, are not based on 

such assumptions. The benefit of the Monte Carlo analysis 

is not to decrease uncertainty in the input statistics, but to 

represent the different combinations of the values of variables 

that determine potential risks for water-quality excursions. 

Simpler methods may provide estimates of mean values, but 

it is commonly the extreme events that are of most interest to 

scientists, engineers, and decisionmakers for evaluating the 

potential for excursions.

A mass-balance approach (fig. 1) is commonly applied 

to estimate the concentrations and loads of water-quality 

constituents in receiving waters downstream of an urban 

or highway-runoff outfall (Warn and Brew, 1980; Di Toro, 

1984; Driscoll and others, 1989; Driscoll and others, 1990b; 

Schwartz and Naiman, 1999). In a mass-balance model, the 

loads from the upstream basin and runoff source area (in 

this case, the highway) are added to calculate the discharge, 

concentration, and load in the receiving water downstream of 

a runoff discharge point. These models commonly are based 

on the assumptions that the runoff and the receiving water 
4 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Upstream Load

Downstream Load

C

Q

US

US

Concentration

Stormflow

C

Q

DS

DS

Concentration

Stormflow

CQ

HRHR

ConcentrationStormflow

Highway-Runoff Load

C DS

Cumulative

Frequency

QDS

Cumulative

Frequency

C HR

Cumulative

Frequency

QHR

Cumulative

Frequency

QUS

Cumulative

Frequency

Random Stormflow

C US

Cumulative

Frequency

Random Concentration

Concentration from Transport Curve with Random Variation Random Stormflow

Random Concentration

Random Stormflow

Random Concentration

US HRDS DS

US HR DS

US HR DS

C C = C Q Q

Q Q = Q

Q Q

Stochastic Mass Balance for Multiple Storm Events

Mass Balance for Each Storm Event

Downstream Stormflow

Downstream Concentration

(Derived from Mass Balance for Each Event)

Downstream Values

(Random Variables)

Highway-Runoff Values

(Variables Random and/or Correlated by Transport Curve)

Upstream Values

=

+

Downstream Load

 US US HR HR DS DS Q C Q C = Q C

+

+

+

CUS

QUS

C = mQ + b + e US US i

where is the slope is the intercept, and is the random variation i

mbe

Figure 1. The stochastic mass-balance approach for estimating stormflow, concentration, and loads of water-quality constituents upstream of a highway-runoff outfall, 

from the highway, and downstream of the outfall.
Theory and Implementation 5

are fully mixed and that there are no reactions that reduce the 

mass of the constituent at the point of mixing.

The diagram in the upper left corner of figure 1 shows a 

hypothetical highway site crossing a stream with inputs from 

the upslope areas on both sides of the stream. The massbalance calculation also may be done for the highway on each 

side of the stream and the bridge as separate contributing areas 

(one area at a time). A highway also may be parallel to the 

stream, in which case the mass-balance calculations could be 

modeled for each discharge point or for a conceptual discharge 

point incorporating the entire contributing area. The massbalance approach also may be used to model any land use 

discharging runoff to a stream. In each case, the mass-balance 

equations in the upper right corner of figure 1 would be used 

to calculate the downstream stormflow, concentrations, and 

loads on the basis of runoff values from the selected areas.

Stochastic estimates of concentrations, stormflows, and 

loads of constituents are needed to use the mass-balance 

approach for estimating the potential for excursions in runoff 

and receiving waters (Warn and Brew, 1980; Schwartz and 

Naiman, 1999). Storm events commonly are defined as 

independent statistical events characterized by a volume, 

intensity, duration, and time between midpoints of successive 

storms for the purposes of planning, analysis, and sampling 

efforts (Driscoll and others, 1979; Athayde and others, 1983; 

Goforth and others, 1983; Driscoll, Palhegyi, and others, 

1989; Driscoll, Shelley, and others, 1989; U.S. Environmental 

Protection Agency, 1992; Wanielista and Yousef, 1993; 

Adams and Papa, 2000; Church and others, 2003). Statistics 

describing the frequency distributions of component 

discharges and concentrations are needed to estimate the 

statistics for downstream discharges, concentrations, and 

loads. For example, Di Toro (1984) used information about 

probability distributions of stormflows and EMCs from the site 

of interest and the upstream basin to construct an empirical 

probabilistic dilution model to develop planning-level 

estimates of downstream EMCs and stormflow volumes. 

The lower half of figure 1 indicates how the random 

distributions of concentration and flow from the upstream 

basin and the highway site are used to calculate downstream 

values in SELDM.

Upstream concentrations may vary randomly or may 

be correlated with stormflows. For example, Di Toro (1984) 

based his method on the assumption that contributing 

stormflows and concentrations are independent and 

lognormally distributed. Warn and Brew (1980), however, 

indicated that upstream concentrations and loads are 

correlated. Schwartz and Naiman (1999) also demonstrated 

the effect of correlation between concentrations and flows in 

receiving waters on the adequacy of planning-level estimates 

of concentrations and loads in runoff. Granato and others 

(2009) developed water-quality transport curves, which are 

regression equations for estimating concentrations from 

streamflow. Their results indicate that concentrations of 

suspended sediment and sediment-associated constituents 

commonly increase with increasing streamflow, whereas 

concentrations of dissolved constituents such as total hardness 

commonly decrease with increasing streamflow. The resulting 

probability distributions of downstream EMCs indicate 

the potential for water-quality excursions and therefore the 

potential need for more information and data that may be used 

to identify suitable mitigation measures. 

To calculate the concentrations, flows, and loads required 

for the mass-balance analyses (fig. 1), SELDM calculates 

values for 17 primary environmental variables, 15 of which 

are modeled as stochastic variables (table 1). For each storm, 

the volume of highway-runoff stormflow is calculated by 

using precipitation and highway-runoff-coefficient statistics. 

The timing of runoff from the highway is calculated as a 

function of site characteristics, a fixed hydrograph-recession 

ratio equal to 1, and storm duration. If BMP modifications to 

the highway runoff are specified, then the timing and volume 

of runoff from the BMP also is calculated. The volume of 

upstream stormflows is calculated by using prestorm-flow, 

precipitation, and runoff-coefficient statistics for the upstream 

basin. The timing of runoff from the upstream basin is 

calculated as a function of site characteristics, a stochastic 

hydrograph-recession ratio, and the storm duration. Dilution 

in the receiving water is calculated by using the volume of 

upstream flow that coincides with untreated highway runoff 

and the BMP discharge. The concentrations of upstream 

constituents are stochastic variables that can be calculated as 

purely random variables, dependent variables (defined as a 

function of another constituent), or as functions of upstream 

flow (the transport curve).

The annual lake-basin analysis also is a stochastic 

mass-balance model. The variables from the storm-bystorm highway-runoff calculations are added by annual-load 

accounting year to generate a population of annual loads. 

Loads from the rest of the lake basin, however, are the sums 

of loads associated with all daily streamflows in each annualload accounting year, rather than just the stormflows, because 

dry-weather base flow can contribute substantially to annual 

loads. Although the concentration-attenuation factor is not 

a stochastic variable, the population of annual average lake 

concentrations is a stochastic variable because it is calculated 

as the sum of all influent loads divided by the sum of all 

influent flows for each annual-load accounting year.

Monte Carlo Simulation Methods

Monte Carlo simulation methods are modeling techniques 

that use random-number generators to repetitively draw 

random samples based on input statistics and then combine 

the different variables to determine the probability distribution 

for model outputs. Use of the term Monte Carlo for such 

simulations was coined during the Manhattan Project of 

World War II in a reference to the games of chance such as 

dice or the Roulette wheel that produce stochastic results 

(Saucier, 2000; Harrison, 2010). In hydrology, a stochastic 

process is commonly defined as a process with deterministic 
6 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0 Table 1. Probability distributions used to model primary environmental variables in the Stochastic Empirical Loading and Dilution Model. [Information on modeling of these distributions is in appendix 1. HRC, highway runoff concentration; HRQ, highway runoff flow; 2PExp, two-parameter exponential distribution; Norm, normal distribu- tion; LogN, lognormal distribution (natural or common logarithms); PIII, Pearson type III distribution; LogPIII, log-Pearson type III distribution of the natural or common logarithms of the variable; Tri, triangular distribution; Trap, trapezoidal distribution] Variable number Environmental variable Probability distributions Censoring Correlation Not stochastic 2PExp Norm LogN PIII LogPIII Tri Trap 1 Precipitation volume X Yes No 2 Precipitation duration X Yes No 3 Time between storm-event midpoints X Yes No 4 Prestorm flow X1 X2 X2 No3 No 5 Upstream runoff coefficient X4 X4 Yes Variable 4 6 Upstream hydrograph recession X5 Yes6 No 7 Highway runoff coefficient X4 X4 Yes Variable 5 8 Highway hydrograph recession X No No 9 Water quality, random X1 X4 X2 X4 X2 Yes7 No 10 Water quality, dependent X8 X8 Yes9 Yes10 11 Water quality, transport curve X8 X8 Yes9 Yes10 12 Water quality, downstream adverse effect ratio X5 X5 Yes6 No 13 BMP, flow modification X5 X5 Yes6 HRQ

14 BMP, hydrograph extension X5 X5 Yes6 HRQ

15 BMP, concentration modification X5 X5 Yes6,11 HRC

16 Lake, daily mean lake-basin streamflow X X No No

17 Lake, apparent annual concentration-attenuation factor X No No

1A Pearson type III distribution with a skew of 2.0 is equivalent to an exponential distribution.2A log Pearson type III distribution with a skew of zero is equivalent to lognormal distribution.3The prestorm flows are not censored, but zero flows are modeled by using conditional probability methods.4A Pearson type III distribution with a skew of zero is equivalent to normal distribution.5The triangular distribution is equivalent to a trapezoidal distribution with equal upper and lower values of the most probable value.6Censoring is accomplished by limiting the lower bound to values that are greater than or equal to zero.7If the distribution is normal or Pearson Type III and the random concentration is less than or equal to zero, the value is fit within a lognormal lower tail with the specified mean and standard deviation.8The stochastic portion of the variable is modeled by using normal (or lognormal) scatter around the regression equation.9If the regression equation is untransformed and results in a concentration that is less than or equal to 0, then 0.002 is used.

10Rank correlation is not used, but correlation is implicit in the regression relations among variables.11Censoring is accomplished by setting the minimum irreducible concentration to a value that is greater than or equal to zero.
Theory and Implementation 7

and random components. For example, when runoff 

discharges to a stream, the sum of the two flow volumes is 

a deterministic calculation, but the flow volume from each 

source area results from the random combination of storm 

properties and the effects of antecedent conditions on this 

runoff from both areas. Similarly, a water-quality transport 

curve may indicate a deterministic relation between flow and 

concentrations (either dilution or washoff), but the data may 

show considerable scatter above and below the regression line. 

It may be said that many environmental variables would be 

deterministic if enough data were available, but such detailed 

descriptions would require a complete characterization on any 

appreciable scale.

Using computers to simulate random processes is 

difficult because computers, by design, are completely 

deterministic machines (Devroye, 1986; Press and others, 

1992; L’Ecuyer, 1999; Saucier, 2000; Gentle, 2005; L’Ecuyer 

and Simard, 2007). Computers are useful tools because they 

will consistently produce the same results when given the 

same starting conditions. Therefore, computers produce 

pseudorandom numbers rather than actual random numbers. 

Pseudorandom numbers are produced deterministically, but 

they should be indistinguishable from a series generated by an 

actual random process (such as Brownian motion, radioactive 

decay, or the rolls of perfect dice or a perfect Roulette wheel). 

The measure of a random-number generator is the ability to 

produce such a series of numbers; many available generators 

do not pass such tests (Press and others, 1992; Hellekalek, 

1998; L’Ecuyer, 1998; Marsaglia and Tsang, 2002; L’Ecuyer 

and Simard, 2007).

With SELDM, data can be modeled by using seven 

probability distributions (table 1), including the two-parameter 

exponential, the normal, the lognormal, the Pearson type III, 

the log-Pearson type III, the triangular, and the trapezoidal 

distribution (appendix 1). In some cases, one distribution 

is a special case of another distribution. For example, the 

exponential distribution is a Pearson type III distribution with 

a coefficient of skew equal to 2.0, and a normal distribution is 

a Pearson type III distribution with a coefficient of skew equal 

to 0. Similarly, the lognormal distribution is a log-Pearson type 

III distribution with a coefficient of skew (of the logarithms of 

data) of 0. The triangular distribution is a special case of the 

trapezoidal distribution with equal upper and lower bounds 

of the most probable value. The probability distributions 

commonly used to model each variable (Athayde and others, 

1983; Di Toro, 1984; Driscoll, Palhegyi, and others, 1989; 

Driscoll and others, 1990b; Van Buren and others, 1997; 

Novotny, 2004; Vogel and others, 2005; Cheng and others, 

2007; Granato and others, 2009; Granato, 2010) were selected 

for use in SELDM (table 1). In some cases—for example, the 

upstream hydrograph recession, the adverse-effects ratio, and 

the BMP-modification variables—no particular distribution is 

commonly used. In these cases, the triangular or trapezoidal 

distributions were selected for use in SELDM (table 1). These 

distributions were selected because they can be used to model 

these processes and are commonly recommended for selection 

when expert judgment is used to model data (Haan, 1977; 

Johnson, 1997; Saucier, 2000; U.S. Environmental Protection 

Agency, 2001; Kacker and Lawrence, 2007). For example, the 

BMP-performance variables are ratios, and Johnson (1997) 

indicates that the triangular distribution is well suited to model 

ratios. Furthermore, BMP-performance statistics currently are 

highly uncertain, and these distributions are recommended 

for such cases (U.S. Environmental Protection Agency, 2001; 

Kacker and Lawrence, 2007).

Ten of the primary variables are generated independently, 

rank correlation coefficients can be specified by the user for 

four variables, the rank correlation coefficient is calculated by 

SELDM for one variable, and two variables may be correlated 

by using regression relations (table 1). Rank correlations can 

be specified between the upstream runoff coefficient and the 

prestorm flow volume, between the highway-runoff-flow 

volume and the BMP flow-modification and hydrographextension variables separately, and between the highwayrunoff concentration and the BMP concentration-modification 

variable. The correlation between the highway-runoff 

coefficient and the upstream runoff coefficient is calculated 

by SELDM as a fixed function of the impervious fraction of 

each source area. The random concentration variables are 

not correlated to storm properties or flows in SELDM; the 

literature on highway- and urban-runoff quality indicates 

that such correlations are weak or nonexistent and do not 

have substantial effects on receiving-water concentrations 

(Warn and Brew, 1980; Athayde and others, 1983; Di Toro, 

1984; Driscoll, Shelley and others, 1989; Driscoll and 

others, 1990b). However, the dependent and transport-curve 

concentration variables are correlated by specifying the 

regression relation and the variability of residuals. 

SELDM uses the MRG32k3a combined multiple 

recursive random-number generator (CMRRNG) algorithm 

by L’Ecuyer (1999) to generate the uniform random numbers 

needed to do the Monte Carlo simulations (appendix 1). This 

algorithm was implemented in Visual Basic® (VB) for use 

with SELDM because the native random-number generators 

in Microsoft® VB and VBA used in the Microsoft Office®

programs fail to meet basic standards for random number 

generators (L’Ecuyer and Simard, 2007; McCullough, 

2008). The MRG32k3a generator produces a series of 

pseudorandom numbers by using the remainder of integer 

division (appendix 1). The initial values are known as the 

random seeds (Devroye, 1986; Press and others, 1992; 

L’Ecuyer, 1999; Saucier, 2000; Gentle, 2005; L’Ecuyer and 

Simard, 2007). MRG32k3a uses two initial seed values 

with preset coefficients to find the remainder of each seed 

and the associated modulus. The remainder values are then 

used to generate the next seed and so forth. The uniform 

random numbers between 0 and 1 are calculated by dividing 

the outputs by the modulus. A random-seed management 

algorithm was developed for SELDM to ensure that each 

runoff-quality analysis would be repeatable.

SELDM uses Monte Carlo methods (appendix 1) to 

model the variables and relations shown in table 1. The 
8 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

uniform random numbers are used as inputs to numerical 

algorithms for generating numbers that fit seven probability 

distributions. The program uses the inverse cumulative 

distribution function (CDF) method to generate numbers from 

exponential, triangular, and trapezoidal distributions (Haan, 

1977; Saucier, 2000; Gentle, 2003; Kacker and Lawrence, 

2007; Cheng and others, 2007). SELDM uses the frequencyfactor method (Chow, 1954; Haan, 1977; Chow and others, 

1988; Stedinger and others, 1993; Cheng and others, 2007) 

to generate numbers from normal, lognormal, Pearson Type 

III, and log-Pearson Type III distributions. SELDM uses 

the modified Wilson-Hilferty algorithm developed by Kirby 

(1972) to adjust the frequency factors to model Pearson 

Type III and log-Pearson Type III distributions. SELDM 

uses a modified frequency-factor method to generate random 

numbers to model regression relations. SELDM models rank 

correlation between selected variables by using an algorithm 

developed by Mykytka and Cheng (1994). SELDM models 

prestorm flows on intermittent or ephemeral streams that have 

a risk for zero prestorm streamflow by using random numbers 

that are adjusted for conditional probability.

Inverse Cumulative Distribution Function Method

The inverse CDF method (also known as the inverse 

transformation method) is a simple, efficient technique for 

generating random numbers from a specified probability 

distribution by using a set of uniform random numbers 

(Press and others, 1992; Saucier, 2000; Gentle, 2003; Cheng 

and others, 2007). If a random variable X has a CDF FX(X), 

then substituting the values of X will yield uniform random 

numbers in the range between 0 and 1 (FX(X)=U01). Thus, 

the inverse CDF can be used to generate values of X from 

U01 variate values (FX

-1(U01)=X). This method is shown 

schematically in figure 2. Although the U01 values that are 

generated are evenly spaced within the interval from 0 to 1, 

the inverse CDF function controls the density of the output 

data. For example, the inverse CDF in figure 2 has low slopes 

in the tails and high slopes in the center of the distribution: 

the X values generated are tightly spaced in the center and 

sparse near the edges of the probability distribution function 

(PDF), even though the U01 values are evenly spaced. 

Implementations of the inverse CDF method commonly use 

the sample statistics within the algorithm to produce output 

values that meet the specified criteria, but a standardized 

distribution also may be used (Press and others, 1992; Saucier, 

2000; Gentle, 2003).

SELDM uses the inverse CDF method with the twoparameter exponential distribution to generate stochastic 

data for the precipitation volume, duration, and time 

between storm-event midpoints (table 1). The two-parameter 

exponential distribution is modeled by using the user-selected 

minimum value and average value for each precipitation 

statistic. These statistics define the location and variability 

of the exponential values that are generated as described in 

appendix 1. The minimum value has the effect of censoring 

U01

F (x): Cumulative distribution function

0

1

f (x): Probability density function

Density

x

x

Figure 2. The inverse cumulative distribution function 

method for generating data that fit a given distribution 

(Modified from Saucier, 2000).

values from a two-parameter exponential distribution because 

the inverse CDF does not include values below the minimum. 

The time between storm-event midpoints is generated 

independently from the event duration and is used only to 

delineate annual-load accounting years and therefore the total 

number of storms generated (appendix 1).

SELDM uses the inverse CDF method with the 

triangular/trapezoidal family of distributions to generate 

stochastic data for the upstream hydrograph-recession 

variable, the adverse-effect ratio, and the three BMP-treatment 

variables (table 1). The upstream hydrograph-recession 

variable is limited to the triangular distribution because it 

is defined by the minimum value, most probable value, and 

maximum value specified by the user. The other four variables 

can be specified by using a trapezoidal distribution. These 

variables are defined by the minimum value, the lower bound 

of the most probable value, the upper bound of the most 

probable value, and maximum value specified by the user. If 

the lower and upper bounds of the most probable value are 

specified as being equal, then SELDM will produce stochastic 

data that fit the triangular distribution. These variables are 

generated using an algorithm developed by Kacker and 

Lawrence (2007), which is described in appendix 1.
Theory and Implementation 9

The Frequency-Factor Method

The frequency-factor method is used to model 

distributions with inverse CDFs that are mathematically 

intractable, which means that they cannot be solved 

analytically by using elementary mathematical functions 

(Chow, 1954; Haan, 1977; Chow and others, 1988; Stedinger 

and others, 1993; Cheng and others, 2007). In both the 

frequency-factor method and the inverse CDF method, a 

random set of U01 values is transformed to a set of data values 

by using an algebraic expression that generates the correct 

probability density (fig. 2). The frequency-factor method is 

an efficient technique for generating random numbers from 

a specified probability distribution with a set of U01 variates 

(appendix 1). Chow (1954) showed that the frequency-factor 

method can be used to calculate values of a variable using 

the average, standard deviation, and standardized variate of a 

specified distribution by using the equation:

X X d d = + S K× , (1)

where

Xd is a value from distribution d, 

is the average value used to generate 

stochastic data,

S is the standard deviation used to generate 

stochastic data, and

Kd is the variate associated with the value Xd for 

the selected distribution (d).

The skew of the data can be modeled by selecting a 

standardized form of a skewed probability distribution. 

The relation between the probability of occurrence and Kd

depends on the distribution being modeled. Kd

 values can be 

generated from the U01 variates by using an interpolation table 

or algebraic approximations for the inverse CDF; SELDM 

uses algebraic approximations for the inverse CDF. The 

AS–241 algorithm by Wichura (1988) is used to generate 

normal variates KN from the U01 values, which are generated 

by using the MRG32k3a algorithm. SELDM uses the modified 

Wilson-Hilferty transformation algorithm developed by Kirby 

(1972) to generate Pearson Type III variates KP from normal 

variates KN, which are generated by using the MRG32k3 and 

AS–241 algorithms. The lognormal and log-Pearson Type III 

variables are generated by using the KN and KP variates, which 

are based on statistics for the logarithms of the data, and then 

retransforming each value.

The frequency-factor method is used directly to generate 

five primary hydrologic variables in SELDM (table 1). 

The prestorm flow and daily mean lake-basin streamflow 

variables can be modeled as lognormal variables if the skew 

of the logarithms of streamflows is specified as being equal 

to 0. These variables will be modeled as log-Pearson Type 

III variables if the skew of the logarithms of streamflows 

is specified as not being equal to 0. Similarly, the runoff 

coefficients can be modeled as normal or Pearson Type 

III variables depending on the specified skew value. The 

runoff coefficients, however, are censored by using standard 

acceptance-rejection methods (Press and others, 1992; Saucier, 

2000; Gentle, 2003) so that values that are less than or 

equal to 0 or greater than 1 will be rejected. Random waterquality variables may be specified by using statistics for the 

normal, Pearson Type III, lognormal, or log-Pearson Type III 

distributions; values less than or equal to 0 are replaced.

Stochastic Regression Method

The stochastic regression method uses statistics defining 

a linear regression equation and the scatter of data above and 

below the regression line to generate data that represents the 

relation between measured data values for the variables of 

interest. Linear regression analysis is the process of fitting 

a straight line to a set of data (or some transformation of 

the data) to obtain a mathematical expression for estimating 

the mean value of the dependent variable from a given 

value of the independent variable (Haan, 1977; Helsel and 

Hirsch, 2002). Two of the primary assumptions that underlie 

regression analyses are that the residuals are normally 

distributed with a local mean value centered on the line, and 

that the residuals have constant variance within the range of 

the independent variable characterized by the line. Given this, 

the stochastic-regression method uses the frequency-factor 

method to calculate values of dependent variable above or 

below the regression line. The regression equation with this 

stochastic component is

Y b i i = + m X × + ( ) KNi r ×σ , (2)

where

Yi is the ith value of the dependent variable, 

b is the intercept of the regression line,

m is the slope of the regression line, 

Xi is the ith value of the independent (or 

predictor) variable,

KNi is the ith value of the random normal variate, 

and

σr is the standard deviation of the residuals from 

the regression analysis.

The intercept, slope, and standard deviation of the 

residuals are input by the user, presumably from analysis of 

available water-quality data. As described in appendix 1, the 

standard deviation of the residuals (σr

) and a normal frequency 

factor calculated from a uniform random variate (U01) are used 

to determine the placement of the generated data point (Yi

) 

above or below the local regression-line value calculated on 

the basis of Xi

. The standard deviation of residuals (σr) 

is characterized by the median absolute deviation (MAD) 

of residuals, which is a nonparametric measure of σr

(Helsel and Hirsch, 2002; Granato, 2006; Granato and others, 

2009). The MAD was selected to represent the variability of 
10 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

residuals without the undue effect that outliers may have on 

the σr value. The independent variable is another stochastic 

water-quality variable, or for a transport curve, the stochastic 

sample of upstream stormflows, which are calculated as the 

sum of prestorm flow and runoff averaged over the storm 

event (table 1). The SELDM interface provides several 

options for defining stochastic regression relations, including 

regression relations with one, two, or three segments and 

regression relations based on logarithmic transformations 

of the independent and dependent variables. The KendallThiel Robust line (KTRLine) program (Granato, 2006) 

developed for the SELDM project can be used to calculate 

these regression statistics for a one-segment or a multisegment 

model. Because the multisegment models generated by the 

KTRLine program are based on nonparametric statistics, 

the potential for gross overestimation or underestimation of 

constituent concentrations is minimized within the limits of 

available data (Granato, 2006; Granato and others, 2009).

Regression relations in SELDM may include one to three 

segments each with positive, negative, or zero slopes and 

segment-specific MAD values. If the population of residuals 

is normally distributed, the regression-line prediction of 

the median Yi

 value will approximate the mean of Yi values 

for a given Xi

, and the MAD will approximate the standard 

deviation (σr

) of the Yi values at that point within the interval 

represented by a given segment. If there is a linear onesegment relation between the predictor (Xi

) and response (Yi) 

variables, the slope of the line will be significantly different 

from 0 (either positive, as in figure 3A, or negative), and 

the population of Yi

 values will have a normal distribution 

of data above and below the line. Thus, these data vary 

with X and have a random error component. If the slope 

is not significantly different from 0 (fig. 3B), the Xi

 term 

in equation 2 will drop out, the intercept of the line will 

represent the median and mean of Yi, and the population of 

values will have a normal distribution of data above and 

below the intercept. The data are random with respect to X

and are described only by the Y-population statistics, which 

are characterized by the intercept and error components of 

the regression analysis. If multiple processes affect relations 

between the predictor (Xi

) and response (Yi) variables, different 

relations might predominate over different ranges of the 

predictor variable. Figure 3C shows the case in which there 

is random variation in the response variable (for example, 

base-flow conditions) until a second process predominates 

(for example, runoff). In this case, each segment will have 

a different slope, intercept, and MAD. Granato and others 

(2009) provide several examples of the development and 

use of regression relations with a stochastic component. 

The water-quality transport curves for suspended-sediment 

concentrations, total phosphorus, and total hardness developed 

by Granato and others (2009) have been preloaded into 

SELDM for each ecoregion.

Logarithmic regression relations may better reflect 

the characteristics of hydrologic data (Helsel and Hirsch, 

2002; Granato, 2006; Vogel and others, 2005). The use of 

logarithmic regression relations also precludes generation 

A. One-segment regression model with 

a positive slope

B. One-segment regression model with 

a zero slope

X

Y

Y

Y

C. Two-segment regression model with 

a zero slope and a positive slope

Figure 3. The normal distribution of data above and below 

the regression lines for A, a one-segment regression model 

with positive slope, B, a one-segment regression model 

with zero slope, and C, a two-segment regression model 

(Modified from Granato and others 2009, p. 16).
Theory and Implementation 11

of data that are less than or equal to 0. SELDM does not 

reject values that are less than 0, so the user must take care 

to specify untransformed regression statistics that will not 

produce concentration values less than 0. If a concentration 

that is less than or equal to 0 is generated, then SELDM resets 

the concentration to equal the arbitrarily selected value of 

0.002. This process may bias output results if the number of 

generated values less than or equal to 0 is substantial. Thus, 

output from SELDM that includes multiple concentration 

values equal to 0.002 may indicate that the input statistics 

produce concentration values less than or equal to 0.

Correlated Random Numbers

Correlation analysis is a method to quantify the type 

and strength of relations between two variables (Haan, 1977; 

Press and others, 1992; Helsel and Hirsch, 2002). SELDM 

uses the rank correlation coefficient known as Spearman’s 

rho (ρ) to model statistical relations between variables. 

Spearman’s rho is calculated by ranking the data and 

calculating the correlation coefficients between the rank values 

rather than the data values (Haan, 1977; Helsel and Hirsch, 

2002). Spearman’s rho indicates the strength of the relation 

regardless of the linearity of the relation between variables. 

Correlation coefficients commonly are dimensionless and 

are scaled to be in the range of -1 to 1, inclusive. The sign 

of the correlation coefficient indicates the type of relation. A 

positive sign indicates that one variable generally increases 

as the other increases; a negative sign indicates that one 

variable generally decreases as the other increases; and a value 

of 0 indicates that variations in the two variables are totally 

unrelated. The strength of relations between two variables 

is indicated by the value of the correlation coefficient. The 

relation between variables goes from random association (no 

relation) to monotonic covariance as the absolute value of the 

correlation coefficient increases from 0 to 1. For example, 

figure 4 shows the results of eight Monte Carlo experiments, 

each using 500 paired uniform random numbers. The results 

of the experiments demonstrate the amount of scatter that may 

be associated with different correlation coefficient values. The 

diagonal line in each graph indicates the perfect one-to-one 

relation that would be evident if the correlation coefficient 

were equal to 1. The graphs indicate the increasing scatter in 

the paired values as the rank correlation coefficient, in this 

case Spearman’s rho (ρ), decreases. It is notable that there is 

considerable scatter in the relation on almost all of the graphs, 

even on the graph representing a rho value as high as 0.85. 

Furthermore, the relation between variables appears to be 

almost random in the graph representing a rho value as high 

as 0.45 (absent the 1:1 line). Relations between variables for 

negative rho values would show the same scatter, but the trend 

would be from the upper left to the lower right in each graph. 

The statistical significance of the correlation coefficient is a 

function of the absolute value of the coefficient and the sample 

size. Figure 4 indicates that it is more difficult to distinguish 

between actual correlation and an accidental correlation 

caused by random sampling as the absolute value of the 

correlation decreases, even with large sample sizes.

In SELDM, four variables are calculated by using rank 

correlation to other variables (table 1). The upstream runoff 

coefficients are generated by using the rank correlation 

to the prestorm flow; the highway-runoff coefficients are 

generated by using the rank correlation to the upstream 

runoff coefficients; the BMP flow-modification and 

hydrograph-extension variables are generated by using the 

rank correlation to the highway-runoff inflow volumes; and 

the BMP-concentration-modification variables are generated 

by using the rank correlation to the highway-runoff inflow 

concentrations. The rank correlation between the highwayand upstream-runoff coefficients is calculated by SELDM, 

but the rank correlations for all the other variables are 

user defined.

Stochastic data generated from a specified rank 

correlation between the primary U01 variate (XU) and 

the secondary U01 variate (YU) are calculated by using 

an algorithm developed by Mykytka and Cheng (1994). 

Implementation of this algorithm is described in appendix 1. 

The secondary variate is calculated from the primary variate 

and an intermediate uniform random variate as a function of 

the rank correlation coefficient. The correlation of the uniform 

random variates is equal to the rank correlation of these 

variates because the ranks are the product of the value of the 

variate plus a plotting-position adjustment and the number of 

variates to be generated plus a plotting-position adjustment. 

The generalized equation for producing correlated uniform 

variates is

 Y U3 i i f XU Y Ui

2

2

2 = × + −1 0 × + 5 1 × − ( ) − −1 

 

 ρ ρ ' ( ') . ' ρ ρ( ') , (3)

where

Y3Ui is the ith value of the output U01 variate, 

ρʹ is the absolute value of the correlation coefficient,

XUi is the ith value of the input U01 variate,

Y2Ui is the ith value an intermediate U01 variate, and

f [ ] is a transformation function that depends on the value of Y2Ui and the sign of ρ.
12 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Uniform random numbers for first variable

Correlated uniform random numbers for second variable

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.3

0.4

0.5

0.7

0.8

0.9

1.0

0.1

0.6

0.0

0.2

0.3

0.4

0.5

0.7

0.8

0.9

1.0

0.1

0.6

Rho = 0.95

1:1 Line

Rho = 0.85

1:1 Line

Rho = 0.75

1:1 Line

Rho = 0.65

1:1 Line

Figure 4. The results of a Monte Carlo analysis to demonstrate scatter of paired uniform random-number samples around a 

one-to-one relation for eight different values of the rank correlation coefficient (Spearman’s rho). Each sample consists of 

500 paired uniform random numbers in the range between 0 and 1.
Theory and Implementation 13

Uniform random numbers for first variable

Correlated uniform random numbers for second variable

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.3

0.4

0.5

0.7

0.8

0.9

1.0

0.1

0.6

0.0

0.2

0.3

0.4

0.5

0.7

0.8

0.9

1.0

0.1

0.6

Rho = 0.55

1:1 Line

Rho = 0.35

1:1 Line

Rho = 0.25

1:1 Line

Rho = 0.45

1:1 Line

Figure 4. The results of a Monte Carlo analysis to demonstrate scatter of paired uniform random-number samples around a 

one-to-one relation for eight different values of the rank correlation coefficient (Spearman’s rho). Each sample consists of 

500 paired uniform random numbers in the range between 0 and 1.—Continued
14 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

The algorithm developed by Mykytka and Cheng (1994) 

was tested for use in SELDM by generating 141,282 sets of 

500 and 1,000 U01 variates. The mean, standard deviation, 

output correlation coefficient, and the probability plot 

correlation coefficient (PPCC) statistics (Vogel and Kroll, 

1989) were calculated for each set of output values. These 

tests showed that the algorithm developed by Mykytka 

and Cheng (1994) preserves the theoretical mean (0.5), the 

theoretical standard deviation (about 0.288675), and the 

theoretical marginal distribution of the U01 variates with small 

random variations that are within theoretical tolerance limits 

for the sample sizes generated. On average, the algorithm 

produces a small degree of bias dependent on the absolute 

value of ρ in the population of output ρ values, but this can be 

overcome by adjusting the input ρ value before doing the rest 

of the calculations (Mykytka and Cheng, 1994). 

Stormflow

Estimates of stormflows are needed to use a mass-balance 

approach for predicting the stormflows, concentrations, and 

loads of constituents of concern in runoff and receiving waters 

(Warn and Brew, 1980; Di Toro, 1984; Schwartz and Naiman, 

1999; Granato, 2010). Highway and urban runoff-quality 

assessments are based on storm-event analyses to characterize 

the potential effects of stormwater discharges on receiving 

waters. The mass-balance approach for storm-event analyses 

is based on estimates of upstream- and highway-runoff 

discharges. The total upstream stormflow component for 

each storm event comprises prestorm streamflow and the 

upstream storm runoff. This runoff is the product of the stormevent characteristics, the drainage area, and the volumetric 

runoff coefficient, which is defined as the ratio of runoff to 

precipitation volume (fig. 5). Similarly, the highway-runoff 

discharge for each storm event is the product of the stormevent characteristics, the drainage area, and the volumetric 

runoff coefficient. The storm-event characteristics and the 

runoff coefficients are stochastic variables in SELDM.

The relative importance of each stormflow component 

in determining downstream stormflow, concentrations, and 

loads depends on storm-event characteristics, upstreambasin characteristics, highway-catchment characteristics, and 

BMP characteristics. At one extreme, runoff from a highway 

catchment may contribute all of the downstream flow from a 

small storm during which rainfall is completely absorbed by 

soils in a pervious rural basin with an ephemeral stream. At 

another extreme, runoff from a highway catchment in a large 

basin with a large perennial stream may cause undetectable 

changes in downstream stormflow and water quality. The 

Downstream-Flow Statistics

Storm-Event Statistics

HighwayRunoff Flow

Upstream Flow

Upstream Basin Highway

PrestormStreamflow

Statistics 

RainfallRunoff

Transformation

Statistics

RainfallRunoff

Transformation

Statistics

Figure 5. The upstream-flow and highway-runoff components that must be estimated for a mass-balance analysis of 

receiving-water quality.
Theory and Implementation 15

general geographic location is important because streamflow 

statistics, which determine prestorm flows, and stormevent statistics vary spatially across the United States. The 

hydrologic basin characteristics determine the volume and 

timing of the runoff component. For example, volumetric 

runoff coefficients are a function of the fraction of total 

impervious area (TIA). 

Estimates of the timing of runoff from the highway site 

and the upstream basin are also needed to estimate potential 

effects of runoff in receiving waters. The timing of runoff is 

a function of the length, slope, and drainage features of the 

highway site and the upstream basin (methods for determining 

these variables are described in detail in appendix 2). These 

drainage features, which include the proportion of curb 

and gutter streets, storm sewers, and engineered drainage 

improvements, are characterized by the Basin Development 

Factor (BDF) developed by Sauer and others (1983). 

Specifically, estimates of upstream stormflows during the 

period when highway runoff is discharging to the stream 

are needed to calculate the potential dilution of highway 

runoff in the receiving stream. The duration of highway 

runoff determines the proportion of the upstream stormflow 

hydrograph that contributes to the concurrent downstream 

flow. Therefore, BMPs that extend the duration of the 

highway-runoff hydrograph within or beyond the upstream 

stormflow hydrograph may increase the total concurrent 

flow and thus the dilution of runoff constituents in the 

receiving water.

SELDM is a lumped parameter model because the 

highway site, the upstream basin, and the lake basin each are 

represented as single homogenous units. Results from SELDM 

are calculated as point estimates at the site of interest. Each of 

these source areas is represented by average basin properties. 

For example, highway-runoff results are produced for the 

highway site and the outlet of a BMP, if a BMP is specified. 

Upstream-basin results are produced for the point at which 

the highway runoff enters the stream. Use of the lumped 

parameter approach provides a number of advantages over 

distributed modeling approaches. First, use of the lumped 

parameter approach facilitates rapid specification of model 

parameters to develop planning-level models. Second, this 

approach also is representative of the detail available in 

most datasets. For example, available datasets for highway 

runoff and BMP performance commonly are limited to a 

small number of representative sites. Similarly, watershed 

studies for stream- or lake-quality monitoring commonly are 

based on data collected at sites that represent multiple land 

covers and different tributaries in a single drainage basin. 

Third, this approach allows for parsimony in the required 

inputs to and outputs from the model. Fourth, this approach 

allows flexibility in the use of SELDM. Because runoff from 

the highway site is calculated with only 5 basic hydraulic 

basin properties (appendix 2), SELDM can be used to model 

runoff from areas with other land uses, including the natural 

predevelopment land use, residential land uses, commercial 

land uses, or mixed land uses. 

Storm-Event Characteristics

SELDM uses Monte Carlo methods to calculate the 

precipitation volume, precipitation duration, and the time to 

the next storm-event midpoint for each storm as a stochastic 

variable (appendix 1). Granato (2010) provides a detailed 

discussion of the methods and data for estimating storm-event 

characteristics for use with SELDM. The statistics available 

in SELDM for storm-event characteristics were based on data 

from the 2,610 selected National Weather Service hourlyprecipitation data stations in the conterminous United States 

shown in figure 6 (Granato, 2010). The synoptic statistics 

characterize each storm as a discrete event over the entire 

basin without regard to within-storm variations. These storm 

events are characterized by the minimum interevent time 

(IET), total event duration, total event volume, and interval 

between storm-event midpoints (fig. 7). This definition of a 

storm event commonly is used for planning-level estimates 

of the quantity and quality of highway and urban runoff, 

the design and evaluation of runoff-quality BMPs, and the 

simulation of runoff flows (Driscoll and others, 1979; Goforth 

and others, 1983; Adams and others, 1986; Strecker, Mayo, 

and others, 2001; Driscoll, Palhegyi, and others, 1989; 

Driscoll and others, 1990a, b; U.S. Environmental Protection 

Agency, 1992; Adams and Papa, 2000; Granato, 2010).

SELDM uses the two-parameter exponential distribution 

to stochastically generate the duration, volume, and time 

between storm-event midpoints (table 1; appendix 1). SELDM 

generates uniform random numbers between 0 and 1 for each 

storm-event variable and uses these values with the inverse 

CDF of the two-parameter exponential distribution. Granato 

(2010) examined the use of several common probability 

distributions for modeling storm-event characteristics: the 

one-parameter exponential, the two-parameter exponential, 

the two-parameter lognormal, the two-parameter gamma, 

and the Pearson type III distribution. The two-parameter 

exponential distribution was selected because it preserves 

the characteristics of input statistics and is an efficient 

method for generating stochastic storm events. The twoparameter exponential distribution is parameterized by the 

mean and minimum values; the standard deviation and skew 

are functions of these values. The minimum volume of 

0.1 inch (in.) and the minimum IET of 6 hours are the values 

defined by the USEPA as a runoff-generating event (Driscoll, 

Palhegyi, and others, 1989; U.S. Environmental Protection 

Agency, 1992; Granato, 2010). Use of the hourly data fixes 

the minimum storm duration at 1 hour, and the minimum time 

between storm-event midpoints (7 hours) is 1 hour plus the 

IET. This approach is consistent with the synoptic statistics 

calculated for most stations in the dataset. These values are 

the default in SELDM, but may be changed by loading a new 

dataset or by using the user-defined statistics option on the 

synoptic storm-event precipitation-statistics form.

