United States
Environmental Protection
Agency
EPA/600/R-08/065
May 2008
Development of Duration-Curve
Based Methods for Quantifying
Variability and Change in
Watershed Hydrology and
Water Quality
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                                                   EPA/600/R-08/065
                                                          May 2008
Development of Duration-Curve Based Methods for
 Quantifying Variability and Change in Watershed
             Hydrology and Water Quality
                       Matthew A. Morrison
                        Research Chemist
                U.S. Environmental Protection Agency
                 Office of Research and Development
             National Risk Management Research Laboratory
                      Cincinnati, Ohio 45268

                         James V. Bonta
                    Research Hydraulic Engineer
                USDA - Agricultural Research Service
              North Appalachian Experimental Watershed
                      Coshocton, Ohio 43812
                         Project Officer
                       Matthew A. Morrison
            Land Remediation and Pollution Control Division
            National Risk Management Research Laboratory
                      Cincinnati, Ohio 45268

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Foreword
The U.S. Environmental Protection Agency (EPA) is charged by Congress with protecting
the Nation's land, air, and water resources. Under a mandate of national environmental laws,
the Agency strives to formulate and implement actions leading to a compatible balance
between human activities and the ability of natural systems to support and nurture life. To
meet this mandate, EPA's research program is providing data and technical support for
solving environmental problems today and building a science knowledge base necessary to
manage our ecological resources wisely, understand how pollutants affect our health, and
prevent or reduce environmental risks in the future.

The National Risk Management Research Laboratory (NRMRL) is the Agency's center for
investigation of technological and management approaches for preventing and reducing risks
from pollution that threaten human health and the environment. The focus of the Laboratory's
research program is on methods and their cost-effectiveness for prevention and control of
pollution to air, land, water, and subsurface resources; protection of water quality in public
water systems; remediation of contaminated sites, sediments and ground water; prevention
and control of indoor air pollution; and restoration of ecosystems. NRMRL collaborates with
both public and private sector partners to foster technologies that reduce the cost of
compliance and to anticipate  emerging problems. NRMRL's research provides solutions to
environmental problems by: developing and promoting technologies that protect and improve
the environment; advancing scientific and  engineering information to support regulatory and
policy decisions; and providing the technical support and information transfer to ensure
implementation of environmental regulations and strategies at the national, state, and
community levels.

This publication has been produced as part of the Laboratory's strategic long-term research
plan. It is published and made available by EPA's Office of Research and Development to
assist the user community and to link researchers with their clients.

                                  Sally Gutierrez, Director
                                  National Risk Management Research Laboratory

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                                     Abstract

During the past decades, U.S. Environmental Protection Agency (EPA), U.S. Department of
Agriculture (USDA) and other Federal program administrative and regulatory agencies spent
considerable amounts of time and money to manage risks to surface waters associated with
agricultural activities,  urbanization and other avenues of nonpoint source pollution. A variety
of best management  practices (BMPs)  exist for this  purpose  and  have been  installed
throughout the country, yet very little is known about their overall effectiveness in reducing
stressors at the watershed scale. The  objective of this research is to  explore and  develop
uniform methods for simple quantification of hydrology and water quality data, focusing on
watersheds containing agricultural BMPs.  A significant motivation for the research  is to
provide tools that can be  used to identify and  quantify the  major  factors that  connect
watershed hydrology  and water quality (such as climate,  soil type, slope, land use). These
connecting factors are important for evaluating the effectiveness  of agricultural and other
BMPs, because they often  determine stream and  stressor management decisions. Research
methods must take into account natural variability and uncertainty in watershed response to
BMP installation and precipitation events. The research project documented in this report is a
collaborative effort,  funded through  an Interagency Agreement, between U.S.  EPA's
National Risk  Management Laboratory and  USDA's  North  Appalachian Experimental
Watershed (NAEW)  in  Coshocton, OH.  Project objectives were achieved  through  an
examination of historical data collected at  the NAEW, with examinations of other related
databases.  As  a result  of this  research, methods  were  developed  to  quantify  BMP
effectiveness, and to  understand how natural systems respond  to watershed changes  over
time. The  research will  benefit  states and other stakeholders faced with assessing the
performance and effectiveness of BMPs within a watershed management framework.

Keywords: Best management practices, BMP, agriculture, hydrology, water quality, duration
curves, effectiveness
Notice
The U.S. Environmental Protection Agency through its Office of Research and Development
collaborated in the research described here. It has been subjected to the Agency's review and
has been approved for publication as an EPA document.
                                                                                   in

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Table of Contents

Acknowledgements	ix
Introduction	1
  Objectives and Scope	1
  Duration Curves	2
Approach and Data Sources	3
  Approach	3
  Data Sources	3
Constructing Duration Curves: Types and Examples	4
  Flow Duration Curves (FDCs)	4
  Relationships between Concentration and Discharge (regression equations)	6
  Concentration and Load Rate Duration Curves (CDCs andLDCs)	8
  Assumptions and Limitations	10
  Concepts for Using Duration Curves to Quantify Changes in Watershed Condition	10
Minimum  Number of  Stream-Flow Samples Necessary for the Construction of Duration
Curves	13
  Introduction	13
  Data	13
  Approach to Regression Equations for WE3 8	13
  Effects of Sample Size Based on Regression Analysis	14
  Effects of Sample Size on CDCs and LDCs	15
  Summary and Conclusions Regarding Data Set Sample Size	17
Quantifying Uncertainty in the use of Flow Averaging, and Daily vs. Instantaneous Flows. 19
  Introduction to the Temporal Component of Flow Data	19
  Method for Data Averaging	19
  Effects of Averaging Time on CDCs andLDCs	20
  Recommendations Based on the Effects of Averaging Time	21
Case  Study:  Evaluating  Monthly,  Seasonal  and  Annual  Period Changes in  Nitrate
Concentration	24
  Objective and Approach	24
  Scenarios and Periods of Record	24
  Seasonal Distribution of Discharge and MVN Data for WE38	24
  Concentration-Flow Rate Regressions (C-Q)	24
  Monthly Duration Curves (Scenario 7)	25
    Flow duration curves	25
                                                                               IV

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    Concentration-flow regressions	27
    CDCs for monthly data	30
    LDCs for monthly data	32
  Seasonal Duration Curves (Scenario 8)	32
    Determining seasons based on hydrology and water quality data for WE38	32
    FDCs for seasonal data	33
    C-Q regressions for seasonal data	34
    CDCs for seasonal data	35
    LDCs for seasonal data	37
  Apparent Changes in Annual Concentration-Discharge Records (Scenario 9)	39
    C-Q regressions for annual periods	40
    CDCs for annual periods	41
    LDCs and an illustration of quantifying changes in watershed conditions	44
  Case Study Conclusions	44
Report Conclusion and Summary: The Utility of Duration Curve-Based Methods	45
References	46

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Figures
Figure  1.  Example  showing the  relationship  between  flow duration curves  (FDC),
     concentration duration curves (CDC) and load duration curves (LDC). Each is plotted as
     a percent of time that  the  condition is exceeded, yielding a graph with high flow
     conditions to the left (low percentile, less frequent occurrence)	3
Figure 2.   Hydrograph  for Watershed  174,  NAEW, Coshocton, OH.  Figure shows how
     duration curves can be constructed with varying flow-rate steps, and  how average daily
     flow rate misrepresents watershed hydrology. (Bonta and Cleland, 2003)	5
Figure 3.  Variability in annual flow duration curves compared with the  composite 40-year
     FDC for Watershed 174, NAEW, Coshocton, OH (Bonta and Cleland, 2003)	6
Figure 4.   Three basic relationships  between discharge and the concentration of a water
     quality constituent	7
Figure 5.   Plot of SO42" concentration  data versus discharge for watershed WE38  (years
     1990-1995). Regression line with 95%  confidence intervals follows the basic  power
     equation C=aQb with values: a=11.97 and b=0.113	8
Figure 6. Flow and concentration duration curves (FDC and CDCs) for SO42" concentration
     data for watershed WE38, years  1990-1995.  CDCs are  shown  for raw  data and
     regression equation based on the positive correlation between SO42" and flow	9
Figure 7. Flow and load duration curves (FDC and LDCs)  for SO42" concentration data for
     watershed  WE38, years 1990-1995. LDCs are shown  for raw  data and regression
     equation based on the positive correlation between SO42" and flow	10
Figure 8. Conceptual depiction of using duration curves to quantify  changes in  water quality
     following implementation of management practices. (A) Concentration duration curves
     show concentration reduction; and (B) Load-rate duration curves show reductions in
     pollutant loading (Bonta and Cleland, 2003)	12
Figure 9. Concentration (C) - discharge (Q) relationships at WE38 for SO4	14
Figure  10.    Variation in  regression parameters (Eqn.  1) with sample  size in random
     regressions for 804: a) coefficient a; b) exponent b	15
Figure 11.  Variation in mean difference between duration  curves  developed from random
     regressions  and baseline duration curves, with sample size  in random regressions for
     SO4:  a) CDC; b) LDC	15
Figure 12. Normal probability plots of CDC for SO4 for sample sizes of: a) n=5; b) n=30; c)
     n=50;d)n=100	16
Figure 13.  Normal probability plots of LDCs for SO4 for sample sizes of:  a) n=5; b) n=30;
     c)n=50;d)n=100	17
                                                                                 VI