Each storm that is generated for an analysis is identified 

by sequence number and annual-load accounting year. The 

model generates each storm randomly; there is no serial 
16 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0 ! !! ! ! ! ! ! ! !! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !!! ! ! !! ! ! !! !! ! ! !! ! ! ! ! !! ! ! ! ! !! !! ! !!! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !!!! ! ! ! !! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !!! ! ! !! ! ! ! ! ! ! !! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! !! ! ! ! ! ! ! ! ! ! !! !! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !!!! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !!! !! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !!!!! ! !! ! ! ! ! !! ! ! ! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! !! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! !!!!! ! !! ! ! !! ! ! ! ! !! ! ! ! ! !! ! ! ! !! ! ! ! !!! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !! !!!!! ! !! !! ! ! ! !!! ! ! ! ! ! ! ! ! !! ! ! ! ! !! ! ! !!! ! !! ! ! ! ! !! ! ! ! ! ! ! !! ! !!! ! ! !!!! ! !! ! ! ! ! !! !! !! ! ! ! ! !! !! ! ! ! ! ! !! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! !! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! !!!! ! ! ! ! ! ! ! ! ! ! !!!! ! ! ! ! ! ! ! ! ! ! !! ! ! !!! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! !!! ! ! ! ! ! !! ! !! ! ! !! ! ! !!! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !! ! ! ! !! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! ! !!! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! !! !! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !!! ! ! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! !! ! ! ! ! ! !! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !! !! ! ! ! ! !!! ! !! ! ! !!! ! ! ! ! !! ! ! !! ! ! ! !! ! ! ! !!!! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! !!! ! ! ! ! ! !! !!! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! !! ! ! ! ! ! ! ! ! ! ! ! ! !!! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !!! ! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! !! !!! !!!! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! !! ! ! ! ! ! !! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! !!! ! !!! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !! ! ! !! ! !! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! !!!! ! !! ! ! ! ! !! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !! ! ! ! !! !!!!! ! ! ! ! ! ! ! ! !! ! ! !! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! ! ! !! ! ! !!!! ! ! !! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !!! ! ! ! ! ! ! ! ! !! ! !!! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! !! !!! ! !! !! ! ! ! !!!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! !! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! !! ! !! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! !!! ! ! !! ! ! ! ! ! 15 14 13 11 12 10 9 8 5 4 6 3 1 2

1. Northeast2. Northeast Coastal3. Mid-Atlantic

 7. East Gulf 8. East Texas 9. West Texas

13. Northwest Inland14. Pacific Central15. Pacific Northwest

4. Central5. North Central6. Southeast

10. Southwest11. West Inland12. Pacific Southwest

Base from U.S. Geological Survey National MapEcoregions from U.S. Environmental Protection AgencyGeographic projection

120° 110° 100° 90° 80° 70°

50°40°30°

7

Rain zones

EXPLANATION

Figure 6. The spatial distribution of 2,610 hourly-precipitation data stations (black dots) with respect to the U.S. Environmental Protection Agency rain zones (Driscoll, 

Palhegyi, and others, 1989) and the U.S. Environmental Protection Agency (2003) Level III ecoregions (colored polygons) that have been discretized to a 15-minute grid in the conterminous United States. Ecoregions are identified on the plate useco.pdf on the CD–ROM accompanying this report.
Theory and Implementation 17

A. Hourly rainfall data

B. Synoptic storm event definition

Time, in hours

Intensity, in inches per hour

Minimum interevent time (IET), in hours

One-hour rain pulse

Average event

intensity, in

inches per hour

Interval between 

storm-event midpoints, in hours 

Event duration, in hours

Event volume, in inches

Figure 7. Schematic diagram showing A, The characterization of hourly rainfall data, and B, the synoptic 

storm-event definition (Modified from Driscoll, Palhegyi, and others, 1989).

correlation. The order of storms does not reflect seasonal 

patterns. The annual-load accounting years, which are random 

collections of events generated with sums of interevent 

times less than or equal to a year, are used to generate annual 

highway flows and loads for both the TMDL and the lakebasin analyses. Granato (2010) checked the results of the 

random-number algorithm and verified that use of the twoparameter exponential distribution would produce the correct 

number and distribution of storms and annual precipitation 

volumes for the selected minimum precipitation volume and 

IET. Selection of runoff-generating events with a minimum 

volume of 0.1 in. and a minimum IET of 6 hours is expected 

to reduce the number of storms and annual precipitation totals 

because a considerable number of very small storms are not 

included, and some storms become aggregated into larger and 

longer storms (fig. 7) (Driscoll, Palhegyi, and others, 1989; 

Adams and Papa, 2000; Granato, 2010).

SELDM offers seven options for selecting stormevent statistics on the synoptic storm-event-precipitation 

statistics form. These options include selecting (1) the mean 

of station statistics by USEPA rain zone, (2) the median of 

station statistics by USEPA rain zone, (3) the mean of station 

statistics by ecoregion, (4) the median of station statistics 

by ecoregion, (5) the mean of statistics from user-selected 

stations, (6) the median of statistics from user-selected 

stations, or (7) entering user-defined statistics. Granato (2010) 

selected the USEPA rain zones and USEPA Level III nutrient 

ecoregions to provide regional planning-level estimates of 

precipitation statistics because these regions are accepted 

for water-quality monitoring and management. Methods for 

selecting these options on the statistics form are described in 

detail in appendix 4. The default rain zone and ecoregion are 

automatically selected by entering the latitude and longitude 

of the highway site. The user, however, can manually select 

an ecoregion that better represents conditions at a site of 

interest. Precipitation statistics can be calculated for the site 

of interest from the mean or median of statistics from selected 

hourly-precipitation data stations at nearby hydrologically 

similar sites. The option for entering user-defined statistics 

can be used to enter site-specific statistics, do a sensitivity 

analysis, or evaluate the potential effects of climate change on 

model results. Regional statistics differ substantially among 

neighboring rain zones (table 2) and among neighboring 

ecoregions (Granato, 2010); they do not necessarily 

characterize precipitation statistics for any particular drainage 

basin within each zone (or region), but are intended to be 

the basis for producing initial planning-level estimates for a 

typical basin within that zone (or region).
18 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0 Table 2. Summary of synoptic storm-event precipitation statistics for the 15 U.S. Environmental Protection Agency rain zones within the conterminous United States. [The statistics are calculated from data collected at selected hourly-precipitation data stations in each rain zone during the 1965–2006 period (Granato, 2010). COV, coefficient of variation is defined as the stan- dard deviation divided by the mean; storm events are defined as having a minimum volume of 0.1 inches of precipitation and a 6-hour minimum interevent time threshold. Rain zones are shown on figure 6.] Rain zone Number of storm events (per year) Annual precipitation volume (inches) Storm-event volume (inches) Storm-event duration (hours) Time between storm-event midpoints (hours) Number Name Number of stations Mean COV Mean COV Mean COV Mean COV Mean COV

1 Northeast 270 55 0.29 30.81 0.30 0.56 0.93 8.4 0.85 148.7 1.19

2 Northeast Coastal 61 53 0.26 37.59 0.30 0.71 1.00 9.3 0.81 153.9 1.04

3 Mid-Atlantic 122 49 0.29 32.67 0.31 0.67 0.99 7.8 0.88 166.2 1.26

4 Central 182 50 0.29 35.57 0.31 0.71 1.00 7.2 0.90 164.5 1.30

5 North Central 506 42 0.30 24.70 0.32 0.59 0.97 7.0 0.88 210.5 1.46

6 Southeast 227 53 0.25 43.19 0.28 0.82 1.04 6.8 0.93 158.8 1.25

7 East Gulf 84 57 0.23 47.56 0.27 0.84 1.11 5.7 1.00 150.1 1.36

8 East Texas 173 37 0.26 28.79 0.30 0.78 1.07 6.5 0.96 239.3 1.34

9 West Texas 63 24 0.30 14.95 0.35 0.62 1.00 5.6 1.00 375.4 1.72

10 Southwest 157 23 0.34 10.15 0.38 0.45 0.87 6.2 0.95 427.7 1.64

11 West Inland 72 17 0.43 11.48 0.51 0.60 0.99 8.6 0.91 675.7 1.70

12 Pacific Southwest 89 17 0.42 14.90 0.54 0.85 1.13 10.4 0.94 498.9 2.07

13 Northwest Inland 353 27 0.34 11.48 0.36 0.42 0.87 7.4 0.91 356.6 1.62

14 Pacific Central 134 35 0.34 29.97 0.41 0.84 1.10 12.0 0.92 255.1 2.05

15 Pacific Northwest 117 62 0.26 46.03 0.31 0.70 1.11 12.6 0.93 159.4 1.74
Theory and Implementation 19

Prestorm Streamflow Volumes

SELDM is designed to calculate prestorm streamflow 

volumes from the basin upstream of the highway-runoff 

mixing point for each storm as a stochastic variable. Prestorm 

streamflow is one component of the total stormflow from the 

upstream basin (fig. 5). Granato (2010) provides a detailed 

discussion of the methods and data used for estimating 

prestorm streamflows for use with SELDM. The prestorm 

streamflow, modeled as the instantaneous flow at the 

beginning of a storm, is added to the current storm runoff for 

the duration of the highway runoff or BMP discharge to model 

the total flow available for dilution in the current storm. In the 

environment, the components of prestorm flow may include 

base flow (generally defined as groundwater discharge) and 

stormflow from a previous storm. In the environment and in 

SELDM analyses, some proportion of prestorm flows may 

equal 0 if the stream is intermittent or ephemeral. Estimates 

of prestorm streamflow in receiving waters are important for 

assessing risks for adverse effects of runoff on water quality, 

because prestorm flow can be a substantial proportion of total 

stormflow. The prestorm-streamflow statistics available in 

SELDM were calculated by using data from the 2,783 selected 

U.S. Geological Survey streamgages in the conterminous 

United States shown in figure 8 (Granato, 2010).

Granato (2010) demonstrated that the population 

of prestorm flows is well represented by the complete 

population of daily mean streamflows. This concept is shown 

schematically in figure 9. This schematic diagram shows that 

the range of prestorm flows might be wide if the definition 

of the minimum time between precipitation events is less 

than the stormflow-recession duration. These patterns are 

apparent in daily mean flow data from many streams in the 

United States. Approved streamflow data are reported as daily 

mean flows by the USGS (Mathey, 1998; U.S. Geological 

Survey, 2011). In comparison, independent storm events are 

commonly defined by using hourly data and by specifying 

an interevent time, which is the minimum number of dry 

hours between independent storm events (Driscoll and others, 

1979; Athayde and others, 1983; Adams and others, 1986; 

Driscoll, Palhegyi, and others, 1989; U.S. Environmental 

Protection Agency, 1992; Wanielista and Yousef, 1993; Guo 

and Adams, 1998a; Adams and Papa, 2000; Granato, 2010). 

The minimum interevent time may differ considerably among 

regions but is generally approximated by an interval of about 

6 hours (Driscoll, Palhegyi, and others, 1989; Granato, 2010). 

Theoretically, there may be as many as four independent 

storm events with an event duration of 1 hour and a minimum 

interevent time of 6 hours during one 24-hour period used for 

reporting one daily mean streamflow value. Runoff events 

commonly are defined by the duration of the stormflow 

hydrograph (Linsley and others, 1975; Chow and others, 

1988). Prestorm flows may include runoff from a previous 

storm because stormflow-recession durations for many basins 

commonly are longer than one or more days (Linsley and 

others, 1975; Sloto and Crouse, 1996). Despite the difference 

between the operational definitions of storm events and runoff 

events, daily mean streamflow statistics commonly are used as 

an approximation for receiving-water flow during storm events 

(Di Toro, 1984; Driscoll, Shelley, and others, 1989; Driscoll 

and others, 1990a, b; Novotny, 2004). Granato (2010), 

however, demonstrated that daily mean streamflow statistics 

may be a better approximation for representing prestorm flow 

than stormflow because continuous-flow records commonly 

include a substantial proportion of dry days.

SELDM uses conditional-probability methods to account 

for the occurrence of prestorm flows equal to 0 and the logPearson Type III distribution to stochastically generate the 

remaining population of nonzero prestorm flows (table 1; 

appendix 1). A uniform random number between 0 and 1 is 

generated to represent the total probability (plotting position) 

of the zero and nonzero prestorm flow for each storm event. 

If this number is less than or equal to the proportion of zero 

flows, then a prestorm streamflow value of 0 is assigned for 

that storm event. If the initial uniform random number is 

greater than the proportion of zero flows, then this number 

is rescaled to the range of 0 to 1 to generate a frequency 

factor that represents the prestorm flow within the probability 

distribution of the nonzero streamflows. 

Although the lognormal distribution is most commonly 

used for highway- and urban-runoff studies, a log-Pearson 

type III distribution was selected for generating stochastic 

planning-level estimates of prestorm streamflow by means of 

the frequency-factor method (equation 1). The log-Pearson 

type III distribution was selected because it is an extremely 

flexible distribution that can assume different shapes such 

as symmetrical, positively skewed, or negatively skewed 

(Haan, 1977; Chow and others, 1988; Bobee and Ashkar, 

1991; Stedinger and others, 1993; Cheng and others, 2007; 

Granato, 2010). The log-Pearson Type III distribution is 

equivalent to the lognormal distribution if the logarithms of 

streamflow have zero skew. The log-Pearson Type III was 

selected because Granato (2010) found that coefficients of 

skew of the logarithms of nonzero flow measured at the 

2,783 selected streamgages ranged from -2.2 to 5.4, and only 

9 percent of the distributions had log-skew values within the 

95-percent confidence limit of a lognormal distribution. The 

mean and standard deviation of the logarithms of nonzero 

streamflow data were used to calculate the location and spread 

of the prestorm flow values (equation 1). The skew coefficient 

was used to adjust the standard normal variates to produce a 

representative sample of data. 

SELDM offers five options for selecting prestorm-flow 

statistics on the streamflow-statistics form. These options are 

selecting (1) the mean of streamgage statistics by ecoregion, 

(2) the median of streamgage statistics by ecoregion, (3) the 

mean of statistics from user-selected streamgages, (4) the 

median of statistics from user-selected streamgages, or 

(5) entering user-defined statistics. Methods for selecting 

these options on the statistics form are described in detail 

in appendix 4. The ecoregion is automatically selected by 

entering the latitude and longitude of the highway site. The 
20 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0 120° 110° 100° 90° 80° 70° 50° 40° 30° Base from U.S. Geological Survey National Map Ecoregions from U.S. Environmental Protection Agency Geographic projection

Figure 8. The spatial distribution of 2,783 selected U.S. Geological Survey streamgages (black dots) with respect to U.S. Environmental Protection Agency (2003) Level 

III ecoregions (colored polygons), which have been discretized to a 15-minute grid in the conterminous United States. Ecoregions are identified on the plate useco.pdf on the CD–ROM accompanying this report.
Theory and Implementation 21

Minimum time between storms

Precipitation

Prestorm streamflow

Streamflow

(Schematic diagram, not to scale)

Time

Streamflow increasing Precipitation increasing

EXPLANATION

Figure 9. The potential variability in prestorm flows that may occur if the definition of the minimum 

time between precipitation events is less than the duration of stormflow recession for a given 

basin. The minimum time between storms for highway- and urban-runoff studies is 6 hours without 

measurable precipitation (Driscoll, Palhegyi, and others, 1989), whereas the stormflow recession for 

many basins may last longer than one or more days (Linsley and others, 1975).

user, however, can manually select an ecoregion that better 

represents conditions at a site of interest. The options for 

calculating prestorm flow statistics for the site of interest from 

the mean or median of statistics from selected streamgages 

are provided so that values from nearby hydrologically similar 

basins can be used. The option for entering user-defined 

statistics can be used to enter site specific statistics, to do a 

sensitivity analysis, or to evaluate potential effects of climate 

change on model results.

The options for calculating statistics for the site of 

interest from the mean or median of statistics for every site 

in an ecoregion are provided to formulate initial planninglevel estimates. For example, Granato (2010) selected USEPA 

Level III ecoregions to provide regional planning-level 

estimates because ecoregions represent areas of hydrologic 

similarity on a large-scale basis. He showed ecoregions can 

be an effective tool for characterizing streamflows in different 

areas of the country. Also, ecoregions are commonly being 

used as a spatial framework for organizing and interpreting 

environmental data (Intergovernmental Task Force on 

Monitoring Water Quality, 1995a, b; Omernik and others, 

2000; Omernik, 2004; McMahon and others, 2001; U.S. 

Environmental Protection Agency, 2003; Brown, 2006). 

Regional statistics differ substantially among neighboring 

ecoregions (table 3); they do not necessarily characterize 

prestorm-flow statistics for any particular drainage basin 

within each ecoregion, but are intended to be the basis for 

producing initial planning-level estimates for a typical basin 

within that ecoregion (Granato, 2010).

If regional statistics or statistics from nearby 

hydrologically similar sites are used, then SELDM uses the 

drainage-area-ratio method to calculate streamflow statistics 

for the site of interest (Granato, 2010). The assumption of 

hydrologic similarity is implicit in the application of the 

drainage-area-ratio method because basin characteristics are 

not explicitly included in the predictive equation. Natural 

factors (such as orographic effects, variations in soils, and 

geology) and anthropogenic factors (such as total impervious 

fraction and water use) should be considered in assessments of 

hydrologic similarity. The general equation for the drainagearea-ratio method is
22 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

, (4)

where

Qy is the estimated streamflow at the site of 

interest in cubic feet per second (ft3

/s),

Qx is the streamflow at the index site in ft3/s, 

Ay is the drainage area for the site of interest in 

square miles (mi2

), 

Ax is the drainage area for the index site in mi2, 

and

Z is the streamflow exponent, commonly 

assumed to equal 1.

The default application of the drainage-area-ratio method 

for generating prestorm flows for the site of interest is based 

on the assumption that the exponent Z is 1. Granato (2010) 

found that this was not an unreasonable assumption for most 

ecoregions and provided regression equations that adjust 

flow statistics for each ecoregion to account for the effects 

of divergent Z values. SELDM uses the drainage area of 

the site of interest multiplied by the normalized geometric 

mean streamflow for the ecoregion (Qx

/Ax), in cubic feet 

per second per square mile (ft3

/s/mi2) for selected sites to 

estimate the necessary mean values. The standard deviation 

and skew of the logarithms of nonzero flows are unitless 

and therefore do not need to be adjusted by drainage area 

under the assumption that Z is 1. If the exponent Z varies 

substantially from 1, then statistics from hydrologically 

similar sites with similar drainage areas may be used to 

estimate the flow statistics at the site of interest. In this case, 

the flow statistics may be estimated by select nearby sites 

in SELDM or by using the methods described by Granato 

(2010) to calculate statistics based on data from other nearby 

sites. Alternatively, the regression equations provided by 

Granato (2010) for predicting the mean, standard deviation, 

and skew of the logarithms of nonzero flows from drainage 

areas may be used to calculate improved regional estimates. 

The results may be entered as user-defined values on the 

streamflow-statistics form.

Granato (2009) developed five programs for obtaining 

and analyzing streamflow data from the USGS National 

Water Information System Web site (NWISWeb) in support 

of the SELDM development project. The Get National Water 

Information System Streamflow (Q) files (GNWISQ) program 

was written for obtaining daily mean streamflow data. The 

Streamflow (Q) Statistics (QSTATS) program was written 

to calculate the statistics used by SELDM. The Streamflow 

Record Extension Facilitator (SREF) program was written 

to provide a long-term record of daily mean streamflows 

(record extension) or estimates of long-term streamflow 

statistics (record augmentation) for sites with limited data. 

The Make Plotting Position file (MkPP) was written to 

facilitate generation of flow-duration curves for assessing 

hydrologic similarity in the flow record among potential index 

streamflow-monitoring sites. The Make U.S. Environmental 

Agency DFLOW3 batch input Files (MkDFlowF) program 

was written to facilitate batch use of DFLOW3 to calculate 

low-flow statistics for many streamflow records; such statistics 

can be used as a measure of hydrologic similarity among 

monitoring sites.

Granato (2010) used streamflow data from 2,783 longterm streamgages to provide prestorm-flow estimates, but 

additional data are available for more detailed regional or 

local studies. Continuous records of daily mean streamflow 

for periods of years to decades are available for almost 

26,000 streamgages across the United States. The USGS 

(2011) indicates that concurrent measurements of stage 

and flow are available for almost 53,000 sites. Data from 

one or more paired measurements of stream discharge and 

concentration measured at more than 43,000 surface-waterquality monitoring stations in the conterminous United States 

are available in NWISWeb (Granato and others, 2009). The 

programs developed by Granato (2009) may be useful for 

establishing new streamflow datasets for the SELDM database 

or providing user-defined statistics for a given site of interest.

Storm-Runoff Volumes

SELDM is designed to calculate runoff volumes for 

each storm as a stochastic variable (appendix 1). Runoff 

coefficients are used with the randomly generated precipitation 

volumes to calculate the volume of runoff from the highway 

site and the upstream basin for each storm (fig. 5). A truncated 

Pearson type III distribution was selected for generating 

stochastic planning-level estimates of runoff coefficients for 

the highway site and the upstream basin by the frequencyfactor method (equation 1) (Haan, 1977; Interagency Advisory 

Committee on Water Data, 1982; Chow and others, 1988; 

Stedinger and others, 1993; Cheng and others, 2007; Granato, 

2010). Runoff coefficient values from a Pearson Type III 

distribution are estimated using the frequency-factor method 

(equation 1) because it is an extremely flexible distribution 

that can assume different shapes such as symmetrical, 

positively skewed, or negatively skewed (Haan, 1977; Chow 

and others, 1988; Bobee and Ashkar, 1991; Granato, 2010). 

The mean and standard deviation of runoff-coefficient data are 

used to calculate the location and spread of the resultant runoff 

coefficients (equation 1). The skew coefficient is used to 

adjust the standard normal variates to produce a representative 

sample of data. If the skew of a population equals 0, the 

frequency factor is the standard normal variate. As skew 

coefficients deviate from 0, the relation between the plottingposition probability and the associated frequency factor 

shifts to reflect the distribution of values above and below 

the median value. Although the Pearson type III distribution 

is not bounded by 0 and 1, modeled runoff coefficients are 

limited to this range because the model generates each storm 

as an independent event. Granato (2010) discusses hydrologic 

conditions and sources of uncertainty in rainfall-runoff data 

that may result in runoff coefficients larger than 1. 
Theory and Implementation 23

Table 3. Medians of selected streamflow statistics for the 84 U.S. Environmental Protection Agency Level III nutrient ecoregions in the 

conterminous United States calculated by using daily mean streamflow data from 2,783 selected U.S. Geological Survey streamgages 

for the period 1960–2003. Statistics include the proportion of zero flows and the mean, standard deviation, and skew of the logarithms of 

nonzero mean daily streamflow measurements.

[No., number; ft3/s/mi2, cubic foot per second per square mile; mi2, square mile; SD, standard deviation. Ecoregions are identified on the plate useco.pdf on the 

CD–ROM accompanying this report]

Ecoregion Number 

of stations

Median of streamflow statistics for each ecoregion

Days with zero 

discharge 

(percent of total 

record)

Statistics for the common logarithms of nonzero discharges

No. Name Geometric mean 

(ft3

/s/mi2)

Geometric SD 

(dimensionless)

Coefficient of skew 

(dimensionless)

1 Coast Range 35 0.00 1.77 3.99 0.11

2 Puget Lowland 28 0.00 2.14 2.37 0.26

3 Willamette Valley 15 0.00 1.21 4.22 -0.03

4 Cascades 72 0.00 2.72 2.59 0.06

5 Sierra Nevada 86 0.00 0.50 3.92 0.41

6 Southern and Central California Plains and 

Hills

113 5.46 0.08 7.18 0.19

7 Central California Valley 8 40.59 0.08 7.46 -0.15

8 Southern California Mountains 17 15.32 0.04 7.20 0.02

9 Eastern Cascades Slopes and Foothills 11 0.00 1.67 2.25 0.43

10 Columbia Plateau 17 0.00 0.22 4.90 0.06

11 Blue Mountains 15 0.00 0.64 2.91 0.50

12 Snake River Basin/High Desert 3 45.56 0.06 5.00 -0.19

13 Northern Basin and Range 30 0.00 0.23 2.75 0.75

14 Southern Basin and Range 6 0.02 0.07 4.50 0.20

15 Northern Rockies 11 0.00 0.72 3.02 0.58

16 Montana Valley and Foothill Prairies 14 0.00 0.46 2.76 0.91

17 Middle Rockies 48 0.00 0.39 2.91 0.63

18 Wyoming Basin 29 0.00 0.28 3.61 0.70

19 Wasatch and Uinta Mountains 40 0.00 0.37 2.83 0.87

20 Colorado Plateaus 25 0.00 0.20 2.75 0.70

21 Southern Rockies 114 0.00 0.39 3.13 0.72

22 Arizona/New Mexico Plateau 22 0.00 0.13 2.71 0.64

23 Arizona/New Mexico Mountains 19 0.00 0.08 2.92 0.82

24 Southern Deserts 5 98.32 0.03 18.40 0.08

25 Western High Plains 15 0.00 0.10 2.49 0.37

26 Southwestern Tablelands 14 0.00 0.03 3.32 0.25

27 Central Great Plains 35 0.97 0.03 5.34 0.18

28 Flint Hills 6 0.40 0.09 5.60 0.07

29 Central Oklahoma/Texas Plains 31 10.20 0.04 8.06 0.31

30 Edwards Plateau 13 2.39 0.14 4.86 -0.31
24 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Table 3. Medians of selected streamflow statistics for the 84 U.S. Environmental Protection Agency Level III nutrient ecoregions in the 

conterminous United States calculated by using daily mean streamflow data from 2,783 selected U.S. Geological Survey streamgages 

for the period 1960–2003. Statistics include the proportion of zero flows and the mean, standard deviation, and skew of the logarithms of 

nonzero mean daily streamflow measurements.—Continued

[No., number; ft3/s/mi2, cubic foot per second per square mile; mi2, square mile; SD, standard deviation. Ecoregions are identified on the plate useco.pdf on the 

CD–ROM accompanying this report]

Ecoregion Number 

of stations

Median of streamflow statistics for each ecoregion

Days with zero 

discharge 

(percent of total 

record)

Statistics for the common logarithms of nonzero discharges

No. Name Geometric mean 

(ft3

/s/mi2)

Geometric SD 

(dimensionless)

Coefficient of skew 

(dimensionless)

31 Southern Texas Plains 7 58.27 0.01 8.29 0.48

32 Texas Blackland Prairies 28 8.77 0.06 9.07 0.10

33 East Central Texas Plains 11 5.90 0.05 8.26 0.00

34 Western Gulf Coastal Plains 29 0.00 0.20 5.08 0.56

35 South Central Plains 40 0.22 0.23 6.75 -0.01

36 Ouachita Mountains 7 2.67 0.23 9.16 -0.29

37 Arkansas Valley 4 2.82 0.20 10.17 -0.54

38 Boston Mountains 6 1.65 0.30 8.20 -0.65

39 Ozark Highlands 24 0.00 0.43 3.39 0.35

40 Central Irregular Plains 38 0.58 0.10 7.52 0.06

41 Canadian Rockies 3 0.02 1.30 3.57 0.07

42 Northwestern Glaciated Plains 12 0.35 0.21 3.38 0.16

43 Northwestern Great Plains 27 0.02 0.12 3.05 0.68

44 Nebraska Sandhills 3 0.00 0.24 1.75 1.76

45 Piedmont 112 0.00 0.66 2.73 0.20

46 Northern Glaciated Plains 22 32.98 0.02 10.81 0.13

47 Western Corn Belt Plains 56 0.02 0.19 4.27 0.01

48 Lake Agassiz Plain 12 3.96 0.03 6.19 0.07

49 Northern Minnesota Wetlands 1 1.25 0.05 9.19 -0.38

50 Northern Lakes and Forests 45 0.00 0.69 2.13 0.74

51 Northern Central Hardwood Forests 14 0.00 0.34 2.65 0.60

52 Driftless Area 13 0.00 0.49 1.88 1.27

53 Southeastern Wisconsin Till Plains 16 0.00 0.39 2.84 0.36

54 Central Corn Belt Plains 71 0.00 0.39 3.65 0.04

55 Eastern Corn Belt Plains 87 0.00 0.34 4.25 0.24

56 S. Michigan/N. Indiana Drift Plains 70 0.00 0.53 2.41 0.13

57 Huron/Erie Lake Plains 11 0.00 0.25 4.13 0.36

58 Northeastern Highlands 107 0.00 1.09 2.90 0.09

59 Northeastern Coastal Zone 79 0.00 1.02 2.90 -0.16

60 Northern Appalachian Plateau and Uplands 31 0.00 0.68 3.20 0.11
Theory and Implementation 25

Table 3. Medians of selected streamflow statistics for the 84 U.S. Environmental Protection Agency Level III nutrient ecoregions in the 

conterminous United States calculated by using daily mean streamflow data from 2,783 selected U.S. Geological Survey streamgages 

for the period 1960–2003. Statistics include the proportion of zero flows and the mean, standard deviation, and skew of the logarithms of 

nonzero mean daily streamflow measurements.—Continued

[No., number; ft3/s/mi2, cubic foot per second per square mile; mi2, square mile; SD, standard deviation. Ecoregions are identified on the plate useco.pdf on the 

CD–ROM accompanying this report]

Ecoregion Number 

of stations

Median of streamflow statistics for each ecoregion

Days with zero 

discharge 

(percent of total 

record)

Statistics for the common logarithms of nonzero discharges

No. Name Geometric mean 

(ft3

/s/mi2)

Geometric SD 

(dimensionless)

Coefficient of skew 

(dimensionless)

61 Erie/Ontario Lake Hills and Plain 21 0.00 0.65 3.20 0.19

62 North Central Appalachians 35 0.00 1.03 2.98 -0.07

63 Middle Atlantic Coastal Plain 23 0.00 0.51 3.85 -0.00

64 Northern Piedmont 94 0.00 0.78 2.58 0.31

65 Southeastern Plains 100 0.00 0.64 3.04 0.28

66 Blue Ridge Mountains 35 0.00 1.84 1.99 0.36

67 Central Appalachian Ridges and Valleys 105 0.00 0.75 2.73 0.39

68 Southwestern Appalachians 15 0.00 0.67 4.87 0.02

69 Central Appalachians 54 0.00 0.78 3.63 -0.05

70 Western Allegheny Plateau 54 0.00 0.54 4.08 -0.18

71 Interior Plateau 59 0.00 0.48 4.70 0.02

72 Interior River Lowland 33 0.55 0.16 7.72 0.02

73 Mississippi Alluvial Plain 7 0.00 0.64 4.32 0.09

74 Mississippi Valley Loess Plains 7 0.00 0.64 2.73 1.52

75 Southern Coastal Plain 93 0.20 0.29 4.55 -0.20

76 Southern Florida Coastal Plain 1 58.48 2.40 4.43 -1.44

77 North Cascades 15 0.00 4.67 2.44 0.11

78 Klamath Mountains 34 0.00 0.80 3.68 0.22

79 Madrean Archipelago 3 3.57 0.004 3.62 0.32

80 Northern Basin and Range 13 0.00 0.10 3.37 0.40

81 Sonoran Basin and Range 11 27.52 0.08 8.10 -0.11

82 Laurentian Plains and Hills 11 0.00 1.07 2.86 0.16

83 Eastern Great Lakes and Hudson Lowlands 43 0.00 0.71 3.04 0.18

84 Atlantic Coastal Pine Barrens 34 0.00 1.04 1.96 0.11
26 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Regression equations relating the mean, standard 

deviation, and skew of runoff coefficients to the total 

impervious fraction were developed to facilitate the selection 

of representative statistics for the highway site and the 

upstream basin (fig. 10). Regression equations that are based 

on Schueler’s (1987) analysis of NURP data also are available 

for use in SELDM, and user-defined runoff-coefficient 

statistics also may be used. The regression equations for 

highway sites were developed on the basis of data from 58 

highway sites across the country with 9 or more storm events. 

The drainage areas of these sites range from 0.05 to 106 acres 

and TIA fractions from 0.27 to 1. Regression equations for 

estimating runoff-coefficient statistics for the upstream basin 

were developed on the basis of data from 167 sites across the 

country with 9 or more storm events (Granato, 2010). The 

drainage areas of these sites range from 0.005 to 93.47 mi2

and TIA fractions from 0.0001 to 0.994. Separate regression 

equations were developed for the two sets of sites because 

highway sites commonly are smaller and more homogenous 

than the other sites. If the impervious fraction of the highway 

site (or upstream basin) is less than about 0.3, then the 

regression equations developed with the 167 nonhighway 

monitoring sites will provide more representative statistics 

than the regression equations developed by using only 

highway sites (fig. 10).

The mean of the runoff coefficients for the highway 

and the upstream basin can be estimated with respect to 

the impervious fraction (IFs) with the regression equations 

developed for use with SELDM. A one-segment regression 

model was developed by using the highway-site data because 

no break in slope was apparent in the mean runoff coefficient 

values for highway sites (fig. 10A). This equation is

. (5)

In this equation, the runoff coefficients (RvMean) and the IF

are unitless and range from 0 to 1. The slope of this regression 

line is significantly different from 0 within a 95-percent 

confidence interval. A two-segment regression model was 

developed by using data from the nonhighway sites to account 

for a change in slope of the relation between Rv

Mean and IF. 

The breakpoint between the best-fit segments is at an IF value 

of 0.55. These equations are 

R Mv ean I = + 0 129 0 225 . . × F (6)

if the IF value is less than or equal to 0.55 and 

R Mv ean I = −0 371 . . + × 1 14 F (7)

if the IF is greater than or equal to 0.55 (fig. 10A). The slopes 

of both segments are significantly different from 0 within a 

95-percent confidence interval. The two-segment regression 

model accounts for the steeper trend in site-mean runoff 

coefficients above an IF value of about 0.55. The equations 

developed for the highway sites and the upstream basins 

both produce mean runoff coefficients that are less than 0.8 

for sites that are completely impervious. A site-mean runoff 

coefficient of 0.8 may seem low for a completely impervious 

area, but studies show that evaporation and infiltration from 

paved surfaces may have mean values in the range from about 

20 to more than 30 percent over many storms (Mansell and 

Rollet, 2006; Ramier and others, 2006; Wiles and Sharp, 2008; 

Wanielista and others, 2010). 

The standard deviations of the runoff coefficients (Rv

SD) 

for the highway and the upstream basin (fig. 10B) also are 

unitless. One-segment regression models were developed from 

the highway-site data and data from the nonhighway sites 

because no breaks in slope were apparent in the Rv

SD values 

for both types of site. The highway-site equation is

R Sv D I = − 0 229 0 0373 . . × F , (8)

and the upstream-basin equation is 

R Sv D I = + 0 099 0 015 . . × F . (9)

Neither of these slopes is significantly different from 0 within 

a 95-percent confidence interval. For the standard deviation, 

use of the median value, which is about 0.21 for highway 

sites and about 0.10 for nonhighway sites, has about the same 

predictive power as the associated regression equation.

The coefficients of skew of the runoff coefficients (Rv

Sk) 

for the highway and the upstream basin (fig. 10C) also are 

unitless. A one-segment regression model was developed for 

the highway-site data because no break in slope was apparent: 

R Sv kew = − 2 1. . 3 3 32 × IF . (10)

The slope of this regression line is significantly different from 

0 within a 95-percent confidence interval. A two-segment 

regression model was developed for the skew of runoff 

coefficients at nonhighway sites from the estimated IF. These 

equations are 

(11)

if the IF value is less than or equal to 0.52 and 

(12)

if the IF is greater than or equal to 0.52 (fig. 10C). The model 

accounts for the steeper trend in the coefficient of skew of 

runoff coefficients above an IF of about 0.52. The slope of 

the first segment is not significantly different from 0, but the 

slope of the second segment is significantly different from 0 

within a 95-percent confidence interval. These slopes indicate 

that the skew coefficients vary randomly below an IF of 0.52 

and generally decrease with increasing IF above this threshold 

(Granato, 2010).

The third predefined set of regression equations for 

calculating Rv

 statistics in SELDM is labeled “Schueler 
Theory and Implementation 27

IF = 0.55

IF = 0.52

Average runoff coefficient, 

dimensionless

Standard deviation of the runoff 

coefficient, dimensionless

Coefficient of skew of the runoff 

coefficient, dimensionless

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

-2.0

4.0

3.0

2.0

1.0

0.0

-1.0

0.5

0.4

0.3

0.2

0.1

0.0

0.6

0.7

0.8

0.9

1.0

0.3

0.4

0.5

0.0

0.1

0.2

Impervious fraction (IF ) of site drainage area, dimensionless

A

B

C

Highway Runoff Monitoring Sites 

Site value Regression Line

Site value Regression Line

Other Storm-Runoff Monitoring Sites 

Figure 10. A, The mean, B, standard deviation, and C, coefficient of skew of runoff coefficients for 58 highway-runoff 

monitoring sites and 167 other storm-runoff monitoring sites with 9 or more storm events. Nonparametric regression lines 

indicate the relation between each statistic and the impervious fraction (IF) in the respective drainage areas.
28 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Trimmed NURP.” This set of equations is based on Schueler’s 

(1987) selection of 43 sites with impervious fractions ranging 

from 0.01 to 1.0 from the USEPA NURP report (Athayde and 

others, 1983). The equations are provided for comparison 

with results from other studies or with average calculations 

developed by using Schueler’s “Simple Method” (Schueler, 

1987). The regression equation for the mean Rv

 value is 

Schueler’s (1987) equation. The regression equation for the 

standard deviation was calculated from the coefficient of 

variation (COV, which is the standard deviation divided by the 

mean) of each of these sites reported by Athayde and others 

(1983). Athayde and others (1983) indicated that the Rv

 values 

for individual NURP sites were lognormally distributed; 

therefore, the skew coefficients were estimated from the COV 

by using a theoretical equation (Stedinger and others, 1993). 

The regression equation was developed to estimate the skew 

of Rv

 values with respect to the impervious fraction based on 

these theoretical skew values.

SELDM can model effects of antecedent conditions 

on the coefficients for runoff from the upstream basin by 

using the prestorm streamflow as the explanatory variable. 

If a nonzero correlation coefficient is entered on the runoffcoefficient statistics form, SELDM uses the methods described 

for generating correlated random numbers to calculate the 

plotting position of the runoff coefficients from the plotting 

positions of the prestorm flows. Granato (2010) calculated 

rank correlation coefficients to evaluate potential relations 

between prestorm streamflow and runoff coefficients. 

Although prestorm streamflow is used as the explanatory 

variable, correlation does not necessarily imply causation. 

For example, antecedent precipitation may saturate soils and 

increase prestorm streamflow. In this case, higher prestorm 

streamflow may indicate wetter antecedent conditions, but not 

necessarily cause more runoff. 

In the storm-event database compiled for the SELDM 

study, 43 sites have at least 7 paired prestorm-streamflow 

and runoff-coefficient values (Granato, 2010). Figure 11 

shows the Spearman’s rho values for these datasets and 

the associated 95-percent confidence intervals, which are 

functions of sample size (Caruso and Cliff, 1997). Three sites 

had very weak negative correlations. Ten sites had positive rho 

values that were less than about 0.3, indicating that variations 

in prestorm streamflow may be associated with less than 

30 percent of the variations in runoff coefficients from storm 

to storm at each of these sites. Seven sites have rho values 

between 0.3 and 0.5, indicating that prestorm streamflow 

may be associated with 30 to 50 percent of the variations 

in runoff coefficients at each of these sites. An additional 

14 sites have rho values between 0.5 and 0.71, indicating 

that prestorm streamflow may be associated with 50 to 

71 percent of variations in runoff coefficients at each of these 

sites. Nine sites have rho values that are greater than 0.71, 

Percentage of sites with rank correlation coefficients less than or equal to the specified value

Rank correlation coefficient between base flow and 

the runoff coefficient, dimensionless

1 2 5 10 80706050403020 9590 9998

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

-0.75

Upper

95th-percentile 

confidence

limit

Lower

95th-percentile 

confidence

limit

Figure 11. The upper and lower 95th-percentile confidence limits and the mean values of the nonparametric rank 

correlation coefficients (Spearman’s rho) for each of the 43 monitoring sites with 7 or more paired base-flow and 

runoff measurements (Granato, 2010).
Theory and Implementation 29

indicating a moderate to strong correlation between prestorm 

streamflow and runoff coefficients at these sites. Differences 

in correlation coefficients among sites may reflect hydrologicbasin characteristics, artifacts in the assembled dataset, such 

as the use of different hydrograph-separation techniques in 

each study, or uncertainty in the samples (Granato, 2010). 

Seventeen sites had 95-percent confidence intervals that 

included 0; this result indicates that the true rho value may not 

be different from 0. Conversely, however, 35 of the sites have 

upper confidence limits that are greater than 0.5, which may 

indicate a substantial correlation between prestorm streamflow 

and the runoff coefficient. Rank correlations are user defined 

because prestorm streamflows vary substantially between 

different areas of the nation, and this dataset was too small to 

develop predictive relations for these rank correlation values 

and prestorm flows on the basis of available basin properties.

SELDM models potential relations between runoff 

coefficients for the upstream basin and the highway site by 

using the rank correlation between runoff coefficients because 

the potential for water-quality excursions depends, in part, 

on the volume of runoff from each source. The antecedent 

conditions that affect runoff from the upstream basin also 

may affect runoff from the highway site. Modeling the runoff 

coefficients without any correlation excludes the fact that 

the sites are adjacent and are subject to similar antecedent 

conditions, whereas modeling the runoff coefficients with 

perfect correlation excludes the fact that runoff-generating 

processes for the highway site and the upstream basin may be 

different, especially if the land use and drainage characteristics 

of these areas are different. An extensive literature search for 

information about correlation between runoff coefficients 

from adjacent sites was conducted, but no information was 

found. Four hydrologic experts were queried, but none of 

the respondents knew of any available literature on this topic 

(Dr. W.C. Huber, Oregon State University, written commun., 

September 2010; Dr. L.W. Mays, Arizona State University, 

written commun., September 2010; Dr. R.H. McCuen, 

University of Maryland, written commun., September 2010; 

Dr. M.P. Wanielista, Water Research Center, University of 

Central Florida, written commun. September, 2010).

An algorithm for calculating the rank correlation 

coefficient between runoff coefficients for the upstream 

basin and the highway site was developed under three 

general assumptions: (1) some degree of correlation would 

be expected because antecedent conditions and storm 

characteristics in nearby source areas would be similar; (2) 

the strength of the correlation would improve with increasing 

hydrologic similarity between the upstream and highway sites; 

and (3) the maximum correlation for basins with the same or 

similar impervious fractions would increase as the impervious 

fractions increase because runoff coefficients for different 

pervious areas are expected to be more variable than for 

different impervious areas for a given storm. Large differences 

in drainage areas between the upstream and highway sites 

are expected to introduce variability in the amount and 

distribution of precipitation. SELDM does not directly account 

for spatial variation in precipitation because it is a lumped 

parameter model, but a reduced correlation between runoff 

coefficients may help to account for such effects. Large 

differences in total imperviousness between the highway 

site and the upstream basin may be a surrogate for large 

differences in drainage area because imperviousness tends 

to decrease with increasing drainage areas in stream basins 

(Granato, 2010). Comprehensive datasets are not available to 

confirm or quantify correlations based on these assumptions, 

but the hydrologic experts who responded to the queries for 

information agreed with these general assumptions (Dr. W.C. 

Huber, Oregon State University, written commun., September 

2010; Dr. L.W. Mays, Arizona State University, written 

commun., September 2010; Dr. R.H. McCuen, University 

of Maryland, written commun., September 2010; Dr. M.P. 

Wanielista, Water Research Center, University of Central 

Florida, written commun., September 2010). 

This algorithm was designed to use the absolute value 

of the difference in impervious fractions to calculate the rank 

correlation coefficient between runoff coefficients for the 

upstream basin and the highway site. Linear interpolation 

between impervious fractions and rank correlation coefficients 

was selected as a simple approximation in the absence of 

experimental data. The assumption of hydrologic similarity 

was implemented by calculating the correlation coefficient to 

generate a local maximum value for two sites with the same 

impervious fraction. The assumption that the correlation 

coefficient will increase as the impervious fractions of 

both the highway site and the upstream basin increase was 

implemented by making the local maximum and minimum 

(the floor) values functions of both the impervious fraction and 

the difference in impervious fractions. The linear functions for 

defining the maximum possible correlation coefficients (the 

ceilings) for the highway site and the upstream basin are

Rho R CH = + hoC RhoC RhoC 0 1 ( ) − 0 × IFH , (13)

and

Rho R CU = + hoC RhoC RhoC 0 1 ( ) − 0 × IFU , (14)

where

IFH is the impervious fraction of the highway site, 

IFU is the impervious fraction of the upstream 

basin, 

RhoC0 is the ceiling when the impervious fraction 

equals 0, 

RhoC1 is the ceiling when the impervious fraction 

equals 1,

RhoCH is the local value of the ceiling associated 

with the impervious fraction of the 

highway site, and 

RhoCU is the local value of the ceiling associated 

with the impervious fraction of the 

upstream-basin site.
30 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Similarly, the linear functions for defining the minimum 

possible correlation coefficients (the floors) for the highway 

site and the upstream basin are

Rho R FH = + hoF RhoF RhoF 0 1 ( ) − 0 × IFH , (15)

and

Rho R FU = + hoF RhoF RhoF 0 1 ( ) − 0 × IFU , (16)

where

RhoF0 is the floor when the impervious fraction 

equals 0, 

RhoF1 is the floor when the impervious fraction 

equals 1,

RhoFH is the floor associated with the impervious 

fraction of the highway site, and 

RhoFU is the floor associated with the impervious 

fraction of the upstream-basin site.

The rank correlation value is then calculated by interpolating 

between the local ceiling and floor values according to the 

difference in impervious fractions and averaging the result for 

the highway and upstream basin:

Implementation of this algorithm to generate RhoRv values that 

are greater than 0 requires that the input values are selected 

so that maximum distance between the ceiling (RhoC) and 

the floor (RhoF) will be less than the ceiling value for the 

whole range of impervious fractions. The ceiling and floor 

values were implemented as default values that can be 

changed in the table named “tblHighwayAnalysis” in the 

SELDM database for a given analysis or reset in the table 

design for all subsequent analyses. This default-value option 

was selected because neither available data nor the published 

hydrologic literature is sufficient to support either fixed values 

in the program code or guidance for user inputs in the model 

graphical user interface (GUI).

SELDM was implemented with ceiling and floor values 

that result in Spearman’s rho values ranging from 0.9875 

to 0.375 with local maxima where the impervious fractions 

are equal. The Spearman’s rho values in figure 12 were 

calculated by using ceiling values of 0.5 and 0.9875 where 

the impervious fractions equal 0 and 1, and floor values of 

0.25 and 0.5 where the impervious fractions equal 0 and 1. 

These values were selected to provide a high correlation 

between highly impervious and hydrologically similar sites 

and reduced but nonzero values as impervious fractions 

diverge. The minimum RhoRv value (0.375) corresponds to the 

maximum difference in impervious fractions (equation 17, 

fig. 12) between completely impervious (IF = 100) and 

completely pervious (IF = 0) basins. Local RhoRv minimums 

along the X and Y axes coincide with local-maximum 

differences in impervious fractions along the axis. The values 

of RhoRv along the axes are equal to the means of the floor 

values (equation 17), which are functions of imperviousness 

(equations 15 and 16). Local maximums coincide with equal 

IF values; the RhoRv value equals the ceiling value for that 

impervious fraction, which in this case is a linear function of 

IF between 0.5 and 0.9875 (equations 13 and 14) The local 

maximums appear as the diagonal ridge on the contour plot 

(fig. 12).

SELDM generates a population of Rv

 values for the 

upstream basin and the highway site by using a four-step 

process: (1) the plotting positions for the Rv

 values of the 

upstream basin are generated by using the specified correlation 

to the plotting positions of upstream prestorm flow; (2) the 

upstream Rv value is calculated from upstream Rv

 statistics by 

using the frequency-factor method (equation 1) with Pearson 

Type III variates; (3) the plotting positions for the Rv

 values of 

the highway site are generated from the specified correlations 

to plotting positions for the Rv

 values of the upstream basin; 

and (4) the highway Rv

 value is calculated from highway Rv

statistics by using the frequency-factor method (equation 1) 

Rho Rho Rho Rho IF IF Rho Rho Rho IF

Rv

CH CH FH H U CU CU FU

= ( ) − − ( ) × − + − ( ) − × H U ( ( ) − IF )

2

. (17)

with Pearson Type III variates. If the resultant Rv

 values are 

less than 0 or greater than 1, then the Rv

 values are set equal to 

0 or equal to 1, respectively.

Storm-Event Hydrographs

SELDM is designed to calculate the event-mean 

concentrations and total storm loads for the entire highwayrunoff event rather than intraevent flows and loads, but 

modeling the timing of flow from the highway, from the BMP 

outfall (if a BMP is selected), and from the upstream basin is 

necessary to calculate the potential amount of dilution during 

the period of discharge to the stream (fig. 13). Granato (2010) 

demonstrated that triangular runoff hydrographs commonly 

are used to model intraevent stormflows in hydraulic and 

water-quality models and are adequate for producing 

planning-level estimates for dilution analyses. It is necessary 

to model the intraevent stormflow hydrographs because 

differences in the locations, sizes, and drainage characteristics 

of the highway catchment and the upstream basin may cause 

differences in the timings and durations of runoff from each 

area. For example, if the highway catchment is small and the 

runoff drains directly to the stream, the duration of appreciable 

runoff from the highway catchment may be approximated 
Theory and Implementation 31

Impervious fraction of the upstream basin, dimensionless

Impervious fraction of the highway site, dimensionless

0.0

0.0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8 0.9

0.9

1.0

1.0

rho = 0.9

rho = 0.8

rho = 0.7

rho = 0.6

rho = 0.5

rho = 0.4

rho = 0.5

rho = 0.4

Figure 12. Calculated Spearman’s rho values showing the relations between the impervious fractions of the 

upstream basin and the highway site. These Spearman’s rho values are calculated by using ceiling values of 0.5 and 

0.9875 for impervious fraction values of 0 and 1 and floor values of 0.25 and 0.5 for impervious fraction values of 0 and 

1, respectively.
32 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Runoff

Prestorm flow

Upstream 

runoff

Total storm flow

Upstream-flow components

Upstream 

prestorm 

flow

Highway runoff with 

10-percent retention and 

extended detention

Unmodified highway runoff

Runoff duration

A. Hypothetical triangular hydrographs

B. Hypothetical cumulative volumes

Percentage of total upstream flow

Duration 2

Duration 1

Duration 1

Duration 2

Runoff flow rate

Time

Time

EXPLANATION

0

90

80

70

60

50

40

30

20

10

100

Figure 13. A, Hypothetical triangular hydrographs, and B, the 

hypothetical cumulative upstream-stormflow volume that would 

coincide with unmodified runoff from a highway (duration 1) and 

with runoff from an extended detention structure (duration 2). This 

diagram shows the hypothetical runoff event with two upstreamflow components (runoff and prestorm flow), an unmodified 

highway-runoff hydrograph, and a highway-runoff hydrograph 

with retention and detention.

by the duration of the precipitation event. If the upstream 

basin is relatively large and more pervious than the highway 

catchment, appreciable runoff from the basin may continue for 

hours or days longer than runoff from the highway catchment. 

In this case, only a small proportion of the upstream runoff 

may be available to dilute highway-runoff constituents in 

the receiving waters. If, however, a structural BMP has been 

installed (or will be used) at the highway site to attenuate and 

extend the highway-runoff hydrograph, then a longer period of 

the upstream runoff may coincide with the period of highway 

runoff, and thus a larger volume of the upstream stormflow 

will be available to dilute highway-runoff constituents in the 

receiving waters.

This concept is demonstrated schematically in figure 13. 

In this hypothetical example, the triangular runoff hydrograph 

for the upstream basin is superimposed on a rectangular 

representation of the prestorm base-flow volume (fig. 13A). 

The durations of highway-runoff hydrographs with and 

without BMP modification are labeled “Duration 1” and 

“Duration 2.” As indicated in the figure, a small increase in 

the duration of runoff from the highway may be accompanied 

by a large increase in the cumulative amount of concurrent 

runoff and base flow from the upstream basin, especially in 

the rising limb of the upstream-basin hydrograph (fig. 13B). If 

the highway-runoff or the BMP-discharge hydrograph extends 

beyond the end of the upstream-runoff hydrograph, then the 

available dilution equals the sum of the storm-event base flow 

(calculated as the product of the prestorm-flow rate and the 

duration of the storm), the storm-event runoff (fig. 13), and 

the poststorm base flow, which is calculated as the product of 

the prestorm-flow rate and the excess duration of the highway 

runoff or BMP discharge.

A triangular runoff hydrograph is used to calculate the 

amount of concurrent stormflow (fig. 13). The triangular 

runoff hydrograph can be fully parameterized with the total 

runoff volume, the start of runoff (To

), the end of runoff (Te), 

and the time to peak (Tp

), respectively (fig. 14). Although 

the triangular hydrograph is a simple linear approximation, 

the CDF is an S-curve that is similar to the CDF for more 

complex approximations (Granato, 2010). Thus, the triangular 

hydrograph is sufficient for modeling the temporal distribution 

of flow during a storm. The proportion of total runoff at time 

Ti

 from the beginning of the storm for a triangular hydrograph 

is expressed as

(18)

if To <= Ti <= Tp and

(19)

if Tp <= Ti <= Te, 
Theory and Implementation 33

D Duration of rainfall-excess increment

BL Basin-lag time

Tp

Tf

Tb

Time to peak (rising-limb time)

Recession (falling-limb time)

Hydrograph base time

T0 Te

To Begin time of the runoff hydrograph

Te End time of the runoff hydrograph

Tf Tp

Tb

D

2 BL

D

(volume not 

to scale)

Runoff flow rate

EXPLANATION

Centroid

Figure 14. Time factors for a triangular storm-event hydrograph (Modified from Kent, 1973).

where

Rc is the cumulative proportion of the total runoff 

at time Ti

,

Ti is any selected time step within the runoff 

hydrograph, 

To is the begin time of the runoff hydrograph, 

Te is the end time of the runoff hydrograph, and

Tp is the peak time of the runoff hydrograph.

If the begin time is set to 0, the end time (Te

) is equal to 

the duration of the runoff hydrograph Tb

 (fig. 14). The time 

to peak is commonly calculated as a function of one-half 

the duration of rainfall-excess increment (D/2) plus a basin 

lagtime (BL, in hours). In SELDM, the duration of rainfallexcess increment is set equal to the stochastic precipitation 

duration. Although there are many definitions of the basin 

lagtime in the literature (Rao and Delleur, 1974; Linsley 

and others, 1975; Chow and others, 1988; Fang and others, 

2005), SELDM defines the basin lagtime as the time from the 

center-of-mass (centroid) of rainfall excess to the centroid 

of the corresponding runoff hydrograph. This definition was 

selected because the regression equations that are used to 

calculate the basin lagtime from drainage-basin properties 

were developed primarily with data from U.S. Geological 

Survey runoff studies, which are based on the center-of-mass 

definition (Granato, 2012). The relation between the time to 

peak Tp

 (in hours) and the time to the centroid Tc (in hours) of 

the runoff hydrograph is a function of the ratio (Rf

, which is 

unitless) of the duration of the falling limb to the rising limb 

of the hydrograph. For a triangular hydrograph this may be 

calculated as

. (20)

The BL in equation 20 commonly is defined as a 

characteristic of the basin rather than a characteristic of 

individual storms, so this variable is fixed in the analysis. 

SELDM, however, uses Monte Carlo methods to generate 

random precipitation-event durations (D) and Rf

; which 

are values calculated using input statistics; as a result, Tp

 is 
34 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

calculated as a random variable, a property which is consistent 

with observed hydrographs (Granato, 2012). Precipitationevent durations (D) are generated from synoptic precipitation 

statistics (selected or input on the synoptic storm-event 

precipitation-statistics form) with a two-parameter exponential 

distribution. The upstream hydrograph recession-ratio values 

(Rf

) are generated by using user-defined values (input on the 

upstream-basin form) with a triangular distribution (table 1) 

(Granato, 2010, 2012). The Rf value for the highway site, 

however, is fixed at a value of 1 (an isosceles triangle with 

equal rising- and falling-limb durations); this method is 

consistent with the rational method for runoff hydrographs for 

small highly impervious sites (Granato, 2010).

Granato (2012) developed multiple linear regression 

equations to estimate the values of basin lagtime used in 

SELDM from drainage-basin properties of the highway site 

and the upstream basin (which are entered on the highway-site 

and upstream-basin forms). These drainage-basin properties 

are defined in detail in appendix 2. The basin lag factor (BLF), 

which is the basin length in miles divided by the square 

root of the channel slope in feet per mile was selected as the 

primary physiographic variable. One multiple linear regression 

equation that included the BLF and the BDF (appendix 2) 

for calculating basin lagtimes was developed on the basis of 

data from 493 sites documented in 22 different studies (the 

primary dataset). Another multiple linear regression equation 

for calculating basin lagtimes from the BLF and the TIA was 

developed on the basis of data from 896 sites documented 

in 37 different studies (the secondary dataset). Both datasets 

are comprehensive representations of the characteristics of 

potential highway sites and upstream basins. Basin drainage 

areas range from 0.000116 mi2

 (about 0.074 acre) to 1,477 mi2

in both datasets. The median drainage areas are 4.1 and 3.8 mi2

in the primary and secondary datasets, respectively. BDF

values range from 0 to 12 with a median of 5 in the primary 

dataset. TIA values in both datasets range from 0 to 100. These 

regression equations are

no physical meaning; entering this value directs the model to 

use TIA (equation 22) rather than the BDF (equation 21) to 

calculate BL.

Triangular hydrograph-recession-ratio (Rf

) statistics also 

are needed to calculate the time to peak Tp

 (equation 20) and 

the duration of the runoff hydrograph Tb

 for each individual 

storm event. The minimum (Rf-Min), most probable (Rf-MPV), 

and maximum (Rf-Max) values may be estimated visually, by 

using literature values, or by calculating values from storm 

hydrographs from nearby hydrologically similar basins. 

Granato (2010) documented a range of values of Rf

 from the 

literature that might be suitable for an analysis. He found that 

hydrograph-recession studies are not common in the literature 

because most stormflow studies focus on the basin lag and 

magnitude of the peak flow to provide information for flood 

control. However, he found that qualitative estimates of Rf

ranged from 1 to 12 in several studies with values that were 

attributed to basin characteristics such as basin size, slope, 

and the degree of development. Granato (2012) developed 

a computer program and several spreadsheets to implement 

methods for fitting triangular hydrographs and calculating 

recession-ratio statistics. These methods were developed to 

estimate Rf

 statistics using instantaneous streamflow data 

available from the USGS instantaneous data archive (U.S. 

Geological Survey, 2012).

Granato (2012) developed triangular hydrographrecession ratio statistics by using instantaneous streamflow 

data from 32 USGS streamgages draining portions of 

Massachusetts and 9 USGS streamgages in other areas of 

the country (fig. 16). In this dataset, the minimum recession 

ratios ranged from 1 to 1.77 with a median of 1.02 and a 

mean of 1.15; thus, the minimum recession ratios are well 

characterized using a value of 1. The most probable value 

(MPV) of recession ratios ranged from 1 to 3.52 with a median 

and mean of 1.85. This median compares well with the median 

of 1.67 developed from average curvilinear flood hydrographs 

from USGS studies in different areas of the country. The 

maximum recession ratios ranged from 2.66 to 11.31 with a 

median of 4.32 and a mean of 4.76.

Granato (2012) found that correlations between recession 

ratios and basin characteristics were weak; this result, 

which was consistent with the findings of other hydrograph 

recession studies in the literature (Shamir and others, 

2005; Shuster and others, 2008), precluded development of 

meaningful predictive equations. Quantitative Rf

 selections 

are not possible through use of the information documented 

by Granato (2012), but qualitative estimates can be made 

on the basis of hydrologic similarity. Comparison of basin 

characteristics at the site of interest with basin characteristics 

for streamgages in the Rf

 dataset compiled by Granato (2012) 

may inform the choice of Rf

 statistics that are greater than 

or less than the median values in figure 16. For example, 

development is weakly associated with decreased MPVs of Rf

values, whereas forested areas and onstream impoundments 

are weakly associated with increased MPVs.

BL BL = × 0 967 F B × − 13 DF 0 571 0 681 . ( ) . . (21)

and

BL BL = × 0 499 F T × − 100 0 99 × IA 0 601 0 443 . ( . ) . . . (22)

The structure of these equations and the selected 

explanatory variables were informed by results of many other 

studies (Granato, 2012), including the USGS nationwide 

regression equations (Sauer and others, 1983). Selected values 

of the regression equations are shown with the source data in 

figure 15. Equation 21 is used to calculate the basin-lagtime 

values for the highway site or upstream basin by SELDM if 

a BDF in the range from 0 to 12 is entered in the highwaysite form or the upstream-basin form. To use equation 22 to 

calculate the basin-lagtime values for the highway site or 

upstream basin as a function of the BLF and TIA, enter a BDF

value of -1 in the appropriate form. A BDF value of -1 has 
Theory and Implementation 35

Dilution Factors

SELDM uses the stochastically generated highway 

stormflows, BMP discharges, and concurrent upstream 

stormflows to calculate dilution factors for each storm. 

The dilution factor is the ratio of highway runoff (or BMP 

discharge) to downstream flow. The dilution factor is 

calculated as

, (23)

where

DF is the dilution factor, which is dimensionless;

HQ is the highway runoff or BMP-discharge 

volume, in cubic feet (ft3

); 

DQ is the downstream stormflow concurrent with 

the highway runoff or BMP discharge, 

in ft3

; and

UQ is the upstream stormflow concurrent to the 

highway runoff or BMP discharge, in ft3

(Driscoll and others, 1990b).

The dilution factor can vary from 0 to 1 as the highway 

runoff increases in proportion to the upstream flow. A dilution 

factor near 0 indicates that highway runoff is a negligible 

portion of the downstream flow. A dilution factor of 1 

indicates that the downstream flow is all highway runoff. 

The dilution factor increases as dilution decreases; Driscoll 

and others (1990b) defined the dilution factor in this way to 

prevent division by 0 errors in the 1990 FHWA runoff model. 

SELDM calculates dilution factors for highway runoff with 

and without BMP modification. A BMP that extends highway 

stormflows will decrease the dilution factor (increasing 

dilution) by incorporating a larger portion of the total upstream 

stormflow as concurrent stormflow (fig. 13). The highway and 

BMP dilution-factor outputs will be equal if there is no BMP 

flow modification.

The dilution-factor output provides a quick initial 

assessment of the risks for water-quality excursions with 

and without BMP treatment. For example, examination of 

the dilution-factor file for each of several highway-stream 

crossings can be used to identify the streams with the 

highest potential for excursions. Similarly, if a highway with 

many outfalls is parallel to a stream, information about the 

cumulative upstream drainage and pavement areas at each 

outfall can be used to run SELDM. The dilution-factor file for 

each outfall can be used to identify the point along the stream 

with the highest potential for excursions. In either case, this 

information can be used to allocate resources for a detailed 

analysis of flows, loads, and concentrations at the most 

critical site(s).

Highway and Upstream Stormwater 

Concentrations and Loads

Estimates of highway and upstream stormwater 

concentrations and loads are needed for using a mass-balance 

approach to predict the potential effects of runoff on receiving 

waters (Warn and Brew, 1980; Di Toro, 1984; Schwartz and 

Naiman, 1999; Granato, 2010). The mass-balance approach for 

storm-event analyses is based on the sum of the loads from the 

highway and the upstream basin. These loads are the product 

of stochastically generated random runoff concentrations 

and flows. The concentration of runoff constituents from 

the highway can be modeled as stochastic random values 

or stochastic dependent values. The concentration of runoff 

constituents from the upstream basin can be modeled as 

stochastic random values, stochastic dependent values, or 

stochastic transport-curve values. The statistics for generating 

these constituent concentrations are based on available data. 

Sources of Water-Quality Data

Water-quality modeling methods in SELDM are designed 

to support the FHWA step-by-step decision tree for waterquality-assessments (Sevin, 1987; Cazenas and others, 1996; 

Federal Highway Administration, 1998). This process starts 

with an initial assessment on the basis of available data and 

proceeds to more detailed analyses if the risk of an adverse 

effect is unacceptable to decisionmakers. SELDM uses 

regional water-quality statistics to facilitate generation of 

initial planning-level estimates. If necessary, initial estimates 

can be refined with water-quality statistics based on available 

data collected at nearby hydrologically similar sites or at 

the site of interest. Basing the initial analyses on available 

data is a prudent approach because collecting sufficient 

water-quality data to effectively characterize conditions 

at a given site is expensive, difficult, and time consuming. 

Furthermore, regional estimates may be more robust for 

predicting environmental variables at unmonitored sites than 

measurements from a relatively short-duration site-specific 

sampling program, unless this program generates enough data 

to represent conditions at the site of interest, characterizes 

the full range of discharges, and is not affected by short-term 

natural or anthropogenic factors (Hughes and Larsen, 1988; 

Hosking and Wallis, 1997; Vogel and others, 1998; Robertson 

and others, 2001; Shirazi and others, 2001; Jenerette and 

others, 2002). The results of short-duration site-specific 

sampling programs also may not characterize conditions that 

are likely to occur after a highway project is complete.

Highway-Runoff-Quality Data

Nationally, the Highway Runoff Database (HRDB) 

is the primary source of highway-runoff statistics and data 

for use with SELDM (Granato and Cazenas, 2009). The 

HRDB application is designed as a data warehouse in which 

to document data and information from highway-runoff 
36 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

100

10

1.0

0.1

0.01

0.001 0.01 0.1 1.0 10 100

200

BDF=12

BDF=0

Primary dataset (493 sites)

Secondary dataset (896 sites)

Regression-equation lines

Basin lagtime = 1.272 x 0.760 x BLF 0.571 x (13-BDF) 0.681

A

Basin lagtime, in hours

Basin lag factor (BLF=Length/Slope0.5)

EXPLANATION

Figure 15. The basin-lagtime data and regression equations based on the basin-lag factor (BLF) and A, the basin development 

factor (BDF) with the primary dataset, or B, the total impervious area (TIA) with the secondary dataset.
Theory and Implementation 37

100

10

1.0

0.1

0.01

0.001 0.01 0.1 1.0 10 100

200

TIA=100

Primary dataset (493 sites)

Secondary dataset (896 sites)

Regression-equation lines

Basin lagtime = 1.306 x 0.382 x BLF 0.601 x (100-0.99xTIA ) 0.443

TIA=90

TIA=0

Basin lagtime, in hours

Basin lag factor (BLF=Length/Slope0.5)

EXPLANATION

B

Figure 15. The basin-lagtime data and regression equations based on the basin-lag factor (BLF) and A, the basin development 

factor (BDF) with the primary dataset, or B, the total impervious area (TIA) with the secondary dataset.—Continued
38 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0 Triangular recession-ratio statistics

Minimum

MPV

Maximum

1

2

3

4

5

6

7

8

9

10

11

12

Maximum

90th percentile

75th percentile

25th percentile

10th percentile

Mean

Median

Minimum

Figure 16. Distributions of the minimum, most probable 

value (MPV), and maximum for the statistics of the bestfit triangular hydrograph-recession ratio estimated from 

20 or more storm-event hydrographs from each of the 

41 streamgages in the multistate dataset documented by 

Granato (2012).

monitoring studies and as a preprocessor for highwayrunoff data for use in the SELDM application. Available 

highway-runoff data provide the basis for defining runoff 

quality and quantity at monitored sites and predicting runoff 

quality and quantity at unmonitored sites. Version 1.0.0 of 

the HRDB included data from 2,650 storms for 39,713 EMC 

measurements of more than 100 water-quality constituents 

monitored at 103 sites in the conterminous United States 

and documented in 7 highway-runoff datasets (Granato 

and Cazenas, 2009). Smith and Granato (2010) added data 

from Massachusetts and Ohio to produce version 1.0.0a 

of the database. This newer version includes 54,384 EMC 

measurements of 194 water-quality constituents monitored at 

117 sites during 4,186 storm events.

The HRDB application also is designed to be a 

preprocessor for use with SELDM (Granato and Cazenas, 

2009). Most common data-manipulation tasks can be 

accomplished with the GUI of the HRDB or several predefined 

queries by users with only a cursory knowledge of Microsoft 

Access®. The database application provides standard and 

robust estimates of population statistics for highway-runoff 

data. The HRDB calculates the mean, standard deviation, and 

skew of the data and the logarithms of data for the random 

water-quality simulations. It uses accepted methods for 

calculating these statistics for datasets with one or more values 

that are below detection limits. The HRDB also is designed 

to export paired water-quality data in a format that can be 

used with the KTRLine program (Granato, 2006) to calculate 

regression statistics defining dependent water-quality relations 

for highway-runoff modeling.

Additional highway-runoff data also may be available 

from six state departments of transportation. The HRDB 

program contains data collected in California through 2004 

and in Washington through 2005, but these states have 

ongoing stormwater-research programs (Gersib, 2011; 

McGowen, 2011). A study of highway runoff is currently 

under way in Oregon (Fletcher, 2011). DOTs in North 

Carolina (Wagner and others, 2011), South Carolina (Conlon 

and Journey, 2008), and Wisconsin (Horwatich and others, 

2011) recently completed comprehensive highway-runoff 

monitoring studies with the USGS. These data could be 

integrated into the HRDB to facilitate runoff-quality analyses. 

Version 3 of the National Stormwater Quality Database 

(NSQD) is another potential source of highway-runoffquality data (Pitt and others, 2008). This database, which is 

in Microsoft Excel® format, includes data from the National 

Pollutant Discharge Elimination System (NPDES) from 

44 highway sites. The highway-runoff-quality data in this 

database include 5,454 EMC or grab-sample concentrations of 

54 constituents from 734 storm events. This is a large dataset, 

but Pitt and others (2008) described serious concerns about the 

reliability and utility of these phase-one stormwater NPDES 

monitoring data because many different experimental designs, 

sampling procedures, and analytical techniques were used in 

the different studies. The NSQD does not include methods 

to query data or generate statistics beyond the standard 

Excel features.
Theory and Implementation 39

Upstream-Water-Quality Data

The national water-quality database compiled by Granato 

and others (2009) may be the primary source of planninglevel upstream-water-quality data that is readily available 

for use with SELDM. The compilation was based on data 

available in the USGS National Water Information System 

Web (NWISWeb), a source of water-quality data that can 

be used to estimate local, regional, or national water-quality 

parameters (Mathey, 1998; U.S. Geological Survey, 2002, 

2011). As of October 2011, NWISWeb included data for 

118,000 stream sites, 2,456 canals, 1,701 ditches, 550 outfalls, 

534 wetlands, 211 storm sewers, 145 combined sewers, and 

141 pavement sites in the conterminous United States. It also 

includes stream water-quality data for 2,540 streams in Alaska, 

1,031 streams on the Caribbean islands, and 1,480 streams 

on the Pacific islands. Review of the data, however, indicates 

that the number of water-quality samples listed for any given 

constituent is small for many sites. For example, in a review of 

sediment data in the NWISWeb database, Turcios and others 

(2010) determined that 30 or more paired suspended sediment 

and streamflow measurements were available for fewer than 

25 percent of the listed monitoring sites. For this reason, 

regionalization or combining data from nearby sites may be 

necessary to produce quantitative estimates of water quality at 

unmonitored sites from available datasets.

Granato and others (2009) selected water-qualitymonitoring stations in the conterminous United States at 

stream sites having a defined drainage area and at least 

one paired streamflow and water-quality measurement in 

NWISWeb. A total of 24,581 stations with drainage areas 

ranging from 0.002 to 1,140 mi2

 were identified and cataloged 

for retrieval of water-quality data from NWISWeb (fig. 17). 

The percentages of the selected sites with drainage areas less 

than 0.1, 1, and 10 mi2

 are 0.7, 5.4, and 28.9, respectively. 

These basins were not screened for land-use characteristics 

or impervious fractions because this analysis was designed 

to provide generalized estimates of ambient receiving-water 

quality by ecoregion without incorporation of site-specific 

characteristics. In comparison, the focus of the USEPA 

analysis effort has been to define reference or minimally 

impacted conditions by selecting reference sites or choosing 

the lowest quartile of data from all sites in a region (U.S. 

Environmental Protection Agency, 2000; Robertson and 

others, 2001). However, relatively few sites have been 

monitored to determine reference conditions. For example, 

Biesecker and Leifeste (1975) characterized water quality 

at 57 unaltered sites. Similarly, Smith and others (2003) 

extrapolated estimates of natural background concentrations 

of nutrients in streams and rivers of the conterminous 

United States from the water-quality data for 63 stations 

120° 110° 100° 90° 80° 70°

30°

40°

50°

Figure 17. The spatial distribution of 24,581 water-quality-monitoring stations (black dots) with drainage areas less than 1,050 

square miles within U.S. Environmental Protection Agency (2003) Level III ecoregions (colored polygons) that have been discretized 

to a 15-minute grid in the conterminous United States (geographic projection). Ecoregions are identified on the plate useco.pdf on 

the CD–ROM accompanying this report.
40 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

that met all of the USEPA criteria for minimally impacted 

reference basins. Use of reference basins that represent 

natural background concentrations, however, may lead to 

underestimation of water-quality constituent concentrations 

commonly found in streams receiving highway runoff. 

Thus, data from reference basins may, on a regional basis, 

underrepresent the potential for adverse water-quality effects 

from highway runoff. Conversely, selection of data from sites 

that are substantially affected by anthropogenic activities may, 

on a regional basis, over represent the potential for adverse 

water-quality effects from highway runoff.

Granato and others (2009) developed methods to derive 

order-of-magnitude planning-level estimates of EMCs in 

receiving waters at unmonitored sites in the conterminous 

United States. The methods also may be used to obtain and 

interpret more quantitative site-specific data. Granato and 

others (2009) used data-mining and analysis techniques to 

identify and compile more than 1,876,000 paired streamflow 

and water-quality measurements that included 21 constituents 

commonly measured in highway and urban runoff studies 

and were made between October 1, 1900 and September 30, 

2004 (table 4). Granato and others (2009) documented these 

techniques and the associated computer programs used to 

obtain and analyze the data (fig. 18). The first step of the 

data-compilation process is to download paired concentration 

and streamflow data from NWISWeb with the National 

Water Information System Wizard (NWiz) developed for this 

project. Water-quality monitoring stations were identified in 

NWISWeb and grouped according to ecoregions by using 

GIS software. Once stations were identified, three programs 

were used to download and filter the data. These programs are 

NWiz, the National Water Information System Site Cleaner 

(NWISSC), and Relational DataBase File Processor (RDBP). 

The data selected for further analysis was imported into the 

Surface-Water-Quality Data Miner (SWQDM), which is a 

relational database that can be used to associate the paired 

concentration and streamflow data with an ecoregion or 

any user-specified site location in the conterminous United 

States. Granato and others (2009) used the SWQDM with 

the KTRLine program to develop the water-quality transport 

curves by ecoregion. Transport curves are provided in SELDM 

for total (unfiltered) phosphorus (parameter code p00665), 

total hardness (parameter code p00900) and suspended 

sediment concentrations (parameter code p80154).

Although the dataset compiled by Granato and others 

(2009) is extensive, a number of limitations apply to use of 

the dataset. Most of the USGS data probably are not EMC 

values because most samples were collected for USGS status 

and trends studies. Methods for collection, processing, and 

analysis of samples have changed during the period of record. 

The data include the effects of trends caused by local changes 

in land use and potentially by a number of regional and 

national processes through time. The variability in regional 

water-quality estimates incorporates such at-site variations as 

well as site-to-site variations within each region. However, 

the data compilation and interpretation methods described 

in this report may be used with other information, such as 

local land-use data, for more selective regional or local data 

analysis.

The NSQD database, which contains a substantial 

amount of NPDES data, also is another potential source 

of water-quality data that can be used to model upstream 

stormflow quality (Pitt and others, 2008). The runoff-quality 

data in this database include 75,291 EMC or grab-sample 

measurements (including 91 water-quality constituents) from 

7,474 storm events monitored at 476 runoff-monitoring sites. 

These data are comparable to highway-runoff values because 

they are EMC or grab-sample measurements rather than the 

instantaneous streamflow and concentration data available in 

NWISWeb. These data, however, may not be representative 

of the water quality in many upstream basins because they are 

limited to relatively small drainage areas with predominantly 

highly impervious and homogenous land uses. For example, 

the nonhighway sites in this database have drainage areas 

ranging from about 0.001 to 16 mi2

, but about 58.5 percent of 

these sites have drainage areas less than 0.1 mi2

 (65 acres), and 

95.25 percent have drainage areas less than 1 mi2

 (640 acres). 

About 72 percent of the NPDES monitoring sites cataloged by 

Pitt and others (2008) have TIA values equal to or exceeding 

30 percent of the drainage area, a value commonly defined as 

the threshold for designating urban areas. About 46 percent of 

these sites have TIA values equal to or exceeding 50 percent of 

the drainage area, a value commonly defined as the threshold 

for designating ultraurban areas (Shoemaker and others, 

2000). About 57 percent of the drainage areas include one 

land-use category, about 23 percent two land-use categories, 

and about 20 percent more than two land-use categories.

Many of the upstream basins that could be analyzed 

in SELDM may be larger, have lower TIA values, and be 

more diverse than these NPDES monitoring basins. For 

example, Falcone and others (2010) cataloged basin properties 

for 6,785 USGS streamgages, many of which are at road 

crossings. Fewer than 1 percent of the upstream basin areas 

to these streamgages are less than 1 mi2

. About 98.6 and 

99.9 percent of these areas have TIA values that are less than 

30 and 50 percent.

Data also may be available from the USEPA STOrage 

and RETrieval (STORET) database (U.S. Environmental 

Protection Agency, 2005b). The STORET Web site includes 

legacy data collected from the 1960s through 1999 and an 

option for a modernized STORET system with data collected 

since 1999. As of 2006, the STORET database did not include 

data for many states, including parts of Alabama, California, 

Idaho, Illinois, Indiana, Massachusetts, Mississippi, Nevada, 

New Mexico, New York, Oregon, Texas, Virginia, and 

Washington. The STORET database also includes data from 

wells, wastewater-treatment plants, USEPA Superfund sites, 

landfills, and mine-discharge points, which are important on 

a local scale but may skew estimates of ambient receivingwater quality for a regional analysis. The USGS NWISWeb 

and the USEPA STORET databases use common definitions 

and formats to provide a common view of data for the two 

systems. The methods described in this report may be adapted 

for selective use with STORET data.
Theory and Implementation 41

Table 4. Paired water-quality and streamflow measurements made between October 1900 and September 2004 

compiled by Granato and others (2009) from data for 24,581 water-quality-monitoring stations throughout the 

conterminous United States.

[Pcode, parameter code; USEPA, United States Environmental Protection Agency]

USEPA 

PCode

Number of 

samples

Number of 

stations PCode definition

Physical properties, major ions, solids and sediment, and bacteria

p00340 15,273 959 Chemical oxygen demand, high level, water, unfiltered, milligrams per liter

p00403 129,666 7,753 pH, water, unfiltered, laboratory, standard units

p00900 107,289 7,290 Hardness, water, unfiltered, milligrams per liter as calcium carbonate

p00940 206,341 10,936 Chloride, water, filtered, milligrams per liter

p00530 84,364 2,397 Solids, suspended, water, milligrams per liter

p00535 28,831 525 Solids, volatile suspended, water, milligrams per liter

p80154 275,950 7,477 Suspended sediment concentration, milligrams per liter

p31501 22,998 1,052 Total coliform, M-Endo MF method, immediate, water, colonies per 100 milliliters

Nutrients

p00600 50,160 2,820 Total nitrogen, water, unfiltered, milligrams per liter

p00630 137,231 4,936 Nitrite plus nitrate, water, unfiltered, milligrams per liter as nitrogen

p00665 246,403 8,169 Phosphorus, water, unfiltered, milligrams per liter

p00671 113,896 5,594 Orthophosphate, water, filtered, milligrams per liter as phosphorus

p00680 83,212 3,513 Organic carbon, water, unfiltered, milligrams per liter

Metals

p01027 38,352 2,986 Cadmium, water, unfiltered, micrograms per liter

p01113 84 8 Cadmium, water, unfiltered, recoverable, micrograms per liter

p01034 40,855 3,244 Chromium, water, unfiltered, recoverable, micrograms per liter

p01118 34 8 Chromium, water, unfiltered, recoverable, micrograms per liter

p01042 40,563 2,978 Copper, water, unfiltered, recoverable, micrograms per liter

p01119 554 28 Copper, water, unfiltered, recoverable, micrograms per liter

p01045 77,899 5,864 Iron, water, unfiltered, recoverable, micrograms per liter

p01051 35,585 2,771 Lead, water, unfiltered, recoverable, micrograms per liter

p01114 408 19 Lead, water, unfiltered, recoverable, micrograms per liter

p01055 69,248 5,511 Manganese, water, unfiltered, recoverable, micrograms per liter

p01067 18,805 1,891 Nickel, water, unfiltered, recoverable, micrograms per liter

p01074 34 3 Nickel, water, unfiltered, recoverable, micrograms per liter

p01092 52,131 3,837 Zinc, water, unfiltered, recoverable, micrograms per liter

p01094 468 27 Zinc, water, unfiltered, recoverable, micrograms per liter
42 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Identify water-quality-monitoring

stations that meet criteria

(National Water Information System) 

Sort by ecoregion

(Geographic Information System)

Sufficient

coverage?

Yes

No Modify

criteria

Download paired concentration

and streamflow data

(NWiz and NWIS)

Filter data retrieval for 

files with paired-data values

(NWISSC)

Extract paired-data values and 

format for data-mining database

(NWIS RDB Processor RDBP)

Select and group paired-data values 

by ecoregion or by proximity 

among stations

(Surface-water-quality data miner database)

Nonparametric regression analysis 

with Kendall-Theil robust line method

(KTRLine Version 1.0)

Application of regression models

Regionalization

and

site selection

Data

compilation

Data

interpretation

Figure 18. The steps (and associated software) used for 

regionalization and site selection, data compilation, and data 

interpretation to define relations between surface-water 

discharge and concentrations of selected water-quality 

constituents.

If the sediment-associated fraction dominates the total 

concentration of a constituent in receiving waters, then the 

suspended sediment concentration can be used with the 

concentration of that constituent in sediment to approximate 

upstream concentrations (Breault and Granato, 2003; 

Mahler and Van Metre, 2003). The USGS National WaterQuality Assessment Program (NAWQA, accessible at 

http://water.usgs.gov/nawqa/) has produced many reports 

characterizing the concentrations of trace elements and organic 

chemicals in fine-grained streambed sediments within study 

areas throughout the United States. The USGS also provides 

extensive datasets characterizing the trace-element chemistry 

of soils and stream sediments in the National Geochemical 

Database (Smith, 2006) and the National Geochemical Survey 

(Grossman and others, 2008). Rice (1999) compares traceelement concentrations in urban and rural areas across the 

conterminous United States and provides this dataset on the 

Internet. These data can be used with SELDM by specifying 

dependent water-quality relations in the model. 

Random Water-Quality Modeling

The SELDM random water-quality option provides the 

means for stochastically generating a population of constituent 

concentrations from statistics calculated from the average, 

standard deviation, and skew of available concentration data. 

Random water-quality definitions can be defined for highway 

runoff and upstream stormflow. For the upstream basin, the 

random water-quality option may be preferred if an analysis 

shows that relations between constituent concentrations and 

streamflow (the transport-curve option) are weak. Stormflow 

water-quality data commonly are characterized and modeled 

as being from a lognormal distribution, but other distributions 

also are used (Athayde and others, 1983; Di Toro, 1984; 

Driscoll, Shelley and others, 1989; Driscoll and others, 1990b; 

Van Buren and others, 1997; Novotny, 2004; Burton and 

Pitt, 2002; Maestre and others, 2004; National Research 

Council, 2008). 

SELDM uses the frequency-factor method (equation 1) to 

generate random constituent concentrations. SELDM can be 

used to generate concentration data approximating a truncated 

normal distribution, a truncated Pearson Type III distribution, 

a lognormal distribution, or a log-Pearson Type III 

distribution. SELDM also can be used to model concentration 

data that fit selected exponential or gamma distributions, 

which are special cases of the Pearson distribution (Bobee and 

Ashkar, 1991). The water-quality statistics may be entered 

in the form of arithmetic values or the natural (base e) or 

common (base 10) logarithms of data. Misspecification of 

the data type on the input form can cause large errors in 

generated values. The average and standard deviation of the 

data or the logarithms of the data are used to generate values 

with a normal or lognormal distribution, respectively; for 

these distributions, the coefficient of skew must be set equal 

to 0. Use of the Pearson Type III or log-Pearson Type III 

distribution is not an explicit choice offered by the interface; 

to use these distributions, a nonzero-skew value must be 

entered with the associated data-transformation option. If 

arithmetic values are used, SELDM produces data following 

a truncated normal distribution or a truncated Pearson Type 

III distribution because the use of untruncated distributions 

can produce negative concentration values, which have no 

physical meaning. SELDM truncates these results by fitting 

such values to a logarithmic lower tail, but the results may not 

accurately model the input statistics. A detailed description 
Theory and Implementation 43

of methods for entering random water-quality statistics is 

provided in the section of appendix 4 on the random waterquality statistics input form.

Selection of statistics from robust and representative 

datasets is important for modeling runoff quality. The input 

statistics that are selected can have a substantial effect on 

the TMDL allocation assigned to a DOT and the potential 

number of water-quality excursions that may be modeled. 

The standard errors of the input statistics are a function of 

the variability of the data and the number of samples in the 

dataset used to calculate the statistics (Haan, 1977; Chow and 

others, 1988; Stedinger and others, 1993; Burton and Pitt, 

2002). Although SELDM will typically generate data for 500 

to 2,500 storms (depending on precipitation statistics and the 

random seed), the uncertainty in input statistics is based on 

the size of the measured dataset, not the size of the modeled 

dataset. Furthermore, because SELDM generates many more 

storms than are characterized in the measured dataset by using 

a theoretical distribution, the range of values in the modeled 

output is expected to exceed the range of the values in the 

measured dataset. Thus, the effect of selected statistics may 

be accentuated in the tails of the data. For example, the effect 

of the skew coefficient on modeled concentration data for 

suspended sediment is shown in figure 19. The statistics in the 

measured dataset were calculated by using the HRDB from 

27 highway-runoff EMC values collected along State Route 2 

in Littleton, Massachusetts (Smith and Granato, 2010). The 

percentiles, which were calculated by using the Cunnane 

plotting-position formula (Helsel and Hirsch, 2002), range 

from 2.21 to 97.79, whereas the range of plotting positions for 

the stochastic populations, 0.038 to 99.962, is larger. 

Two stochastic populations were generated on the basis 

of the average and standard deviation of the logarithms 

of data. The first stochastic population was modeled on a 

lognormal distribution with a skew of 0; the second was 

modeled as a log-Pearson Type III distribution with the 

skew value calculated for the measured dataset. The skew, 

which is negative, produces a probability plot that is concave 

downward leading to reductions in the lower-tail and uppertail values with respect to the lognormal population values. 

The measured data range from 6 to 406 mg/L, the lognormal 

values range from 1.83 to 5,020, and the log-Pearson type 

III values range from 0.447 to 1,000. If the skew of the 

logarithms of data were positive, the result would be a 

probability plot that is concave upward leading to potentially 

large increases in the lower-tail and upper-tail values with 

respect to the lognormal population values. The number of 

allowable excursions would depend on the magnitude of 

highway flows; the effectiveness of the specified BMP; and 

the magnitudes of upstream concentrations, flows, and loads. 

All else being equal, however, one would expect higher runoff 

concentrations to produce a greater number of excursions. 