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Figure 14.  FDC, CDC, and LDC for 864 data from  watershed WE38, computed using
     instantaneous flow (black lines) and average daily flows (red lines)	20
Figure 15. Monthly distribution of flow rate (upper) and NO3-N concentration (lower) for
     WE38	25
Figure 16. Flow-duration curves for the twelve monthly periods of scenario 7	27
Figure 17.  Concentration -flow-rate graphs for NOs-N for September through February  data
     atWE38	29
Figure 18.  Concentration -flow-rate graphs for NOs-N for March through August data at
     WE38	30
Figure 19.  Concentration-duration curves for the 12 periods (months) of scenario 7 for NOs-
     N	31
Figure 20. Load-rate-duration curves for the 12 periods (months) of scenario 7 for NOs-N. 32
Figure 21. Normalized plots of monitoring well levels and stream  flow data for WE38; used
     to identify seasons for scenario 8 (Bil Gburek (2006) personal communication)	33
Figure 22. Flow-duration curve for the three periods (seasons) of scenario 8	34
Figure 23. Piecewise-linear concentration-flow rate regressions for scenario 8. A refers to
     the graph of empirical data and B refers to the regression for each scenario	35
Figure 24.  Concentration-duration  curve for the three periods  (seasons)  of scenario 8
     computed from regression equations	36
Figure 25.  Concentration-duration  curves and superimposed  data for  the three  periods
     (seasons) of scenario 8	37
Figure 26. Load-rate-duration curve for the three periods (seasons) of scenario 8	38
Figure 27.  Load-rate-duration curves and superimposed data for the three  periods (seasons)
     of scenario 8	38
Figure 28.  Concentration-flow-rate graphs for NCVN for  each year to identify periods of
     similar relationships for scenario 9. Open circles depict the remainder of the data set in
     each graph for comparison purposes	39
Figure 29. Flow-duration curve for the three periods of scenario 9	40
Figure 30.  Piecewise-linear concentration-flow rate  regressions  for the three periods for
     scenario 9 for NCVN	41
Figure 31. Concentration-duration curve for the three periods of scenario 9	42
Figure 32.  Concentration-duration curves and superimposed data for the three periods of
     scenario 9	42
Figure 33. Load-rate-duration curve for the three periods of scenario 9	43
Figure 34. Load-rate-duration curves and superimposed data for the three periods of scenario
     9	43
                                                                                   vn

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Tables
Table 1.  Average constituent concentrations and load rates for the data set computed using a
    range of flow averaging times	22
Table 2.   Effect of flow averaging times on the maximum flow rate, and on concentration
    and load rates	23
Table 3.  Scenarios and periods of data for comparing duration curves	26
Table 4.  Concentration-flow rate regression parameters for 1- and 2-equation piecewise fits
    using the form of equation 2 for scenario 7	28
Table 5.  Concentration-flow rate regression parameters for a 2-equation piecewise fit using
    the form of equation 2 for scenario 8	35
Table 6.  Concentration-flow rate regression parameters for a 2-equation piecewise fit using
    the form of equation 2 for scenario 9	41
                                                                                 Vlll

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Acknowledgements


We would like  to thank support personnel at the USDA-Agricultural Research  Service
facility at the North Appalachian Experimental Watershed (NAEW) near Coshocton, Ohio
for the significant efforts to collect and prepare data sets, and the USDA-ARS facility at the
Pasture Systems and Watershed Management Research Unit in University  Park,  PA for
making available the WES 8 data set, and for their helpful insights to the watershed and data.
Thanks also to Ms. Phyllis Dieter (USDA-ARS, NAEW) for the cover design. In addition,
we would like to thank Mike Borst and Karen Blocksom for careful review and  comments
that we hope will improve the clarity of the report for other readers. The work was supported
through an Inter-Agency Agreement (IAG#  DW12990301-3) between the USDA-ARS and
USEPA-ORD.
                                                                             IX

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Introduction

Objectives and Scope
The project objective is to explore and develop uniform methods, based on the duration
curve  concept,  for  comparing  hydrology and  water quality data from  watersheds
subjected to agricultural and other best management practices (BMPs). The objective will
be met using  historical watershed hydrology  and water  quality data, which take into
account risk and natural, observed variability and uncertainty in watershed response to
BMPs and precipitation events.
       The stated objective of  the  Clean Water  Act is to restore and maintain  the
chemical, physical,  and biological integrity of the Nation's waters. What follows this
statement are laudable  goals and policies, but we have learned in the intervening years
that clear success is often  difficult to demonstrate. In part this difficulty lies with the
goals  of the Clean Water Act that focus on source control rather than the improvement of
ambient water quality (Brady 2004).  Sources of pollution, both point and nonpoint, are
integrated within the boundaries of the watershed to yield an ambient water-quality
condition, but are addressed separately by Clean Water Act programs. In recent years, the
U.S.  Environmental Protection  Agency  (EPA)  began  addressing the  difference in
emphasis         by        advocating        a        watershed         approach
(http://www.epa.gov/owow/watershed/approach.html) to managing water resources, and
states followed suit via implementation of a rotating basin approach (National Research
Council 2001) to  monitoring. The U.S. EPA supports  the watershed  approach with
concrete documentation regarding  state monitoring  programs  (U.S.  Environmental
Protection Agency 1997, 2003, 2005), but implementation authority rests with state and
local entities. Progress  has also been  made in monitoring technology and the centralized
collection of data on  the internet (see,  for example U.S.  EPA's STORET database:
http://www.epa.gov/storet/). It remains the case,  however,  that most  water-resource
managers do not have access to  the comprehensive water quality and quantity data
needed to assess changes in condition for particular watersheds with confidence.
       In addition to the regulatory  and administrative challenges outlined above,  the
challenge with acquisition  of watershed-monitoring data  is  twofold.   First, the natural
science of watershed management is complex (Montgomery et al. 1995, Black  1997,
Leopold 1997, National Research Council 1999). Surface-water quality  at any given time
is  a combination of atmospheric sources (wet/dry deposition), groundwater exchange,
runoff  from  land  sources, municipal  and industrial point sources,  and  in-stream
constituents from the bed and bank material. Precipitation alters the balance of sources on
daily, monthly and annual time  scales. Terrestrial and aquatic ecosystems alter  water
quality  on  seasonal and annual time scales  via nutrient cycling, microbial activity,
primary production  and other inputs  of waste  and  organic  material.  In  addition to
temporal concerns, the  size and location of a watershed will influence its response to each
of these variables through climate, vegetation, slope and soil type. Second, anthropogenic
sources are  often unpredictable  and dynamic. Both point and nonpoint sources vary
according to  daily, seasonal and annual time scales.  Municipal  water use has daily
maxima, fertilizer and tillage practices contribute pollutants according to seasonal cycles,
and development patterns  are erratic.  The project detailed  in this report provides no

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answers to these complex questions, but it does  aim to provide simple methods  for
interpreting monitoring data that take these variables into account.