A sensitivity analysis using the standard errors of the input 

statistics will reveal the effect of inputs on water-quality 

excursions and perhaps the need for a better input dataset. For 

example, Burton and Pitt (2002) indicate that 25 to 50 EMC 

samples may be needed on the basis of the variability of urban 

and highway-runoff concentrations from numerous datasets.

Specification of statistics for some constituents 

requires special consideration to ensure that output values 

are representative of water quality in the environment. For 

example, the nonlinear pH scale is the negative logarithm of 

the hydronium ion concentration. pH can range from 0 to 14 

(excluding concentrated acid or base solutions), and the pH of 

most natural river waters is expected to be in the range from 

6.6 to 8.5 (Hem, 1992). Similarly, 95 percent of pH values in 

highway runoff ranged from 5.25 to 8.24 in version 1.0.0a of 

the HRDB (Smith and Granato, 2010).

Dependent Water-Quality Modeling

The SELDM dependent water-quality option provides 

the means for stochastically generating a population of 

constituent concentrations from statistics calculated on the 

basis of available monitoring data and regression equations 

characterizing the relations between datasets for two 

constituents (fig. 20). Dependent water-quality relations can 

be defined for highway runoff and upstream stormflow. The 

dependent water-quality option is intended for use if data are 

limited for a constituent of interest, but the relations between 

the constituent of interest and other constituents that are well 

characterized by random statistics or water-quality transport 

curves are good. SELDM uses the concentration values from 

the stochastic population of the selected independent variable 

to generate concentration values for the dependent variable. 

For highway-runoff constituents, the independent variable 

must comprise a population of random highway-runoff 

concentrations. For the upstream constituents, a population of 

random upstream concentrations or upstream concentrations 

generated from a water-quality transport curve can be used.

For dependent relations, the input regression-equation 

statistics are used to calculate a regression result, and Monte 

Carlo methods for regression equations are used to reproduce 

the scatter of data above and below the regression line 

(equation 2). These numbers are stochastically generated on 

the basis of the assumptions that the residuals are centered 

on the regression value and are normally or lognormally 

distributed above and below the regression-line value. If 

arithmetic regression statistics are used, SELDM produces 

residual values that follow a truncated normal distribution 

because use of the lower-tail values can produce negative 

concentration values, which have no physical meaning. If the 

calculated concentration is less than or equal to 0, SELDM 

returns the arbitrarily selected value of 0.002. Thus, output 

from SELDM that includes multiple concentration values 

that are equal to 0.002 may indicate that the input statistics 

produced concentration values less than or equal to 0. 

Regression equations developed from the logarithms of data 

are commonly more robust for runoff-quality analysis than 

regression equations based on untransformed values (Glysson, 

1987; Vogel and others, 2005; Granato, 2006; Granato 

and others, 2009). Misspecification of the data type on the 
44 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Percentage of concentrations less than or equal to the specified value

Suspended sediment concentration (parameter code P80154), milligrams per liter

EXPLANATION

0.1

1.0

10

100

1,000

10,000

0.10.01 1 2 5 10 20 30 40 50 60 70 80 0.5 99 99.5989590 99.9 99.99

 Stochastic concentrations modeled by using the lognormal distribution

 Stochastic concentrations modeled by using the log-Pearson Type III distribution

 Measured-concentration data (Smith and Granato, 2010)

Statistic Average Standard deviation Skew

Measured data 1.88 0.519 -0.693

Lognormal 1.88 0.535 0.039

Log-Pearson type III 1.89 0.506 -0.711

Statistics of logarithms

Measured data 126 111 1.04

Lognormal 166 303 6.34

Log-Pearson type III 133 140 2.19

Arithmetic statistics

Stochastic model generated

Stochastic model generated (retransformed)

Statistics for measured and modeled data

Figure 19. An example of random water-quality data generated by using statistics from 27 suspended sediment 

concentrations measured in highway-runoff samples collected along State Route 2 in Littleton, Massachusetts (Smith and 

Granato, 2010). Stochastic model-generated suspended sediment concentrations were generated by applying lognormal 

and log-Pearson Type III distributions to statistics from the highway data.
Theory and Implementation 45

Suspended sediment concentration (SSC, parameter code P80154), milligrams per liter

Total suspended solids (TSS, parameter code P00530), milligrams per liter

EXPLANATION

1

10

100

1,000

10,000

1 10 100 1,000 10,000

 Stochastic model-generated estimates of suspended sediment 

 concentrations calculated by using the median absolute deviation

 Pair of measured suspended sediment concentration

 and total suspended solids concentration

 1:1 line

 SSC = 0.9413 * TSS 1.1292

Figure 20. An example of dependent water-quality data generation using a regression relation for predicting suspended 

sediment concentrations from total suspended solids concentrations measured in 94 highway-runoff samples (Granato 

and Cazenas, 2009). Stochastic suspended sediment concentrations are generated by using statistics for stochastic total 

suspended solids concentrations for nonurban highways (Driscoll and others, 1990a).
46 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

input form can cause large errors in generated values. For 

example, if arithmetic regression statistics are entered, but the 

logarithmic data type is selected, then data that are orders of 

magnitude too high may be generated. A detailed description 

of methods for entering dependent water-quality statistics is 

provided in the section of appendix 4 on the dependent waterquality statistics and the transport-curve input form. Special 

constituents like pH require evaluation to ensure that output 

values are representative of water quality in the environment.

The output-file format of the KTRLine program is 

designed to facilitate specification of input statistics for 

SELDM (Granato, 2006). Selection of regression statistics 

from robust and representative datasets is important for 

modeling runoff quality because of potential effects on 

the TMDL allocation assigned to a DOT and the potential 

number of water-quality excursions that may be modeled. 

The KTRLine program (Granato, 2006) produces robust 

regression statistics because it is a nonparametric method that 

is not heavily influenced by outliers, nonconstant variance 

of residuals (heteroscedasticity), or assumptions about 

values of data less than one or more detection limits (Helsel 

and Hirsch, 2002; Granato, 2006). The KTRLine program 

calculates the intercept of the line so that the line passes 

through the medians of both the independent and dependent 

values. For some datasets, this method may result in residuals 

that are not centered on the regression line; in this case, the 

median deviation can be added to the intercept to center 

the residual values. The KTRLine program also produces a 

robust nonparametric measure of the variability of residuals 

to eliminate or reduce the effects of a few far outliers on the 

residual statistics: the median absolute deviation (MAD). The 

MAD statistic is about two-thirds of the standard deviation 

and about one-half of the interquartile range (IQR) for a 

population of residual values with a normal (or lognormal if 

transformed values are used) distribution (Helsel and Hirsch, 

2002; Granato, 2006; Granato and others, 2009). SELDM, 

however, uses the MAD value as an approximation of the 

standard deviation of residuals. If it is warranted, the standard 

deviation of residuals can be substituted for the MAD value on 

the dependent water-quality statistics and transport-curve 

input form.

Granato and Cazenas (2009) provide an example of 

a dependent water-quality relation to generate suspended 

sediment concentrations (SSC) from total suspended solids 

concentrations (TSS) from 94 paired highway-runoff samples. 

This dependent water-quality relation, which is included 

as an option in SELDM, is an example of an application of 

the dependent water-quality method. Granato and Cazenas 

(2009) derived this relation because most highway-runoff 

studies collect TSS samples but not SSC samples, and 

because concerns about the representativeness of TSS 

concentrations in receiving waters and in highway and 

urban runoff are substantial. TSS concentrations have been 

shown to underrepresent the true concentration of sediments 

in natural waters. Using results from 17,701 paired SSC 

and TSS samples from across the country, the USGS has 

determined that TSS analyses are “fundamentally unreliable 

for the analysis of natural-water samples” (Gray and others, 

2000; U.S. Geological Survey, 2001; Bent and others, 2003). 

Different methods are used for TSS analyses, but none of the 

methods provide consistent and repeatable results for naturalwater samples. Furthermore, TSS concentrations have been 

shown to substantially under represent the true concentration 

of sediments in natural waters. These results also have been 

substantiated in a number of highway- and urban-runoff 

studies (for example, U.S. Environmental Protection Agency, 

2005a; Guo, 2006, 2007; Landers and others, 2007; Kim and 

Sansalone, 2008; Ying and Sansalone, 2008; Granato and 

Cazenas, 2009). 

Figure 20 shows the 94 paired-data samples used by 

Granato and Cazenas (2009), the regression line between TSS 

and SSC, and a population of 1,571 stochastically generated 

sediment concentrations. Random water-quality values for 

TSS were generated by using statistics for data from sites 

designated as “non-urban” by Driscoll and others (1990a). The 

measured TSS concentrations range from 9.25 to 1,230 mg/L 

with a median of 136.5 mg/L, and the associated SSC 

concentrations range from 12 to 8,580 mg/L with a median of 

242.5 mg/L. About 14 percent of the SSC concentrations in 

the measured data are less than or equal to the associated TSS 

values. The modeled TSS concentrations range from 0.765 to 

2,450 mg/L with a median of 39.5 mg/L, and the associated 

SSC concentrations range from 0.667 to 11,900 mg/L with 

a median of 57.7 mg/L. (The graph shown in figure 20 was 

truncated to values between 1 and 10,000 mg/L). About 

18.5 percent of the SSC concentrations in the modeled data are 

less than or equal to the associated TSS values.

The variability of stochastic suspended sediment 

concentration estimates above and below the regression 

line in figure 20 is less than the variability of measured 

concentrations. The unadjusted MAD value was selected to 

estimate the standard deviation of residuals because there was 

some heteroscedasticity in the residuals. Some variation in the 

residuals may have been an artifact of combining two datasets. 

Also, the modeled period of record was longer than the 

modeled period, so modeled values were extrapolated beyond 

the range of data used to develop the equation. If the MAD is 

multiplied by 1.5 to represent the full standard deviation of 

residuals, then the modeled SSC concentrations would range 

from 0.653 to 16,300 mg/L with a median of 57.2 mg/L, and 

about 26.5 percent of the SSC concentrations in the modeled 

data would be less than or equal to the associated TSS values.

The regression line on figure 20 is drawn so that it 

extends only as far as the range of measured TSS values 

that were used to develop the regression equation. The use 

of regression equations to generate data beyond the range 

of the explanatory-variable data that were used to develop 

the equations (extrapolation) can produce biased results 

(Haan, 1977; Helsel and Hirsch, 2002; Granato, 2006). For 

example, in developing regression equations between specific 

conductance and concentrations of chloride, Granato and 

Smith (1999) found that the regression relation was affected 
Theory and Implementation 47

by the presence of other major ions in solution if chloride 

concentrations were less than 130 mg/L. They also found that 

the regression relation was affected by physical and chemical 

changes in solution properties with increasing concentrations 

of solutes if chloride concentrations were above 580 mg/L. 

For these reasons, evaluation of the applicability of dependent 

relations and the potential physical and chemical processes 

that may affect relations beyond the range of available data 

would be prudent for users of SELDM. The use of data 

from other sites or other regions may be instructive for 

such comparisons.

The regression relation in figure 20 appears to be 

consistent with the theory of sediment transport. The 

regression equation predicts that TSS concentrations 

will be greater than SSC concentrations for TSS values 

below 1.6 mg/L. Theoretically, this is impossible because 

SSC is a measure of all solids in the sample, but the two 

measures of suspended solids are expected to converge at 

low concentrations (Gray and others, 2000); in this case, 

the differences are well within measurement errors. The 

regression relation predicts increasing bias with increasing 

sediment concentrations. This trend is consistent with theory 

because the higher energy flows that are necessary to transport 

high sediment loads also tend to mobilize more of the largegrained fraction of sediment that is not well characterized by 

TSS methods (Gray and others, 2000).

Upstream Water-Quality Transport-Curve 

Modeling

The SELDM transport-curve option provides the means 

for stochastically generating a population of constituent 

concentrations from upstream stormflow values by using 

regression-equation statistics (fig. 21). Water-quality transport 

curves are an accepted method for characterizing waterquality variation with streamflow (Biesecker and Leifeste, 

1975; O’Connor, 1976; Glysson, 1987; Vogel and others, 

2005; Granato, 2006; Landers and others, 2007; Granato and 

others, 2009). The transport-curve option was developed for 

use in SELDM because concentrations of many constituents 

commonly vary as a result of washoff and dilution processes 

in receiving waters. Concentrations of sediment and 

sediment-associated constituents (selected nutrients, organic 

compounds, and trace elements) commonly increase with 

increasing streamflow above a base-flow threshold because 

these constituents are mobilized from the stream or land 

surface during storms (washoff). For example, Landers 

and others (2007) found that concentrations of sediment, 

total phosphorus, total metals, and bacteria increased with 

increasing stormflow. Concentrations of dissolved constituents 

commonly decrease with increasing streamflow above a 

base-flow threshold because these constituents, which are 

commonly associated with groundwater and point-source 

discharges, are diluted by less concentrated storm runoff. 

Landers and others (2007) showed that total dissolved solids 

decreased with increasing stormflows. Some constituents with 

multiple sources may show both increasing and decreasing 

trends with increasing flow. For example, Landers and others 

(2007) showed that total phosphorus was diluted by increasing 

base flows but increased with increasing stormflows in one 

basin with a wastewater-treatment plant. O’Connor (1976) 

used data to estimate the base-flow threshold for different 

sites, but Landers and others (2007) showed that the ranges 

of base flow and stormflow may overlap because some base 

flows during wet periods of the year may exceed flows from 

small storms during dry parts of the year.

SELDM uses the stochastic population of upstream 

stormflows with a water-quality transport curve to generate 

concentration values for the specified constituent. A waterquality transport curve, however, will provide only one 

unique value of concentration for each streamflow value 

rather than the random distribution of concentrations 

that are characteristic of water-quality data above and 

below the transport curve. The input regression-equation 

statistics are used to calculate the regression relation, and 

Monte Carlo methods for regression equations are used to 

reproduce the scatter of data above and below the regression 

line (equation 2). Arithmetic regression statistics can be 

used, but relations between streamflow and constituent 

concentrations commonly are modeled with the logarithms 

of data because these data can vary by orders of magnitude, 

and logarithmic regression equations commonly improve 

linearity, reduce heteroscedasticity, and will not produce 

constituent concentrations that are less than 0 (Glysson, 

1987; Vogel and others, 2005; Granato, 2006; Granato and 

others, 2009). For this reason, regression equations developed 

from the logarithms of data are commonly more robust for 

runoff-quality analysis than regression equations developed 

from arithmetic values. A detailed description of methods 

for entering dependent water-quality statistics is provided 

in the section of appendix 4 on the dependent water-quality 

statistics and transport-curve input forms. Special constituents 

like pH require evaluation to ensure that output values are 

representative of water quality in the environment.

Granato and others (2009) used the KTRLine program 

to develop water-quality transport curves for total phosphorus 

(parameter code p00665), total hardness (parameter code 

p00900), and SSC (parameter code p80154) for each of the 

84 ecoregions in the conterminous United States to facilitate 

generation of robust multisegment regression relations. Total 

phosphorus and suspended sediment concentrations were 

selected for analysis because these constituents are commonly 

identified as constituents of concern. SSC were selected rather 

than TSS concentrations because the USGS has determined 

that TSS analyses are not appropriate for characterization 

of sediment concentrations in receiving waters (Gray and 

others, 2000; U.S. Geological Survey, 2001; Bent and others, 

2003). Total hardness was selected because it is a dissolved 

constituent that is important for calculating water-quality 

criteria for metals (U.S. Environmental Protection Agency, 
48 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

1

10

100

1,000

1

10

100

1,000

0.001 0.01 0.1 1 10 100

A. Development of the transport curve from water-quality data

B. Use of the transport curve for stochastic water-quality data generation

Total hardness concentration, in milligrams per liter as calcium carbonate

Water discharge, in cubic feet per second per square mile

Measured total hardness data (n = 4,057)

Regression line

Regression line

Stochastic total hardness

estimates (n = 1,000)

Minimum 1st 5th 25th 50th 75th 95th 99th Maximum

Mean

Percentiles of 1,000 lognormal random-discharge values generated with an average of 

1.38 cubic feet per second per square mile and a coefficient of variation of 1.7

Percentiles of 4,057 instantaneous

discharge measurements 

made with total hardness samples

Figure 21. Examples of A, development, and B, use of a three-segment water-quality transport curve for stochastic generation 

of total hardness concentration data from regional-average daily-flow statistics for ecoregion 67 by use of a random-error 

component based on the median absolute deviation of water-quality data in each discharge range.

2002a). These three transport curves are provided as a default 

selection in SELDM. 

The interpretive methods used to generate these regional 

planning-level estimates are based on two assumptions. 

The first assumption is that data from multiple stations in 

an ecoregion can be combined to develop a water-quality 

transport curve that represents ambient water quality at 

unmonitored sites in that ecoregion. This assumption, 

however, does not imply that a single regional transport 

curve is adequate to characterize local variability that 

may be caused by explanatory factors within the region, 

but rather that differences among regions of the United 

States may be characterized by separate transport curves. 

The ecoregion-level estimates are intended for an initial 

screening-level analysis, which may be followed by more 

detailed analysis with site-specific data if needed. The 

second assumption is that these transport curves, which were 

developed on the basis of instantaneous streamflow and 

concentration data available in NWIS, can be used for orderof-magnitude planning-level estimates of ambient EMCs in 

receiving waters. Concentrations of dissolved and suspended 

constituents commonly vary with streamflow in hysteresis 

loops during a runoff event (O’Connor, 1976; Glysson, 1987; 

House and Warwick, 1998). Constituents from washoff 

commonly exhibit higher instantaneous concentrations for 

a given discharge on the rising part of the hydrograph than 
Theory and Implementation 49

on the falling part, whereas constituents from groundwater 

discharge, which are diluted by increasing runoff, commonly 

have lower concentrations for a given discharge on the 

rising part of the hydrograph than on the falling part. A 

transport curve fitted through instantaneous measurements 

of streamflow and constituent concentrations collected on 

both parts of the hydrograph will, by definition, represent 

the central tendency of measured concentrations (Glysson, 

1987; Granato and others, 2009). Therefore, use of the 

event-mean flow (calculated as the total upstream stormflow 

volume divided by the duration of the highway runoff or 

BMP discharge) as the explanatory variable in the transport 

curve for a given constituent will produce a planning-level 

estimate of the EMC for that constituent. These assumptions 

were necessitated by the scale of a national synthesis, the 

scope of this study, and limitations in available data. The 

data compilation and interpretation methods described by 

Granato and others (2009), however, may be used with other 

information, such as EMC values and local land-use data, for 

more selective regional or local data analysis.

Granato and others (2009) provide an example of the 

development and application of a three-segment water-quality 

transport curve for total hardness on the basis of data from 

ecoregion 67 (the Ridge and Valley ecoregion used in the 

I–81 highway-site example provided in SELDM). In this 

example (fig. 21A), paired measurements of total hardness 

and streamflow (solid gray dots) were analyzed to determine 

a three-segment water-quality transport curve (solid black 

line) with the KTRLine program (Granato, 2006). The total 

hardness dataset for ecoregion 67 includes 4,057 paired 

measurements of instantaneous streamflow and total hardness 

concentrations. The boxplot along the X-axis of figure 21A 

indicates that samples were collected over more than four 

orders of magnitude of streamflow. The different segments 

in the three-segment model also indicate that the statistical 

properties of a water-quality dataset may depend on the 

streamflows at which samples were collected (fig. 21A). 

If samples are not collected throughout the full range of 

streamflows, regression-line estimates and residual statistics 

may not be adequate to quantify the range of concentrations 

that can be found in a region. 

Granato and others (2009) used this regional waterquality transport curve with regional flow statistics to 

generate a population of concentration values for ecoregion 

67 (fig. 21B). In this case, 1,000 streamflow values were 

generated in a lognormal distribution from regional 

streamflow statistics, which are characterized by the boxplot 

along the X-axis of figure 21B. Application of the waterquality transport curve (solid black line) and a lognormal 

distribution of residuals, generated by using the MAD value 

for each segment, yields a population of total hardness 

estimates (open circles) above and below the line. Comparison 

of the regression line, the stochastic data estimates, and the 

data from ecoregion 67 (figs. 21A and B) indicates that use of 

this total hardness transport-curve with a stochastic component 

may be sufficient for generating planning-level estimates of 

EMC values that are similar to most of the ambient waterquality measurements. 

Granato and others (2009) used total hardness as an 

example because variations in total hardness with streamflow 

may affect the application of water-quality criteria for the 

hardness-dependent trace elements (U.S. Environmental 

Protection Agency, 2002a). The criteria maximum 

concentrations or acute criteria (CMCs) for copper may 

decrease by a factor of 3 (from about 14.7 to 4.67 μg/L) 

because of the change in hardness between the first and second 

inflection points on this three-line transport curve. However, 

the potential dilution over this range of flows increases by 

a factor of 45 between these streamflow values. Random 

variation in total hardness concentrations may be an important 

factor in ecoregion 67 because total hardness and associated 

CMC values for copper can vary substantially at any given 

streamflow value. For example, the randomly generated values 

of total hardness vary from almost 400 to about 30 mg/L for 

streamflows below 1.0 ft3

/s/mi2; the corresponding CMC 

values for copper could range from about 50 to 4 μg/L at any 

given streamflow in this range.

Runoff Modification by Best Management 

Practices (BMPs)

SELDM uses a simple stochastic statistical model of 

BMP performance to develop planning-level estimates of 

runoff-event characteristics rather than a complex theoretical 

or physical model. This statistical approach can be used to 

represent a single BMP or an assemblage of BMPs. The 

SELDM BMP-treatment module has provisions for stochastic 

modeling of three stormwater treatments: volume reduction, 

hydrograph extension, and concentration reduction. Volume 

reduction is modeled to represent how BMPs can affect flows 

and loads from the highway site. Hydrograph extension is 

modeled to represent how BMPs can increase dilution in 

receiving waters by extending the duration of runoff from the 

highway site. Concentration reduction is modeled to represent 

changes in constituent concentrations that may result from 

different treatment options. 

Using the BMP runoff-control options alters the highway, 

upstream, and downstream outputs from the model. If BMP 

volume-reduction statistics are specified, the highway-runoff 

flows and loads will be affected accordingly. If BMP volume 

reductions are specified but concentration reductions are not, 

then the highway-runoff and BMP discharge concentrations 

will be the same, but the BMP discharge loads and the 

concurrent downstream loads and concentrations will all 

be different. If BMP hydrograph extension is specified, the 

concurrent upstream and downstream flows and loads will be 

different than for the untreated runoff because the discharge 

period will be extended to include more of the upstream 

flow and loads. If BMP concentration-reduction statistics are 

specified, BMP discharge concentrations and loads will be 

affected as well as downstream concentrations and loads.
50 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Application of results from BMP monitoring studies is 

highly uncertain; few studies provide reliable predictions of 

treatment performance even with large datasets and complex 

models (Strecker, Quigley, and others, 2001; Wong and others, 

2006; Park and others, 2011). Uncertainties arise because 

of the many categories of BMPs, wide variations in design 

and construction of BMPs within each category, and wide 

variations in the operation and maintenance of BMPs once 

they are installed. Similar BMPs are used at sites with widely 

varying site characteristics including different precipitation, 

site hydraulics, constituent characteristics and loads, and total 

stormwater loads. For example, local soil characteristics can 

influence the amount of runoff generated by a given storm, 

the concentrations of sediment in runoff, and the settling rate 

of the sediments within a BMP. Variations in BMP design 

also can affect actual and modeled BMP effectiveness. For 

example, BMP structures may have overflow or bypass 

structures that have a substantial effect on performance once 

the BMP volume has been filled. These design features may 

affect performance only during large storms or storms that 

occur in rapid succession (Strecker, Quigley, and others, 

2001). Uncertainties in effectiveness also arise because 

BMP monitoring is a complex endeavor that requires a high 

degree of expertise. Although BMP monitoring protocols 

have become more standardized, many BMP studies still are 

conducted individually with different protocols and datareporting standards rather than as part of a large consistent 

and coordinated monitoring effort (Jane Clary, Civil Engineer, 

International BMP Database Project, written commun., 

May 2011). 

Uncertainties in results also are compounded by available 

sample sizes. Driscoll and others (1979) recommend the 

collection of 20 to 40 EMC samples to characterize runoff 

on the basis of the variability of commonly measured runoff 

constituents. Similarly, Burton and Pitt (2002) indicate that, 

at a minimum, 25 to 50 EMC samples may be needed. The 

California Department of Transportation (2009) provides 

similar examples in their BMP monitoring handbook. These 

examples indicate that 50 to 113 paired samples may be 

needed just to detect differences in mean concentrations. In 

comparison, Leisenring and others (2011) looked at TSS data 

for 10 types of BMPs in a recent summary of solids-removal 

data in studies in the International BMP Database. Although 

TSS is one of the most widely monitored constituents in BMP 

studies, the average number of paired samples per category 

ranged from 6 to 16 with a median of about 12 per study. 

Schneider and McCuen (2006) calculated that monitoring 

data from about 90 storms would be necessary to fully 

quantify the hydraulic performance of a stormwater-detention 

cistern in Maryland on the basis of local precipitation-event 

characteristics. In comparison, Poresky and others (2011) 

looked at volume-reduction data from the International 

BMP Database; they found that the number of storm events 

ranged from 5 to 173 with a median of about 11 per study. 

As with other hydrologic data, uncertainty in data related 

to BMP performance increases when data from one site are 

extrapolated to estimate conditions at a different site. In 

addition, small sample sizes limit the ability to select and 

parameterize statistical distributions for modeling BMPs 

with data.

In SELDM, volume reduction, hydrograph extension, 

and water-quality-treatment variables are modeled by using 

the triangular/trapezoidal distribution and the rank correlation 

with the associated highway-runoff variables. This family of 

distributions was selected for modeling BMP performance 

measures because it can be parameterized by using expert 

judgment or by fitting the distribution to data if good data 

are available (Johnson, 1997; Back and others, 2000; U.S. 

Environmental Protection Agency, 2001; Scherer, 2003; 

Kacker and Lawrence, 2007). The triangular distribution, 

which is a special case of the trapezoidal distribution, 

commonly is suggested when uncertainties in input data that 

may be used to define a parametric distribution are large (U.S. 

Environmental Protection Agency, 2001). The triangular/

trapezoidal distribution is bounded by a selected minimum 

and maximum value. When data are uncertain or are limited 

in scope, use of a bounded distribution reduces the chance that 

unrealistic output values will be generated by extrapolating a 

distribution beyond the range of available data.

Volume Reduction

Volume reduction is the practice of retaining, detaining, 

or routing runoff flows to increase the amount of infiltration, 

evapotranspiration, or diversion between the pavement and the 

outfall (Goforth and others, 1983; Schueler, 1987; Urbonas 

and Roesner, 1993; Wanielista and Yousef, 1993; Young and 

others, 1996; Adams and Papa, 2000; Burton and Pitt, 2002; 

National Cooperative Highway Research Program, 2006; 

Poresky and others, 2011). Volume reduction commonly is 

a design criterion for BMPs to reduce flood flows, instream 

erosion, and runoff loads. Features such as flow lengths (for 

swales) or design volumes commonly are used with moistureretention estimates and data on local infiltration rates to 

estimate the volume-reduction capacity of BMPs. Expected 

storm-event characteristics also are considered in BMP 

designs because the volume, intensity, and duration of events 

and the time between storms affect the capacity of the BMP to 

reduce runoff volumes. 

SELDM uses a simple stochastic representation of 

the net volume reduction from a BMP or series of BMPs. 

SELDM models flow-volume reductions for generic 

BMPs as stochastic ratios of inflow to outflow volumes by 

using the triangular/trapezoidal distribution and the rank 

correlation with the highway stormflow volume. Volumereduction statistics for the triangular/trapezoidal distribution 

can be estimated by using expert judgment or by fitting 

the distribution to data. Rank correlation with the highway 

stormflow volume is included in the input statistics because 

the ratio of outflow to inflow volumes would be expected to 

increase with increasing storm volumes. For large storms, a 

smaller fraction of the total inflow may be lost to infiltration or 
Theory and Implementation 51

evapotranspiration than for small storms. The different storm 

characteristics that affect flow reduction, however, also depend 

on the reduction methods and the design characteristics of the 

BMP. Because there are many uncertainties in the analysis 

and application of available BMP data, expert judgment based 

on knowledge of the hydraulic properties of a BMP design 

may be the best available information for parameterizing the 

distribution of volume-reduction ratios.

Many studies have been done to measure and model 

volume reduction, but accurate categorical determination is 

hampered by the stochastic nature of antecedent conditions, 

precipitation, and runoff (Goforth and Heany, 1983; Adams 

and others, 1986; Driscoll and others, 1986; Schueler, 1987; 

Driscoll, Shelley, and others, 1989; Urbonas and Roesner, 

1993; Wanielista and Yousef, 1993; Young and others, 1996; 

Adams and Papa, 2000; Huber and others, 2006; Poresky 

and others, 2011). For example, Emerson and Traver (2008) 

attributed seasonal two-fold variations in infiltration rates 

during a four-year period at BMPs in Maryland to changes in 

the viscosity of ponded water with changes in temperature. 

Most statistical and deterministic modeling studies are 

designed to estimate flow reductions for only one type of 

BMP at a time because of differences among the reduction 

mechanisms in different BMP designs. Also, most modeling 

efforts do not characterize the effects of treatment trains 

(which comprise two or more BMPs in series) because of 

these complexities. 

Despite decades of BMP-monitoring efforts, data are 

limited for some BMP categories, and substantial uncertainties 

in the volume-reduction performance of many BMPs remain. 

In an analysis of flow data in the International BMP Database, 

Poresky and others (2011) noted that because many older 

studies were designed to monitor reductions in concentration 

instead of volume, measurements of volume were made only 

during the collection of flow-weighted water samples. Thus, 

flow-duration and volume data may include only the period 

used for water-quality sampling rather than the complete 

duration of inflows and outflows. They also noted that the 

inflow and outflow data from some BMP studies could not be 

truly paired because these studies measured BMP inputs at 

only one of many inlets to the BMP or at a reference site that 

was not associated with the monitored BMP outlet. Poresky 

and others (2011) emphasize that data are not available for 

many types of BMPs, and that the level of uncertainty of the 

available data is high. 

The July 2011 version of the International BMP Database 

(http://www.bmpdatabase.org/) was queried to examine the 

potential for developing volume-reduction statistics by using 

available BMP monitoring data. Stormflow data are available 

for 328 BMP sites; storm-event data indicating the start and 

end times of inflows and outflows are available for 91 of 

these sites. The range in the number of storms is 2 to 384, 

and the median number of storms recorded at each site is 11. 

A negative flow-extension duration indicates that the period 

of inflow exceeded the period of outflows. The minimum, 

average, and maximum flow-extension durations are negative 

for about 60, 12, and 3 percent of these 91 sites, respectively. 

If monitoring data do not include the complete runoff 

period for inflows or outflows, then the volume-reduction 

statistics will be biased by the truncated flow volumes. These 

anomalous results may be caused by the sampling artifacts 

described by Poresky and others (2011), or they may be the 

result of data-entry errors. These anomalous results also may 

reflect the fact that BMP performance estimates also are 

affected by substantial difficulties in accurately monitoring 

runoff flows (Strecker, Mayo, and others, 2001; Church and 

others, 2003; Poresky and others, 2011). Thus, the feasibility 

and defensibility of transferring BMP monitoring results from 

one site to a different site is hampered by many uncertainties. 

Selected stormflow data from the International BMP 

Database were used as an example to demonstrate the process 

for estimating the statistical parameters of the trapezoidal 

distribution for use with SELDM. First, the inflow and outflow 

volumes were retrieved from the BMP Database, and the ratios 

of the outflow to inflow volumes were calculated. Figure 22A 

is a scatterplot of the inflow and outflow volumes for the 

San Rafael biofilter, which is a grass strip next to a highway. 

Inflow volumes range from 68 to 3,424 ft3

, with a median 

of 424 ft3

. Outflow volumes range from 6 to 1579 ft3, with 

a median of 211 ft3

. The ratios of outflow to inflow volumes 

range from 0.022 to 1.68, with a median of 0.38 (fig. 22B). 

Then the inflow volumes and the ratios were sorted and ranked 

to calculate the rank correlation (Spearman’s rho) between 

inflow volumes and the associated ratios (Haan, 1977; Helsel 

and Hirsch, 2002). The ranks show a weak downward trend 

with considerable scatter; Spearmans’s rho for this dataset is 

about -0.23 (fig. 22C). Thus, volume-reduction ratios decrease 

with increasing runoff volumes. This pattern makes sense 

physically because reductions in flow volumes by BMPs are 

commonly limited by hydrologic conditions; most BMPs are 

not capable of removing a large fraction of an extremely high 

inflow volume. This pattern also makes sense mathematically 

because larger denominators would tend to reduce ratio 

values. Although the correlation between the inflow volume 

and the ratio of outflow to inflow volumes is, by definition, a 

spurious correlation (Haan, 1977), use of the rank correlation 

for stochastic data generation reduces the chance that 

unreasonably large volume reductions may be modeled.

The volume-reduction ratios were then fit to the 

trapezoidal distribution. The ranks of the ratios were used to 

calculate the plotting-position percentile for each ratio; for 

this example, the Cunnane plotting-position formula (Helsel 

and Hirsch, 2002) was used. The cumulative distributionfunction equations for the trapezoidal distribution (Kacker 

and Lawrence, 2007; appendix 1) were used to calculate 

theoretical trapezoidal values for each percentile on the basis 

of trial statistics (fig. 23A). The absolute difference between 

the data value and the theoretical trapezoidal value was 

calculated for each percentile. Optimization methods were 

used to minimize the sum of the absolute differences between 

the data and the trial statistics. In this example, the Microsoft 

Excel Solver® in the Analysis Toolpak® was used to optimize 
52 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Inflow volume, in cubic feet

Inflow volume, in cubic feet

Outflow volume, in cubic feet Ratio of outflow to inflow volume, 

dimensionless

Rank of ratio of outflow to inflow volume, 

dimensionless

Rank of inflow volume, dimensionless

C

B

A

1

10,000

10

100

1,000

1 10 100 1,000 10,000

0.1

1

10

0.01

15

35

20

25

30

10

5

0

1050 15 20 25 30 35

10 100 1,000 10,000

Figure 22. Scatterplots showing A, Inflow and outflow 

volumes from the San Rafael biofilter, B, inflow volumes 

and the ratios of outflow-to-inflow volumes, and C, the 

ranks of the inflow volumes and the ratios of outflow-toinflow volumes. Monitoring data are from the July 2011 

version of the International Best Management Practices 

Database.

the fit. The regression equations provided by Johnson (1997) 

also could be used to generate a triangular approximation 

to the data. Figure 23A shows the data, the statistics for the 

best fit trapezoidal distribution, and the CDF for the best fit 

trapezoidal distribution.

The inflow-volume statistics, the trapezoidal-distribution 

statistics, and the rank correlation values were used with 

a random-number generator to produce two populations 

of inflow and outflow values (fig. 23B). The stochastically 

generated flow values compare well to the to the measured 

inflow and outflow volumes. Although the populations of 

outflow volumes look slightly different from the sample 

of measured outflow volumes, the median and mean of 

the stochastic values are within the 95-percent confidence 

limits of the respective sample values. The median value of 

the measured outflow data is about 202 ft3

, the 95-percent 

confidence interval for the median ranges from 84 to 319 ft3

(calculated by methods described by McGill and others, 1978), 

the mean value is 328 ft3, and the 95-percent confidence 

interval for the mean ranges from 187 to 469 ft3

 (calculated 

by methods described by Haan, 1977). In comparison, the 

median values of the two stochastic populations are about 164 

and 161 ft3, and the mean values are about 356 and 348 ft3, 

both are well within the 95-percent confidence limits of the 

statistics for the measured outflow volumes. The stochastic 

inflow and outflow populations have greater ranges than the 

data because they represent estimates for about 1,500 storms, 

whereas the monitoring data represent samples from 

31 storms. Although the use of the average percent removal 

has been shown to create problems for predicting BMP 

performance (Strecker, Quigley, and others, 2001), it is clear, 

in this example and other test applications, that the use of 

stochastic ratios with rank correlation to inflows can predict 

realistic outflows for a site of interest under the assumption 

that the design features from the example would be properly 

scaled for the hydrologic characteristics of a site of interest.

The stochastic approach used in SELDM is warranted 

because there are large uncertainties in available information, 

and the level of effort required to develop detailed simulation 

models may be beyond the scope of an initial planning-level 

estimate. If, however, the initial analysis done with SELDM 

indicates the potential need for mitigation, then detailed 

simulation models such as those described by Huber and 

others (2006) or detailed statistical models such as those 

described by Adams and Papa (2000) may be used to develop 

the performance statistics used by SELDM. Furthermore, 

if the initial analysis without BMP treatment indicates the 

potential need for mitigation, then SELDM can easily be used 

to develop the BMP-performance statistics needed to reduce 

storm loads or the frequencies of water-quality excursions 

in receiving waters to an acceptable level. This analysis can 

be done by varying BMP flow-reduction statistics to meet 

water-quality objectives. Such an analysis may indicate that 

it is impossible to meet water-quality objectives by using the 

volume-reduction capabilities of feasible BMP designs.
Theory and Implementation 53

1

10

100

1,000

10,000

Inflow to and outflow from biofilter, in cubic feet

Maximum

Mean

Median

Minimum

75th percentile

25th percentile

5th percentile

95th percentile

99th percentile

1st percentile

Measured

inflow

Measured

outflow

Stochastic 

outflow 1

Stochastic 

inflow 1

Stochastic 

inflow 2

Stochastic 

outflow 2

1 2 5 10 20 30 40 50 60 70 80 90 95 98 99

0.00

0.25

0.50

0.75

1.00

1.25

1.50

Percentage of ratios that are less than or equal to the given value

Ratio of outflow to inflow volume, dimensionless

Minimum

Lower bound of the most probable value

Upper bound of the most probable value

Maximum

Rank correlation 

0.00

0.00

0.00

1.43

-0.23

B

A

Trapezoid statistics

Figure 23. Statistics for estimating stochastic flow reductions for the San Rafael biofilter. The probability plot 

A, demonstrates how the trapezoidal statistics are estimated by fitting the data to the distribution. The boxplots in 

B, showing the measured data and two independent stochastic datasets demonstrate how the stochastic populations of 

outflow values generated by using the parameterized trapezoidal distribution compare to the original data. Monitoring 

data from the July 2011 version of the International Best Management Practices Database.
54 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Hydrograph Extension and Concurrent Upstream 

Flows

Hydrograph extension by BMPs is the practice of slowing 

the discharge of runoff flows and releasing these flows to the 

stream over an extended period of time. Hydrograph extension 

commonly is a design criterion for BMPs to reduce flood 

flows, to reduce instream erosion and, more recently, to mimic 

predevelopment stormflow hydrographs. Historically, attempts 

to optimize detention were done to maximize sediment settling 

time while minimizing the chance of untreated overflows 

from subsequent storms (Goforth and others, 1983; Driscoll 

and others, 1986; Schueler, 1987; Wanielista and Yousef, 

1993; Adams and Papa, 2000; Chen and Adams, 2005, 2007; 

National Cooperative Highway Research Program, 2006). 

These efforts commonly resulted in extension of the outflow 

hydrograph. Hydrograph extension also has the added benefit 

of increasing dilution of runoff from small highly impervious 

sites. As illustrated in figure 13, extending the duration of the 

highway-runoff hydrograph can make a big difference in the 

amount of dilution in a receiving stream, especially in the 

rising limb of the upstream storm-event hydrograph. SELDM 

should not be used as a hydraulic design tool, but the output 

from the program may indicate how the BMP design will 

affect potential dilution. This information is summarized in the 

dilution-factor output file from each SELDM analysis, which 

lists the fraction of highway runoff in the downstream flow 

with and without BMP treatment. 

Hydrograph extension is defined as the duration in 

hours of discharge from the BMP that occurs after runoff 

from the highway site has ceased. In theory, runoff from a 

highway site or a BMP may continue to trickle forth for an 

extended period of time. In practice, however, the duration 

of runoff should be defined so that it is truncated at some 

measurable and meaningful value. For example, minimum 

precipitation-monitoring depths commonly are about 

0.01 in. per hour, which would yield about 0.01 ft3/s/acre 

(Church and others, 2003). This threshold, however, may 

not be measurable at small sites. For example, Smith and 

Granato (2010) used a storm-monitoring threshold of about 

0.009 ft3/s to distinguish the presence of flow because it was 

the minimum value that was reliably discernible for a level 

sensor to detect the presence of flow in 8-in. pipes draining 

12,000 to 24,000 square feet (ft2) of pavement (about 0.03 and 

0.016 ft3/s/acre, respectively). 

SELDM models hydrograph-extension times (in 

hours) as a stochastic variable by using the triangular/

trapezoidal distribution and the rank correlation with the 

highway stormflow volume. Hydrograph-extension times 

can be estimated by using expert judgment or by fitting the 

distribution to data. Knowledge of the hydraulic properties 

of a BMP design may provide the information needed to 

parameterize the distribution of hydrograph detention times. 

Alternatively, data from BMP studies in hydrologically similar 

areas may be used to fit a distribution to available data by 

using the methods described for volume reduction.

Hydrograph extension is an explicit design feature for 

some BMPs. For example, detention ponds commonly are 

designed with outfall weirs and orifices to control the rate 

of outflow (Goforth and others, 1983; Driscoll and others, 

1986; Schueler, 1987; Wanielista and Yousef, 1993; Young 

and others, 1996; Adams and Papa, 2000; Park and others, 

2011). Design standards for BMPs commonly specify drain 

times of 24–72 hours for a design storm that fills the detention 

structure. For BMPs with drainage-control structures, 

therefore, the hydrograph-extension duration would be a 

function of the volume of water in the BMP at the end of the 

highway-runoff time. For this type of BMP, the maximum 

flow extension may be estimated from the design drainage 

time, and the minimum flow extension may be estimated from 

the flow formula for the weir or orifice in the drainage-control 

structure and the minimum highway-runoff volume calculated 

during a preliminary SELDM modeling run. The upper and 

lower most probable flow-extension values may be estimated 

by using total runoff volumes from user-selected flow statistics 

from the preliminary SELDM modeling run. For example, 

the mean and median flow volumes may be used as the most 

probable values of the BMP flow extension. If larger volumes 

increase the outflow duration, then the correlation between 

these statistics would be strong and positive. Decreasing 

the rank correlation coefficient from 1 to 0 will increase the 

random variation in extension times with respect to storm 

volume. Using negative correlations between runoff volume 

and flow extension would invert the expected relation between 

these variables for a flow-controlled BMP because the flowextension time would tend to decrease with increasing inflows. 

Hydrograph extension is an implicit design feature for 

some BMPs. If the baseline condition for comparison is a 

curb-and-gutter highway with a drop inlets and a stormsewer system draining directly to the stream, then design 

measures that increase the flow length, lower the drainage 

slope, increase flow-path roughness, increase flow-path 

storage, or provide flow resistance at some point in the flow 

path will extend the outfall hydrograph. For example, a 

simple application of Manning’s equation may indicate the 

effect of converting a sewer design to a grassy swale (Linsley 

and others, 1975; Chow and others, 1988). Alternatively, 

the basin lagtime graph in figure 15 indicates that the basin 

lagtime for a small highway site (with a basin lag factor of 

about 0.01) would increase by a factor of about 5 (from about 

4 to 20 minutes) as the BDF is reduced from 12 to 3. If the 

duration of the rising limb of the unit hydrograph equals the 

duration of falling limb for such sites, bounding estimates 

for the flow-extension parameters may be 0 to 0.5 hours. 

Although this is a small increase in duration, it may result in 

a substantial amount of additional dilution. For example, the 

average of median daily flows among the 2,783 streamgages 

in the 2010 SELDM dataset (Granato, 2010) is 0.67 ft3

/s/

mi2

. This flow rate, which is expected to be lower than a large 

percentage of instantaneous stormflows, would result in an 

additional 1,206 ft3

 (about 34,150 liters) of stormflow per 

square mile of upstream drainage area. Check dams added 
Theory and Implementation 55

to such a swale may further increase the extension time in a 

predictable manner (Young and others, 1996). Flow extension 

would be weakly correlated with flow volume for such 

modifications. Sand filters also may extend the hydrograph 

in predictable ways: the falling limb of the hydrograph can 

be estimated for such systems by using Darcy’s equation 

(Linsley and others, 1975; Chow and others, 1988; Young and 

others, 1996).

The triangular/trapezoidal hydrograph-extension 

parameters may be fit by using good data if good data 

are available for a given BMP design. Methods used for 

estimating flow-extension statistics are similar to methods 

described for estimating flow-reduction statistics. The 

July 2011 version of the International BMP Database 

(http://www.bmpdatabase.org/) was queried to examine the 

potential for developing hydrograph-extension statistics for 

BMPs. Stormflow data is available for 328 BMP sites; data 

indicating the start and end times of inflows and outflows are 

available for 91 of these sites. Stormflow-duration data are 

available for 2 to 384 storms per site; the median number of 

storms per site is 11. However, flow extension estimates made 

using data from the International BMP Database may under 

represent actual BMP performance because, as Poresky and 

others (2011) noted, measurements of volume were made 

only during the collection of flow-weighted water samples 

in many older studies. Thus, flow-duration data may include 

only the period used for water-quality sampling rather than the 

complete duration of inflow and outflow hydrographs. These 

concerns are supported by analysis of duration data for the 

91 sites: the minimum, average, and maximum flow-extension 

durations are negative for about 60, 31, and 3 percent of the 

sites, respectively. Therefore, careful screening of duration 

data is advisable for developing hydrograph-extension 

statistics from these data.

Water-Quality Treatment

Water-quality treatment is the use of physical and 

chemical processes in an attempt to reduce the concentration 

of runoff constituents in stormflow. Hundreds of BMP studies 

have focused on water-quality treatment during the past 

40 years. Settling and filtration commonly are the primary 

water-quality-treatment mechanisms that form the basis for 

reductions in influent concentrations for many constituents in 

commonly used BMP designs (National Cooperative Highway 

Research Program, 2006; Clary and others, 2010, 2011; 

Leisenring and others, 2010, 2011). Increasingly, however, 

chemical and biological processes are being incorporated into 

BMP designs to enhance treatment of runoff constituents. 

Historically, process modeling (for example, methods 

described by Huber, 2006; and Park and others, 2011), 

theoretical statistical modeling (for example, Adams and Papa, 

2000), and data analysis (for example, Strecker, Quigley, and 

others, 2001; Barrett, 2005, 2008; and Leisenring and others, 

2010, 2011) have been used to examine BMP performance. 

The water-quality-treatment module in SELDM is designed to 

be used with the results from the statistical analysis of BMP 

data. Alternatively, however, process modeling or theoretical 

statistical modeling can be used to estimate the statistical 

parameters used to model BMP performance with SELDM.

Interpretation of available data is complex. Although 

data are plentiful, uncertainties abound. Differences in site 

characteristics, BMP designs, precipitation characteristics, 

influent characteristics, and many other factors affect 

how BMPs perform from storm to storm and over long 

periods. Differences in monitoring design, execution, and 

documentation also introduce uncertainties in available BMPperformance results. For example, Poresky and others (2011) 

note that volume control can be estimated from BMP-design 

details, but water-quality-treatment efficiencies commonly 

show little or no dependence on design details within the 

random variation of available data.

Two primary measures are commonly used for 

characterizing and comparing water-quality treatment for 

different BMPs: the average efficiency and the minimum 

irreducible concentration. There are various ways for 

calculating each of these parameters (Strecker, Quigley, and 

others, 2001; Barrett 2005, 2008; Wright Water Engineers and 

Geosyntec Consultants, 2007; and Park and others, 2011). 

The average efficiency is commonly calculated by dividing 

the average influent concentration by the average effluent 

concentration or by averaging the individual concentration 

ratios for sample pairs. Efficiencies for groups of BMPs 

have been calculated by averaging the average values for 

individual BMPs or by pooling data and then calculating 

averages. In some cases, loads are used to calculate flowweighted averages. Use of average efficiencies to compare 

BMPs, however, can lead to substantial errors in performance 

estimates over the range of concentrations expected for a 

single site (Strecker, Quigley, and others, 2001; Barrett 2005, 

2008; Wright Water Engineers and Geosyntec Consultants, 

2007; Park and others, 2011). Furthermore, use of average 

efficiencies can compound errors in performance estimates 

when the averages are applied to sites with different 

influent characteristics. 

The minimum irreducible concentration is commonly 

defined as the lowest concentration achievable for a welldesigned example of each type of BMP (Schueler, 1996; 

Barrett, 2005, 2008; Geosyntec Consultants and Wright 

Water Engineers, 2009). Use of a minimum irreducible 

concentration reflects the fact that most BMPs will not 

produce distilled water, so there will be some lower limit that 

can be achieved with normal BMP unit operations. Although 

the concept of the minimum irreducible concentration is 

sound, determining such a value from data may be difficult 

especially if data are limited, the selected BMP is not 

characteristic of design standards, or a substantial proportion 

of the effluent concentrations are below historical detection 

limits. Schueler (1996) examined available data and settling 

times to estimate minimum irreducible concentrations. Barrett 

(2005) developed regression relations between influent and 

effluent concentrations and interpreted the intercept as a 
56 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

good estimate of the minimum irreducible concentration. The 

average efficiency and minimum irreducible concentration 

concepts are linked because an individual BMP receiving 

runoff that is close to the minimum irreducible concentration 

may exhibit low efficiencies because the ratios of inputs to 

outputs are small.

The effluent probability method (EPM) suggested by 

Strecker, Quigley, and others (2001) is a good diagnostic tool 

for evaluating BMPs; however, the EPM was not designed 

to be a predictive tool for estimating effluent concentrations. 

Furthermore, with a few exceptions, the results of an EPM 

analysis are not directly transferable to sites with substantially 

different influent concentrations (Park and others, 2011). 

For example, an EPM may show a statistically significant 

performance benefit at sites with high influent concentrations, 

but no discernible concentration differences at sites with low 

influent concentrations (Strecker, Quigley, and others, 2001; 

Barrett, 2005; Park and others, 2011). 

Barrett (2005) used regression analysis to determine 

relations between influent and effluent concentrations 

and provided 90-percent confidence intervals for effluentconcentration estimates to build predictive models for BMPs 

and compare performances. He also used the intercept of the 

equation to estimate the minimum irreducible concentration. 

This method is useful as an analysis tool, but with parametric 

regression techniques, one or more erroneous measurements 

can define a regression relation (Helsel and Hirsch, 2002; 

Granato, 2006). Furthermore, the regression approach is based 

on the assumption that data are paired. Strecker, Quigley, and 

others (2001) indicate that outflow for a particular BMP may 

have little or no relationship to the inflow for that same event.

Barrett (2008) used the ratio of average influent 

concentrations to average effluent concentrations as a 

predictive tool to measure performance and a means 

of comparing different types of BMPs. His approach 

differed from the average efficiency approach because 

he presented results as a series of ellipses that bounded 

expected performance for different types of BMPs at sites 

with varying influent concentrations. He demonstrated that 

BMP performance was a function of influent concentration, 

so that site characteristics must be considered in the 

evaluation process. 

SELDM uses a simple stochastic representation of the 

net concentration reduction from a BMP or series of BMPs. 

SELDM models concentration reductions for a generic BMP 

as a stochastic ratio of inflow to outflow concentrations by 

using the triangular/trapezoidal family of distributions, the 

rank correlation of concentration-reduction ratios with influent 

highway stormflow concentrations, and a minimum irreducible 

concentration. Given a population of influent concentrations 

selected on the basis of highway characteristics, the 

stochastic ratio provides the variation in outflow ratios 

and concentrations that are seen in monitoring data. Rank 

correlation provides the capability to model the relations 

between influent and effluent concentrations. For example, 

Barrett (2005, 2008) notes that the effluent concentrations of 

TSS from sand filters are fairly constant and independent of 

influent concentrations. In this case, a strong negative rank 

correlation between influent concentrations and the ratio of 

effluent to influent concentrations would tend to assign high 

ratios to low influent concentrations and low ratios to high 

influent concentrations and thus reduce the magnitude and 

variability of effluent with respect to influent concentrations. 

Applying the minimum irreducible concentration would 

provide a lower bound for calculated effluent concentrations.

The triangular/trapezoidal concentration-reduction 

parameters may be estimated by using expert judgment or 

by fitting the distribution to good data if they are available 

for a given BMP design. Unlike a regression approach, use 

of a bounded distribution reduces the chance that unrealistic 

output values will be generated by extrapolating a relation 

beyond available data. Although a series of equal lower-bound 

concentrations set to a minimum irreducible concentration 

would not be lognormal, such data (above the detection limits) 

are apparent in many of the probability plots in the statistical 

summaries from the international BMP database for nutrients 

(Leisenring and others, 2010) and solids (Leisenring and 

others, 2011). 

Total phosphorus concentration data from the 

International BMP Database were used to demonstrate the 

process for determining the statistical parameters of the 

trapezoidal distribution for use with SELDM. First, the 

inflow and outflow concentrations were retrieved from 

the BMP Database, and the ratios of the outflow-to-inflow 

concentrations were calculated. Figure 24A is a scatterplot 

of the inflow and outflow concentrations from the Lake 

Ridge detention pond. Inflow concentrations range from 

0.04 to 1.3 mg/L, with a median of about 0.31 mg/L. 

Outflow concentrations range from 0.04 to 0.43 mg/L, with 

a median of about 0.21 mg/L. The ratios of outflow to inflow 

concentrations range from 0.11 to 1.75, with a median ratio 

of 0.7 (fig. 24B). Then the inflow values and the ratios were 

sorted and ranked for determination of the percentiles of the 

ratios and the rank correlation between inflow concentrations 

and the associated ratios. The Cunnane plotting-position 

formula (Helsel and Hirsch, 2002) was used to calculate the 

plotting-position percentile for each ratio for this example. 

Spearmans’s rho (Haan, 1977; Helsel and Hirsch, 2002) 

was used to calculate the rank correlation between inflow 

concentrations and the associated ratios. The scatterplot 

of the inflow concentrations shows that the ratios decrease 

with increasing inflow concentration; Spearmans’s rho for 

this dataset is about -0.76 (fig. 24C). This makes sense 

physicochemically because many BMPs are better able to 

substantially reduce very high than very low concentrations 

and because it is difficult to improve the runoff quality 

substantially if inflow concentrations are low (the “clean water 

in = clean water out” phenomenon described by Leisenring 

and others (2010, 2011)). This result also makes sense 

mathematically because larger inflow concentrations (the 

denominators) would tend to reduce ratio values. Although 

the correlation between the inflow concentrations and the 

ratio of outflow to inflow concentrations is, by definition, a 

spurious correlation (Haan, 1977), use of the rank correlation 
Theory and Implementation 57

Rank of ratio of outflow to inflow 

concentration, dimensionless

Rank of inflow concentration, dimensionless

C

15

35

20

25

30

10

5

0

1050 15 20 25 30 35

Inflow concentration, in milligrams per liter

Ratio of outflow to inflow concentration, 

dimensionless

B

0.1

1

10

0.01

0.01 0.1 1 10

Inflow concentration, in milligrams per liter

Outflow concentration, in milligrams per liter

A

0.1

1

10

0.01

0.01 0.1 1 10

Figure 24. Scatterplots showing A, Inflow and 

outflow concentrations of total phosphorus from the 

Lake Ridge detention pond in milligrams per liter, B,

inflow concentrations and the ratio of outflow-toinflow concentrations, and C, the ranks of the inflow 

concentrations and the ratios of outflow-to-inflow 

concentrations. Monitoring data are from the July 2011 

version of the International Best Management Practices 

Database.

for stochastic data generation reduces the chance that 

unreasonably large concentration reductions may be modeled.

The CDF equations for the trapezoidal distribution 

(Kacker and Lawrence, 2007; appendix 1) were used to 

calculate values for each percentile on the basis of trial 

statistics. The absolute difference between the data value 

and the theoretical trapezoidal value was calculated for each 

percentile. Optimization methods (in this case, the Microsoft 

Excel® Solver) were used to minimize the sum of the 

absolute differences between the data and the trial statistics. 

The regression equations provided by Johnson (1997) 

also could be used to generate a triangular approximation 

to the data. Figure 25A shows the data, the statistics for 

the best fit trapezoidal distribution, and the CDF for this 

trapezoidal distribution.

The inflow water-quality statistics, the trapezoidaldistribution statistics, and the rank correlation values were 

used with SELDM to generate two populations of paired 

inflow and outflow values. The stochastically generated total 

phosphorus concentrations compare well to the measured 

inflow and outflow concentrations (fig. 25B). Although the 

populations of outflow values look slightly different from the 

sample of total phosphorus concentration data, the median 

and mean phosphorus concentrations for the stochastic values 

are within the 95-percent confidence limits of the respective 

measured values. The median concentration value (0.21 mg/L) 

is slightly higher than the mean value (0.20 mg/L) in the 

measured outflows. The 95-percent confidence interval for 

the median value of the measured data, however, ranges from 

0.16 to 0.26 mg/L (calculated by using methods described 

by McGill and others, 1978), and the 95-percent confidence 

interval for the mean value ranges from 0.01 to 0.38 mg/L 

(calculated by using methods described by Haan, 1977). 

In comparison, the median values of the two stochastic 

populations are about 0.17 mg/L, and the mean values 

are about 0.20 mg/L; both are well within the 95-percent 

confidence limits of the statistics for the measured outflow 

concentrations. The stochastic populations have greater ranges 

than the data because they represent estimates for about 

1,600 storms (30 years of record), whereas the monitoring data 

represent samples from only 32 storms. 

The approach used in SELDM is warranted because of 

the uncertainties in available information and because the level 

of effort required to develop deterministic simulation models 

may be beyond the scope of an initial planning-level estimate. 

If, however, the initial analysis done with SELDM indicates 

the potential need for mitigation, then detailed deterministic 

simulation models such as those described by Huber and 

others (2006) or detailed statistical models such as those 

described by Adams and Papa (2000) may be used to develop 

the performance statistics used by SELDM. Furthermore, if 

the initial analysis indicates the potential need for mitigation, 

then SELDM could easily be used to develop the BMPperformance statistics for concentration reduction needed 

to reduce storm loads or the frequencies of water-quality 

excursions to acceptable levels.
58 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0 Total phosphorus concentration, in milligrams per liter

Measured

inflow

Measured

outflow

Stochastic 

outflow 1

Stochastic 

inflow 1

Stochastic 

inflow 2

Stochastic 

outflow 2

0.11

0.11

0.23

1.94

-0.76

1 2 5 10 20 30 40 50 60 70 80 90 95 98 99

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.01

0.1

1

10

Maximum

Mean

Median

Minimum

75th percentile

25th percentile

5th percentile

95th percentile

99th percentile

1st percentile

Minimum

Lower bound of the most probable value

Upper bound of the most probable value

Maximum

Rank correlation 

Trapezoid statistics

Percentage of ratios that are less than or equal to the given value

Ratio of outflow to inflow concentration, dimensionless

B

A

Figure 25. Statistics for estimating stochastic reductions in total phosphorus by the Lake Ridge detention pond. The 

probability plot A, demonstrates how the trapezoidal statistics are estimated by fitting the data to the distribution. 

The boxplots B, demonstrate how the stochastic populations of outflow values generated by using the parameterized 

trapezoidal distribution compare to the original data. Monitoring data are from the July 2011 version of the International 

Best Monitoring Practices Database.
Theory and Implementation 59

Downstream Stormwater Concentrations and 

Loads

Downstream water quality is defined by the mixing 

of highway-runoff loads (concentration times flow) with 

upstream loads (fig. 1). SELDM generates a random 

population of flows, concentrations, and loads for the highway, 

with and without BMP treatment, and for the upstream basin. 

Highway and upstream loads are added and then divided by 

the combined flow to calculate downstream concentrations. 

The upstream loads and flows used for the calculations are 

the values that are either concurrent with discharge from 

the highway site or with discharge from the selected BMP. 

If BMP-treatment options are not defined or selected for a 

given water-quality pair, then the downstream water-quality 

values are calculated with the upstream flows and loads 

that occur during the period of highway runoff. If, however, 

BMP-treatment options are defined and selected, then the 

downstream water-quality values are calculated with the 

upstream flows and loads that occur during the period of 

BMP discharge.

Water-Quality Pairs

SELDM uses water-quality pairs to specify how the 

downstream concentrations are calculated. To define a pair, 

the water-quality constituent must be defined and selected 

for both the highway runoff and upstream flow. The same 

highway and upstream constituent selections may be 

defined as multiple pairs with different short names. To do a 

sensitivity analysis within one run of SELDM, multiple pairs 

could be defined by using the same highway and upstream 

water-quality constituent selections. For example, the same 

highway and upstream constituent selections could be used 

to calculate downstream concentrations, flows, and loads as 

one downstream constituent-pair with BMP treatment and a 

second downstream constituent-pair without BMP treatment. 

Similarly, the same highway and upstream selections can 

be used to calculate different downstream concentrations of 

concern by defining different constituent pairs with different 

adverse-effect concentrations. Similarly, the sensitivity of 

annual average lake concentrations to estimated apparent 

annual average attenuation factors can be tested by defining 

different constituent pairs with the same highway and 

upstream constituent selections but different attenuation 

factors. SELDM outputs are accompanied by a description of 

the characteristics of the pair. This description may be used to 

document the assumptions and sources of data used to define 

the selections. 

Concentration of Concern

SELDM deliberately defines the term “concentration 

of concern” in a vague manner because different methods 

may be used for identifying a concentration of concern, and 

SELDM may be used to address different kinds of waterquality problems. The concentration of concern, however, is 

designed to represent that portion of the measured or modeled 

constituent concentration that may have an adverse effect on 

natural or human water use in the receiving water downstream 

of the highway outfall. SELDM uses adverse-effect ratios 

to calculate the concentration of concern as a fraction of 

the calculated downstream EMC for each storm event. 

Thus, the concentration of concern is not a single regulatory 

threshold defining water-quality excursions that would 

require mitigation measures for runoff control, but instead the 

component of each modeled EMC that would apply to a single 

regulatory threshold. 

For example, if dissolved nutrients in the water column 

are of concern, then the adverse-effect ratios may be selected 

to represent the dissolved (filtered) portion of the modeled 

total whole-water concentration. If, however, only one 

component of the dissolved nutrients is of concern, then the 

adverse-effect ratio would represent the fraction expected 

to be contributed by only that component. Similarly, if trace 

metals in the water column are of concern then, depending 

on the regulatory approach, the adverse-effect ratios may be 

selected to represent the entire dissolved (filtered) portion of 

the total whole-water concentration, the filtered portion minus 

the colloidal portion as a fraction of the total whole-water 

concentration, or just the free ionic portion of the total wholewater concentration (Tipping, 1994; U.S. Environmental 

Protection Agency, 1996c; Tipping and others, 1998; Di Toro 

and others, 2001; Santore and others, 2001). If, however, the 

dissolved concentrations are being modeled directly, then 

the adverse-effect ratios would be calculated to represent the 

fraction of the dissolved concentration that accounts for the 

free ionic portion. Similarly, if local deposition of metals or 

nutrients to bed sediments is of concern, and whole-water 

concentrations are being modeled, then the adverse-effect 

ratios may be selected to represent the proportion of metals 

or nutrients expected to be associated with coarse sediments 

(Breault and Granato, 2003; Buckler and Granato, 2003; 

Lopes and Dionne, 2003). 

An alternate approach to use of the adverse-effect-ratio 

statistics is to model each component of the constituent of 

interest as a separate water-quality constituent in SELDM. 

For example, if the adverse effects of copper toxicity were 

a concern, then statistics for measured dissolved (filtered) 

concentrations could be used rather than, or in addition to, the 

whole-water concentrations of copper. Similarly, if the adverse 

effects of fine sediment in the water column were a concern, 

then statistics for measured concentrations of fine sediment 

could be used. If sediment deposition on the streambed were 

an issue, then statistics for measured concentrations of course 

sediment could be used. However, use of the adverse-effect 

ratios to model the concentrations of concern by using wholewater concentrations has at least six important advantages over 

the alternate approach for many constituents: 
60 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

• Data needs—Modeling individual components of a 

whole-water constituent requires data to calculate the 

statistics to characterize upstream stormflow quality, 

highway-runoff quality, and BMP performance for 

each component. 

• Storm-by-storm mass balance—The results 

from SELDM cannot be used to compare total 

concentrations with component concentrations on 

a storm-by-storm basis unless adverse-effect ratios 

or dependent water-quality relationships are used. 

Although SELDM will produce an output-value 

population that represents input statistics for each 

constituent definition, these constituent concentrations 

will be generated independently unless a dependent 

water-quality relationship is defined. The use of 

adverse-effect ratios, however, will ensure that the 

component of interest is some fraction of the total 

concentration modeled for each storm event.

• Data-quality concerns—Processing water samples to 

isolate the components of a water-quality constituent 

increases the probability of introducing physical and 

chemical bias in the results of the analyses (Patterson 

and Settle, 1976; Shiller and Boyle, 1987; Windom 

and others, 1991; Horowitz and others, 1992; Benoit, 

1994; Benoit and others, 1997; Breault and Granato, 

2003; Lopes and Dionne, 2003). For example, 

Benoit and others (1997) showed that ultraclean 

monitoring methods and materials were necessary 

to prevent order-of-magnitude increases in some 

dissolved trace element concentrations measured in 

natural water samples. Horowitz and others (1992) 

demonstrated that the measured concentration of 

dissolved (filtered) metal was dependent on the pore 

size, type, and diameter of the filter; the filtration 

method; and the volume of sample processed. The 

characteristics of the sample, including suspended 

sediment concentration, suspended sediment grain-size 

distribution, concentration of colloids and colloidassociated trace elements, and concentration of organic 

matter, also affected the proportion of the trace element 

measured as dissolved in these controlled experiments. 

Methods used to separate sediment grain-size fractions 

for analysis also may introduce bias and variability 

in reported concentrations. Thus, use of statistics for 

whole-water concentrations may be more robust than 

use of constituent fractions because of the potential 

effects of sample-processing artifacts. 

• Detection-limit issues—The proportion of qualified 

values (most commonly values below detection 

limits) is commonly lower for the total concentration 

of many constituents than for the concentrations of 

the dissolved (filtered) components. For example, the 

number of qualified values in the filtered fractions 

of data for trace metal samples in the International 

BMP Database is 1.12, 1.28, 1.71, 2.03, 1.87, 1.26, 

and 1.96 times the number of qualified values in the 

unfiltered fractions for Cd, Cr, Cu, Fe, Pb, Ni, and Zn, 

respectively. In the HRDB, the number of qualified 

values in the filtered fractions of data for trace metal 

samples is 2.61, 3.47, 0.392, 30.9, 6.69, 3.38 and 

3.22 times the number of qualified values in the same 

sequence of unfiltered trace-metal fractions. Increasing 

the proportion of qualified values complicates analysis 

of water-quality data, increases uncertainties in 

calculated input statistics, and therefore increases 

uncertainties in model results (Helsel and Hirsch, 

2002; Helsel, 2004; Granato and Cazenas, 2009). Thus, 

the use of statistics for whole-water concentrations 

may be more robust than the use of constituent 

fractions because of the potential effects of qualified 

data on sample statistics.

• Data availability—Data for the total concentrations 

of many constituents commonly are more plentiful 

than data for individual components. For example, 

Turcios and others (2010) found half as many sediment 

concentrations separately defined for sand or fines as 

the number of total suspended sediment concentration 

measurements in the USGS National Sediment 

Database. Similarly, queries to the July 2011 version 

of the International BMP Database indicated that the 

ratios of unfiltered to filtered analyses measured in 

BMP inflows and outflows were about 1.96, 1.28, 

1.71, 2.03, 1.87, 1.26 and 1.96 for Cd, Cr, Cu, Fe, Pb, 

Ni, and Zn, respectively. Queries to the 2010 version 

of the HRDB (Smith and Granato, 2010) generated 

similar results with the ratios of unfiltered to filtered 

analyses measured in highway runoff of about 1.5, 

1.45, 2.03, 4.7, 2.1, 1.22, and 1.96 for the same 

sequence of trace metals, respectively. For nutrients, 

the ratios of unfiltered to filtered concentrations 

of nitrite plus nitrate and phosphorus are about 26 

and 8.5, respectively. Data-quality issues also limit 

the availability of data. For example, much of the 

dissolved trace element data collected by the USGS 

prior to the mid-1990s is in question (Windom 

and others, 1991; Smith and others, 1993). Similar 

limitations apply to most dissolved trace element, trace 

organic, and low-level nutrient data collected without 

use of clean sampling methods. Limits in available 

data reduce the opportunity to select input statistics 

based on data from hydrologically similar sites.

• Runoff-quality transformations—Modeling the 

dissolved fraction of a constituent or a single grainsize fraction of sediment is difficult because physical 

and chemical changes can occur in runoff. Sediment 

concentration and the geochemistry of runoff is 

expected to change as runoff moves over the pavement, 

through a drainage system, through a BMP structure, 

and into a receiving stream (Breault and Granato, 
Theory and Implementation 61

2003), especially if the receiving stream has sediment 

concentrations and geochemical characteristics that 

substantially differ from those in the highway runoff 

(Breault and Granato, 2003; Bricker, 2003). Use of 

the adverse-effects ratio, however, requires data or 

information about the fractionation of the constituents 

only in downstream receiving waters. 

However, despite these limitations, modeling the concentration 

of concern as a separate water-quality constituent may be 

warranted if the results from this approach are deemed to 

be more defensible than the use of the adverse-effect ratio 

approach for a given situation.

SELDM uses the triangular/trapezoidal family of 

distributions for generating a stochastic sample of adverseeffect ratios for each specified downstream stormwater 

constituent. This family of distributions was selected 

for modeling the adverse-effect ratios because it can be 

parameterized by using expert judgment or by fitting the 

distribution to data (Johnson, 1997; Back and others, 2000; 

U.S. Environmental Protection Agency, 2001; Scherer, 2003; 

Kacker and Lawrence, 2007). The adverse-effect ratios 

must be greater than or equal to 0 and less than or equal 

to 1. SELDM uses the adverse-effects ratios rather than a 

chemical or physical constituent-speciation modeling module 

to estimate the concentration of concern. SELDM does not 

include a speciation-modeling module because different 

constituents and the associated concentrations of concern 

may be governed by different processes and, therefore, 

different variables. 

The use of adverse-effect ratios, however, does require 

enough data or information to calculate statistics or to 

inform expert judgment. Results from chemical or physical 

constituent-speciation models may be used to estimate 

the parameters of the triangular/trapezoidal distribution 

of the adverse-effect ratios. For sediment, the grain-size 

distribution of bed sediments near highway outfalls and 

estimates of the grain-size distribution of the receiving waters 

will provide estimates of the concentration of concern. For 

nutrients, metals, and organic chemicals, the distribution 

coefficients from the literature can be used to estimate the 

proportions of constituent fractions in the receiving stream 

(U.S. Environmental Protection Agency, 1985a; Thomann 

and Mueller, 1987; U.S. Environmental Protection Agency, 

1996c; Young and others, 1996; Tipping and others, 1998; 

Konstantinos, 2001; Allison and Allison, 2005). For example, 

Allison and Allison (2005) used published coefficient values 

and geochemical-modeling techniques to estimate statistics 

for the distribution coefficients of trace elements. Adverseeffect ratio statistics could be calculated on the basis of the 

values developed by Allison and Allison (2005), and the 

statistics for downstream suspended sediment concentrations 

could be calculated by using results from a preliminary run 

of SELDM for the site of interest. Geochemical-speciation 

models also could be used to estimate the concentrations of 

concern (Allison and others, 1990; Tipping, 1994; Allen and 

Hansen, 1996; Young and others, 1996; Di Toro and others, 

2001; Bricker, 2003; Allison and Allison, 2005). SELDM 

can be used to calculate downstream water-quality statistics 

for pH and the concentrations of sediment, dissolved organic 

compounds, major ions, and other data needed to run such 

models. Although the concentrations for different constituents 

are randomly generated from storm to storm in SELDM, 

population statistics for each constituent can be used to 

represent a range of conditions for selective use as input to the 

geochemical models. Alternatively, for regulatory applications 

of SELDM, the adverse-effect ratio statistics could be 

estimated as part of an expert-judgment process that includes 

input from regulators and other stakeholders.

Mixing

SELDM calculates the concentrations, flows, and loads 

for the highway discharge (either direct runoff or BMP 

discharge), the upstream stormflow, and the downstream 

stormflow with the assumption that the downstream 

stormflow components are rapidly mixed (fig. 1). This is 

the standard assumption for almost all stormwater-quality 

models, including the 1990 FHWA model and the complex 

watershed models supported by the USEPA Basins program 

(Di Toro, 1984; Driscoll and others, 1989; Driscoll and 

others, 1990a, b; Adams and Papa, 2000; Zoppou, 2001; 

Walker, 2007; Rossman, 2010; Kuzin and Adams, 2010; 

U.S. Environmental Protection Agency, 2011). For example, 

Zoppou (2001) reviewed eight watershed models commonly 

used to evaluate the effects of stormwater runoff. He 

found that only one of these complex models included the 

advective-diffusion equations necessary for calculating 

mixing in the receiving stream. The output from SELDM 

provides bounding conditions for evaluating receiving-water 

quality. The highway-discharge concentrations represent 

the most conservative estimate of the risk for water-quality 

excursions. The fully mixed downstream concentrations, 

however, represent more realistic estimates of the risk for 

instream water-quality excursions; the estimates are somewhat 

conservative because the mass-balance calculations do not 

account for the potential effects of losses caused by physical, 

chemical, or biological attenuation; longitudinal dispersion; 

or downstream-flow accretion. These losses are expected 

to reduce instream concentrations of many constituents of 

potential concern as a pulse of stormwater moves downstream 

(Athayde and others, 1983; U.S. Environmental Protection 

Agency, 1985a, b, 1987; Thomann and Mueller, 1987).

The fully mixed runoff-quality approach was developed 

by the USEPA during NURP because the intermittent 

occurrence of runoff is expected to reduce adverse effects 

of excursions in comparison to steady-state discharges such 

as industrial or wastewater outfalls, and because the spatial 

and temporal distribution of runoff to receiving waters 

during storm events is complex (Athayde and others, 1983). 

For example, NURP evaluated the effects of short-term 

exposures that would result from intermittent stormwater 

runoff and estimated that acute (CMC) standards for runoff 
62 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

concentrations could be about twice those for steady-flow 

conditions while providing a similar degree of protection 

(Athayde and others, 1983). The intermittent occurrence 

of stormflow discharges can be assessed by using the 

precipitation statistics in table 2. If the highway-runoff 

duration is assumed to be about equal to the storm-event 

duration, then the average storm-event duration can be 

multiplied by the average number of runoff-producing events 

per year and divided by the number of hours in a year (about 

8,766) to estimate the percentage of time during which 

highway runoff is discharging to the stream. On average, 

the duration of runoff flows would range from about 1.5 to 

8.9 percent of the year in different rain zones under these 

assumptions. Alternatively, if it is assumed that a structural 

BMP is employed to extend the highway-runoff duration to 

24 hours for each storm, then, on average, the duration of 

runoff flows would range from about 4.7 to 17 percent of the 

year in different rain zones. 

The duration of runoff-producing events also can be used 

to examine the conceptual use of a mixing zone for stormwater 

discharges. If stormflow velocities in natural channels are 

on the order of 1 ft/s in low-slope basins (Chow and others, 

1988), and the average runoff duration is on the order of 6 to 

12 hours (table 2), then the pulse of runoff from the highway 

during such a storm would be distributed over a stream length 

of about 22,000 to 43,000 ft. These distances are expected 

to be well beyond the estimated hydrologic mixing zone for 

many streams of potential concern (Day, 1977; Fischer and 

others, 1979; Heard and others, 2001; Jeon and others, 2007; 

Divine and others, 2007; Gualtieri and Mucherino, 2008; Dow 

and others, 2009). 

The fully mixed runoff-quality approach should meet 

DQOs for runoff-quality analyses because the uncertainties 

in runoff monitoring and modeling are large, and sophisticated 

hydrologic mixing-zone models do not represent conditions 

during storm events (Athayde and others, 1983; U.S. 

Environmental Protection Agency, 1985a). Mixing-zone 

analysis methods commonly do not include the complexities 

necessary to address wet-weather flows because these methods 

were largely developed for continuous point sources such 

as wastewater-treatment plants and industrial sources under 

steady-state extreme low-flow conditions (Fischer and others, 

1979; U.S. Environmental Protection Agency, 1991; Dupuis, 

2002; Divine and others, 2007; Oregon State Department 

of Environmental Quality, 2007; McCorquodale, 2007; 

Washington State Department of Ecology, 2010). 

The USEPA and many states recognize the need to 

develop wet-weather criteria because highway and urban 

runoff commonly discharges during highly variable stormflow 

conditions. However, formal criteria have not been developed 

to date (Dupuis, 2002; Rachel Herbert, U.S. Environmental 

Protection Agency, Office of Water, Municipal Stormwater 

Program, written commun., 2012). Furthermore, mixingzone equations may not represent conditions in small streams 

in which highway and urban runoff may be a substantial 

proportion of the downstream flows. Most mixing-zone 

studies and analyses have been limited to large low-gradient 

rivers or laboratory conditions that simulate conditions in 

such rivers (Day, 1977; Fischer and others, 1979; Heard and 

others, 2001; Jeon and others, 2007; Divine and others, 2007). 

Mixing-zone analyses commonly are based on assumptions 

that include simplified (far-field) flow distributions, minimal 

interaction of flow with irregularities on the streambeds and 

banks, and steady flow conditions without lateral inflows. 

Changes in slope and irregularities in streambeds, banks, 

and cross sections can accelerate transverse mixing so that 

the stream is rapidly mixed within about 2–25 stream widths 

(Day, 1977; Heard and others, 2001; Divine and others, 2007). 

Also, the rate of transverse diffusion increases substantially 

with increasing flows even with steady uniform flow in a 

clear, straight channel (Cotton and West, 1980). The hydraulic 

variables used to estimate mixing lengths also change with 

streamflow, distance along the stream, and time elapsed 

between large storms (Mackey, 1998; Heard and others, 2001).

The fully mixed runoff-quality approach also should meet 

DQOs for runoff-quality analyses because, unlike commonly 

used mixing-analysis methods, the fully mixed approach 

accounts for the characteristics of stormflow quality. Mixing 

models commonly are formulated with the assumption that 

background concentrations are negligible, and the tracer 

studies used to validate such models are designed so that 

background concentrations are negligible. For example, users 

of the moderately complex mixing model CORMIX are 

instructed to subtract the background concentrations from the 

effluent concentrations and the regulatory target concentrations 

(Jirka and others, 1996). This assumption may be warranted 

for wastewater discharges under steady-state low-flow 

conditions, but concentrations of constituents in stormwater 

from the upstream basin in many areas may be comparable 

to concentrations in highway runoff. As concentrations in the 

upstream stormwater and highway runoff converge, the effects 

of mixing are less distinct.

If, however, mixing zones are an issue for a given runoff 

study, then there are two types of mixing zones to consider, the 

regulatory mixing zone and the hydrologic mixing zone. The 

regulatory mixing zone is a limited area of the stream within 

which water-quality criteria may be exceeded. The USEPA 

has delegated the definition and regulation of constituent 

concentrations in mixing zones to the states, which use number 

of different approaches and definitions for assessing potential 

effects of effluents on receiving waters (U.S. Environmental 

Protection Agency, 1991; Dupuis, 2002; Divine and others, 

2007; McCorquodale, 2007; Washington State Department 

of Ecology, 2010). The assumption is that the overall aquatic 

resource of the receiving water will be protected despite 

water-quality excursions within mixing zones because they are 

limited in size. For stormwater applications, such excursions 

also would be limited in duration. Different states may define 

the point of interest for regulating constituent-concentration 

excursions in the discharge itself (end-of-pipe concentrations), 

in a small area near the outfall (commonly known as a zone 

of initial dilution), or in a defined area downstream of the 
Theory and Implementation 63

outfall but above the zone of complete mixing. Some states 

that regulate end-of-pipe concentrations adjust the target to a 

final acute value (FAV), which is generally the acute criterion 

multiplied by two. This FAV approach is similar to the NURP 

approach for concentrations of runoff constituents in receiving 

waters (Athayde and others, 1983). Generally, mixing-zone 

guidance approaches provide for routine acute water-quality 

excursions within all or part of a mixing zone and chronic 

excursions within and near the edge of the zone. Mixing-zone 

approaches commonly define a minimum zone of passage 

within the stream cross section where acute excursions are not 

allowed because the contaminant is assumed to be repulsive 

to mobile aquatic life. Regulatory mixing zones commonly 

are established by a rule-of-thumb based approach using 

one or more measures of channel geometry (Colorado State 

Department of Public Health and Environment, 2002; Oregon 

State Department of Environmental Quality, 2007; Washington 

State Department of Ecology, 2010). 

The hydrologic mixing zone is the area in the stream 

between the discharge location and the location where the 

stream water is fully mixed with the discharge. Analysis of 

hydrologic mixing zones commonly centers on two processes, 

near-field and far-field mixing (Fischer and others, 1979; 

McCorquodale, 2007; Washington State Department of 

Ecology, 2010). Near-field mixing is the rapid and irreversible 

turbulent mixing of the water discharging from an outfall 

with the receiving water around the point of discharge. Nearfield mixing is primarily caused by the momentum-induced 

velocity of the effluent and differences in the densities 

of the effluent and receiving waters. Near-field mixing 

characterization may help define zones of initial dilution. 

McCorquodale (2007) provides a summary of methods and 

models for estimating near-field mixing. Far-field mixing is 

generally caused by advection and dispersion and is generally 

much slower than the initial near-field mixing. Far-fieldmixing analyses commonly are done to estimate the extent of 

regulatory mixing zones. If a mixing analysis is necessary, the 

development of the far-field plume can be estimated by using 

analytical equations (for example, equations by Fischer and 

others, 1979), simple models based on analytical equations 

(for example, FARFIELD and RIVPLUM5 developed by the 

Washington State Department of Ecology, 2010), or more 

complex models (for example, CORMIX developed by Jirka 

and others, 1996; or Visual Plumes developed by Frick and 

others, 2001). If analytical models are to be used, then it 

may be necessary to incorporate substantial adjustments to 

the transverse mixing coefficients published by Fischer and 

others (1979) to reflect advances in quantifying this variable 

for different instream conditions (for example, Cotton and 

West, 1980; U.S. Environmental Protection Agency, 1985a; 

Jeon and others, 2007; Divine and others, 2007; Gualtieri and 

Mucherino, 2008).

If the runoff-quality approach based on the assumption 

of complete mixing is deemed insufficient for a given 

application, then two conceptual modeling methods also can 

be applied by using SELDM to evaluate the risks for waterquality excursions. The first method is upstream-flow 

reduction analysis, in which the SELDM analysis is copied 

(by using the copy option on the analysis form) and the 

upstream drainage area is reduced in proportion to the amount 

of stormflow in the regulatory mixing zone (William Fletcher, 

Oregon Department of Transportation, written commun., 

March 2012). The second method is the downstream flowaccretion analysis, in which the SELDM analysis is copied 

and the upstream drainage area is increased to account for the 

additional dilution caused by diffuse inflows of stormwater 

along the length of a stream even in the absence of a tributary 

stream or additional storm outfall. The downstream flowaccretion analysis also may be useful if there is a sensitive 

receptor, such as a water-supply intake, some distance 

downstream of the highway outfall to estimate the effect 

of any additional dilution between the outfall and the 

sensitive receptor.

The upstream flow-reduction analysis method is designed 

to assess additional risk for excursions in the regulatory 

mixing zone in comparison to the assumption that the 

inflow and the stream water are fully mixed. For example, 

in Oregon, the State Department of Environmental Quality 

(2007) established the width of a regulatory mixing zone to 

be less than 25 percent of the cross-sectional area of a river 

or stream. In this case, the risk of water-quality excursions 

could be assessed in the SELDM analysis by reducing the 

upstream drainage area to 25 percent of the actual value to 

proportionally reduce upstream stormflow volumes. However, 

the other hydraulic variables (drainage length, mean basin 

slope, impervious fraction, and BDF) and the hydrographrecession statistics should not be modified to ensure that 

the timing of runoff from the upstream basin is not altered. 

Although it may be a concern, the population of transportcurve concentrations will remain the same because they were 

developed for normalized flow (in cubic feet per second per 

square mile) to scale stormflows to the basin size (Granato 

and others, 2009). Because the downstream flow volume is 

the sum of the upstream volume and the highway-runoff (or 

BMP-discharge) volume, this upstream-flow reduction may 

not result in a proportional reduction in the dilution factor or 

a proportional increase in the risk for excursions in all storms; 

especially in small or arid basins, to which the highway 

may contribute a large proportion of the stormflows. The 

downstream water-quality and dilution-factor output from the 

upstream-flow reduction analysis with SELDM can be used 

in conjunction with the modeling results for the fully mixed 

condition to assess the potential for water-quality excursions 

downstream of the site of interest.

The downstream flow-accretion analysis method is 

designed to calculate the reduction in risk for water-quality 

excursions caused by additional dilution that is expected to 

occur in the hydrologic mixing zone downstream of an outfall 

(as defined by a far-field mixing analysis) during storm events. 

The ratio of the downstream area to the upstream area is the 

downstream flow-accretion factor that can be used to estimate 

dilution at the point of interest. Analyses of relations between 
64 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

basin length (defined as the length of the main channel from 

the outlet to the watershed divide) and drainage area show 

strong correlation for the 845 sites with drainage areas greater 

than 0.1 mi2

 in the secondary dataset compiled by Granato 

(2012) (fig. 26). Use maps, GIS, or Streamstats (Ries and 

others, 2008) to delineate the downstream basin, or add the 

estimated mixing length to the upstream basin length and use 

the equations in figure 26 to estimate the total downstream 

area. If there are tributaries or substantial stormwater outfalls 

that enter the stream within the estimated downstream mixing 

length, use the delineation method. Otherwise, the equations 

in figure 26 can be used to (1) calculate an equivalent 

length of the upstream basin at the highway outfall by using 

the (predetermined) upstream drainage area, (2) add the 

length of the mixing zone to this estimated basin length, (3) 

calculate the downstream drainage area by using the second 

regression equation with the total estimated basin length, 

and (4) calculate the flow-accretion factor by dividing the 

downstream by the upstream drainage areas. If figure 26 is 

used in this way, it is necessary to use the lengths and areas 

calculated by using the line of organic correlation on the 

graph, because (as evidenced by the scatter of data above and 

below the line) use of the actual upstream basin length plus the 

mixing length may result in a calculated downstream area that 

is less than the measured upstream area.