Duration Curves
The purpose of this study is to explore and develop methods based on duration curves for
quantifying change in watershed hydrology and water quality.  The reader is referred to
Bonta and Cleland (2003), for a more in-depth treatment of the basic concepts of duration
curves. Duration curves (DCs) are plots of the percent of time that a given value of a
variable, such as flow rate, is exceeded. The most widely applied type of DC is the flow
duration curve  (FDC). Flow duration curves  have been used  since the late 1800s to
characterize the  duration of watershed flows for a variety of water-resource  purposes.
Miller (1951) showed how FDCs could be combined with sediment concentration flow
curves to estimate total loads of sediment for several rivers in the western United States.
Searcy (1959) suggested  FDCs  could be used  for other chemical constituents.  More
recently, Vogel  and  Fennessey (1995) outlined  a variety of applications of FDCs  for
water resource problems, which include assessment of water quality.
Other investigators have used DCs in water  quality studies to compute total yields and
average load rates on  an  annual or period-of-interest basis  (e.g., Miller 1951, Searcy
1959, Ledbetter  and Gloyna 1964, Bourodimos et al. 1974, Steele  et al. 1974,  Sherwani
and Moreau 1975, Goolsby et al. 1976, Larson et al. 1976, Lettenmaier 1977,  Simmons
and Heath 1979, Harned et al. 1981,  Smith et al. 1982, Kircher et al. 1984, Leib  et al.
1999, Bonta  2000, Bonta and Dick 2003).  The calculations in these studies  result in
constituent distributions that are typically integrated to calculate an  average concentration
or load rate. However, the intermediate DCs contain information  that is useful  for
evaluating change in condition over time and for total maximum daily  load (TMDL)
applications (Cleland 2003, U.S. Environmental Protection Agency  2007).
Three distinct relationships can be derived from the duration curve  concept, as illustrated
in Figure 1. FDCs, as described above, establish the basic relationship between flow (also
called discharge) and the  percent of time that a given flow is  exceeded for a  specific
stream or river location (termed "exceedance level" in this report).  High flows  occur
infrequently  and are thus  exceeded a small percent  of time,  while  low  baseflow
conditions are exceeded frequently (90% and above; Cleland, 2003; U.S. Environmental
Protection Agency, 2007).  Concentration-duration curves (CDCs) show the concentration
of a given water-quality constituent (e.g., copper, sediment) for each corresponding point
on a FDC. The  shape and utility of the CDC depends on the  relationship between  the
constituent concentration and stream flow. When flow is multiplied by concentration to
calculate the  load  for a given constituent, the resulting data may be  plotted  as a load
duration curve (LDC); this formulation can be particularly useful in TMDL applications.

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watershed in the nonglaciated Appalachian Valley and Ridge Physiographic Province.
Land use consists of roughly 57% cropland, 35% forest and woodlots and 8% pasture.
Elevations in the watershed range from 787 to 1575 ft above sea level, and the average
annual precipitation is 39.4 inches.  The characteristic that makes these data valuable for
analyses in this report is that they are a continuous, long-term,  short-time increment
stream flow data set that spans  the period from 1968 through 2003.  Monitoring well
data, precipitation data, and water-quality data from grab samples for many constituents
from 1984 through 2003 (-2500 samples) are other ideal characteristics of the data set.
The data were contributed by the Pasture Systems and Watershed Management Research
Unit of the USDA-Agricultural Research Service.
Other data  for the project came from the North Appalachian Experimental Watershed
(NAEW), which was established in 1935 in the uplands area of Coshocton County. The
NAEW is a  1050-acre outdoor  laboratory facility (experimental watershed) that was
initiated to develop methods for the conservation of soil and water resources. The NAEW
is located near the town of Coshocton in east central Ohio, an unglaciated  portion of the
state with rolling uplands. Underlying  bedrock includes sandstone, shale, limestone, clay,
and coal. Soils are medium textured and range from well-drained, with no  impeding soil
horizon, to soils that have a clay horizon. Average annual rainfall is 37.4 inches.
At the NAEW, historic hydrology and water quality data have been collected from small
experimental watersheds that range in size from  1 to 300 acres. These data include
hydrology and meteorological data collected over the last 70 years, water quality data
collected over the last 25+ years, and  other  data with shorter  records such  as soil
moisture. Runoff and water quality data have been collected continuously on  several
watersheds  using  a network of weather  stations, rain  gauges, lysimeters,  automated
samplers and flumes.

Constructing Duration Curves: Types and Examples

Flow Duration Curves (FDCs)
A flow duration curve is a plot of the percent of time that flow rates are exceeded, and it
removes information on the sequence of recorded flows. There  are two methods for
developing  FDCs, using average flows (e.g., average daily flow), and using short-time-
increment, "instantaneous" flows (e.g., "breakpoint" data).  Breakpoint data are recorded
when there  is a break in  the slope of stage  hydrograph.   This  is the  most accurate
representation of hydrograph traces. Alternatively, breakpoint data can be  approximated
by short sampling interval  data (e.g., every 5 min).  WE38 data have a 5-min sampling
recording frequency.  FDCs using  averaged data are constructed by ranking available
flow data (high to  low) and using the rank position to calculate a plotting position, or
exceedance probability. This is accomplished using an equation such as the following for
annual data containing average daily observations (Fennessey and Vogel 1990):
where pt is the exceedance probability or plotting position, and /' is the rank number for a
given number of observations l,2,3,...,365w where n is the number of years of record for

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the data set. FDCs may be constructed using any number of observations, in which case
the denominator would equal the total number of observations plus one.  A FDC  using
instantaneous flow data is constructed by determining the fractions of durations of a flow
(e.g., segments D4, D5, and D6 within total time, T,  for flow rate Ch in Figure 2).   This
is repeated for many Q;.
With either method, the graph is often plotted on a log-normal probability grid as shown
in Figure 3 but may also be plotted using a linear percentile for the x-axis. Flow-duration
curves characterize the range of flow rates for the period over which data were collected,
and can change with the occurrence of persistently dry or wet periods. Annual variability,
due to wet and  dry years,  is illustrated in Figure  3, and compared with the  40-year
composite FDC for Watershed 174 at the NAEW (Bonta and Cleland 2003).
                 15001
                 1200
                  900-
                  600-
                  300
                  0
                   Q,
                                      Total Time (T)
                                 - Peak flow = 1238 L/s
                                  Average daily flow
                                   rate=63.8 L/s
                                                  Incremental difference
                                                  in Flow Rate, G,j
                   2400  0300 0600  0900   1200  1500  1800   2100 2400
                               Time on 8 June 1990
   Figure 2. Hydrograph for Watershed 174, NAEW, Coshocton, OH. Figure shows how
  duration curves can be constructed with varying flow-rate steps, and how average daily
         flow rate misrepresents watershed hydrology. (Bonta and Cleland, 2003)

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                    4
                   10
                   10
                  10 -
                    -1
                  10 1
                  10
                         ._/-—i- Composite FDC
                                                  Annual FDCs
                               0.01 0.1 0.5   5 10 20 40 60 80  95  99 99.8  99.99
                                 0.050.2 1       30 50  70  90  98 99.599.9
                               Normal Percentile (% greater than)

 Figure 3. Variability in annual flow duration curves compared with the composite 40-year
       FDC for Watershed 174, NAEW, Coshocton, OH (Bonta and Cleland, 2003)
FDCs characterize a watershed's response to precipitation and other inputs, integrating
multiple factors that affect stream flow at a point (topography, soil distribution, climate,
land use, flow controls such as dams, etc.). A flat FDC implies a greater level of storage
in the basin and a steeper FDC implies  a flashy watershed, where streamflow increases
quickly following  precipitation.  Some  investigators have  developed  relationships
between basin parameters and FDC characteristics to yield synthesized FDCs where flow
data are not available (Quimpo  et al. 1983, Fennessey and  Vogel 1990, Franchini and
Suppo 1996, Smakhtin 2001). Mathematically FDCs have the appearance of a log-normal
distribution, but interpretation of them is limited due to non-independence of flow rates.