Once the flow-accretion ratio is calculated, this value 

can be used to assess the risk of water-quality excursions at 

the point downstream of the outfall by adjusting the estimated 

input drainage area in a SELDM analysis or the model 

output directly. To run the scenario in SELDM, the modeled 

upstream drainage area would be increased to proportionally 

increase stormflow volumes. As in the flow-reduction analysis, 

however, the other hydraulic variables (drainage length, 

mean basin slope, impervious fraction, and BDF) and the 

hydrograph-recession statistics should not be modified to 

ensure that the timing of runoff from the upstream basin is not 

altered. To adjust simulation results outside SELDM, apply 

the flow-accretion factor to the upstream flows to adjust the 

dilution factors and the fully mixed concentrations.

The flow-accretion factors in figure 27 demonstrate 

that the additional stormflow contributions that occur over 

relatively short mixing zones may be substantial, especially 

for small upstream drainage basins. For example, inflows 

from the additional contributing area within a 500-ft mixing 

zone downstream of a highway outfall would be expected to 

produce about 40 percent more dilution than is available at the 

highway outfall for a 0.1-mi2

 upstream basin and 10 percent 

more for a 1-mi2

 basin (if the quality of inflows is assumed 

to be comparable to the quality of upstream stormflows). The 

additional drainage area associated with a 2,500-ft mixing 

zone is expected to provide about 14 percent more dilution for 

a 10-mi2

 basin. The magnitude of lateral inflows calculated by 

using the flow-accretion curves in figure 27 also indicates the 

potential for turbulent transverse mixing caused by additional 

stormflow outfalls or diffuse lateral inflows along the stream 

during storm events. The substantial accretion of lateral 

inflow volumes over short mixing distances as evidenced by 

these curves also indicates that steady state (dry-weather) 

mixing-zone models may substantially underrepresent 

dilution downstream of a stormwater outfall during wetweather conditions.

Flow-accretion factors also may be used without a 

detailed mixing-zone analysis. For example, if the quality 

of water at a water-supply intake 1 mi downstream of 

a highway outfall is of concern, then the fully mixed 

downstream concentrations at the intake could be estimated 

by calculating the flow-accretion factor. If the drainage area 

of the upstream basin at the highway outfall is 1 mi2

, then the 

potential downstream dilution at the intake would be about 

2.2 times the potential dilution at the outfall; if the area of 

the upstream basin at the highway outfall is 10 mi2

, then the 

potential downstream dilution at the intake would be about 

1.4 times the potential dilution at the outfall (figs. 26, 27). 

If the duration of the runoff pulse from a highway or BMP 

during the average storm is considered, the flow accretion over 

the 20,000- to 40,000-ft length of the pulse in the absence of 

large downstream tributaries would be about 8 to 20 times 

the potential dilution at the outfall if the drainage area of the 

upstream basin at the outfall is 1 mi2

 and about 2 to 4 times 

the potential dilution at the outfall if the drainage area of 

the upstream basin at the outfall is 10 mi2

 (fig. 27). The 

concentrations calculated by assuming instantaneous mixing 

at the outfall may be considered a conservative estimate of 

the risk for water-quality excursions at downstream locations 

because of the additional dilution and mixing of runoff from 

contributing areas between the outfall and the downstream 

point of interest.

Lake-Basin Analysis

The SELDM lake-basin module produces a population 

of average annual constituent concentrations in a lake 

that receives highway runoff. This component of SELDM 

is designed to provide planning-level estimates for an 

assessment of the potential effects of highway runoff on the 

water quality in a downstream lake or pond. This lake-basin 

module, a lumped-parameter model, treats the highway, the 

lake basin, and the lake as simple control volumes rather 

than detailed, spatially distributed features. This lake-basin 

module is a simple implementation of the Vollenweider massbalance model, which is well established in the literature 

(Vollenweider, 1975; Reckhow, 1979; U.S. Environmental 

Protection Agency, 1983, 1985a, c, 1986a; Driscoll, 1990b; 

Thomann and Mueller, 1997). The Vollenweider mass-balance 

model is commonly used for planning-level estimates of 

annual average lake concentrations for TMDL studies (U.S. 

Environmental Protection Agency, 2000, 2007b) and is based 

on eight basic assumptions:
Theory and Implementation 65

Basin length, in miles

Drainage area, in square miles

Coefficient of determination R2 = 0.925

Drainage area = 0.426(length)1.74

Length = 1.63(drainage area)0.575

Regression equation statistics

0.1 1 10 100 200

0.1

1

10

100

2,000

1,000

Figure 26. Relations between the basin length, defined as the length of the main channel in miles from the outlet to the 

drainage divide, and the basin drainage area in square miles for 845 sites that have drainage areas greater than 0.1 mi2 in 

the secondary dataset compiled by Granato (2012). The regression equations were calculated by using the line of organic 

correlation because it provides a unique line for using either variable as the explanatory variable (Helsel and Hirsch, 2002).
66 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Initial drainage area, in square miles

Downstream flow-accretion factor, dimensionless

1

10

100

0.1 1 10 100

Mixing zone length = 40,000 feet

20,000 feet

10,000 feet

5,000 feet

2,500 feet

1,000 feet

500 feet

250 feet

100 feet

Downstream flow-accretion factor = 

0.426 (1.63 (initial drainage area)0.575 + (mixing zone length/5280))1.74

(initial drainage area) 

Figure 27. The downstream flow-accretion factors, which are the ratios of downstream flows at the ends of the specified 

mixing-zone lengths to the flows immediately downstream from the highway outfall.
Theory and Implementation 67

• the lake is a fully mixed control volume;

• storage does not change on an annual basis so inflows 

equal outflows;

• regional streamflow statistics, which include areas 

of groundwater recharge, discharge, and lakes, are 

representative of the total flux of water through the 

lake on an annual basis; 

• each year in the analysis represents an independent 

“event” for the purpose of calculating annual average 

concentrations;

• annual runoff volumes and loads from the drainage 

areas to one or more highways in a basin can be 

modeled as output from a single site representing the 

combined area for all highway sites; 

• annual runoff volumes and loads from the rest of 

the basin can be modeled as output from a single 

area representing the total lake-basin area minus the 

drainage areas of the highway(s); 

• bed sediments are not a substantial source of solids 

or water-quality constituents in the part of the water 

column being modeled to estimate the average 

concentration; and 

• reductions in concentrations of each water-quality 

constituent in the lake can be modeled as a single 

attenuation factor that represents applicable first-order 

decay constants.

Without such assumptions, the lake-modeling effort 

would be complex and require detailed spatial and temporal 

data and the use of a complex model such as the Water Quality 

Analysis Simulation Program (WASP), which is developed 

and maintained by the U.S. Environmental Protection Agency 

(2010). In addition, the results of a complex modeling effort 

may not be substantially better than a planning-level estimate 

without a long and detailed set of calibration data (Granato, 

2010). For example, Bhavsar and others (2008) compared 

results from two models that included coupled fate and 

transport of metals in lakes and found that, despite a complex 

modeling effort with detailed characterization data, simulated 

concentrations differed from measured concentrations by as 

much as 400 percent. Bhavsar and others (2008) indicated that 

the results were constrained by a lack of detailed calibration 

data rather than the structure of the models.

Mass Balance Approach

The lake analysis is based on two independent 

components for annual lake output—the highway-runoff 

component and the lake-basin component—for all downstream 

water-quality pairs identified as part of the lake analysis. A 

process-flow diagram is shown on figure 28. The highwayrunoff-load component is generated storm-by-storm in the 

Generate annual highway-site flows and loads from 

storm-event flows and loads by using the main SELDM 

package for the runoff-constituent pairs selected for 

lake-basin analysis

Generate the long-term record of daily flows by using the 

selected lake-basin drainage area and the selected 

streamflow statistics

Sum daily flows to calculate annual totals (lake fluxes), 

and rank the annual totals to match the ranks of annual 

precipitation totals

Generate a long-term record of daily concentrations and 

loads for each defined random water-quality constituent 

selected for lake-basin analysis 

Generate a long-term record of daily concentrations and 

loads for each defined transport-curve water-quality 

constituent selected for lake-basin analysis 

Generate a long-term record of daily concentrations and 

loads for each defined dependent water-quality constituent 

selected for lake-basin analysis 

Sum the paired highway-runoff-quality and lake-basin input 

loads and fluxes to calculate total annual loads 

Divide the annual loads by the total annual flux and apply the 

first-order decay coefficient to calculate the average annual 

concentration

Calculate plotting-position values for each annual flux, 

concentration, and load to estimate the probabilility of 

exceedence for each value

Figure 28. The steps used by the Stochastic Empirical 

Loading and Dilution Model to estimate lake-basin fluxes, 

loads, and concentrations.
68 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

main SELDM module. The lake-basin flows, concentrations, 

and loads are generated for each day during the entire period, 

aggregated into annual sums, and paired with the highwayrunoff component. The selected streamflow statistics, which 

are adjusted for the entire lake-basin drainage area, are used 

to generate the daily lake-basin flow values. Individual daily 

values generated for the lake-basin analysis should not be 

exported and used for further analysis because SELDM 

generates each value randomly without serial correlation or 

seasonal patterns; the annual-load accounting years are just 

random collections of 365 or 366 sequential daily values. 

Lake-flux years are assigned to highway-runoff-analysis years 

by pairing values that are ranked respectively by total annual 

streamflow volume and total annual precipitation volume. The 

annual average steady-state concentrations are estimated from 

measures of lake volume, annual loads, flux through the lake, 

and an attenuation factor (which may include physical settling 

and other attenuation mechanisms).

The properties of the lake, including drainage area, 

surface area, and mean depth, are entered on the lake-basin 

form in SELDM. The lake-basin drainage area (in square 

miles) is the entire contributing area to the lake upstream 

of the lake outfall (if there is a surface-water outfall) minus 

the area of the highway. The lake-basin drainage area is 

used to calculate the total annual flux of water through the 

lake (exclusive of highway runoff) based on regional, local, 

or site-specific streamflow data. In most cases, the surfacewater divide of the lake basin may be used to approximate 

the contributing area to the lake. In some cases, however, 

streamflow data or groundwater maps may indicate that the 

surface-water and groundwater divides do not coincide. In 

such cases, the analyst may adjust the streamflow statistics or 

the lake-basin drainage area to best represent the total annual 

flux through the lake-aquifer system. The highway drainage 

area (in acres) can be determined for a given stretch of 

highway or for the total length of all highways in the basin. If 

runoff from the highway site discharges into a single tributary 

within a lake basin, then the fluxes through the stream 

analysis and the lake analysis may be done together by using 

a physical basin-delineation method, shown schematically in 

figure 29A. In this case, the total lake-basin area is the sum 

of the upstream and downstream basin areas, including the 

lake, but not the highway. If, however, a highway or highways 

intersect multiple tributaries, then the total highway area can 

be modeled conceptually as a single contributing area and the 

lake-basin area as the total watershed area minus the area of 

the modeled highways (fig. 29B). In this case, the SELDM 

output for downstream concentrations for individual storm 

events would not represent conditions in any of the individual 

tributaries because the highway and upstream areas do not 

correspond with the drainage areas at any particular stream 

crossing. In either case, the highway contributions are entirely 

from storm-event runoff, and the lake-basin contributions are 

from total annual runoff (base flow plus stormflow).

The SELDM lake-basin model is a fully mixed control 

volume because it does not account for variations in vertical 

or spatial concentrations within the lake. The surface area 

of the lake, in acres, is defined as the annual mean surface 

area, which can be estimated from topographical maps, 

a geographic information system, or other sources of 

information. The surface area is multiplied by the mean depth 

to estimate the total volume of the lake. The total volume of 

the lake is used with the total annual flux of water through 

the lake to estimate the hydraulic residence time of the lake. 

On an annual basis, this approach is suitable for lakes with 

large surface-area-to-depth ratios with many small inlets and 

may be used as a simplified approximation for many lakes 

(U.S. Environmental Protection Agency, 1985c, 1986a). If a 

lake is stratified, this approach may be used to approximate 

conditions in the active layer of the lake if the modeled 

volume and depth represent the volume and depth of this 

layer. Defined as such, the hydraulic residence time is defined 

as the average time for any given year during which a water 

molecule stays in the lake rather than the time required to 

exchange all water in the lake:

, (24)

where

Tw is the mean hydraulic residence time of water, 

in years; 

V is the volume of the lake in cubic feet; and

Q is the annual flux through the lake, in cubic 

feet per year.

If it is assumed that there is no change in storage from year 

to year, then the lake volume (V) is modeled as the surface 

area times the mean depth (a constant), and the annual mean 

residence time of water in the lake (Tw) is the parameter that 

changes with variations in annual flux (Q) so that annual 

inflows will equal outflows. 

Estimates of the mean depth, however, are not as readily 

available as of the surface area of the lake. Bathymetric 

maps commonly are available from Federal and state land 

and water-management agencies (for example, Guthrie and 

Stolgitis, 1977; New Jersey Division of Fish and Wildlife, 

2008; Connecticut Department of Environmental Protection, 

2010; Massachusetts Division of Fisheries and Wildlife, 2010; 

Rhode Island Department of Environmental Management, 

2010; University of Florida Lakewatch, 2010). Water-supply 

agencies may have compiled bathymetric maps for active and 

inactive water-supply reservoirs. For example, Waldron and 

Archfield (2006) provide lake stage, volume, and surface area 

data and relations for 44 reservoirs in Massachusetts as part 

of a water-supply analysis. Currently (2012), instruments that 

combine global positioning system and sonar sensors have 

been integrated for use by recreational boaters. Such systems, 

when coupled with GIS or mapping software, are able to 

provide mean depths that have a high degree of accuracy. 

An initial estimate of average depth can be made by using a 

modification to a regression equation developed by Bartsch 
Theory and Implementation 69

Downstream 

area

Upstream 

area

Highway

Lake

Lake basin area = downstream area + upstream area

= total basin area - highway area

Lake basin area = total basin area - highway area

Highway area = highway 1 + highway 2

A. Physical lake-basin example

B. Hypothetical lake-basin example

Lake

Lake basin area

Highway 1

Highway 2

Figure 29. Examples of lake-basin areas defined by A, a physical delineation (used when there is a clear distinction 

between areas that are upstream and downstream of the highway) and B, a hypothetical delineation (used if the highway 

area does not divide the basin, or if there are multiple sections of highway).
70 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

and Gakstatter (1978) for predicting the mean hydraulic 

residence time in days from the ratio of drainage area to 

surface area for 36 lakes and reservoirs in the northern United 

States. If the mean hydraulic residence time is expressed as 

in equation 24, and the volume is divided by the surface area 

of the lake, the equation developed by Bartsch and Gakstatter 

(1978) becomes

D Q L = × × × SA DA− 32 7 0 177 1 177 . . . , (25)

where

DL is the mean depth of the lake, in feet;

Q is the flux through the lake, in cubic feet 

per year;

SA is the surface area of the lake, in square feet; 

and

DA is the drainage area of the lake, in square feet.

Equation 25 was tested for nine lakes in Massachusetts 

with basin areas, surface areas, and mean depths that had 

been documented for TMDL analyses. The error in resulting 

predictions ranged from -40 to 180 percent with a median error 

of 8 percent and a median absolute deviation of 41 percent. 

For this reason, estimates based on equation 25 should only be 

used for an initial rough planning-level estimate.

The SELDM lake-basin module uses regional streamflow 

statistics to estimate the surface-water and groundwater flux 

through the lake (Q) in each model year. These streamflow 

statistics are based on data from streamgages downstream 

from areas of groundwater recharge and discharge and 

lakes; thus, streamflow data is assumed to be an acceptable 

surrogate for modeling the total amount of surface water and 

groundwater flowing through a lake on an annual basis. This 

assumption is commonly used for regional hydrologic studies 

(Granato, 2010). For example, regional and national contour 

maps produced by the USGS of average annual runoff (total 

streamflow) provide contours of equal annual streamflows in 

inches, which are equivalent to the long-term average of daily 

mean streamflows (Langbein, 1949; Busby, 1966; Gebert and 

others, 1987; Randall, 1996). The FHWA used streamflow 

estimates that were generalized from a national annual-runoff 

map to estimate the effects of highway runoff on streams 

and lakes (Driscoll and others, 1990a, b). The USEPA also 

has developed runoff-contour maps for use in planning-level 

analyses of water-quality data (Bishop and Church, 1995). 

However, annual flux estimates based on normalized 

streamflow statistics can be inaccurate for areas where 

surface and groundwater divides are substantially different 

if the total contributing area is not included. For example, 

Granato and others (2003) found a large difference in 

streamflows caused by differences in groundwater and 

surface water divides in a simulation of the hydrogeology of 

the Big River Area, Rhode Island. In this study, the surfacewater drainage area of the Mishnock River at Route 3 

was measured as 0.29 mi2, and streamflow measurements 

indicated that the long-term average flow would be about 

14.9 ft3

/s/mi2. However, the simulation indicated that the 

groundwater contributing area to Lake Mishnock was 1.6 mi2. 

The normalized average flow from the Lake Mishnock 

groundwater contributing area was about 2.7 ft3

/s/mi2, which is 

much closer to the average for nearby streamgages.

The SELDM user can specify streamflow statistics based 

on the latitude and longitude of the highway site and regional 

statistics or designate statistics from nearby streamgages on 

the streamflow-statistics input form. User-defined statistics for 

modeling site-specific conditions can also be entered on the 

form. SELDM, however, does not currently include the option 

to model different flows, concentrations, and loads from 

different areas of the lake basin. This approach is consistent 

with much of the available water-quality and flow data, which 

represent the effects of multiple land uses instead of only one 

land use within an upstream drainage area, and is evidenced 

by the large drainage areas of sampling sites associated 

with available water-quality data. For example, Granato and 

others (2009) obtained data from 24,581 USGS surfacewater-quality-monitoring stations in the conterminous United 

States and found that the median upstream area to sites where 

data for total nitrogen, total phosphorus, total hardness, and 

suspended sediment concentrations had been collected was 

about 50 mi2

 and ranged from about 0.01 to 1,000 mi2. Fewer 

than 25 percent of these basins had drainage areas less than 

10 mi2

. Examination of the available surface-water-qualitymonitoring data for pH, nitrate plus nitrite, chloride, total 

copper, total lead, and total zinc showed similar drainage-area 

statistics for each of these constituents. With the exception of 

wilderness areas, rangeland, and farmland, land uses would be 

homogeneous in few large basins.

If, however, the SELDM lake-basin analysis is being 

done to assess the effect of a highway on lake-water quality 

in support of a wider study that includes load estimates for 

different land-use areas, then concentration statistics for the 

entire lake basin can be derived from the loads assigned to the 

other land uses in the basin. For a planning-level analysis, the 

average flow-weighted concentration from all nonhighway 

land uses in the basin is the sum of the annual loads divided 

by the sum of the annual flows from each area. The variability 

of the background water-quality component can be estimated 

by using published values of the COV for different waterquality constituents (for example, Athayde and others, 1983; 

Pitt and Vorhees, 2003). The NURP values for the COV of 

TSS, chemical oxygen demand (COD), total phosphorus, total 

copper, total lead, and total zinc range from 0.85 to 2.92, 0.39 

to 0.78, 0.69 to1.66, 0.81 to 1.32, 0.68 to 1.52, and 0.66 to 

1.07, respectively (Athayde and others, 1983). In the NURP 

study, the least developed land uses were associated with the 

highest COV values, and the COV values for urban land uses 

were similar for most constituents. Once the average and COV 

of flow-weighted concentrations have been selected from the 

literature or from available data, the lognormal statistics to 

be used in SELDM can be calculated by using the standard 

methods (Chow, 1954; Driscoll and others, 1990b; Stedinger 

and others, 1993) described in appendix 1.
Theory and Implementation 71

Under steady-state conditions, the sum of the input loads 

equals the sum of the output loads and the attenuated loads. 

If the attenuation factor is expressed as a first-order decay 

process, then the mass balance equation for the lake can be 

expressed as 

Q Cin in = + Q Cout lake lake lake KC V , (26)

where

Qin is the total of inflows to the lake, in units of 

volume per time;

Cin is the average inflow concentration, which is 

calculated by dividing the sum of the input 

loads by the sum of the input flows, in 

units of mass per volume;

Qout is the total of outflows from the lake, in units 

of volume per time;

K is the attenuation factor, in units of the 

reciprocal of time;

Clake is the average concentration in the lake, in 

units of mass per volume; and

Vlake is the volume of the lake

With the assumption of steady-state lake volume, Qin is 

equal to Qout, which is defined as the flux through the lake 

and denoted as Qflux, equation 26 can be rearranged to solve 

for Clake. To obtain the equation as a function of the mean 

residence time of water in the lake, the right side of the 

equation is divided by the flux through the lake:

 C Q C

Q KV

Q C

Q

Q

Q lake

flux in

flux lake

flux in

flux

flux

flux

= + =











 ÷ +

KV

Q

lake

flux











 , (27)

and finally

, (28)

where

Tw is the mean residence time of water in the 

lake.

Equation 28 is the form of the equation used in many 

implementations of the Vollenweider model (U.S. 

Environmental Protection Agency, 1983, 1985c, 1986a; 

Thomann and Mueller, 1997) and for calculation 

of primary settling of sediment for water- and 

wastewater-treatment analyses.

Attenuation Factors

The water-quality parameters to be modeled in the 

SELDM lake module and the associated attenuation factors 

are selected on the water-quality constituent-pair specification 

form. The attenuation factor is a first-order rate coefficient 

defining the change in concentration with time. For sediment, 

the attenuation factor is calculated by dividing the net 

effective settling velocity (in units of length per time) by 

the mean depth of the lake (in units of length). For other 

constituents, the attenuation factor can be modeled as a 

first-order decay rate, which is the sum of the attenuation 

factors for different processes including sedimentation, 

volatilization, biodegradation, photolysis, and hydrolysis 

(U.S. Environmental Protection Agency, 1985a, b, c; 1986a; 

1987; Thomann and Mueller, 1987). The attenuation factor can 

be expressed as

, (29)

where

KT is the total attenuation factor;

KS is the attenuation factor from particulate 

settling;

KV is the total outflow from the lake, in units of 

volume per time;

KB is the attenuation factor from biodegradation, 

which is the transformation of one 

constituent into others as a result of 

biological metabolic processes;

KP is the attenuation factor from photolysis, 

which is the transformation of one 

constituent into others as a result of solar 

radiation; and

KH is the attenuation factor from hydrolysis, 

which is the transformation of one 

constituent to others as a result of 

interaction with water molecules.

SELDM uses one attenuation factor for each defined 

constituent because the state of knowledge for specifying 

these numbers on an annual basis does not warrant a more 

complex approach. The user can specify multiple constituent 

pairs or rerun the analysis with different attenuation factors 

to do a sensitivity analysis for this variable. If data are 

available, the user also may model different fractions (for 

example, the dissolved or sediment-associated fraction or 

different grain-size fractions) as separate constituents with 

different attenuation factors. The user also may estimate the 

long-term average concentrations and attenuation factors 

for each fraction, calculate the resulting annual average 

concentration for each fraction, and then back calculate the 

long-term-average bulk attenuation factor from the sum of the 

component concentrations.

Nutrients

The published attenuation factors for nutrients are 

commonly accepted because the effects of nutrients on lakes 

have been studied extensively. For example, Thomann and 

Mueller (1987) report net effective settling velocities of 32 

to 41 ft/yr and literature values ranging from about 2 to 

656 ft/yr for total phosphorus. Canfield and Bachmann 
72 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

(1981) report mean net effective settling velocities for total 

phosphorus of 79 ft/yr (with values ranging from -85 to 

148 ft/yr) for 290 natural lakes and 407 ft/yr (with values 

ranging from -951 to 1,610 ft/yr) for 433 artificial lakes. 

Walker (1987) presents data from a literature review indicating 

that the net effective settling velocity for total phosphorus 

in seven urban lakes ranged from 29 to 162 ft/yr. Bachmann 

(1981) reports mean net effective settling velocities for total 

nitrogen of 16 ft/yr (with values ranging from -27 to 

427 ft/yr) for 95 natural lakes and 29 ft/yr (with values 

ranging from -164 to 1,290 ft/yr) for 384 artificial lakes. 

Negative values of the mean net effective settling velocities 

may indicate net production of nutrients, conditions that 

were not steady state during the sampling period, or errors of 

estimation (Bachmann, 1981; Canfield and Bachmann, 1981). 

Researchers relate nutrient attenuation factors to loading 

rates, hydraulic residence time, mean depth, and other factors 

(Bachmann, 1981; Canfield and Bachmann, 1981).

Sediment

Attenuation factors for sediment are based on 

the effective settling velocity of suspended sediments. 

Estimating attenuation factors for suspended sediment is 

important because particulate settling is a large part of the 

total attenuation factor for many water-quality constituents. 

Commonly accepted attenuation factors for suspended 

sediments are not readily available because of large temporal 

and geographic variations in the concentrations and physical 

properties of sediment. Suspended sediment includes all 

particulate matter, including mineral grains, plant matter, small 

aquatic organisms such as bacteria, and trash. The attenuation 

factor used for the lake module is for the sediment that is 

transported to the lake and thus depends on many natural 

and human factors. Sampling, sample handling, and analysis 

methods can have a substantial effect on measured values 

of the concentration, density, and grain size of sediment in 

the water column (Guy, 1977; Edwards and Glysson, 1999; 

Bent and others, 2003). Information from the literature can 

guide initial estimates, but knowledge of local conditions is 

necessary to refine such estimates. For many areas, such data 

are available from the USGS NWISWeb database. Turcios and 

others (2010) cataloged the amount of grain-size data available 

for the United States in the USGS NWISWeb database 

as of January 2000 and found that tens of thousands of 

measurements had been collected at thousands of sites in the 

United States. The water-quality data-mining techniques and 

software developed by Granato and others (2009) can be used 

to obtain data on the concentration and grain size of sediment 

from the USGS NWISWeb database.

Attenuation factors for suspended sediment commonly 

are calculated from measurements of settling velocities of 

equivalent spheres in a quiescent fluid (U.S. Environmental 

Protection Agency, 1987; Thomann and Mueller, 1987). 

Hallermeier (1981) indicates that the settling velocities for 

spheres are a good approximation for the settling velocities of 

sand grains. Gibbs and others (1971) developed an equation 

based on the viscosity and density of water and the density of 

the spheres to calculate the settling velocities of spheres in a 

quiescent fluid (in centimeters per second). This equation is 

useful for the lake analysis because it is valid for the full range 

of sediment sizes, whereas the commonly used Stokes law 

diverges from experimental data under standard conditions for 

grain sizes larger than about 20 microns (a grain size in the 

range from median to coarse silt). When converted to feet per 

year for use in calculating the SELDM attenuation factor, the 

equation is

 , (30)

where

Vs is the settling velocity of a sphere, in feet per year (ft/yr);

η is the dynamic viscosity of the fluid, in poises (0.01002 at 20°C);

g is the acceleration caused by gravity, in centimeters per second per second (980.665 cm/s2);

r is the sphere radius, in centimeters;

Pf is the density of the fluid, in grams per cubic centimeter (0.9997 g/cm3 at 20°C);

Ps is the density of the sediment, in g/cm3; and

CF is the conversion factor from centimeters per second to feet per year (1,035,354 s ft/cm yr).
Theory and Implementation 73

Settling velocities calculated by use of the equation by 

Gibbs and others (1971) for spheres with different grain-size 

diameters and densities are shown in figure 30. The velocities 

range over nine orders of magnitude and are expressed in feet 

per year to facilitate calculation of annual attenuation factors 

with average lake depth from SELDM.

In comparison, Driscoll and others (1986) summarized 

settling velocities measured during NURP in five equal size 

classes by mass as about 263, 2,630, 13,100, 61,400, and 

570,000 ft/yr, but they did not provide information on the 

grain-size or density distributions of these particles. Dorman 

and others (1996) measured settling velocities in highway 

runoff by the same methods used by Driscoll and others 

(1986) and found that highway-runoff sediments had a lower 

proportion of particles in the first four settling-velocity classes 

(about 17 percent of total mass in each class) and a larger 

proportion (about 28 percent of total mass) in the category 

with the highest settling velocity (570,000 ft/yr). Dorman 

and others (1996) also did not provide information on the 

grain-size or density distributions of these particles. Grainsize information is needed, however, to calculate size-specific 

attenuation factors or a single weighted-average attenuation 

factor. The grain-size diameters used to calculate fall velocities 

displayed on figure 30 range five orders of magnitude and are 

shown with corresponding sediment-size classes.

Studies of highway and urban runoff indicate large ranges 

in the masses associated with measured grain sizes in runoff 

sediments from storm to storm and site to site (Sansalone and 

Tribouillard, 1999; Li and others, 2006; Smith and Granato, 

2010). For example, Sansalone and Tribouillard (1999) 

showed that about 0, 10, 60, and 30 percent of the sediment by 

mass in pavement runoff is in the clay, silt, sand, and gravel 

categories, respectively. Li and others (2006) collected 16 

grab samples from 3 sites where runoff exited a drainage pipe; 

they did particle counts and measured the mass of 11 size 

fractions between 2 and 1,000 microns and found that more 

than 90 percent of the particles were less than 10 microns in 

diameter, but this fraction contributed only about 10 percent of 

the mass of particulates in runoff. About 5, 30, and 60 percent 

of the particulates in the samples described by Li and others 

(2006) were in the diameter ranges for clay, silt, and sand, 

respectively. Fowler (2008) studied 13 samples of parkinglot runoff and found that the median diameters for the 10th, 

50th and 90th percentiles were 4, 50, and 1,400 microns, 

respectively. Kim and Sansalone (2008) showed grain-size 

distributions in runoff from seven studies, including their 

own, indicating that grain sizes in noncolloidal fractions 

range from 1 to larger than 24,500 microns with median grain 

sizes ranging from 20 to 800 microns. Kim and Sansalone 

(2008) also indicated that many studies of urban and highway 

runoff show similarities in grain-size distributions. Smith and 

Granato (2010) documented the percentages of the masses for 

three sediment-size fractions—less than 62.5 microns (clay 

and silt), 62.5 to 250 microns (very fine to medium sand), and 

greater than 250 microns (coarse sand and gravel)—in 162 

highway-runoff samples from 15 sites in Massachusetts. They 

found that the average mass percentages for each fraction 

were 61, 15, and 24, respectively. The percentage ranges for 

the three size fractions were 4–99, 0–48, and 0–87 percent, 

respectively. Selbig and Bannerman (2011) provided detailed 

data for the particle-size distribution in 20–90 samples of 

runoff from a roof, three parking lots, a feeder street, a 

connector street, an arterial street, and a mixed land-use 

basin in the area around Madison, Wisconsin. They measured 

the percentages of the mass of particulates in 10 grain-size 

intervals from less than 2 to less than 500 microns.

Grain-size distributions in receiving waters vary 

substantially. Vice and others (1969) studied a basin 

undergoing development in Virginia and found that the 

percentages of clay, silt, and sand and gravel by mass during 

the 1961–64 period were 26, 60, and 14 percent, respectively. 

The percentages of clay, silt, and sand and gravel transported 

in stormflows were 25, 61, and 14 percent, respectively. The 

percentages of the masses of clay, silt, and sand and gravel 

transported in nonstorm flows were 74, 26, and 0 percent, 

respectively, but nonstorm sediment discharges accounted 

for less than 1 percent of the total sediment discharge during 

the study period. Similarly, Horowitz and others (2008) 

intensively monitored the water quality in 10 urban streams 

in Atlanta, Georgia, and found that stormflows accounted 

for more than 65 percent of annual streamflows and more 

than 94 percent of suspended sediment loads. Yorke and 

Herb (1978) measured the particle-size distribution in 

19 medium- to high-flow samples collected during the period 

1960–73 in a basin in Maryland. Sediment transport in this 

basin was being studied because of rapid suburban growth; 

development on about 3 percent of the undeveloped land 

surface each year during the study period resulted in an 

increase in total impervious area from about 4 to 9 percent of 

the watershed area. The mean percentages by mass of clay, 

silt, and sand and gravel measured in this study were 35, 45, 

and 20 percent, respectively. The percentage ranges of the 

clay, silt, and sand and gravel fractions were 11–73, 21–62, 

and 3–38, respectively. Yorke and others (1985) calculated 

loads of sediment in the Schuylkill River in Pennsylvania 

from 1954–79 and reported that suspended sediments 

comprised about 40, 54, and 6 percent of clay, silt, and sand, 

respectively. Slattery and Burt (1997) indicated that about 13, 

65, and 22 percent of suspended sediments in a stream in an 

agricultural area comprised clay, silt, and sand, respectively. 

Holnbeck (2005) measured suspended sediment on the upper 

Yellowstone River in Montana and found that, on average, 

about 51 percent of suspended sediment was silt or clay, and 

the remaining 49 percent was sand; these percentages varied 

with streamflow.

Grain-size distributions in lake sediments indicate the 

characteristics of sediment in flows from contributing areas. 

These grain-size distributions, however, vary substantially 

both spatially and temporally (Gottschalk, 1961; Guy, 1970). 

Gottschalk (1961) indicated that the percentage ranges of the 

masses of clay, silt, and sand in reservoir deposits were 9–85, 

14–78, and 0–77, respectively. The average percentages by 
74 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Fall velocity of a sphere, in feet per year

Sphere diameter, in microns

Specific gravity 1.05Specific gravity 1.25Specific gravity 1.50Specific gravity 2.00Specific gravity 2.50Specific gravity 2.65Specific gravity 3.00Specific gravity 10.0

EXPLANATION

Gravel Sand Silt Clay

0.1

1E81E71E61E51E4

1,000

100

10

1.0

0.1 1E4 1,000 100 10 1.0

Figure 30. The fall velocities of spheres with various specific gravities and diameters in quiescent water at 20° Celsius. Fall velocities were calculated by using the 

equation by Gibbs and others (1971). 1E4 is scientific notation for 10,000.
Theory and Implementation 75

mass in the table presented by Gottschalk (1961) were about 

39, 39, and 22 percent for clay, silt, and sand, respectively. 

Gottschalk (1961) also indicated that grain-size distributions 

can differ by several orders of magnitude in a reservoir with 

a high proportion of sand near the inlets to the reservoir and 

high proportions of silt and clay in other parts of the reservoir. 

The settling velocities of suspended sediments are a 

function of the densities of the water and of the particulates 

(equation 30). The density of water is a function of 

temperature, but the change in density over the temperature 

range common in rivers and streams is almost negligible 

(Lide, 1997). Dissolved and suspended solids also may affect 

the density of solution, but these effects are not usually of 

concern unless concentrations reach 7,000 mg/L for dissolved 

solids (Hem, 1992) and 8,000 mg/L for suspended solids 

(American Society for Testing and Materials, 2009). Figure 30 

includes calculated fall velocities for spheres in quiescent 

water with densities ranging from 1.05 to 10 g/cm3

. 

Reported sediment-density values in the literature range 

considerably, indicating that the fall velocities of runoff 

sediments will range considerably. For example, the densities 

of many natural minerals range from 2 to 3 g/cm3

 (Perry, 

1963). Sansalone and Tribouillard (1999) found that sediment 

densities in pavement runoff ranged from about 2.75 to 

2.9 g/cm3

 over a range of grain sizes from 20 to 5,000 microns. 

Similarly, Li and others (2006) reported densities of sediment 

samples from 1.89 to 2.86 g/cm3

 and differences among grain 

sizes. Lin (2003) examined the density of sediment in samples 

of runoff from Baton Rouge and New Orleans, Louisiana, and 

Little Rock, Arkansas; he found consistent densities of about 

2.5 g/cm3

 for urban and highway sediments in a grit chamber. 

Densities of sediments in a subsequent collection chamber 

increased from about 1.5 to 2.6 g/cm3

 with decreasing grain 

sizes in the range of 9,500–425 microns and then remained 

constant at about 2.6 g/cm3

 for grain sizes down to 25 microns. 

Lin (2003) noted that much of the large size fraction in runoff 

was composed of organic matter such as leaves and other 

plant materials, wood pieces, straw, and tire debris, and noted 

that such materials have densities of about 1.9 g/cm3

. Butler 

and others (1996) found that the densities of organic solids 

in storm sewers are in the range of 1.1 to 2.5 g/cm3

. Edil 

and Bosscher (1994) reported densities of tire shreds in the 

range of 1.02 to 1.36 g/cm3

 depending on the amount of glass 

belting or steel wire in the tire particles. Fowler (2008) found 

that sediment in parking-lot runoff samples had bulk specific 

gravities of 1.86 to 2.96 with a median of 2.04. Asphalt has a 

density of about 1.1 to 1.5 g/cm3

 (Lide, 1997). The densities 

of commercial-grade aluminum, bronze, copper, iron, nickel, 

lead, tin, and zinc are about 2.8, 8.15, 8.94, 7.86, 8.89, 11.34, 

7.35 and 7.14 g/cm3

, respectively (Perry, 1963; Lide, 1997). 

Thus, the metal particulates in runoff will settle more rapidly 

than quartz particles of equivalent size and shape because the 

densities of commercial-grade metals are larger.

Temperature can have a substantial effect on the 

fall velocities of sediments, mainly because of the effect 

of temperature on the viscosity of water. Two factors in 

equation 30, the dynamic viscosity and the density of fluid, 

are functions of temperature. The dynamic viscosity of water 

at 0 °C (0.01793 poise) is about 1.37, 1.8, 2.25, and 2.7 times 

the viscosity of water at 10, 20, 30, and 40 °C, respectively 

(Lide, 1997). The density of water, however, changes by less 

than 1 percent over the range from 0 to 40 °C (Lide, 1997). 

Figure 31 indicates the effects on the fall velocity of spheres 

in quiescent water of changes in dynamic viscosity and water 

density at temperatures ranging from 0 to 30 °C. The smaller 

grain sizes are affected more than the larger sizes, and the 

curves converge toward common values for sediment in the 

range of coarse sands. Fang and Stefan (1999) report the 

minimum, maximum, and average lake temperatures at 209 

locations in the conterminous United States. They provide 

temperatures for these lakes at 1 m below the surface and 1 m 

above the deepest point in the lake. Nationally, latitude and 

elevation are explanatory variables for the range and mean 

values for these temperatures. Fang and Stefan (1999) report 

maximum daily near-surface temperatures in the range of 

19.5 to 32.8 °C, minimum daily near-surface temperatures 

in the range of 0 to 19.6 °C, maximum daily near-bottom 

temperatures in the range of 19.5 to 31.8 °C, and minimum 

daily near-bottom temperatures in the range of 4 to 7.7 °C.

Fall velocities of sediment and solids under natural 

conditions also may be affected by particle shape and by 

oscillations in the water column. Guy (1977) provides data on 

the effect of variations in particle shape and water temperature 

on fall velocities. Carmichael (1982) studied different particle 

shapes and found that settling velocities of nonspherical 

particles were 67 to 97 percent of the settling velocity of an 

equivalent sphere. The largest reductions of about 30 percent 

of the settling velocity were measured for disk-like particles. 

Reductions of 10 to 20 percent were common for cylindrical 

particles falling with the long axis vertical. Velocities became 

maximal as the aspect ratio of the particle approached 1. 

Similarly, Dietrich (1982) studied fall velocities of sand 

particles with different shape factors and found that the fall 

velocities of the natural sand particles used in his study were 

about 68 percent of the fall velocities of equivalent spheres. 

Relations between fall velocities and oscillations in the water 

column are complex, but reductions in velocities can be as 

much as 50 percent of the fall velocity in a quiescent fluid 

(Granato, 1992).

A single attenuation factor is needed for use with the 

SELDM lake module. A size and density-weighted attenuation 

factor can be calculated on the basis of the proportion of 

each sediment class and a representative settling velocity. 

If equation 30 is applied with the fall velocity of each size 

fraction, and the sums of the input concentrations and of the 

estimated lake concentrations are calculated, then an effective 

attenuation factor can be calculated:
76 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Fall velocity of a sphere, in feet per year

Sphere diameter, in microns

EXPLANATION

Specific gravity 2.5, 0 °CSpecific gravity 2.5, 10 °CSpecific gravity 2.5, 20 °CSpecific gravity 2.5, 30 °CSpecific gravity 1.5, 0 °CSpecific gravity 1.5, 10 °CSpecific gravity 1.5, 20 °CSpecific gravity 1.5, 30 °C

Gravel Sand Silt Clay

0.1

1E81E71E61E51E4

1,000

100

10

1.0

0.1 1E4 1,000 100 10 1.0

Figure 31. The fall velocities of spheres in quiescent water at 0, 10, 20, and 30° Celsius with specific gravities of 1.5 and 2.5 grams per cubic centimeter and different 

grain-size ranges. Fall velocities calculated by using the equation by Gibbs and others (1971). 1E4 is scientific notation for 10,000.
Theory and Implementation 77

K

C

C

T Effective

in i

lake i

w

=

− 









 ∑

∑

( )

( )

1

, (31)

where

KEffective is the effective attenuation factor;

Cin(i) is the ith proportion of the input 

concentration;

Clake(i) is the ith proportion of the lake concentration 

calculated for each fraction by using 

equation 28; and

Tw is the mean hydraulic residence time of water 

in years, calculated by using equation 24.

Table 5 shows a hypothetical example that demonstrates 

the method for using equation 31 to calculate a bulk sedimentattenuation factor. The spreadsheet Table5Example.xls 

is provided on the CD–ROM accompanying this report 

to facilitate analysis of attenuation factors. First, list the 

properties of the lake that are needed for use in equations 24, 

28, and 30. Next, list the properties of the sediment that are 

needed for use in equation 30. In this case, we are modeling an 

annual average particle-size distribution. For this example, the 

average grain-size distribution published by Selbig and others 

(2011) for stormwater runoff from a mixed-use area was used 

as the basis for calculation of the inflow-proportion values 

(Cin). The particle diameter is selected to be at or near the 

minimum diameter associated with each fraction so that the 

calculated attenuation factors will be conservatively low. In 

this example, the fall velocities for spheres in quiescent water 

are calculated by using equation 30, and then these values are 

reduced by 30 percent to approximate the potential effects of 

nonideal conditions (non-spherical shapes and water-column 

oscillations). The adjusted fall velocity for each size class 

is multiplied by the mean depth of the lake to calculate the 

attenuation factor. The proportion of each size class remaining 

in the lake (Clake) is calculated by using equation 28. In the first 

example shown in table 5, the sand fractions are effectively 

removed because of the settling velocity, depth, and hydraulic 

residence time of the lake. The inflow and outflow proportions 

are added, and the effective attenuation factor (K) is calculated 

by using equation 31. 

This method could be applied to a less detailed grainsize distribution, and entries could be included for different 

particle densities within each grain-size fraction. Such an 

approach may or may not be warranted based on available data 

and the variability of particle-size distributions and particle 

densities in the inflows to the lake. This user-defined effective 

attenuation factor is used in SELDM with the stochastically 

generated population of annual highway-runoff and lakebasin loads to calculate a population of annual average 

lake concentrations.