Relationships between Concentration and Discharge (regression equations)
In  many  watersheds  a statistically-significant correlation  exists  between  chemical
concentrations (C) and flow rate  (Q) (see the following for an in-depth discussion: Lewis
Jr. and Grant 1979, Tasker and Granato 2000, Helsel and Hirsch 2002). Three types of
linear relationships are illustrated in Figure 4. A positive trend indicates that the largest
concentrations occur at high flow rates.  For constituents with positive trends, the supply
in the watershed is available for transport by runoff from a terrestrial  source, and/or may
be mobilized via in-stream sediment transport processes associated with increased stream
velocities  and higher flows from precipitation. A negative correlation (inverse) trend
implies that constituent supply is limiting,  and/or dilution  occurs during precipitation
events,  and indicates  that the largest concentrations occur at  lower flow rates. Larger
concentrations may occur at lower flow rates, for example, because baseflow is derived
from stored water having long contact times within the aquifer, or because of continuous
discharges that dominate at low flow (e.g., point sources).

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             O
             o
             c
             0)
             o
             C
             o
             O
                          negative slope
                          (e.g., TDS, conductivity)
                                                            x
                      x
                                                    X
                                               X
                                            X •
                 positive slope
              •  (e.g., suspended solids)
<8?          Q§5    '.e'8-»sotne cations.
no significant correlation
(e.g., soi
metals)
                              Discharge (Q)
  Figure 4.  Three basic relationships between discharge and the concentration of a water
                                 quality constituent.
Because of possible correlations between C and Q ("C-Q regressions"), simple averages
of concentrations and loads may not accurately characterize the variability of C  and Q
that occurs  naturally,  and  regressions  must  be  used.  However,  there  is  often no
statistically significant correlation between C and Q, and a simple average concentration
can be used to characterize the concentration for different stream flows. Incorrect use of
averaging in the place  of regression analysis will not allow proper estimates of water-
quality-load changes when BMPs are implemented that may change either the hydrology
or supply  of constituents in a  watershed.  Smith  et  al. (1982)  suggest that relations
between C and Q can be related linearly, logarithmically, or inversely. These equation
forms are special cases of the simple power equation,
                                                                             (2)
where a and b are parameters. A more general form of the power equation is
(C + d) = a (Q + ej
                                          (3)
where d and e are parameters that straighten a single convex or concave curve on a log-
log grid. All parameters are fitted by traditional nonlinear regression techniques. Smith et
al. (1982) also suggest a hyperbolic form

-------
                                                                            (4)
where/is a parameter. These equations are often referred to as constituent rating curves.
Relations between C and Q can exhibit much scatter, and regression confidence intervals
supply a measure of uncertainty. While  monotonic relations  are preferred (fitting the
equations given above), it is not a requirement for developing derived duration curves.
            100
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                                                     Cso  = 11.97*Q0'113
                                                       r = 0.51, p < 0.001
               0.1
                                            10
                                                         100
                                                                       1000
                                    Discharge (cfs)
 Figure 5. Plot of SO42 concentration data versus discharge for watershed WE38 (years 1990-1995).
 Regression line with 95% confidence intervals follows the basic power equation C=aQb with values:
                                 a=11.97 and b=0.113.
Nonmonotonic forms add complexity to the method and  are not pursued in the present
study.  Statistically significant correlations  can be screened  by  computing  the  rank
correlation coefficient (RCC) and selecting a threshold significance probability. The RCC
is not  dependent  upon  an  underlying  regression-equation  form.  It  is sometimes
appropriate for data to be fitted continuously in a piecewise manner over different ranges
of Q with different equations. An example is the piecewise, simultaneously constrained
curve fitting in Bonta (2000) for a sediment rating curve using equation 2 (simple power
relationship) for two ranges of Q.  The piecewise-linear approach is also used in the last
section of this report (Case Study) using MVN data.

Concentration and Load Rate Duration Curves (CDCs andLDCs)
There are three basic forms of concentration duration curves (as noted above, refer to
Figure 4) - those developed from C-Q correlation regressions that have a positive slope,
those having  a negative  slope, or those that are  statistically independent (i.e.,  no

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relationship). The following discussion will be based on the simple power equation (C =
a Q6; Eqn. 2) where the slope, and therefore the exponent (&), is greater than 0.  This
relationship is illustrated  in  Figure 5, for SC>42"  (sulfate)  concentration  data  from
watershed WE38, for the years 1990 to 1995. The regression line is significant (p<0.001,
r2=0.51), yielding values for a and b equal to 11.97 and 0.113, respectively. Confidence
limits are computed at the 95 percent level. CDCs and LDCs may be constructed in two
ways, using the SC>42" data.  The empirical concentration and load data (i.e., raw data) can
be sorted according to flow ranking to yield a rough CDC or LDC, as shown in Figure 6
and Figure 7. Also shown, the regression equation can be used to calculate both the CDC
and LDC, providing 95% confidence intervals. For calculating load rates (LR) from Eqn.
2, the flow (Q) is first converted to units of liters per second (L/sec) and the following
equation is used:
LR (kg/sec) = CQ*10'6 = aQ6Q*10"6 = aQ(1+6)*10-6
                                                   (5)
where C is concentration in mg/L, and 10"6 converts mg to kg. The reader is referred to
Bonta and Cleland (2003) for a more complete treatment.
           100 -
            10 -
        o
           0.1
           Flow Duration Curve
           Concentration Duration - Ranked raw data
           Concentration Duration - Regression line
           with 95% confidence intervals
                                                                     100
                                                                         o
                                                                   -10   £
                                                £
                                                
-------
           100 -
£,  10 -

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     1 -
           0.1
                                 Flow Duration Curve
                                 Load Duration - Ranked raw data
                                 Load Duration - Regression line
                                 with 95% confidence intervals
                                                                  10-'
                                                                  10-
                                                                 - 10-5
                                                                      o:
                                                                      •o
                                                                      «
                                                                      o
                                                                      It
                                                                      O
                                                                  10-5
                  10
                       20
                            30   40    50   GO   70
                              Duration IntervalI0/
                                                      80
                                                               100
 Figure 7. Flow and load duration curves (FDC and LDCs) for SO42~ concentration data for
 watershed WE38, years 1990-1995. LDCs are shown for raw data and regression equation
                based on the positive correlation between SO42" and flow.
Assumptions and Limitations
Using DCs with regression  relationships requires  several assumptions regarding data,
analysis, and  physical conditions. Extrapolation of data beyond  observed data limits,
using the form of the C-Q equation chosen, is assumed to be valid. Assuming validity
beyond observed data is critical because most data sets consist of a limited number of
observations,  and  very few (if any) contain the maximum and minimum values for a
given  population (e.g.,  all flow  conditions for a given watershed).   All of the basic
assumptions underlying regression analysis must be met. The  FDC is assumed to  be
stable so that errors due  to  FDC characterization of watershed flows are minimized.
However, an analysis of error in FDCs can be made with uncertainty analysis and the
derived distribution method  as outlined in  Bonta and  Cleland  (2003). The underlying
flow and concentration data need to be of high quality. The watershed is assumed to be
stable,  and there  should be  no factors  that would  change  C-Q relations  (e.g.,
anthropogenic factors, etc.). Best management practices implementation is allowed, but
the initiation  of BMPs begins a new  set of data  with which  to  compare baseline
conditions.  Furthermore,  the  C-Q  relationships  are  assumed to be  stable for  all
precipitation events (i.e.,  non-uniform precipitation and runoff over a basin  does not
significantly alter C-Q relationship).