Trace Elements

Attenuation factors for trace elements, including the 

highway-associated metals, are dependent on many complex 

and interrelated factors. Trace elements in the aquatic 

environment commonly are described as being in a four-phase 

system in association with dissolved constituents, suspended 

sediment, bed sediment, and biological tissue (Breault and 

Granato, 2003). Partitioning among these fractions depends 

on the physical and chemical characteristics of the water, 

sediments, and biota. Trace elements can move back and 

forth between these phases depending on the geochemistry, 

concentration of sediments, and properties of sediments in the 

solution. The total concentration of a trace element and the 

amount in each phase may be heavily influenced by the sample 

collection, processing, and analysis methods as well as the 

methods used to interpret available data (Breault and Granato, 

2003). The results of sampling and partitioning studies of trace 

elements also may be affected by changes in the geochemistry 

of the solution between collection and processing. If the 

time period for the lake analysis is assumed to be sufficient 

to produce geochemical and biochemical equilibrium with 

respect to annual average concentrations, then a planninglevel attenuation factor can be calculated. An attenuation 

factor can be calculated by estimating the proportion of the 

total concentration of the trace element associated with each 

settleable fraction of particulates.

The dissolved fraction of a trace element commonly is 

operationally defined as the portion associated with the water 

that passes through a 0.45-micron filter. This size fraction 

includes fine clays, colloids, and microscopic organisms; the 

fall velocities of such particulates, however, are negligible 

(figs. 30 and 31). The concentrations of dissolved trace 

elements at geochemical and biochemical equilibrium are 

approximately constant, so the attenuation factor for this 

portion could be modeled as equal to 0.

Highway- and urban-runoff studies have presented a 

wide range of the dissolved and particle-associated fractions 

for each of the highway-associated trace elements (Breault 

and Granato, 2003). For example, Sansalone and Buchberger 

(1997) indicated that the dissolved percentages of Cd, Cu, Ni, 

Pb and Zn in highway runoff were about 54–95, 31–71, 47–78, 

27, and 54–96, respectively. Pitt and Vorhees (2003) indicated 

that the dissolved percentages of Cu, Pb and Zn in urban 

runoff were about 1–86, 2–20, and 1–100, respectively. Li 

and others (2008) summarized the results of different studies 

indicating that the dissolved percentages of Cd, Cu, Ni, Pb and 

Zn in highway runoff had ranges of 0.8–96, 1–71, 9–61, 0–45, 

and 4–96, respectively. 

Studies of natural waters also indicate large potential 

differences in dissolved and particle-associated fractions, 

both spatially and temporally. For example, Shafer and 

others (1999) studied the dissolved and sediment-associated 

fractions of Cd, Cu, Pb, and Zn in 14 rivers in four ecoregions 

in Wisconsin. They determined that the average dissolved 

fractions were about 61, 76, 25, and 35 percent, respectively, 
78 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0 Table 5. Example for calculating the effective sediment-attenuation factor for a hypothetical lake. [The example grain-size distribution is for stormwater runoff from mixed land use (adapted from Selbig and others, 2011). Concentrations are normalized by total concentration. The nonideal settling velocity adjustment is used to account for potential reductions caused by nonsphericity of sediment grains and by water-column oscillations. cm, centimeters; ft, feet; ft3/s/mi2, cubic foot per second per square mile; g, grams; mi, miles; mi2, square miles; r, radius; s, second] Lake water temperature (degrees Celsius) Water density (Pf, in g/cm3

)

Dynamic viscosity 

(η, poises)

Lake surface area 

(SA, in acres)

Mean lake depth 

eq. 25 

(D, in ft)

Basin drainage 

area 

(DA, in mi2

)

Long-term annual 

average flux of water 

(ft3/s/mi2

)

Mean hydraulic 

residence time eq. 24 

(Tw, in years)

10 0.9997 0.01307 25 16 1 1 0.5521

Sediment grain size Initial propor- tion of total Estimated particle diameter (2r, in microns) Sediment density (Ps, in g/cm3

)

Sediment settling 

velocity eq. 30 

(Vs, in ft/year)

Nonideal settlingvelocity adjustment 

30.0%

Sediment-attenuation factor 

eq. 31

(KEffective, in 1/year)

Remaining 

proportion eq. 28 

(normalized Clake)

Medium clay 0.06 1 2 43 30 2 0.02851

Coarse clay 0.07 2 2 173 121 8 0.01292

Very fine silt 0.10 5 2 1,079 755 47 0.00371

Fine silt 0.18 8 2 2,762 1,933 121 0.00265

Medium silt 0.15 14 2 8,451 5,916 370 0.00073

Coarse silt 0.05 32 2 43,926 30,748 1,922 0.00005

Very fine sand 0.09 63 2 167,189 117,032 7,315 0.00002

Fine sand 0.08 125 2 615,975 431,183 26,949 0.00001

Medium sand 0.11 250 2 2,020,946 1,414,662 88,416 0

Coarse sand 0.11 500 2 5,363,586 3,754,510 234,657 0

Sum (normalized Cin): 1.00 Sum (normalized Clake): 0.04860

Effective K eq. 31: 35
Theory and Implementation 79

but found order-of-magnitude differences in the speciation 

of trace elements from site to site. Shafer and others (1999) 

also include datasets from the northeastern United States and 

Texas with dissolved-fraction ranges of about 60–82, 10–18, 

and 20–49 percent for Cu, Pb, and Zn, respectively. Horowitz 

and others (2008) intensively monitored the water quality in 

10 urban streams in Atlanta, Georgia, and found that about 

72, 97, 84, 90, 98, 85, 56, and 85 percent of the annual loads 

of Cd, Cr, Cu, Ni, Pb, total N, total P, and Zn, respectively, 

were associated with suspended sediments. Therefore, about 

28, 3, 16, 10, 2, 15, 44, and 15 percent of the annual loads of 

these elements may be operationally defined as being in the 

dissolved fraction.

The concentrations of sediment-associated trace elements 

commonly are estimated by measuring the concentrations in 

the sediment and multiplying these values by the concentration 

of suspended sediment in the water (Breault and Granato, 

2003). Smith and Granato (2010) demonstrated that measured 

concentrations of trace elements and organic chemicals in 

highway sediments could be used with suspended-sediment 

concentrations in runoff to estimate the total concentrations 

in runoff. Although these estimates were biased low, 

the differences between calculated and measured runoff 

concentrations were consistent with the expected amount 

of the dissolved fraction. This approach could be used with 

the many sources of data documenting the concentrations 

of trace elements in natural and contaminated soils and 

sediments (Shacklette and Boerngen, 1984; Smith, 2006; 

Rice, 1999; Breault and Granato, 2003; Mahler and others, 

2006; Chalmers and others, 2007). For example, Rice (1999) 

documented analysis of trace-element concentrations from 541 

streambed-sediment samples collected from study areas across 

the conterminous United States and found that concentrations 

of trace elements in fine-grained sediments (grain sizes 

less than 63 microns) ranged across three to four orders of 

magnitude with maximum values that were 90 to 4,500 times 

the minimum values. The median concentrations of Cd, Cr, 

Cu, Pb, Ni, and Zn documented in this study were 0.4, 64, 27, 

27, 27, and 110 micrograms per kilogram (μg/kg) of sediment, 

respectively. Rank correlations in this dataset between 

concentrations of these metals in bottom sediments and 

population densities are 0.25, 0.33, 0.49, 0.61, 0.32, and 0.49 

for the same six elements, respectively. Chalmers and others 

(2007) then used these data to develop relationships between 

land-use characteristics and sediment quality.

Information about the masses of trace elements 

associated with different grain-size fractions is useful for 

calculating trace-element attenuation factors because of the 

large differences in settling velocities of different fractions. It 

is generally accepted that the fine fractions of sediments are 

associated with higher concentrations of trace elements than 

are the larger grain-size fractions. Horowitz and Elrick (1987) 

measured metal concentrations and the properties of sediment 

in the grain-size intervals 0 to less than 2, 2 to less than 16, 

16 to less than 63, and 63 to less than 125 microns in samples 

collected from rivers and lakes across the United States. 

They found that correlations with metal concentrations were 

strongest in the fractions representing 16 to less than 63 and 

63 to less than 125 microns and attributed this to the organic 

fraction, surface area, and surface chemistry of these grainsize fractions. Data from Li and others (2006) indicate that 

concentrations of copper, lead, and zinc in the fine fractions 

(clay and silt) of urban runoff were about four times the 

concentrations in the sand fraction, but because of the grainsize distribution, about 20, 30, and 50 percent of the metals 

(by mass) could be associated with the clay, silt, and sand 

fractions, respectively. 

The data in table 6 provides a hypothetical example that 

demonstrates the method for calculating a bulk attenuation 

factor for a total trace-element concentration, which is defined 

as including dissolved and particle-associated fractions. The 

spreadsheet Table6Example.xls is provided on the CD–ROM 

accompanying this report to facilitate analysis of attenuation 

factors. First, an effective sediment attenuation factor is 

calculated by using equation 31. This example is similar to 

the first example in table 5, but only four grain-size fractions 

were calculated for simplicity. The dissolved fraction is 

assumed to be conservative and therefore has an attenuation 

factor of 0. The proportion of the trace element in each 

particle-size fraction remaining in the lake (Clake) is calculated 

by using equation 28. The proportions of the total trace 

element associated with the sand fractions in this example are 

effectively removed because of the settling velocity, depth, 

and hydraulic residence time of the lake. The inflow and 

outflow proportions are added, and the effective attenuation 

factor (Keff) is calculated by adding the dissolved portion to 

the remaining sediment fractions and using equation 31. The 

user-defined effective attenuation factor for the trace element 

can be used in SELDM with the stochastically generated 

population of annual highway and lake-basin loads to calculate 

a population of annual average lake concentrations.

Organic Chemicals

Attenuation factors for organic chemicals 

depend on many factors including particulate settling, 

volatilization, biodegradation, photolysis, and hydrolysis 

(U.S. Environmental Protection Agency, 1985a, b, c, 1986a, 

1987; Thomann and Mueller, 1987). Lopes and Dionne (2003) 

provide a review of the occurrence and distribution of organic 

chemicals that are of primary concern for many highway 

and urban-runoff studies. Smith and Granato (2010) provide 

data on the concentrations of organic chemicals in highway 

runoff, suspended sediment, and plant matter from highways 

in Massachusetts. Many organic chemicals of concern in urban 

and highway runoff are associated primarily with suspended 

particulates. Several regional and national studies provide 

information on the concentrations of many organic chemicals 

in stream and lake-bed sediments in areas of different land 

use throughout the United States with explanatory variables 

for estimating sediment concentrations (Yorke and others, 

1985; Van Metre and others, 2000; Van Metre and Mahler, 
80 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0 Table 6. Example for calculating the effective sediment- and trace-metal-attenuation factors for a hypothetical lake. [The example grain-size distribution is for stormwater runoff from mixed land use (adapted and simplified from Selbig and others, 2011). Concentrations are normalized by total concentration. The nonideal settling velocity adjustment is used to account for potential reductions caused by nonsphericity of sediment grains and by water-column oscillations. cm, centimeters; ft, feet; ft3/s/mi2, cubic foot per second per square mile; g, grams; mi, miles; mi2, square miles; r, radius; s, second] Lake water temperature (degrees Celsius) Water density (Pf, in g/cm3

)

Dynamic viscosity 

(η, poises)

Lake surface 

area 

(SA, in acres)

Mean lake depth 

eq. 25 

(D, in ft)

Basin drainage 

area 

(DA, in mi2

)

Long-term annual 

average flux of water 

(ft3/s/mi2

)

Mean hydraulic 

residence time eq. 24 

(Tw, in years)

10 0.9997 0.01307 25 16 1 1 0.5521

Sediment grain size Initial propor- tion of total Estimated particle diameter (2r, in microns) Sediment density (Ps, in g/cm3

)

Sediment settling 

velocity eq. 30 

(Vs, in ft/year)

Nonideal 

settling-velocity 

adjustment 30.0%

Sediment-attenuation

factor eq. 31

(KEffective, in 1/year)

Remaining 

proportion eq. 28 

(normalized Clake)

Clay 0.26 1 2 43 30 2 0.12356

Silt 0.60 8 2 2,762 1,933 121 0.00885Fine sand 0.08 63 2 167,189 117,032 7,315 0.00002Medium sand 0.06 250 3 2,772,380 1,940,666 121,292 0

Sum (normalized Cin): 1.00 Sum (normalized Clake): 0.13243

Effective sediment K eq. 31: 12

Trace metal fraction Initial propor- tion of total Sedimentation attenuation factor K (1/year) Remaining proportion equation 28

Dissolved 0.2 0 0.20000Clay 0.4 2 0.19009Silt 0.2 121 0.00295Fine sand 0.15 7,315 0.00004Medium sand 0.05 121,292 0.00000Sum (normalized Cin): 1.00 Sum (normalized Clake): 0.39308

Effective trace element K eq. 31: 3
Theory and Implementation 81

2005; Chalmers and others, 2007). A large proportion of 

the total attenuation factor (equation 29) for these organic 

compounds may be associated with sedimentation. The U.S. 

Environmental Protection Agency (1987), however, provides 

guidance and values for estimating general first-order decay 

rates, as well as the solubility, potential volatilization, 

biodegradation, photolysis, and hydrolysis of many organic 

chemicals for calculating a total attenuation factor.

Interpreting the Results of an Analysis

Water-resource managers are concerned about the 

frequency, magnitude, and duration of concentrations and 

loads that may have an adverse effect on the quality of 

receiving waters (Driscoll and others, 1979, 1989; Athayde 

and others, 1983; Di Toro, 1984; Driscoll, Shelley, and 

others, 1989; U.S. Environmental Protection Agency, 

1996b, 2002b, 2007a; Smith and others, 2001; Borsuk and 

others, 2002; Bonta and Cleland, 2003; Gibbons, 2003; 

Novotny, 2004; Elshorbagy and others, 2007; Brouwer and 

De Blois, 2008; Langseth and Brown, 2011). There has 

been a growing awareness that statistical approaches and 

Monte Carlo methods are needed to address these concerns. 

SELDM generates a long record of storm-event and annual 

concentrations, flows, and loads that match input statistics 

so that scientists, engineers, and decisionmakers can assess 

the potential risks of water-quality excursions and the 

potential effects of mitigation measures to reduce those 

risks. SELDM also calculates the relative contribution of 

the highway runoff to such excursions. The Monte Carlo 

methods used by SELDM to generate storm-event statistics 

are necessary for quantitative analysis of risk because simple 

methods are not sufficient to characterize the interplay of 

different distributions for precipitation, prestorm flow, runoff 

coefficients, concentrations, and BMP-performance measures. 

Simpler methods may provide estimates of mean values, but it 

is commonly the extreme events that are, or should be, of most 

interest to scientists, engineers, and decisionmakers.

SELDM produces one documentation file, eight streampackage output files, and one lake-package output file that 

document the inputs to each analysis and provide the results 

of analysis. The numerical outputs are in a tab-delimited 

text format that is readily copied from the output files and 

into spreadsheet programs, graphing packages, statistical 

software, and word-processing tables. This output is designed 

to facilitate post-modeling analysis and presentation of results. 

Details about the format and content of each file are described 

in the section of this manual on model-output files.

The benefit of the Monte Carlo analysis is not to decrease 

uncertainty in the input statistics, but to represent the different 

combinations of the variables that determine potential risks of 

water-quality excursions. Uncertainty in input statistics can be 

expressed as confidence limits for sample statistics, correlation 

coefficients, and regression relations used to estimate such 

statistics (Haan, 1977; Chow and others, 1988; Press and 

others, 1992; Helsel and Hirsch, 2002). Statistical confidence 

limits most commonly are a function of 1 over the square root 

of the number of samples. For example, the standard error of 

the skew coefficient is equal to the square root of the quotient 

of six divided by the number of samples (Press and others, 

1992). If data are lognormal, the real coefficient of skew 

would be equal to 0, but the 95-percent confidence intervals 

for this statistic would be plus or minus 2.15, 1.52, 1.24, and 

1.1 if the sample sizes were 5, 10, 15, and 20, respectively. 

SELDM is designed to generate about 800 to 2,400 runoffproducing storm events based on the average number of storm 

events per year to stabilize the statistics of output results 

(appendix 1). 

The 95-percent confidence limits of the SELDM outputs, 

however, are not more precise than the inputs because the 

outputs are based solely on inputs. For example, the average, 

standard deviation, and skew of the logarithms of total 

phosphorus data measured in 18 highway-runoff samples 

collected at USGS station 423027071291301 along State 

Route 2 in Littleton, Massachusetts, were -1.05, 0.423, and 

-0.679, respectively (Smith and Granato, 2010). SELDM 

results based on these statistics indicate that 0.5 percent 

of storms would have concentrations exceeding 0.6 mg/L. 

However, the theoretical 0.5-percent exceedances calculated 

by the frequency-factor method with the Pearson Type III 

variates would be 0.42, 1.1, and 3.02 for skews of -1.13 (the 

lower 95-percent confidence limit for a sample of 18 values 

from a lognormal distribution), 0 (the expected skew of 

a lognormal distribution), and 1.13 (the upper 95-percent 

confidence limit for a sample of 18 values from a lognormal 

distribution), respectively. 

SELDM does, however, provide a method for rapid 

assessment of information that is otherwise difficult or 

impossible to obtain because it models the interactions among 

hydrologic variables (with different probability distributions) 

that result in a population of values that represent likely longterm outcomes from runoff processes and the potential effects 

of different mitigation measures. SELDM also provides the 

means for rapidly doing sensitivity analyses to determine the 

potential effects of different input assumptions on the risks for 

water-quality excursions.

SELDM produces a population of storm-event and annual 

values to address the questions about the potential frequency, 

magnitude, and duration of water-quality excursions. The 

output represents a collection of random events rather than 

a time series. Each storm that is generated in SELDM is 

identified by sequence number and annual-load accounting 

year. The model generates each storm randomly; there is no 

serial correlation, and the order of storms does not reflect 

seasonal patterns. The annual-load accounting years, which are 

just random collections of events generated with the sum of 

storm interevent times less than or equal to a year, are used to 

generate annual highway flows and loads for TMDL analysis 

and the lake basin analysis.

Probability plots (sometimes called duration curves, 

quantile plots, or percentile plots) provide one of the simpler 
82 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

methods for evaluating and presenting the results of statistical 

analyses (Riggs, 1968; Haan, 1977; Chow and others, 1988; 

Helsel and Hirsch, 2002; U.S. Environmental Protection 

Agency, 2007a). A probability plot is a scatterplot in which 

the magnitudes of data values are graphed with respect to 

a statistical estimate of the probability of occurrence. The 

statistical estimate of the probability of occurrence may 

be expressed as an exceedance probability, a cumulative 

probability, a normal variate, or a return period. 

An exceedance probability indicates the proportion 

or percentage of values that equal or exceed a given value. 

Data are sorted in descending order to graph the values 

with respect to the exceedance probabilities. A cumulative 

probability indicates the proportion or percentage of values 

that are less than or equal to a given value. Data are sorted 

in ascending order to graph the values with respect to the 

cumulative probabilities. Exceedance probabilities are equal to 

1 minus the associated cumulative probability (100 minus the 

associated cumulative probability if percentiles are used). 

The normal variates indicate the distance, measured as 

the number of standard deviations, of each percentile from 

the mean under the assumption that the median equals the 

mean in a standard normal distribution. Return periods are the 

reciprocals of the probability of occurrence. Return periods 

commonly are expressed as a number of years; however, a 

return period is not a literal expectation of the time between 

values of the given magnitude. For example, a three-year 

concentration value has a 33-percent chance of occurring in 

any given year but may be equaled or exceeded many times in 

any given year.

Probability plots may be constructed by using a linear 

axis or a probability axis. A probability axis can be constructed 

for a particular probability distribution so that the probability 

plot of data that fit the selected distribution will fall on a 

straight line (Haan, 1977); however, the normal probability 

distribution is the one most commonly implemented in 

graphing software. If data are lognormally distributed, then 

the plotted logarithms (or untransformed data plotted on a 

logarithmic scale) will form a straight line. Probability axes 

facilitate examination of the extreme percentiles that are of 

interest for hydrologic applications. 

The example problem that was included in the SELDM 

database is used to demonstrate inputs to and outputs from 

the model. This example is based on the highway-runoffmonitoring site described by Gupta and others (1981) on a 

section of I–81 near Harrisburg, Pennsylvania, because the 

highway site and the upstream basin are well documented in 

that report. The site is in ecoregion 67 (Central Appalachian 

Ridges and Valleys) and the mid-Atlantic rain zone (no. 3). 

The area of the highway site is 18 acres, the impervious 

fraction of the area is 0.27, and the drainage length is about 

2,000 ft. The area of the upstream basin is about 0.5 mi2

, the 

impervious fraction of this area is 0.007, and the basin length 

is about 6,500 ft. 

Total phosphorus EMCs (USEPA parameter code 

p00665) were selected to demonstrate the analysis and 

presentation of storm-event outputs and annual outputs from 

SELDM. Total phosphorus EMCs measured in 18 highwayrunoff samples collected at USGS station 423027071291301 

along State Route 2 in Littleton, Mass. (Smith and Granato, 

2010), were used to represent recent (2005–07) runoff quality 

at the I–81 site because the Route 2 and I–81 sites have similar 

highway characteristics. Upstream concentrations at the test 

site were modeled by using the planning-level transport curve 

developed by Granato and others (2009) with data from 

this ecoregion.

Storm-Event Results

This storm-event example is used to demonstrate general 

concepts for developing and interpreting the probability plots 

that can be used to interpret results of a SELDM analysis. 

This example focuses on interpretation of concentration 

outputs from SELDM; similar methods would be employed 

to interpret results for storm-event characteristics, prestorm 

flows, runoff coefficients, stormflows, and loads. Figure 32 

demonstrates construction of probability plots for the 

highway-runoff EMC outputs with three different probability 

axes. Concentration data are plotted on a logarithmic axis 

because the population can be characterized by a log-Pearson 

type III distribution based on the mean, standard deviation, 

and skew of the logarithms of the highway-runoff samples. 

Figure 32A was constructed by using a linear probability axis 

because commonly used spreadsheet programs do not provide 

the option to use a probability axis. Statistical and graphing 

programs, however, do commonly provide this option. A linear 

axis can be used to view most data in the simulated population 

and the range of data, but details of extreme values outside the 

2nd and 98th percentiles can be difficult to distinguish. This 

is because probability plots of exponential, lognormal, and 

log-Pearson type III distributions are S-curves plotted on a 

logarithmic data axis and a linear probability axis and J-curves 

plotted on an arithmetic data axis and a linear probability axis. 

Extreme probability values, which commonly are of greatest 

interest in hydrologic studies, are difficult to distinguish 

because of the short range of extreme values at the ends of the 

linear probability axis.

Figure 32B was constructed by using a probability 

axis with percentile values that have been converted to 

standard normal variates. This method is used to create a 

probability axis in commonly used spreadsheet programs 

that do not provide the option to use a probability axis. The 

simulated concentration data in figure 32B are linearized 

by this transformation. If the simulated data were from a 

lognormal distribution, they would plot on a straight line. The 

logarithms of the concentration data have a slight negative 

skew, so the simulated data plot has a slight curve that is 

concave downward (Haan, 1977; Chow and others, 1988; 

Granato and others, 2010). The Microsoft Excel® spreadsheet 

software has a function named “NORMSINV” that will 

convert the percentiles (scaled from 0 to 1) to the associated 

standard normal variates. If this function is not available, the 
Theory and Implementation 83

Percentage of storms in which the event mean concentration was equaled or exceeded 

(percentiles on a linear axis)

Total phosphorus concentration (parameter code P0665), in milligrams per liter

Percentage of storms in which the event mean concentration was equaled or exceeded 

(percentiles transformed to a standard normal variate axis)

Percentage of storms in which the event mean concentration was equaled or exceeded 

(percentiles on a normal probability axis)

1.0

0.1

0.01

0.001

A

B

C

1.0

0.1

0.01

0.001

1.0

0.1

0.01

0.001

0 10 20 30 40 50 60 70 80 90 100

-4 -3 -2 -1 0 1 2 3 4

0.01 0.50.1 1 2 5 20 30 40 50 60 70 80 90 95 98 99 99.9 99.99 10 99.5

Figure 32. The stochastic population of total phosphorus concentrations calculated with statistics from highway-runoff 

data collected on State Route 2 in Littleton, Massachusetts (Smith and Granato, 2010). The graphs show concentrations 

plotted by A, percentiles on a linear axis, B, percentiles transformed to the associated standard normal variates, and 

C, percentiles on a normal probability axis.
84 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

conversion can be done by a two-step algebraic approximation 

(Abramowitz and Stegun, 1965; Chow and others, 1988). 

First, an intermediate variable w is calculated from the 

probability p (scaled from 0 to1) as

(32)

if p is less than or equal to 0.5 and

(33)

if p is greater than or equal to 0.5. Second, w is used 

to calculate the normal variate Z with the algebraic 

approximation

 . (34)

If p is greater than 0.5, then Z is given a negative sign; the 

maximum error in this approximation is 0.00045 (Abramowitz 

and Stegun, 1965; Chow and others, 1988). 

The standard normal variate axis can be converted to 

a probability axis manually by methods described by Haan 

(1977). The data in figure 32C are plotted on a probability axis 

with percentiles spaced according to the normal distribution. 

Comparison with figure 32B shows that the spacing of 

percentile values is the same. Whereas both plots clearly show 

the extreme values between 0 and 2 percent and between 

98 and 100 percent, the total extent of these ranges along the 

X axis—about 50 percent of the entire length—may give the 

appearance of a greater number of extreme points to a 

casual observer.

The EMCs are plotted in descending order, so the 

graphs indicate the percentage of storms in which any given 

concentration was equaled or exceeded, whereas most 

hydrologic probability plots indicate the percentage of time 

a given concentration, flow or load was equaled or exceeded. 

This is an important distinction if SELDM outputs are to be 

evaluated with respect to water-quality criteria that are based 

on the frequency and duration of water-quality excursions. 

Evaluation of the SELDM outputs indicates that, for one 

Monte Carlo run, the runoff-producing storm-event durations 

for 1,586 storms represented about 4.67 percent of the total 

time during a 30-year period, and highway runoff flowed 

for about 5.29 percent of the time (these percentages would 

be expected to vary from run to run at one site and from site 

to site). Thus, if highway-runoff quality exceeds a waterquality target during 10 percent of storms, this exceedance 

might occupy only about 0.5 percent of the time during a 

30-year period. This analysis reveals a paradoxical effect 

for implementation of BMPs: if a BMP cannot substantially 

reduce the concentration of a constituent but does extend the 

runoff hydrograph to attenuate the adverse effects of peak 

flows, then the BMP may contribute to a higher percentage of 

time-based excursions.

Figure 33 demonstrates construction of a probability 

plot for the same highway-runoff EMC outputs with a returnperiod axis (sometimes called a recurrence-interval axis). The 

return period is defined as the reciprocal of the probability 

of occurrence, commonly in units of years. The longer the 

return period, the lower the probability of recurrence. Returnperiod values, however, must be interpreted with care. For 

example, the 100-year flood is a commonly used statistic for 

hydrologic design and planning. The 100-year flood, however, 

is not the flood that occurs only once in 100 years; it is the 

magnitude of maximum annual streamflow that, statistically, 

is expected to have a 1-percent chance of occurring in any 

given year. SELDM randomly produces the times between 

storm-event midpoints and groups storms into annual-load 

accounting years (defined as total times between storm-event 

midpoints equal to or greater than 1 year, equal to 8,760 hours 

for normal years and 8,784 hours for leap years). By chance, 

all the extreme events generated by SELDM in a Monte 

Recurrence interval, in years

Total phosphorus concentration (parameter code P0665), 

in milligrams per liter

0.01 0.1 1.0 10 100

1.0

0.1

0.01

0.001

Figure 33. The stochastic population of total phosphorus 

concentrations calculated with statistics from highway-runoff 

data collected on State Route 2 in Littleton, Massachusetts 

(Smith and Granato, 2010). The graph shows concentrations in 

runoff from 1,586 simulated storm events over 30 years plotted by 

recurrence intervals on a logarithmic axis.
Theory and Implementation 85

Carlo run could occur within one annual load year. This is an 

unlikely but plausible occurrence in real precipitation records 

and in Monte Carlo simulations, given a long enough period 

of record. Return periods commonly are calculated by using 

the Weibull plotting position formula (Haan, 1977; Chow and 

others, 1988; Helsel and Hirsch, 2002). If the SELDM output 

were time based rather than storm based, then the recurrence 

interval could be calculated as 1/(N+1), where N is the number 

of days or years in the record. SELDM, however, is stormbased, and the percentiles are calculated by using the Cunnane 

plotting position formula because it results in more unbiased 

values for normal percentiles (Helsel and Hirsch, 2002). 

The recurrence intervals are calculated from the Cunnane 

percentiles as 

Y

A P N

N

R

y

c

= + +

+





 





1

02 04

1

( . ) . (35)

if the data are sorted in descending order and

Y

A P N

N

R

y

c

=

− + +

+





 





1

1 02 04

1

( . ) . (36)

if the data are sorted in ascending order, where

YR is the return period for storms, in years;

Pc is the Cunnane plotting position percentile 

(between 0 and 1);

N is the number of storms generated in a given 

run of the Monte Carlo model; and

Ay is the average number of annual-load 

accounting years in the generated record 

(equal to the number of storms N divided 

by the number of annual-load accounting 

years).

Two equations are necessary because return-period 

plots for storm-event concentrations, flows, and loads should 

be constructed so that higher values have longer return 

periods. The concentration data in figure 33 are plotted 

on a logarithmic axis because of the selected probability 

distribution for total phosphorus concentrations. The return 

periods are plotted on a logarithmic axis to expand the data 

over the measurement range. The maximum return period for 

this example is about 30 years, and about 98.1 percent of the 

storms (the bulk of the stochastic data) have return periods 

that are less than 1 year. In this example, on average, the total 

phosphorus EMC that is expected to be equaled or exceeded 

once every 3 years is about 0.6 mg/L.

SELDM does not explicitly provide comparisons to 

water-quality target concentrations because standards may 

differ from state to state or represent different interpretations 

of national standards, may depend on water use and water 

quality, and may change with time. SELDM does, however, 

provide the information necessary to evaluate results 

quantitatively. For example, if a standard for total phosphorus 

in stormwater discharge was 0.5 mg/L, then the output shown 

in the probability plot indicates that this standard would be 

exceeded by only about 1.6 percent of storm EMCs over a 

long period of time (fig. 32C), equivalent to a return-period 

estimate of about 1.2 year (fig. 33). If, however, the standard 

for stormwater discharge was set at 0.1 or 0.01 mg/L, then 

it would be expected that EMCs may exceed these values 

in about 50 percent or 97.5 percent of storms (fig. 32C). 

Return-period values for these percentiles are about 0.04 and 

0.02 years (fig. 33). 

SELDM also is designed to be a tool for evaluating the 

potential effectiveness of mitigation measures for reducing 

water-quality excursions in highway-runoff discharges and 

in receiving waters. SELDM can be run to evaluate the 

quality and quantity of discharge to the receiving water with 

and without use of a BMP. Figure 34 is a probability plot 

showing total phosphorus EMCs in highway runoff and in the 

effluent from a hypothetical BMP. The EMCs in BMP effluent 

were calculated from the highway-runoff EMCs by using 

the trapezoidal-distribution statistics for total phosphorus 

reductions in the Lake Ridge detention pond (fig. 25A). In this 

example, the estimated minimum irreducible concentration of 

total phosphorus was estimated to be 0.005 mg/L. 

In general, the duration curve of the BMP-effluent 

concentrations is shifted downward from the curve for the 

highway-runoff discharges, indicating the effectiveness of 

the BMP for reducing concentrations. The treatment statistics 

for the Lake Ridge detention pond (fig. 25) result in a 

decrease in the maximum concentration from about 0.92 to 

0.23 mg/L (fig. 34). These treatment statistics also indicate 

that a substantial reduction in water-quality excursions 

could be achieved for high allowable effluent-concentration 

limits, but only modest improvements could be made for low 

concentration limits. For example, if the effluent-concentration 

limit was 0.1 mg/L, then the percentage of storms exceeding 

this limit for untreated highway runoff would be about 

50 percent, but the percentage of storms exceeding this limit 

for BMP effluent would only be about 3.8 percent, which is 

a major improvement. If the effluent limit were 0.01 mg/L, 

however, the percentage of storms exceeding this limit for 

untreated highway runoff would be about 97.5 percent, 

but the percentage of storms exceeding this limit for BMP 

effluent would only be about 95.4 percent, which is an almost 

imperceptible improvement. The effect of the minimum 

irreducible concentration of 0.005 mg/L is apparent because 

the BMP-effluent probability plot plateaus at this value, 

even if highway-runoff concentrations are lower. Thus, if the 

effluent concentration limit was 0.004 mg/L, this BMP could 

not achieve the effluent limit in this example.

A problem with probability plots is that it is easy to 

assume that individual storms correspond to one another 

on the probability axis. This would be the case if a constant 

factor were applied to all values. If the average effectiveness 
86 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0 Total phosphorus concentration (parameter code P0665), in milligrams per liter

Percentage of storms in which the event mean concentration was equaled or exceeded 

(percentiles on a normal probability axis)

0.01 0.50.1 1 2 5 20 30 40 50 60 70 80 90 95 98 99 99.9 99.99 10 99.5

1.0

0.1

0.01

0.001

Storm 288

BMP-discharge concentrations

Highway-runoff concentrations

Storm 288

Storm 566

Storm 566

Figure 34. The stochastic population of total phosphorus concentrations in highway runoff and in discharge from a best 

management practice (BMP). The highway-runoff values were calculated with statistics from data collected on State Route 

2 in Littleton, Massachusetts (Smith and Granato, 2010). The BMP-discharge values were calculated by using the trapezoidal 

distribution statistics for the Lake Ridge detention pond (fig. 25) and a minimum irreducible concentration of 0.005 milligrams

per liter.

of the BMP were used, then the BMP-effluent-concentration 

curve would be parallel to the runoff-concentration curve on a 

logarithmic scale because the outlet concentration is a constant 

proportion of the inlet concentration. Use of the stochastic 

treatment variable, however, has the effect of shuffling 

individual storms along the probability axis. The probability 

plots in figure 34, however, reflect patterns in measured BMPperformance data. The lines diverge at low percentiles because 

the negative correlation between the ratio of outflow to inflow 

concentrations and runoff concentrations indicates better 

treatment at higher concentrations. 

The storms identified in figure 34 demonstrate the effects 

of the stochastic treatment ratios. These storms represent the 

maximum differences in plotting-position percentiles between 

the stochastic highway-runoff quality population and the 

stochastic BMP-effluent quality population. Storm 566 in 

this example output had a runoff concentration of about 

0.062 mg/L (with a plotting-position percentile of 69 percent) 

and an effluent ratio of about 0.94, so the BMP-effluent 

concentration for this storm was 0.058 mg/L (with a plottingposition percentile of 19.4 percent). The statistics for storm 

288 include a low effluent ratio of (about 0.11) with a runoff 

concentration of about 0.18 mg/L (with a plotting-position 

percentile of 24.7 percent) and a BMP-effluent concentration 

of 0.021 mg/L (with a plotting-position percentile of 

79.3 percent).

If receiving-water concentrations are the primary 

concern rather than effluent concentrations, then the quality 

of upstream stormflows and highway (or BMP) effluent are 

of interest for evaluating potential mitigation measures. To 

estimate the quality of upstream stormflows, Granato and 

others (2009) developed a two-segment water-quality transport 

curve relating total phosphorus concentrations to streamflows 

for ecoregion 67. They found that total phosphorus 

concentrations varied randomly with a median value of 

0.05 mg/L for streamflows less than 1.5 ft3/s/mi2 and increased 

substantially with increasing flow above this threshold. 

This relation indicates that total phosphorus in the streams 

in ecoregion 67 was mobilized by stormflows (Granato and 

others, 2009). Use of this transport curve to estimate upstream 

EMC values resulted in a population of concentrations that 

ranged from 0.007 to 2.0 mg/L, with a median of 0.104 
Theory and Implementation 87

and an average of 0.156 mg/L (fig. 35A). These upstream 

EMCs exceeded concentrations for both highway runoff 

and BMP effluent. Highway-runoff EMCs ranged from 

0.002 to 0.92 mg/L, with a median of 0.1 and an average 

of 0.13 mg/L (fig. 35B). BMP-effluent EMCs ranged from 

0.005 to 0.23 mg/L, with a median of 0.037 and an average of 

0.041 mg/L (fig. 35B). Although the EMCs for the upstream 

stormflows were generally higher than those for highway 

runoff and BMP effluent in this example, the upstream EMCs 

for an individual storm may be higher, lower, or equal to the 

EMCs for highway runoff or BMP effluent.

The downstream EMCs are a function of the 

concentration and flow of the upstream-stormflow and 

highway-runoff or BMP effluent (fig. 1). The simulated 

dilution factors, which are defined as the highway (or BMP) 

discharge volumes divided by the concurrent downstream 

stormflows, indicate that runoff from the I–81 site commonly 

was a small portion of the downstream flow for most 

storms in this example. Dilution factors ranged from 0.02 to 

61.2 percent, with a median of 7.6 percent and an average of 

9.9 percent. Downstream concentrations ranged from 0.008 

to 1.97 mg/L with a median of 0.11 mg/L and an average 

of 0.16 mg/L if highway runoff was discharged directly to 

the stream (fig. 35C). Downstream concentrations ranged 

from 0.007 to 1.97 mg/L with a median of 0.10 mg/L and 

an average of 0.15 mg/L if the BMP effluent was discharged 

to the stream instead. In this example, the BMP was highly 

effective for reducing high concentrations of total phosphorus 

in highway runoff (fig. 34), but had an almost negligible effect 

on the downstream EMCs for most storms. BMP reductions 

had a small effect on receiving-water concentrations for most 

storms at this site because the background EMCs were large 

and the highway-runoff volumes small in comparison to 

upstream flows. 

In this example, use of the BMP at the I–81 site could 

have a substantial effect on downstream concentrations 

after some storms (fig. 35C). The results for storm 15 were 

atypical in comparison to most storms but indicate how 

random combinations of input variables can have a large 

effect on the outcome. Upstream stormflows for storm 15 

were low (equaled or exceeded 99.5 percent of the time), 

and therefore the upstream EMC, which was calculated by 

using the water-quality transport curve, also was low (about 

0.007 mg/L, which was equaled or exceeded 99.7 percent 

of the time) (fig. 35A). By chance, the highway-runoff 

volume was higher than the median (runoff volumes in about 

43 percent of storms equaled or exceeded the runoff during 

storm 15). The difference between the EMC values for the 

highway runoff and BMP effluent also was large for this 

storm (fig. 35B). The combination of low upstream stormflow 

and EMC, substantial highway stormflows, and the large 

differences among the highway EMC, the BMP EMC, and the 

upstream EMC resulted in a substantial difference between the 

downstream EMCs with and without BMP treatment for storm 

15 (fig. 35C).

Overall, however, the probability plots in figure 35C 

indicate that implementation of this BMP mitigation method 

will not make a substantial difference in water-quality 

excursions of total phosphorus at this site. For example, 

upstream EMCs equaled or exceeded 0.01, 0.1 or 1.0 mg/L 

during about 99.2, 51.6, and 0.349 percent of storm events, 

respectively, (fig. 35A). In downstream runoff, the EMCs that 

resulted from untreated highway-runoff discharge equaled or 

exceeded 0.01, 0.1 or 1.0 mg/L during about 99.8, 53.9, and 

0.343 percent of storm events, respectively (fig. 35C). The 

EMCs that resulted from treated BMP discharge equaled or 

exceeded 0.01, 0.1 or 1.0 mg/L during about 99.5, 48.7, and 

0.339 percent of storm events, respectively. In this example, 

the highway BMP substantially reduced concentrations in 

discharges to the stream, but upstream controls would be 

needed to make substantial changes in stream-water quality. 

The changes in downstream quality caused by BMP treatment 

are so minor that they could be altered by changing the 

Monte Carlo seed values. Changing the seed values would 

shuffle the random combinations of flows, concentrations, 

and treatment efficiencies among the simulated storm events. 

In this example, changes in downstream water quality could 

be altered randomly and systematically by doing a sensitivity 

analysis with different input statistics. 

Annual Results

SELDM produces two files with annual results, the 

annual highway-runoff-loads output file and the lake-basinanalysis output file. The annual highway-runoff-loads output 

file is designed to facilitate analysis of TMDLs for the 

highway site and to record the annual highway contributions 

to the lake-basin analysis. The population of annual runoff 

loads from the highway indicates the potential highway 

contribution and the uncertainty of such estimates. The 

population of annual runoff loads from the BMP discharge 

indicates the potential for reducing the highway loads to meet 

any proposed load allocations. The lake-basin-analysis output 

file is designed to document annual inputs from the highway 

(with or without a BMP) and from the rest of the lake basin. 

SELDM uses the random samples of highway discharges 

and loads with random samples of discharges and loads from 

the entire lake basin to generate a random sample of total 

annual loads from the entire lake basin and average annual 

concentrations in the lake. The lake-basin output is based on 

the highway-runoff results for storm events but calculates 

daily loads from the rest of the lake basin for each day during 

the entire simulation period because a substantial proportion 

of many constituents can be transported during dry-weather 

base flows. 

SELDM does not calculate annual loads or produce 

annual output files for the upstream stormflow or the 

downstream receiving water in stream-basin analyses because 

the storm-event analyses do not include dry periods between 

storms. Also, instead of recording the flows and loads for 

the entire upstream-runoff hydrograph, SELDM calculates 
88 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0 Total phosphorus concentration (parameter code P0665), in milligrams per liter

Percentage of storms in which the event mean concentration was equaled or exceeded 

(percentiles on a normal probability axis)

A.

B.

C.

0.01 0.50.1 1 2 5 20 30 40 50 60 70 80 90 95 98 99 99.9 99.99 10 99.5

1.0

0.1

0.01

0.001

10

1.0

0.1

0.01

0.001

10

1.0

0.1

0.01

0.001

10

Upstream stormflow quality

Highway-runoff and BMP-discharge quality

Downstream stormflow quality

Storm 15

Concentrations with BMP treatment 

Concentrations without BMP treatment 

Concentrations with BMP treatment 

Concentrations without BMP treatment 

Storm 15 (without BMP treatment)

Storm 15 (with BMP treatment)

Storm 15 (without BMP treatment)

Storm 15 (with BMP treatment)

Figure 35. The stochastic population of total phosphorus concentrations in A, upstream stormflow, B, highway runoff and 

discharge from best management practices (BMPs), and C, downstream stormflows generated for the I–81 example problem.
Theory and Implementation 89

the flows and loads concurrent to highway runoff and BMP 

discharges. In some cases, the duration of highway runoff 

or BMP discharge may exceed the duration of upstream 

stormflows. This is more likely if the upstream basin is small, 

the hydrograph extension by the BMP is large, or both. If the 

duration of highway runoff or BMP discharge exceeds the 

duration of upstream stormflows, then SELDM uses the entire 

upstream stormflow and the prestorm base flow that continues 

for the duration of highway runoff or BMP discharge after 

the upstream stormflow has ended. Thus, the sum of annual 

stormflows from the upstream-basin output may not represent 

the total annual stormflows and loads for the upstream and 

downstream sites.