Concepts for Using Duration Curves to Quantify Changes in Watershed Condition
Duration curves can be used  to quantify changes in flows, load rates, and concentrations
over  time  due  to the  implementation   of  management  practices (BMPs).  This
quantification can apply at specific discharges or over intervals of flow  that might be
                                                                                10

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important. For example, Figure 8A diagrammatically  shows a concentration reduction
value of CR mg/L after BMP implementation compared with the baseline data for the 1
percent exceedance  (points A to B). Similarly, Figure 8B shows a load-rate reduction
(LRR) in kg/day (points C to D). The exceedance reduction (ER; equals risk reduction)
can also be  obtained from Figure 8. For example,  starting at point C in Figure 8B (1
percent), the risk for  the  same  baseline daily load rate is  at  point E  after  BMP
implementation (0.02 percent, a difference of 0.98 percent, which happens to be a 98
percent reduction in exceedance). Although Figure 8 is a simplification of quantifying the
impact of a BMP, it  illustrates the potential for using duration curves for tracking changes
in watershed response after BMP implementation. A reduction  in concentration or load
rate can be obtained by a reduction in flow rates and/or a change in the C-Q  relation.  For
example, Bonta (2003) documents  how C-Q  regression parameters change due to
changing land disturbances caused by geology and mining and reclamation activities. The
DC approach to quantifying stream water changes can be used in planning by estimating
a change in regression line  parameters, and  constructing CDCs  and LDCs. However,
parameter estimation for changing land uses is beyond the scope of the present study.
Using DCs for BMP evaluation is useful beyond evaluating measured data, as they can be
used to evaluate watershed model  outputs as well. A case study using the CR and LRR
approach for quantifying changes in watershed  water quality  is presented in the  last
section of this report.
                                                                              11

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                                 0.01 0.05 0.1 0.2 0.5
                               Normal Percen
                  (B)  LDC
                                 Rec uctiolv
                                E
                              Load-Rate
                              Reduction (LRR)
                                           D
                                                  5  10  20  30 40 50 60 70
:ile (% greater than)
                                 0.01 0.05 0.1 0.2 0.5 1
                                                     10  20  30 40 50 60 70
                                Normal Percentile (% greater than)

   Figure 8.  Conceptual depiction of using duration curves to quantify changes in water
  quality following implementation of management practices. (A) Concentration duration
curves show concentration reduction; and (B) Load-rate duration curves show reductions in
                      pollutant loading (Bonta and Cleland, 2003).
                                                                                   12

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Minimum  Number  of Stream-Flow  Samples Necessary
for the Construction of Duration Curves

Introduction
While DCs can supplement watershed analysis, their  limitations and utility must be
explored to provide guidance on their use.  In particular, the minimum number of water
samples that must be collected to provide reliable CDCs and LDCs is unknown.  This is
important for  new data-collection programs and also when using historic data sets.  The
high cost of  field collection of water  samples  and  laboratory  water quality analyses
requires guidance on the minimum number of stream samples necessary to obtain a
desired level  of confidence in a given analysis of condition.  Obtaining  more samples
than necessary can be costly, and the value of additional data is questionable.  This is
especially important in developing watersheds and BMP implementation studies because
the duration of pretreatment conditions is often short, due to budget, time, and physical
watershed constraints.  Having guidance on the minimum number of samples necessary
to obtain reliable water quality  and quantity condition assessments allows practitioners to
allocate human and fiscal resources more efficiently. The question can only be answered
by  exploring  data  sets  for  which there is  a long stream-discharge  record  and
corresponding set of water quality  samples.  The question addressed in  this  section
concerns whether the minimum number of samples should be based on C-Q regression
stability or  on the convergence of DCs to an underlying watershed characteristic.  An
exploratory study was conducted into these two approaches to determine the minimum
number of water samples required for characterization  of concentration (C) - flow rate
(Q) regressions using a power equation, and use of CDCs and LDCs.

Data
Constituent SO4 from WES8 is used in the present study, for which there are 2290 water
samples.  Sulfate data exhibited a positive correlation with flow rate, yielding values for a
= 8.32 and b = 0.109 for the regression (Figure 9).

Approach to Regression Equations for WE38
The approach to evaluating the minimum number of samples was to use Monte Carlo
simulation from subsets of the  data.  Subsets of water quality samples were obtained by
randomly sampling, without replacement, from all available 864 data.   Target  sample
sizes of 5,  10,  15, 20, 30,  50,  75,  100, 200, 300, 500,  1000,  1600, 2000, and  2200
observations were obtained.  Fifty replicates of a target sample size were generated for a
total of 700 subsets.  Resulting regressions are referred to as "random regressions".  The
estimates of parameters  a and b from Eqn.  2  are compared with the  corresponding
regression parameters  estimated from the  entire baseline data set.  A regression  was
considered statistically significant if the significance probability was less than or equal to
0.10.
                                                                            13

-------
               100-4
                10-
                0.1-
               0.01-
                   1
                  0.1
                                                    C=8,32Q
                                                          0.109
                                                      r =0.278
                                  10     100    1000    10000   100000
                                  Stream Discharge (Q), L/sec
       Figure 9. Concentration (C) - discharge (Q) relationships at WE38 for SO4.
 Baseline  CDCs  and LDCs  were developed using parameters in Eqn.  2,  with the
"instantaneous" FDC providing the flow data used in regressions.  The baseline CDCs
and LDCs were the benchmarks against which all CDCs and LDCs were compared.  The
difference in  concentration,  CDC (random  regression)  minus  CDC  (baseline),  was
computed at each flow and the mean  difference in concentration computed. Differences
and means for LDCs were similarly computed. The mean differences were examined to
estimate a minimum sample size.  If a random regression for SO 4 had a negative slope
(b), the normal percentile for the CDC was not corrected even though the slope for the
baseline CDC is positive,  thus illustrating the potential for error associated with small
samples sizes.

Effects of Sample Size Based on Regression Analysis
Visually, plots of regression parameter variation are characterized by three  regions
(Figure lOa): 1) wide variability for smaller samples tending toward narrow variability as
sample size  increases; 2) narrow but  constant variability for larger sample sizes;  and 3)
smaller but approximately constant variability for the largest sample sizes.  After about
25 samples (parameter V) to 35 samples (parameter a) for SO 4, both a and b stabilize to a
constant narrow variation up to  about 400  samples for a and about 150 samples for b
(Figure lOa and b). Coefficients in this range of sample sizes are within +/- 0.2 units  (2%
to 3%) of that found using all  data (Figure lOa). Most exponents are within +/- 0.02 units
(16% to 17%) of that found using all  data (Figure  lOb). Sample sizes greater than about
400  samples (a, Figure lOa)  and about 150 samples for b (Figure lOb) result in little
variation in regression parameters.
                                                                               14

-------

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            10        100       1000
          Number of samples used in regression
                                       10000
                                                        10
                                                                 100
                                                                          1000
                                                                                   10000
                                                       Number of samples used in regression
    Figure 10. Variation in regression parameters (Eqn. 1) with sample size in random
                  regressions for SO4:  a) coefficient a; b) exponent b.
Not  all regressions were statistically significant for sample sizes  less than 30 (Figure
lOa).  For a sample size of 5, only 26% of regressions were statistically significant for
SO/t.  For a sample size of 10, 76% of regressions were significant.  These percentages
increased rapidly with  increasing sample size to 30 samples after which  100% of all
regressions were significant.  The baseline slope parameter (b) for SO4 was positive,
however, the  random regressions showed that slope can  vary between negative  and
positive values for both constituents for  sample sizes of 30 or less (Figure lOb).
                                                0.08 H
                                                0.06-
                                                0.04
                                                0.02-
                                                o.oo-
                                                -0.02--:
             10
                               1000
                                       10000
                   Sample Size
                                                                     100       1000
                                                                   Sample Size
 Figure 11. Variation in mean difference between duration curves developed from random
 regressions and baseline duration curves, with sample size in random regressions for SO4:
                                  a) CDC; b) LDC.

Effects of Sample Size on CDCs andLDCs
Mean  differences are  quantified in Figure 11 between the baseline  CDC (and  LDC)
computed from the regression developed using all C-Q  data and CDCs (and LDCs)
computed using random regressions developed with smaller sample sizes.  For sample
sizes less than about 30, the range of mean differences decreased rapidly with increasing
sample size.  For n=5, large mean differences plotted  off the graphs in Figure 11. SCv
                                                                                       10000
                                                                                 15

-------
mean-difference plots for CDCs and LDCs (Figure lla and lib, respectively) show that
for sample sizes between 30 and 100 that mean differences do not vary much.  After
about  100 samples,  there  is little variability  in  mean differences  for  SC>4.   LDC
differences are of the order of only ± 0.01 to 0.03 (greater than 30 samples; Figure 1 Ib).
The variability  in mean  differences for CDCs and LDCs for  SC>4  are  selectively
illustrated in Figure 12 and Figure 13, respectively.  CDCs for sample sizes less than 30
for 864 all  showed  some  positively  sloped  CDCs because of individual  random
regressions with a negative  exponent in Eqn.  1  (e.g., Figure 12a for n=5).  Negative
exponents are apparent in smaller  sample sizes in Figure 12b.  CDCs with regression
sample sizes larger than 100  quickly approach the baseline CDC (Figure 12d).  The trend
toward reducing CDC  variability about the baseline CDC is apparent in Figure  12a
through d as sample size increases.  For LDCs, the small mean differences in Figure 13
are illustrated  by  the  small variability in LDCs about the baseline LDC.   LDCs
corresponding to CDCs in Figure 13, show much less visual variability than apparent in
Figure 12 for concentrations.
          0001 001 0.1 05
            QC05 005 02  1
5  20 40 60 80   95
 10  30 50  70  90  98 99.5
OO31 001 01 05
  0005 005 02  1
             Normal Percentile (.% greater than)
                                     Normal Percentile (% greater than)
          QC01 001 01 05
            0005 005 02  1
   20 40 60 80   95  99
 10  30 50  70  90  98 99.5
QC01 001 01 05
  0005 005 02 1
5  20 40 60 80   95  99 99.8
 10  30 50  70  90  98 99.5
              Normal Percentile (% greater than)
                                       Normal Percentile (% greater than)
Figure 12. Normal probability plots of CDC for SO4 for sample sizes of:  a) n=5; b) n=30; c)
                                   n=50; d) n=100.
                                                                                  16

-------
  0.0001-
 O.OOOOIn
0.000001-1
             OD01 001 0.1  OS    5  20 40 60  80  95 99 99.8
              0005 0.05 02  1   10  30 50 70  90  98  99.5
                  Normal Percentile (% greater than)
                                                   0.01-
                                                  O.OOli
                                                 0.0001-
                                                 O.OOOOIn
                                                0.000001-1
            OD01 OD1 0.1 05
              0005 0.05 02  1
                 Normal Percentile (% greater than)
    1.0-

    0.1-

   0.01-

  0.001-

  0.0001-

 0.00001-

0.000001-
            0.001 0.01 0.1 05   5  20  40 60  80  95  99  .
              OflB 0.05 02  1   10  30 50 70   90  98 99.5
                 Normal Percentile (% greater than)
  0.0001-
 0.00001-
0.000001-1
            0001 001 0.1 OS
              0.005 0.05 02  1
5  20 40  60 80   95
 10  30 50 70  90  98 99.5
                 Normal Percentile (% greater than)
 Figure 13. Normal probability plots of LDCs for SO4 for sample sizes of:  a) n=5; b) n=30;
                                  c) n=50; d) n=100.

Summary and Conclusions Regarding Data Set Sample Size
Three regions were  found in plots of regression parameters and differences between
duration curves versus  sample  size  variability.   The boundaries  of the regions were
different for the two approaches.  The  three regions resulted in recommendations for
minimum  and maximum sample  sizes.   The  regression-FDC and  duration-curve
approaches to determining minimum sample sizes result in similar minimum sample size
recommendations.  However, they differ in the number of samples, beyond which there is
no noticeable improvement in variability in parameter estimates  or differences between
baseline and randomly developed CDCs and LDCs.

The results suggest that a sample size of no less than about 35 samples  is needed to
minimally characterize the C-Q  data for 864 for watershed WES 8  to  avoid improper
regression slopes and non-statistically  significant regressions.   Beyond  35 samples,
variation  in  parameters  is roughly constant with slow convergence toward baseline
values. The different sample  sizes at this point could be related to constituent behavior in
the watershed, magnitudes of the parameters, snowmelt effects, effects of season of year,
                                                                                   17

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hydrograph  position of the  samples,  and possible  mischaracterization  of the C-Q
relationship by equation 2.  Separating the data according to position of the samples on
the hydrograph (i.e., rising/falling limbs, etc.) and by growing and dormant seasons may
reduce the variability observed in Figure  10 (see Case Study section for evaluation by
season).  Smaller minimum required sample sizes might result because of the smaller
expected variability.  Based on the data evaluated by  regression analysis in the present
study,  50 samples is the minimum sample size suggested.  After about  150 to  400
samples, the variability  in parameter estimates decreases to a low value,  and the value of
additional data is questionable.

The  exploratory  study  in this section into the minimum of number of water samples
required for adequate characterization of  concentration (C) - flow rate (Q) regressions
and subsequent development of concentration and load-rate duration curves (CDCs and
LDCs) led to the following conclusions:

•  A minimum  of about 25-35  samples is  required to reach an acceptable level of
   coefficient and exponent variability for SCv Narrow parameter variability results
   with sample numbers larger than about 150-400 samples for SC>4.
•  Differences between CDCs and LDCs suggest that 30 samples are adequate for  864
   with no  noticeable  improvement in trends of the duration  curves  after about  100
   samples.
•  The regression and  duration-curve approaches result in similar recommendations for
   minimum sample sizes, but the duration curve approach suggests a lesser maximum
   number of samples,  beyond which there is no noticeable improvement in differences
   between baseline and randomly  developed CDCs and  LDCs compared with the
   regression approach.
•  Based on the  combined approaches, 50 samples is the suggested minimum sample
   size to  reduce variability  of regression parameters and  differences for CDCs  and
   LDCs.  There is little benefit derived from  obtaining more than about 100 samples,
   and the value  of obtaining additional data is questionable.  In the present study, 50
   samples  represent only about 2%  of the  total number  of samples available for
   analysis. One hundred samples represent only 4% of the total data set available.  The
   additional 96% to 98% of the C-Q data  do not appear  to be valuable for general
   watershed characterization from strictly regression, CDC, or LDC  points of view.
   However, the  additional data can be valuable for other objectives such as  studies of
   seasonal variations,  sources of  chemical  constituents, modeling,  climate change,
   nonstationarity, and  land-management effects, etc.
                                                                              18

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Quantifying Uncertainty in the  Use of Flow Averaging,
and Daily vs. Instantaneous Flows

Introduction to the Temporal Component of Flow Data
Flow duration curves are characterized  by the time base of the data  used in their
development.  Given the same watershed flow data, FDCs developed using minute, daily,
weekly, etc., time-step data will have different characteristics (Searcy, 1959). Serial flow
data, sampled at regular intervals, is important because it provides some assurance -
especially with larger data  sets and longer periods of time - that the full  range and
distribution of flow conditions is represented by the duration curve.  The  best available
source for flow data is the U.S. Geological Survey, which provides online,  downloadable
access   to    data   sets   with   both   30   minute   and   daily  time   steps
(http://waterdata.usgs.gov/nwis/sw). The U.S. Department  of  Agriculture, Agricultural
Research  Service (ARS), also  provides online access  to hydrology and climate data
(http ://hydrolab .arsusda.gov/wdc/arswater.html). Duration curves can be constructed  by
using monthly or  opportunistic  sampling approaches, but  the  results need  to  be
interpreted cautiously because the limited data set may not accurately represent the full
range of flow  conditions or their incidence of occurrence.
In the routine development of duration curves, water-resource managers need to be able
to judge whether they can use commonly  available average daily flow data to develop
FDCs, even when water-quality samples are obtained instantaneously. Samples of runoff
are often taken instantaneously by automatic samplers or manually, and  do not represent
the chemistry  for average daily flow rates. Instantaneous flow data, measured at the time
water quality  samples are taken,  provides a more accurate FDC and resultant CDC or
LDC. Errors  are  more likely to occur for high flow conditions in  smaller, flashier
watersheds where peak discharge occurs quickly and recedes on an hourly time scale, as
depicted in Figure 2.  Figure 2 shows that the average daily flow rate for  WS174 at
Coshocton, Ohio, of 63.8 L/s was only 5 percent of the instantaneous peak flow rate
(1,238 L/s) for the  day. For data from WS174, streamflow is recorded  in breakpoint
format.  This  data recording format captures the most detail, but for some applications
data  of equal time intervals is all that is available (e.g., WE38).  Mean daily discharge is
likely to  be  least accurate at representing high flows but  should be  adequate for
representing the flow regime under non-flood conditions. A separate question, addressed
in detail in the following section, is whether the FDCs based on averaged flows can be
substituted for FDCs based on  instantaneous  flows as  the  independent variable  in
regressions to develop CDCs and LDCs without compromising their accuracy for all flow
conditions.