Like the storm-event outputs, the outputs for the highway 

site and lake basin are associated with plotting-position 

percentiles that can be used to assess the magnitudes and 

frequencies of water-quality excursions with or without 

implementation of the BMPs. The stochastic populations 

that are generated by this Monte Carlo model can be used to 

address concerns about the need for probabilistic approaches 

for defining TMDLs (Smith and others, 2001; Borsuk and 

others, 2002; U.S. Environmental Protection Agency, 2002b, 

2007a; Bonta and Cleland, 2003; Novotny, 2004; Elshorbagy 

and others, 2007; Langseth and Brown, 2011). For example, 

figure 36 shows the plotting positions of 30 annual-load 

values for highway runoff and BMP discharge. The range of 

plotting-position percentiles in the annual graphs is smaller 

than the range in the storm-event graphs (for example, fig. 35) 

because the number of years and the rank of each year is 

used to calculate the percentiles rather than the number of 

storm events. In this example, there are 30 annual values and 

1,586 storm-event values.

The load data in figure 36 are plotted on a linear 

axis because annual loads are the sum of loads for all the 

independent storm events during each annual-load accounting 

year. Thus, the annual flows and loads may be better 

approximated by a normal or Pearson Type III distribution 

rather than a lognormal or log-Pearson Type III distribution 

even though the individual storm-event concentrations, flows, 

and loads may follow a lognormal or log-Pearson Type III 

distribution. The normal approximation may be sufficient 

for the loads in figure 36 because the pattern of the data can 

Total phosphorus loads (parameter code P0665), 

in pounds per year

Percentage of years in which the annual load was equaled or exceeded 

(percentiles on a normal probability axis)

1 2 5 10 20 30 40 50 60 70 80 90 95 98 99

8

6

4

0

Annual BMP-discharge loads

Annual highway-runoff loads

Year 1

7

5

3

1

2

Year 1

Year 17

Year 17

Figure 36. The stochastic population of annual total-phosphorus loads in highway runoff and in discharge from a best 

management practice (BMP). The highway-runoff values were calculated with statistics from data collected on State 

Route 2 in Littleton, Massachusetts (Smith and Granato, 2010). The BMP-effluent values were calculated by using the 

trapezoidal distribution statistics for the Lake Ridge detention pond (fig. 25) and a minimum irreducible concentration of 

0.005 milligrams per liter.
90 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

be approximated by a straight line with the arithmetic load 

axis and the probability axis. If the line shows significant 

curvature (substantial nonzero skew), the Pearson Type III 

distribution may be a better approximation for annual loads. 

Although both distributions can produce meaningless negative 

concentrations, flows, and loads, the COV of annual values 

commonly is small. To test the applicability of the distribution, 

multiply -1 by the reciprocal of the COV; the result will be the 

normal or Pearson Type III variate. The probabilities of loads 

less than or equal to 0 can be calculated by using commonly 

available tables of the standard normal or Pearson Type III 

distributions. Granato (2010) provides electronic tables of the 

Pearson Type III variates.

SELDM calculates and prints the plotting-position 

percentiles in the output files, but return periods can be 

calculated from the outputs if necessary (fig. 37). Unlike 

the stormflow results, the annual results from SELDM are 

time based because they account for the total flows and 

loads from runoff-producing events from the highway site 

and the total daily loads from the rest of the lake basin. The 

Cunnane plotting-position formula is used to calculate annual 

percentiles in the output from SELDM because this formula 

results in more unbiased values for normal percentiles (Helsel 

and Hirsch, 2002). If, however, return periods are desired, then 

it is necessary to convert the plotting positions from Cunnane 

to Weibull to produce unbiased return-period values (Haan, 

1977; Chow and others, 1988; Helsel and Hirsch, 2002). 

Calculate the return period of annual values from the Cunnane 

percentiles as

Y

P N

N

R

c

= + +

+





 





1

02 04

1

( . ) . (37)

if the data are sorted in descending order and

Y

P N

N

R

c

=

− + +

+





 





1

1 02 04

1

( . ) . (38)

if the data are sorted in ascending order, where

YR is the return period for annual sums, in years;

Pc is the annual Cunnane plotting-position 

percentile (scaled to the range 0 to 1); and

N is the number of annual-load accounting years 

generated in a given run of the Monte 

Carlo model.

Two equations are necessary because recurrence-interval 

(or return period) plots for annual sums of storm-event 

concentrations, flows, and loads should be constructed so 

that larger values have longer return periods. The recurrence 

intervals are plotted on a logarithmic axis to expand the data 

over the measurement range. The maximum return period for 

this example is about 31 years. In this example, on average, 

the total annual phosphorus loads that are expected to be 

Recurrence interval, in years

Total phosphorus loads (parameter code P0665), 

in pounds per year

1 10 50

1

0

2

3

4

5

6

8

7

Annual highway runoff loads

Annual BMP-discharge loads

Figure 37. The stochastic population of annual totalphosphorus loads calculated with statistics from highway-runoff 

data collected on State Route 2 in Littleton, Massachusetts 

(Smith and Granato, 2010). The values for effluent from best 

management practices (BMPs) were calculated by using the 

trapezoidal distribution statistics for the Lake Ridge detention 

pond (fig. 25) and a minimum irreducible concentration of 0.005 

milligrams per liter. The graph shows loads in highway runoff 

and BMP discharges from a simulated 30-year period plotted by 

recurrence intervals on a logarithmic axis.

equaled or exceeded once every 3 years (an exceedance 

percentile of about 31.8 percent) are about 5.29 pounds (lb) for 

highway runoff and 1.63 lb for BMP effluent; for a recurrence 

interval of 10 years (an exceedance percentile of about 

8.61 percent), the loads are about 6.62 lb for highway runoff 

and 1.86 lb for BMP effluent. Similar plots and analyses can 

be made for the other annual highway-output variables and the 

lake-basin output variables.

Use of the Stochastic Empirical 

Loading and Dilution Model Interface

SELDM was developed as a Microsoft Access® database 

software application to facilitate storage, handling, and use of 

the hydrologic dataset with a simple graphical user interface 

(GUI). The program’s menu-driven GUI uses standard 

Microsoft Visual Basic for Applications® (VBA) interface 

controls to facilitate entry, processing, and output of data. 

Program source code for the analytical techniques is provided 

within SELDM and also in electronic text files accompanying 
Use of the Stochastic Empirical Loading and Dilution Model Interface 91

this report. Program source code that is specific to Microsoft 

Access® and to the SELDM GUI and that is intended for data 

handling is provided in the database. This section of the user’s 

manual focuses on installing the model, use of the GUI, and 

the content and formats of the output files. 

Installing the Model

SELDM was designed and implemented using the 

Microsoft Windows® XP operating system and Microsoft 

Access® 2003. These versions of Windows® and Access®

were used because about 94 percent of the model testers were 

using the XP operating system, and about 44 percent were 

using the 2003 version of Microsoft Office® at the time the 

model was being developed and tested (2010 and 2011). These 

model testers were from the USGS, FHWA, USEPA, and 

from state departments of transportation and state regulatory 

agencies. The model was successfully tested for forward 

compatibility with the Windows Vista® and Windows 7®

operating systems, with Office 2007® and with Office 2010®. 

A compiled version of the model (SELDMv.01.00a.mde) is 

provided as an independent file and as part of an installation 

package on the computer media accompanying this report. 

An uncompiled version of the model (SELDMv.01.00a.mdb) 

is provided on the computer media accompanying this report 

to help document the model code and to help facilitate future 

maintenance of or modifications to the model code. Users may 

port the uncompiled version to newer versions of Access®, but 

before doing so, should test the new uncompiled version to 

ensure that the model-interface code is fully compatible with 

new Access® reference libraries and data-access models. The 

compiled (mde) version will run faster than the uncompiled 

version (mdb) and will be more robust for common use.

The installation package installs the Microsoft Access®

runtime program and a compiled version of the model 

(SELDMv.01.00a.mde). Installing the compiled version 

of the model allows users who do not have a copy of 

Microsoft Access® to run the model and also ensures that 

SELDM has all the proper reference libraries to run the model 

and interface code. Someone having administrative rights 

on the user’s computer must install this software. The folder 

named “Install” on the computer disk accompanying this 

report contains a readme.txt file with installation instructions; 

the installation files are in the same folder. The installation 

package is a standard Microsoft installation wizard that is 

likely to be familiar to the user or to the system administrator. 

Follow the standard choices for software installation to 

minimize potential problems. The model software and its 

support files can be uninstalled by using the standard add 

or remove program wizard found on the control panel for 

Microsoft Windows®. The installation program creates the 

directory C:\Program Files\FHWA\SELDM\ and includes the 

runtime software in the computer’s registry. The installation 

program also creates shortcuts to the compiled version of 

the model.

Establishing the FHWA-SELDM Output Directory

SELDM is designed to write the model output to a 

directory named “FHWA-SELDM” on the root drive of 

the computer. SELDM uses a standard directory to help 

establish traceability of model results. This output directory 

must be distinct from the program-files directories used to 

install SELDM and the accompanying programs that were 

developed to facilitate analysis of hydrologic and waterquality data (Granato, 2006, 2009, 2010; Granato and 

Cazenas, 2009; Granato and others, 2009). Running the 

model without administrator privileges may cause errors if 

permissions to the root drive are locked. If so, the system 

administrator must give the user full control (permission to 

read, write, and modify) of the “FHWA-SELDM” output 

directory because these permissions are needed to run the 

model. The system administrator can create the directory 

manually or can run SELDM to have the model create the 

directory. If administrator privileges are necessary, the system 

administrator must set these permissions for the user after 

SELDM is installed and tested. To test the model, the person 

installing SELDM can quickly click through the hypothetical 

example problem that is preloaded in the model.

Using the Source Code or Modifying the Model

The VBA code in SELDM is open source and may be 

exported and used for other applications designed for private 

study, scholarship, and research (U.S. Copyright Office, 

2000). The VBA subroutines and functions used to implement 

the Monte Carlo methods, to perform statistical operations, 

and to manipulate data may also be used in other programs. 

Some subroutines and functions were developed in VBA from 

other published works that, if used, must be cited. Some of 

these sources may require copyright permission for use in 

commercial software. The rest of the source code for SELDM 

was developed exclusively by the USGS in cooperation with 

the FHWA and therefore is in the public domain. If, however, 

this code is used, the user must cite and annotate it so that any 

modifications the user has made to the code can be identified.

The SELDM user interface and model code manipulate 

input and output data by using the database objects. 

Commonly, data are manipulated by using predefined 

structured query language (SQL) statements that explicitly 

manipulate tables and data fields by name according to 

established database relationships between key fields. 

The design of the database is documented in appendix 3. 

Any database-design changes to a compiled version of the 

model will likely render the model inoperable. Any change 

in the table names, field names, or database relationships 

must be accompanied by changes in the source code of an 

uncompiled version of the model in each place where the 

VBA code interacts with that portion of the database. These 

changes must be extensively tested to ensure that the proper 

working relationships between the database and the code are 
92 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

maintained. Changes to the database design may result in 

unanticipated consequences in the GUI and in model results. 

Thus, the database design should not be modified at all unless 

such modifications are made as part of a comprehensive and 

systematic redevelopment effort. Any code changes should be 

thoroughly documented by comment statements in the code. 

These comments should include information about the author 

of such changes, the date on which changes were made, and 

the reason for making the changes.

Navigating the Graphical User Interface

The SELDM user interface has one or more GUI 

forms that are used to enter four categories of input data, 

which include documentation, site and region information, 

hydrologic statistics, and water-quality data (fig. 38). The 

documentation data include information about the analyst, the 

project, and the analysis. The site and region data include the 

highway-site characteristics, the ecoregions, the upstreambasin characteristics, and, if a lake analysis is selected, 

the lake-basin characteristics. The hydrologic data include 

precipitation, streamflow, and runoff-coefficient statistics. The 

water-quality data include highway-runoff-quality statistics, 

upstream-water-quality statistics, downstream-water-quality 

definitions, and BMP-performance statistics. There also is a 

GUI form for running the model and accessing the distinct set 

of output files. The SELDM interface is designed to populate 

the database with data and statistics for the analysis and to 

specify index variables that are used by the program to query 

the database when SELDM is run. It is necessary to step 

through the input forms (appendix 4) each time an analysis 

is run.

To facilitate use of the GUI, most of the forms display 

a similar layout and use the same controls for entering 

data and the same options for processing them. Individual 

forms that require exceptions to these standards are noted 

where applicable. An example of the standard components 

and appearance of most forms is provided by the Analysis 

Identification Form (fig. 39). The first line of the form, at the 

upper left, is the title that describes the form’s contents. At the 

upper right, an Information button opens a form that provides 

text explaining the purpose and scope of the current form. The 

selection combobox near the top of the form lets users choose 

from previously entered data. Figure 39A displays the title 

of the selection combobox as Select Analysis, and it displays 

Example Analysis (I–81) as the current selection. To select the 

desired data, left-click on the down arrow in the combobox, 

and left-click again on the desired choice. On most forms, the 

default selection for the previously entered data is the last 

data entered. 

The next component on most forms is the optionselection frame (fig. 39), which controls the appearance and 

operation of the form. Many of the forms have the option to 

Select Current, Edit Current, and Enter New data. Some forms 

also have the Copy Current option. The Select Current option 

is the default choice when each form opens. The data-entry 

fields (beneath these options on the form) are disabled and 

locked (as their grayed-out fields indicate): the data are visible, 

but cannot be changed when the form is in the Select Current 

mode (fig. 39A); this prevents the inadvertent modification of 

data. Changing the selection in the combobox will change the 

data in the fields, but the data-entry fields will remain locked 

and disabled. Clicking the Edit Current option unlocks and 

enables the fields (fig. 39B); the color of the field will change 

from gray to white, and you will be able to click in the field 

and modify or enter data. Although the Select Current option 

is the default option in SELDM, the figures showing the GUI 

were made using the Edit Current option selected in figure 

39B to better show the contents of the available fields. If you 

click the Enter New option, this selection unlocks, enables, 

and clears the data-entry fields, and it also disables the 

selection combobox. Some forms have a Copy Current option, 

which unlocks and enables the data-entry fields and the Copy 

button. The Copy Current option can be used create a new 

analysis with the same input values as the original. Copying 

an analysis and modifying selected variables can be used for 

doing a sensitivity analyses or for scenario testing.

The command buttons on the bottom of the form 

(fig. 39) let you finalize data entry and navigate between 

forms. Most forms have an Exit SELDM button to close the 

form and exit the database application. Most forms also have 

an Accept Updates button to check the inputs and save the data 

to the SELDM database. Most forms have a Proceed button 

to go to the next form, and a Go Back button to return to the 

previous form. 

Choosing different options in the option-selection frame 

at the top of the form affects the status of the command 

buttons on the bottom of the form. Clicking the Edit Current 

or Enter New selections in the option-selection frame enables 

the Accept Updates button and disables the Proceed button 

so that you can save changes to data. If Copy is an available 

option clicking on this option will disable the Proceed button 

and enable the Copy button on the bottom of the form. You 

can discard any changes by clicking the Select Current option 

before clicking the Accept Updates or Copy buttons. Clicking 

the Select Current option will enable the Proceed button and 

disable the Accept Updates and the Copy buttons.

On most forms, the data-entry fields are arranged 

on a group of tabbed pages (fig. 39). The top tabbed 

page commonly is for entering introductory information. 

Underlying tabbed pages are for associated data. On the 

Analysis Identification Form there is one tabbed page for 

associated data labeled Analysis Options. The last tabbed page 

on seven of the data-entry forms has a large text field designed 

to let you record explanatory information (for example, citing 

a report as the source of your input data). On the Analysis 

Identification Form, this tab is labeled Analysis Description.

Mandatory fields on all the forms are marked by an 

asterisk. Mandatory data in these fields are necessary for 

populating the interface and running the model. The titles of 
Use of the Stochastic Empirical Loading and Dilution Model Interface 93

Documentation

Hydrology

Analyst Project Analysis

Highway Grid Ecoregion

Upstream Basin Lake Basin

Site and Region

Precipitation Streamflow Runoff Coefficient

Water Quality

Highway Upstream Downstream

BMP

RUN SELDM

Figure 38. The flowchart for entering input data and for running an analysis. The component “Lake Basin” is italicized 

because this form is not used in all analyses. BMP, best management practice.
94 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

A

B

Figure 39. The Analysis Identification Form in the A, select current 

and B, edit current modes showing the basic components of many 

SELDM data input forms.

data-input tabs that contain mandatory input fields also are 

marked by an asterisk. 

Each type of data is assigned a short, meaningful, and 

distinct name that is used in the selection combobox on each 

form. The short name must be of limited length so it can be 

displayed properly in the combobox. The short name should 

be meaningful and distinctive to facilitate selection using the 

combobox. Only primary ASCII characters, including letters 

(uppercase and lowercase), numbers, dashes, underscores, and 

parentheses, are permitted for short names. The short names 

are printed to the output files with the other data.

Most forms have one or more context-sensitive help 

buttons denoted with a question mark “?” (fig. 39). These 

buttons are placed adjacent to related form controls and fields. 

Left-clicking these context-sensitive help buttons with the 

mouse cursor launches an Information Form or a message box 

with relevant information. 

All the forms use two Microsoft® form-control help 

features—the tool-tip text and status-bar text—which provide 

information about each control on the form. These help 

features summarize the content of each data-entry feature. 

Tool-tip text messages appear as popup messages when you 

position the mouse cursor over any control on the form for a 

few seconds. Status-bar text messages appear in the lowerleft corner of the Microsoft Access® application window. The 

content of each of these features is limited to 255 characters. 

The amount of status-bar text that can be displayed, however, 

may be limited by your choice of a Windows font size and 

the way you resize the Access window, because Access can 

display only as much text as will fit in the status bar.

The two methods for navigating within a form are using 

the mouse and using the Tab key. These methods can be used 

in combination. Using the mouse to click from feature to 

feature is the most commonly used method because you can 

click on any control in any order with this method. Using the 

Tab key to step through a form’s controls is useful but less 

flexible because you must follow the predetermined tab order, 

and you are limited to fields within the active tabbed page. 

You must click on the page with the mouse before tabbing 

through the fields on that page. However, the tab key can be 

the most efficient way to step through a series of text fields to 

enter data.

This user’s manual is focused on modeling runoff-quality 

rather than using the SELDM GUI. The SELDM GUI is 

designed to lead you through the flow chart shown in figure 38 

so that you may enter or edit the data and statistics that are 

described in the theory and implementation section of this 

report. Detailed information about each form is described in 

detail in appendix 4. 
Use of the Stochastic Empirical Loading and Dilution Model Interface 95

Model-Output Files

The results of each SELDM analysis are written to 5 to 

10 output files, depending on the options you select during 

the analysis-specification process (table 7). The five output 

files that are created for every model run are the output 

documentation, highway-runoff quality, annual highway 

runoff, precipitation events, and stormflow file. The output 

options you chose on the Analysis Identification Form (fig. 39) 

determine which of the remaining five output files are also 

created. If the Stream Basin or Stream and Lake Basin output 

options are selected, then the prestorm streamflow and dilution 

factor files also are created. If these same two output options 

are selected and, in addition, one or more downstream waterquality pairs are defined by using the water-quality menu 

(appendix 4, fig. 4–20), then the upstream water-quality and 

downstream water-quality output files also are created by 

SELDM. If the Stream and Lake Basin Output or Lake Basin 

Output option is selected (appendix 4, fig. 4–6), and one or 

more downstream water-quality pairs are defined by using 

the water-quality menu (appendix 4, fig. 4–20), then the Lake 

Analysis output file is created when the Lake Basin Analysis 

is run by clicking on the Run The Lake Package button on 

the Run SELDM form (appendix 4, fig. 4–29). The name of 

each output file is the Analysis short name with the associated 

output-file suffix (table 7).

The output files are written as tab-delimited ASCII 

text files in a relational-database (RDB) format that can be 

imported into many software packages (Manis and others, 

1988). The SELDM RDB file format begins with comment 

lines that describe the content of the file, the analysisspecification options, and the output-value columns. A pound 

symbol (#) in the first character location of each line denotes 

these comment lines. The comments are followed by a line of 

variable names for each type of output value in the file. These 

documentation and format lines in these files are essential 

for documenting model outputs in a meaningful and efficient 

format. The remaining output lines in the RDB files contain 

the numerical results of the analysis. The tab-delimited 

format of the numerical-result output tables is designed to 

be copied and pasted into a spreadsheet or graphing package 

for presentation of results or for further analysis. If an output 

file includes more than one block of numerical outputs, 

then each block of numerical outputs will be preceded by 

a block of comment lines that explain the specific outputs. 

Example output files are provided in the directory named 

“ExampleOutput” on the CD–ROM containing this manual.

All the output files that contain numerical simulation 

results have similar heading information. The first line of 

text in all the output files identifies the contents of the file as 

SELDM output and lists the date and time (in hours, minutes, 

and seconds) that the model was run. This date stamp can be 

used with the file-system date stamp to verify that the contents 

of the file are unaltered SELDM outputs. The date stamp is the 

same for all output files except the lake-analysis output file, on 

which the stamp indicates the time that the lake analysis was 

Table 7. Output files from the Stochastic Empirical Loading and Dilution Model (SELDM) associated with the stream-basin, streamand lake-basin, and lake-basin analyses.

[Type 1, analysis with downstream-pair definitions; Type 2, analysis without downstream-pair definitions; X, has the specified type of output; Y, has the 

highway-stormflow output, but not the upstream-stormflow output; --, does not have the specified type of output]

File Stream-basin analysis Stream- and lake-basin 

analysis Lake-basin analysis

Type Suffix Type 1 Type 2 Type 1 Type 2 Type 1 Type 2

Output documentation file Out.txt X X X X X X

Precipitation events -PE.txt X X X X X X

Prestorm streamflow -PS.txt X X X X -- --

Stormflows -SF.txt X X X X Y Y

Dilution factor -DF.txt X X X X -- --

Highway runoff quality -HQ.txt X X X X X X

Annual highway runoff -Annual.txt X X X X X X

Upstream water quality -UQ.txt X -- X -- -- --

Downstream water quality -DQ.txt X -- X -- -- --

Lake analysis -Lake.txt -- -- X -- X --
96 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

run. The second line of text is a title that indicates the content 

of each output file. The next block of text lists the version 

number, the version date, and the citation for the version of 

SELDM being used. The fourth block of text briefly describes 

the file contents and identifies the output-documentation file 

(Out.txt) as the source for further information and disclaimers. 

The fifth block of text briefly describes the annual and storm 

sequences as a random series of values that are generated 

regardless of sequence or seasonality. The sixth block of text 

explains the fact that the plotting position for each variable 

reflects the results of sorting the values of the individual 

variables rather than assigning a plotting position to all the 

values generated for a specific storm. The seventh block of 

text is the analysis information, which includes the short title, 

short description (if it is specified), type of analysis, status of 

analysis, and full description (if it is specified). The remainder 

of the information in these output files is context sensitive, 

because the output depends on the type of information in the 

output file. 

The first numerical column of each storm-event output 

file contains the storm-event sequence number. Individual 

storm events are listed in the order in which they were 

generated by the model and are identified by sequence number. 

SELDM generates each storm randomly; there is no serial 

correlation. The order of storms does not reflect seasonal 

patterns. Flows and loads are not propagated from storm to 

storm. Different highway-runoff constituents generated for 

each storm are not correlated unless a dependent water-quality 

relationship is specified. Different upstream constituents 

generated for each storm are not correlated unless a dependent 

water-quality relationship is specified or the constituents 

are generated from stormflow data by using a water-quality 

transport curve. The storm-event sequence numbers are 

provided to facilitate generation of paired-value scatterplots 

and examination and analysis of the output to check and verify 

the results of model calculations.

The second numerical column of each storm-event output 

file and the first numerical column of the annual highwayrunoff and lake-analysis output files contain the year number. 

The year number denotes an annual-load accounting year 

rather than a historical time series. Annual-load accounting 

years are just random collections of events generated with 

storm durations and inter-event times that sum to a value that 

is less than or equal to one year (8,760 to 8,784 hours).

The remaining output columns consist of paired plottingposition and hydrologic values. The plotting-position values 

are calculated by sorting the associated hydrologic variable 

and calculating the plotting position as a fraction or a 

percent. Options for sorting and calculating plotting positions 

are selected by using the Run SELDM form (appendix 4, 

fig. 4–29). Plotting positions for storm-event values are 

calculated by using the total number of events generated. 

Plotting positions for the annual values are calculated by using 

the total number of annual-load accounting years generated.

Output-Documentation File

The SELDM output-documentation file (Out.txt) 

provides basic information about the analysis and is, in effect, 

an annotated outline for the selected output files created for 

a given analysis (table 7). This Out.txt file lists the name of 

each output file with a description of the type of information 

in each output file. This file has 15 blocks of text documenting 

inputs to the model, standard model disclaimers, and basic 

information about the results of the analysis. This file also 

documents the properties of the BMP selected for highwayrunoff control. If a BMP is selected, the short name for the 

selected BMP is listed with other information; otherwise, the 

text “The BMP information is not defined” will appear in its 

place. The full name and description for the selected BMP will 

be included if they are specified. Depending on user inputs, 

the BMP flow-reduction specifications, hydrograph-extension 

specifications, and water-quality-modification definitions 

will be listed in this output file. Detailed information about 

the BMP performance options is described in the section of 

this report titled “Runoff Modification by Best Management 

Practices (BMPs).” 

Precipitation-Event Output File

The Precipitation-Event output file (-PE.txt) includes 

information about precipitation selections that were input to 

the model and numerical outputs for each storm event in the 

random sample that SELDM generates. This file documents 

the precipitation dataset that was used in the analysis, the 

selection used to calculate statistics, and the calculation 

method. Unless the user-defined option was selected on 

the Synoptic Storm-Event Precipitation Statistics form 

(appendix 4, fig. 4–13), this file also contains the list of hourlyprecipitation data stations that were used to calculate the 

event statistics. The numerical outputs include a table listing 

the interval between storm-event midpoints, the precipitation 

volume of the each generated storm event, the duration of each 

generated storm event, and the paired plotting-position values 

for each variable. The approach for defining and generating 

precipitation events is briefly described in the section of this 

report titled “Storm-Event Characteristics.” Granato (2010) 

provides a detailed discussion of methods for calculating 

precipitation statistics for use with SELDM.

Prestorm-Streamflow Output File

The prestorm-streamflow output file (-PS.txt) includes 

information about the upstream basin and streamflow statistics 

that were input to the model and numerical outputs for the 

upstream prestorm streamflows for each storm event in the 

random sample that SELDM generates. This file documents 

the streamflow dataset that was used in the analysis, the 

selection used to calculate statistics, and the calculation 

method. Unless the user-defined option was selected on 
Use of the Stochastic Empirical Loading and Dilution Model Interface 97

the Streamflow Statistics form (appendix 4, fig. 4–16), this 

file also contains the list of streamgages that were used to 

calculate the flow statistics. The numerical outputs include 

a table of upstream-flow rates for each storm and the paired 

plotting-position values. The prestorm streamflow is used as 

an approximation for base flows during each modeled runoff 

event. The approach for defining and generating prestorm 

streamflow is briefly described in the section of this report 

titled “Prestorm Streamflow Volumes.” Granato (2010) 

provides a detailed discussion of methods for calculating 

streamflow statistics to estimate prestorm streamflow for 

SELDM.

Stormflow Output File

The stormflow output file (-SF.txt) includes information 

about the highway site, the upstream basin, stormflow 

selections, and numerical outputs for the components of the 

stormflow analysis for each storm event in the random sample 

that SELDM generates. This file documents the input runoffcoefficient statistics, BMP hydrograph-extension statistics, and 

BMP flow-reduction statistics that were used in the analysis. 

The components of the stormflow analysis for each modeled 

runoff event include runoff coefficients, storm-event durations, 

and discharges from the highway site and the upstream basin 

for each storm event in the random sample generated by 

SELDM. The stormflow output file contains four tables of 

numerical data that include runoff coefficients, storm-event 

durations, highway-runoff and BMP discharges, upstream 

stormflow, and the paired plotting-position values for each 

variable. The runoff-coefficient table has runoff coefficients 

for the highway and the upstream basin. The storm-eventduration table has durations for highway runoff, BMP 

discharge, and the total upstream streamflow for a storm event. 

The highway-runoff and BMP-discharge table has the volume 

of runoff from the highway and the volume discharged from 

the BMP; these values may be equal if there is no BMP, or if 

the selected BMP does not produce discernible reductions in 

flow. The upstream-stormflow table has the upstream runoff, 

the total upstream flow, the total upstream flow concurrent 

to the highway-runoff duration, and the total upstream flow 

concurrent to the BMP-discharge duration. The total upstream 

flow concurrent to the highway-runoff duration and the total 

upstream flow concurrent to the BMP-discharge duration may 

be equal to the highway-runoff duration if there is no BMP, 

or if the selected BMP does not produce discernible extensions 

to the highway-runoff durations. The approach for defining 

and generating stormflows from the highway site and upstream 

basin are briefly described in the section of this report titled 

“Stormflow.” Granato (2010) provides a detailed discussion 

of methods for calculating stormflow statistics for SELDM. 

The approach for defining and generating flow-modification 

statistics is briefly described in the section of this report 

titled “Runoff Modification by Best Management 

Practices (BMPs).”

Dilution-Factor Output File

The dilution-factor output file (-DF.txt) includes 

information about the highway site, the upstream basin, 

and numerical outputs for the dilution factor for each storm 

event in the random sample that SELDM generates. The 

dilution factor is the total highway discharge divided by the 

concurrent downstream stormflow for each storm event in the 

random sample generated by SELDM. The dilution-factor 

output provides a simple assessment of the potential for 

adverse effects from highway runoff. A dilution factor of 1 

indicates that all of the downstream flow is highway runoff, 

and a dilution factor near 0 indicates that highway runoff is 

a negligible portion of the downstream flow. The numerical 

outputs include the highway-runoff and BMP-discharge 

dilution factors for each storm event and the paired plottingposition values for each variable. The highway runoff and 

BMP discharge dilution factors will be equal if there is no 

BMP, or if the selected BMP does not produce discernible flow 

reductions. More detailed information about the calculation 

and interpretation of dilution factors is described in the 

section of this report titled “Dilution Factors.” SELDM does 

not calculate the dilution factors at the end of a mixing zone 

downstream of the highway outfall, but it can be used to 

estimate these factors if the upstream-basin area is adjusted. 

Detailed information about this downstream flow-accretion 

analysis method is described in the section of this report 

titled “Mixing.”

Highway-Runoff-Quality Output File

The highway-runoff-quality output file (-HQ.txt) 

includes information about the highway site, input runoffquality specifications, and numerical outputs for each selected 

highway-runoff constituent for each storm event in the 

random sample generated by SELDM. Each highway-runoffquality output table is preceded by constituent-definition 

information including the short name of the constituent, the 

USEPA parameter code (PCode), the parameter name (which 

includes the water-quality units), the method for generating 

the output, the random or dependent water-quality statistics, 

and the water-quality abstract (if this information is entered). 

The runoff-quality output table for each selected constituent 

includes the event mean concentration of highway runoff, 

the total highway-runoff load, the event mean concentration 

of discharge to the stream, the total runoff load discharged to 

the stream, and the paired plotting-position values for each 

variable. The highway-runoff and discharge concentrations 

and loads will be equal if there is no BMP, or if the selected 

BMP does not produce discernible reductions in constituent 

concentrations. The approach for stochastic generation of 

highway-runoff-quality constituents is described in the section 

of this report titled “Highway and Upstream Stormwater 

Concentrations and Loads.” Granato and Cazenas (2009) 

provide a detailed discussion of methods for calculating the 

input highway-runoff-quality statistics.
98 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

Annual Highway-Runoff Output File

The annual highway-runoff quality output file 

(-Annual.txt) includes input information about the highway 

site, input runoff-quality specifications, and annual numerical 

outputs generated by SELDM for each annual-load accounting 

year. These annual numerical outputs include total annual 

precipitation volumes, flows and loads of highway runoff, 

and flows and loads of BMP discharge for each selected 

constituent. Output values and the associated plotting-position 

values are produced for each annual-load accounting year in 

the random sample generated by SELDM. Each highwayrunoff-quality output table is preceded by constituentdefinition information including the short name of the 

constituent, the PCode, the parameter name (which includes 

the water-quality units), the method for generating the output, 

the random or dependent water-quality statistics, and the 

water-quality abstract (if this information is entered). This file 

also contains the total annual highway-runoff flow and BMP 

discharge in volumetric units (cubic feet or cubic meters) 

and in normalized units (in watershed inches or watershed 

centimeters for the highway drainage area) for each selected 

runoff-quality constituent. The annual highway-runoff-quality 

output file also contains the total annual highway-runoff and 

BMP-discharge loads for each selected constituent. These 

flows and loads will be equal if there is no BMP, or if the 

selected BMP does not produce discernible reductions in 

runoff flows or constituent concentrations. The annual flows 

and loads for each annual-load accounting year are the sums 

of each storm-event value assigned to that year, but the annual 

flows and loads may not equal the sum of individual values 

in the highway-runoff-quality output file because of rounding 

to the specified number of significant digits. More detailed 

information about the calculation and interpretation of annual 

flows and loads is described in the section of this report titled 

“Interpreting the Results of an Analysis.”

Upstream Runoff-Quality Output File

The upstream runoff-quality output file (-UQ.txt) includes 

information about the upstream basin, input stormflowquality specifications, and numerical outputs for each selected 

upstream water-quality constituent for each storm event in the 

random sample generated by SELDM. Each upstream runoffquality output table is preceded by constituent-definition 

information including the short name of the constituent; the 

PCode; the parameter name (which includes the water-quality 

units); the method for generating the output; the random, 

dependent, or transport-curve statistics; and the waterquality abstract (if this information is entered). The upstream 

runoff-quality output table for each selected constituent 

includes the event mean concentrations of upstream runoff 

and loads concurrent to the highway runoff, the event mean 

concentrations of upstream runoff and loads concurrent to 

the highway BMP discharge, and the paired plotting-position 

values for each variable. These upstream-runoff concentrations 

and loads will be equal if there is no BMP, or if the selected 

BMP does not produce discernible reductions in flows and 

in constituent concentrations. The approach for stochastic 

generation of upstream runoff-quality constituents is described 

in the section of this report titled “Highway and Upstream 

Stormwater Concentrations and Loads.” Granato and 

others (2009) provide a detailed discussion of methods for 

calculating the input upstream stormflow-quality statistics.

Downstream Runoff-Quality Output File

The downstream runoff-quality output file (-DQ.txt) 

includes information about the highway site, the upstream 

basin, input stormflow-quality specifications, and numerical 

outputs for each selected downstream water-quality pair for 

each storm event in the random sample generated by SELDM. 

Each downstream-runoff-quality output table is preceded by 

constituent-definition information including the short name 

of the constituent; the PCode; the parameter name (which 

includes the water-quality units); the method for generating the 

output; the random, dependent, or transport-curve statistics; 

and the water-quality abstract (if this information is entered). 

This information is provided for both the highway-runoff 

and the upstream-basin constituents that are used to define 

a pair. The downstream runoff-quality output table includes 

the fully mixed downstream event mean concentration, the 

associated adverse-effect concentration, the downstream 

stormflow, the downstream load for each selected constituent 

pair, and the plotting-position values for each variable. These 

concentrations and loads are calculated for the storm volumes 

that are concurrent to the highway runoff or to the BMP 

discharge as defined for the selected water-quality pair. If 

the option for using BMP flow and water-quality treatment 

is selected when a downstream pair is defined (appendix 4, 

fig. 4–24), then the downstream water-quality variables will 

be labeled “concurrent to the highway BMP discharge.” If 

the BMP option is not selected, the downstream water-quality 

variables will be labeled “concurrent to the highway runoff.” 

The adverse-effect concentrations will equal the downstream 

concentration if the Adverse-Effect Concentration Statistics 

option is not selected (appendix 4, fig. 4–24). The approach 

for stochastic generation of downstream water-quality pairs, 

including the adverse-effect concentration, is described in 

the section of this report titled “Downstream Stormwater 

Concentrations and Loads.”

Lake-Analysis Output File

The lake-analysis output file (-Lake.txt) includes 

information about the highway site, the lake basin, input 

lake-analysis specifications, and annual numerical outputs 

generated by SELDM for each annual-load accounting year. 

These annual numerical outputs include total annual flows, 

total annual loads, the annual detention time, the average 
Summary 99

annual lake concentration and the associated plotting-position 

values for each output value. The annual flows and loads are 

calculated for the highway site and the whole lake basin with 

and without the highway-site contributions. Each lake-analysis 

output table is preceded by constituent-definition information, 

including the short name of the constituent, the PCode, the 

parameter name (which includes the water-quality units), the 

method for generating the output, the random or dependent 

water-quality statistics, and the water-quality abstract (if this 

information is entered). The lake-analysis output table for each 

selected constituent includes the flows, loads, concentrations, 

and lake detention times associated with that constituent 

definition. The annual flows and loads from the highway for 

each annual-load accounting year are the sums of the stormevent values assigned to that year, but the annual flows and 

loads may not equal the sums of individual values in the 

highway-runoff quality output files because of rounding to the 

specified number of significant digits. The annual flows and 

loads from the lake basin for each annual-load accounting year 

are the sums of the 365 or 366 daily flows and loads generated 

for the lake-basin analysis. Both dry-weather and storm-event 

flows and loads from the lake basin are included because dryweather flows and loads may represent a substantial proportion 

of the annual totals. More detailed information about the 

calculation and interpretation of annual lake-basin flows and 

loads is described in the section of this report titled “LakeBasin Analysis.”

Each variable in the output file is associated with a 

plotting-position variable from the stochastic analysis that 

indicates the probability of exceedance of that variable. 

As with the main SELDM module, the user can specify 

whether the plotting positions are expressed as fractions or as 

percentiles and are assigned to output values in ascending or 

descending order. The sorting order affects the interpretation 

of the plotting positions. If an output variable is sorted 

in ascending order, which is the default, then the plotting 

position indicates the magnitude of values less than or equal 

to the value associated with the selected plotting position. For 

example, if the 20th-percentile value (or a plotting position 

of 0.2) is equal to 10, then there is a 20-percent chance that 

the average annual value will be less than or equal to 10 and 

an 80-percent chance that it will be greater than 10 for any 

given year. If the output variable is sorted in ascending order, 

a value of 10 would be assigned to the 80th percentile (or a 

plotting position of 0.8). The SELDM lake module does not 

output target concentrations or loads because such values 

commonly are context sensitive, or they are established for 

a political jurisdiction. The plotting-position output can be 

used to assess the risk for exceeding any target value. The 

SELDM lake-basin module can be run to simulate the effects 

of different mitigation measures, and the plotting positions can 

be used to estimate the potential effects of such measures on 

the probability of exceeding target values.

Summary

The Stochastic Empirical Loading and Dilution Model 

(SELDM) is a tool that can be used to transform disparate and 

complex scientific data into meaningful information about 

the risk for adverse effects of runoff on receiving waters, 

the potential need for mitigation measures, and the potential 

effectiveness of such management measures for reducing 

these risks. The U.S. Geological Survey developed SELDM 

in cooperation with the Federal Highway Administration 

to generate planning-level estimates of event mean 

concentrations, flows, and loads. This information is needed 

for highway engineers, regulators, and decisionmakers to 

evaluate the potential effects of highway runoff on receiving 

waters and, if necessary, to help guide the choice of potential 

mitigation strategies. SELDM uses information and data about 

a highway site, a receiving-water basin, precipitation events, 

stormflow, water quality, and the performance of mitigation 

measures to produce a stochastic population of runoff-quality 

variables. SELDM provides input statistics for precipitation, 

prestorm flow, runoff coefficients, and selected water-quality 

constituents derived from National datasets. These input 

statistics may be selected on the basis of the latitude and 

longitude of the site of interest and the characteristics of the 

highway site and the upstream basin. The user also may derive 

and input statistics that are specific to a given site or a given 

area for each variable. 

SELDM is a stochastic model because it uses Monte 

Carlo methods to produce the random combinations of 

input-variable values needed to generate the stochastic 

population of values for each component variable. SELDM 

uses concentrations, flows, and loads from a highway site 

and a receiving-water basin to calculate the dilution of runoff 

in the receiving waters and the resulting downstream event 

mean concentrations and annual lake concentrations. Results 

are ranked, and plotting positions are calculated, to indicate 

the risk that runoff concentrations, flows, and loads may 

cause adverse effects on receiving waters by storm and by 

year. Unlike deterministic hydrologic models, SELDM is not 

calibrated by changing values of input variables to match a 

historical record of values. Instead, input values for SELDM 

are based on site characteristics and representative statistics 

for each hydrologic variable. Thus, SELDM is an empirical 

model based on data and statistics rather than theoretical 

physiochemical equations.

SELDM is a lumped parameter model because the 

highway site, the upstream basin, and the lake basin each 

are represented as single homogenous units. Each of these 

source areas is represented by average basin properties, and 

results from SELDM are calculated as point estimates at 

the site of interest. For example, highway-runoff results are 

produced for the highway site and the outlet of a BMP, if a 

BMP is specified. Upstream-basin results are produced for the 

point at which the highway runoff enters the stream. Average 

annual lake-basin results are produced for the entire (well 
100 Stochastic Empirical Loading and Dilution Model (SELDM) Version 1.0.0

mixed) lake. Use of the lumped parameter approach provides 

a number of advantages over distributed modeling approaches. 

First, use of the lumped parameter approach facilitates rapid 

specification of model parameters to develop planning-level 

models. Second, this approach also is representative of the 

detail available in most available datasets. For example, 

available datasets for highway runoff and BMP performance 

commonly are limited to a small number of representative 

sites. Similarly, watershed studies for stream- or lake-quality 

monitoring commonly are based on data collected at sites 

that represent multiple land covers and different tributaries 

in a single drainage basin. Third, this approach allows for 

parsimony in the required inputs to and outputs from the 

model. Fourth, this approach allows flexibility in the use of 

SELDM. For example, because the highway site is defined 

by a few hydraulic parameters, the site definition also can be 

used to model runoff from various land covers by using the 

appropriate impervious fraction and representative runoff 

concentrations. In addition, the lumped parameter highwaysite definition can be used to form conceptual runoff models 

with a unit drainage area and an impervious fraction to 

calculate annual loads for different highway sites or land 

covers. For example, if data for constituent concentrations 

representing runoff from different road classes or land covers 

are generated for a generic one-acre site, then the estimated 

annual loads per unit area can be used with a geographic 

information system (GIS) program to estimate total loads from 

the sum of different contributing areas in a watershed.

This report is a user’s manual for the model. It provides 

information about the theory and implementation of the model 

(including an appendix describing Monte Carlo methods), 

deriving model inputs (including an appendix describing the 

basin properties needed to characterize the highway site and 

the upstream basin), and interpreting model outputs. It also 

provides a detailed discussion of the graphical user interface 

and the format of output files.

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