Method for Data Averaging
The  effects of averaging period used to compute FDCs on subsequent CDCs  and LDCs
are examined  by using the regression equations (Eqns. 2 and 5) developed from all data
for SC>4 (instantaneous flow rate at time of stream-water sampling) with averaged flow
rates calculated using different periods of time.  The baseline CDC and LDC computed
using all C-Q  data with the instantaneous  FDC is used as the  benchmark. An average
                                                                             19

-------
flow rate is computed by dividing the accumulated flow volume during the averaging
time periods by the length of the time period. FDCs were developed using average flow
rates over averaging periods of 5 min, 10 min, 15 min, 20 min, 30 min, 1 hr, 2 hr, 4 hr, 6
hr, 12 hr, and 24 hr.  These average flow rates are used to develop CDCs and LDCs from
equations 2 and 5, and to explore trends of the results with averaging time. The mean of
the ratios of the individual CDC  and LDC values to the  baseline CDC and LDC at
selected flow points  is plotted against flow averaging time to quantify the effects of using
averaged FDCs.  Maximum flows  are examined  separately.  Average time-weighted
concentrations and  load rates using the  weights  from the duration curves for each
averaging time are compared with instantaneous values, and selected FDCs, CDCs, and
LDCs are plotted.
Effects of Averaging Time on CDCs and LDCs
Average time-weighted concentrations and load rates computed from the duration curves
for SO4 are practically the same for all averaging times compared with the instantaneous
duration curves (Table 1) as seen by the ratios near unity. Mean weighted concentration
ratios range from only 1.000 to 1.003 of instantaneous averages and load rates range from
0.969 to 1.000.  The small  differences in averages are in part due to the small exponents
in Eqn. 2. Plots for the two most extreme cases (instantaneous and average daily flows)
show the curves lie almost entirely on top of one another (Figure 14; 864 only).  The
curves tend to diverge slightly at larger flow rates where maximum averaged flows are
not representative of instantaneous flows as discussed next.
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                             Normal Percentile (% greater than)

   Figure 14. FDC, CDC, and LDC for SO4 data from watershed WE38, computed using
           instantaneous flow (black lines) and average daily flows (red lines).
While there is little effect of averaged flow rates on the overall structure of CDCs and
LDCs in Figure 14, the largest  flows  computed for the  averaged FDCs are reduced
because of the averaging. This can be seen in Table 2 where the maximum flow for the
                                                                             20

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instantaneous (raw) data is 27533 L/sec, whereas the maximum  average daily flow is
18201  L/sec, 66% of the  maximum  recorded flow.  This does not have a noticeable
impact on SC>4 concentrations where it can be seen that SC>4 concentration for daily flows
is 96% of that associated with instantaneous flow (Table 2).  The small differences are
due to small exponents  (Eqn. 2).  The maximum load rates, however,  are significantly
affected.  For example, maximum load rates computed with average daily flows are only
63% (804) of that of the  maximum instantaneous flow.
FDCs developed  from different flow-averaging times (when  used  as  the independent
variable in Eqns. 2 and 5 to develop CDCs and LDCs) do not appear to affect CDCs and
LDCs using the 864 data from Watershed WE38. This suggests  that using commonly-
available  average daily flows for  water quality  analyses may allow a  reasonable
characterization  of chemical concentrations for watersheds  of  7.3 km2  and larger.
However, as shown in  Figure 2 for a gauged 21.4-ha watershed at the  NAEW  in
Coshocton, Ohio, average  daily flow  was only 5% of the measured peak runoff for one
large monitored runoff event. In the present study, peak average daily flow was 66% of
measured instantaneous flow  rate.    The flashy character of runoff from  "smaller
watersheds" may  preclude the substitution of average daily  flows for  measured
instantaneous flow  rates.   However, guidance on watershed  size limitations, and  the
behavior of water quality  constituents that behave differently  in response to flow rate,
requires further study.

Recommendations Based on the Effects of Averaging Time
One advantage of the duration-curve  approach to evaluating water quality data is that it
has the potential  to convert concentration data  collected  in the field to load rate  data
required  by regulatory agencies, such as  for TMDLs (mass/day; Bonta and Cleland,
2003). The nearly identical load rates found for all averaging times (that include average
daily flows) suggests that  the conversion is facilitated by assuming that water samples
collected instantaneously can be used with average daily flows to yield  a mass/day
(TMDL).   Average daily  flows  are a  common form  of flow data available to a
practitioner. A further advantage is that the mass/day value also  has a percent time of
exceedance associated with it.  While there was near equivalence of CDCs and LDCs
developed from instantaneous and average daily  flows in the  present  study, other
parameters with  larger parameter values may show larger differences requiring further
study.
Time-weighted averages for the entire range of flows were used in the present study, but
it may be desirable for a practitioner to censor the duration curve for specific purposes as
suggested by Bonta and  Cleland (2003).  Selecting flow or percentile ranges in this way
may yield different results.  This is particularly true if the focus  is on larger flow rates. It
is not likely that a BMP will be effective for the entire  range of flows that can occur (e.g.,
extreme  flooding),  and  the  larger flow  rates may  be excluded in  analyses using  the
duration curve approach  if chemical loads and concentrations cannot be controlled by the
BMP.  The larger flows and load rates occur infrequently and provide little weight in
time-weighted averages.  However, the results of the present study  suggest that midrange
and  small instantaneous flow rates may be  represented  by average daily flows.   The
practitioner should be aware that maximum concentrations can occur at the smallest flow
                                                                              21

-------
rates if the exponent in Eqn. 2 is negative. However, the load rates will be larger at the
larger flows within the constraints of Eqn. 5. Recognizing this can affect a decision on
potential BMPs that are feasible. The use of average daily flows to characterize very
small  watersheds  could introduce  significant error depending  on  constituents  and
regression-parameter magnitudes.  Errors are more pronounced at larger or smaller flows
depending on the sign of the exponent in Eqn. 2. A positive correlation results in larger
concentration errors for larger flows, and a negative correlation results in larger errors for
smaller flows.  For larger exponents, the differences will be larger.
Other constituents associated with different erosion and transport mechanisms may reveal
limitations  on using average daily flows.  A simple power relationship (Eqn.  2)  was
assumed in the present study to characterize the C-Q  relationship.  Other constituents in
the WES8 data show piecewise monotonic and non-monotonic behavior which requires
special  treatment   using  the  duration  curve  method.    These  more  complicated
representations will likely affect both guidance on minimum required samples sizes and
the errors resulting from using average daily flows. An example of a piecewise analysis
is shown in the Case Study section of this report.
Table 1.  Average constituent concentrations and load rates for the data set computed using
a range of flow averaging times.
                      Concentration                 Load Rate
                                              Average     Ratio to
                                                SO4,     instantaneous
                                               kg/sec        (SO4)
Averaging
Time
instantaneous
5 min
10 min
15 min
20 min
30 min
60 min
2hr
4hr
6hr
12 hr
24 hr
Average
SO4, mg/L
12.860
12.860
12.860
12.861
12.861
12.861
12.862
12.863
12.868
12.871
12.881
12.902
Ratio to
instantaneous
(S04)
1.00000
1.00001
1.00000
1.00001
1.00002
1.00003
1.00009
1.00023
1.00055
1.00084
1.00158
1.00322
                                             0.001896    1.00000
                                             0.001896    0.99981
                                             0.001896    0.99975
                                             0.001895    0.99969
                                             0.001895
                                             0.001895
                                             0.001894


                                             0.001891
                                             0.001886
                                             0.001881


                                             0.001865
                                             0.001838
0.99964
0.99939
0.99871


0.99723
0.99467
0.99205


0.98384
0.96925
                                                                               22

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Table 2.  Effect of flow averaging times on the maximum flow rate, and on concentration
and load rates
Flow Rates
Averaging
Time
instantaneous
5 min
10 min
15 min
20 min
30 min
60 min
2hr
4hr
6hr
12 hr
24 hr
Concentrations
Max. Max.
Flow, Ratio to SO4 ,
L/sec instantaneous mg/L
27,533
27,082
26,666
26,522
26,301
26,134
25,525
24,477
22,486
20,940
18,912
18,201
1.00
0.98
0.97
0.96
0.96
0.95
0.93
0.89
0.82
0.76
0.69
0.66
25.3
25.2
25.2
25.2
25.2
25.1
25.1
25.0
24.7
24.5
24.3
24.2
Ratio to
instantaneous
(S04)
1.000
0.998
0.997
0.996
0.995
0.994
0.992
0.987
0.978
0.971
0.960
0.956
Load Rates
Max.
SO4,
kg/sec
0.696
0.683
0.672
0.668
0.662
0.657
0.640
0.611
0.556
0.514
0.459
0.440
Ratio to
instantaneous
(S04)
1.000
0.982
0.965
0.959
0.951
0.944
0.919
0.878
0.799
0.738
0.659
0.632
                                                                               23

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Case Study:  Evaluating  Monthly,  Seasonal  and  Annual
Period Changes in  Nitrate Concentration

Objective and Approach
The overall objective for this case study is to illustrate how information can be inferred
from  DCs  to  supplement water-quality  investigations in river  basins.  This section
provides an example of how the concepts described in Figure 8 can be applied to quantify
changes in a water  quality constituent. The case study utilizes the unique long-term,
WES 8 data set and will examine seasons and associated instantaneous FDCs, CDCs, and
LDCs, changes in hydrology and water quality during different periods, and possibilities
for further analyses.  The case study is not fully developed in terms of causality,
watershed characterization and the use of complimentary and supplementary data sources
and analyses that watershed  managers would use in a real world application of the
duration curve  concept.

Scenarios and Periods of Record
Long, continuous runoff records with concurrent discrete  samples of water quality are
generally not readily available. However, watershed WES 8 flow and NCVN data have
characteristics  that enable the utility of duration curves for comparing changes in water
quality to be illustrated.  These characteristics include monthly and seasonal differences
in the C-Q relationship for NCVN, and apparent changes in watershed hydrology within
the period of record. FDCs, CDCs, and LDCs were  compared within three scenarios:
monthly DCs for the entire period of record, DCs for seasons identified from stream flow
and ancillary hydrological data, and DCs for three periods  of apparent change in the
WES8 record.  The three scenarios are outlined in Table 3 and are described more fully in
subsequent sections.   Periods were intervals of time selected according to the three
scenarios.   Individual  combinations  of  scenarios and periods are referred to as  a
"scenario.period". For example, scenario 7, period 5 is "7.05".  A 10 mg/L reference line
is used on concentration graphs for NCVN because of its significance as a drinking water
standard.
FDCs were  constructed using the 5-min data to make instantaneous DCs - no flow
averaging was  performed (e.g., average daily flows). Regressions between concentration
and discharge used instantaneous flows at the time the samples were taken.

Seasonal Distribution of Discharge and NO3-N Data for WE38
Discharge and NCVN concentrations  have  similar monthly trends, except early in the
year, when NCVN tends to decrease while discharge increases (Figure 15).  CDCs and
LDCs are able to capture these two trends separately because flow is expressed through
the FDC and concentrations are expressed through C-Q regressions.  LDCs  combine the
effects of both hydrology and water-quality processes.  The need to consider seasons of
the year is apparent from this figure.

Concentration-Flow Rate Regressions (C-Q)
Preliminary plots of C-Q, for NCVN data on  a log-log grid for the three scenarios
                                                                            24

-------
suggested that a 2-equation, piecewise regression, using two power equations (Eqn. 2),
would fit the data best for developing CDCs and LDCs.  However, sometimes  a single
power equation was satisfactory. The nonlinear, 2-equation fit was constrained to ensure
that the two equations would intersect at a flow rate chosen by inspection of the C-Q data
during the fitting process (Q;).  The flow-intersection point varied from period to period
within a scenario,  highlighting its  dependence on  watershed conditions and NOs-N
availability  and  transport conditions.   Individual  regressions  are  presented  in  the
following sections.
                     1000
                      100 -
                       10 -
                       1 -
                      0.1
                      0.01
                       20
                       18 -

                       16 -

                       14 -

                    3  12
                    CD
                    I  10-
                    cf  8
                    •z.  8
                       e -

                       4

                       2 -

                       0
                                        6  7
                                        Month
                                                    10  11  12
                                        6  7
                                        Month
                                                    10  11  12
 Figure 15. Monthly distribution of flow rate (upper) and NO3-N concentration (lower) for
                                      WE38.
Monthly Duration Curves (Scenario 7)
Flow duration curves
Flow data were extracted from the WES 8 stream-flow record by whole months (scenario
7, Table 3), and monthly DCs were developed by collapsing all data across years for an
individual month to develop a single monthly FDC (Figure 16). Seasonality (by month)
                                                                               25

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is apparent from the FDCs (Figure 16), with August having generally the smallest stream
flows and March having the largest.  The thinner lines cover the first six months of the
year and the thicker lines  cover the  last  six months.   The first six months are
characterized as having generally larger flows than the second half of the year.
Table 3. Scenarios and periods of data for comparing duration curves.

Scenario
Number

7












8



9


Period
Number

1
2
o
6
4
5
6
7
8
9
10
11
12

1
2
o
6

i
2
3
Scenario
Description

Monthly
DCs












Seasonal
DCs



Period DCs


Beginning
Date*

1/1/1984












1/1/1984



1/1/1984
1/1/1993
1/1/1999
Ending
Date*

12/31/2002












3/31/2003



12/31/1992
12/31/1998
12/31/2002
Beginning*
Month

1
2
o
5
4
5
6
7
8
9
10
11
12

4
9
12

o**
0
0
Day

1












1
1
1

0
0
0
Ending*
Month

1
2
o
5
4
5
6
7
8
9
10
11
12

8
11
o
5

0
0
0
Day

31
28
31
30
31
30
31
31
30
31
30
31

31
30
31

0
0
0
inclusive dates

**A zero month and day implies the entire period of time between beginning and ending dates was used.

It is apparent that the lines tend to graph nearly as straight lines on the  log-normal plot,
with some concavity,  suggesting a possibility for curve fitting using the equation for the
normal distribution.  The extremes do not tend toward straight lines because insufficient
data are available to develop stable FDCs in these regions. The linear tendencies of most
of the flows suggest that curve fitting with an extra parameter (z) added to flow rates
prior  to plotting may straighten the lines (Q+z).  This extra parameter, along with the
mean and standard deviation suggest possibilities for relating these three parameters with
basin characteristics.  For example, earlier it was mentioned that the slope  of the FDC
was a measure of the flashiness of a watershed.  The slope  of a FDC is the standard
                                                                                26

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deviation and may be related to channel and/or overland-flow steepness or other basin
properties.  Even  the low flow extreme of the  FDCs could be related  to  drainage
characteristics  (e.g.,  floodplain soil  texture,  meandering,  geology,  etc.).   Detailed
examination of such relationships is beyond the scope of the present report.
Figure  16 suggests that  seasons could be  identified from  the  range of FDCs.  For
example, the FDCs for the 2-month periods for March-April (7.04-7.05) and July-August
(7.07-7.08) (the extreme FDCs), and Nov-Dec (7.11-7.12),  could be grouped together to
minimize analyses and to allow more  water-quality data to be collapsed into the two 2-
month periods.  This is important because water-quality data  are generally scarce and
grouping the data for similar hydrological conditions would aid data interpretation.
      100000.00-
       10000.00
       1000.00
     Pi
        100.00
        10.00
         0.10
               0.001  0.01  0.1 0.5 1
                  0.005 0.05 0.2
5 10  20   40  60
       30   50  70
                                                        95
                99  99.8   99.99
               ;  99.5  99.9
                                      Normal Percentile
                         7.01
                         7.07
 7.02
 7.08
7.03
7.09
7.04
7.10
7.05
7.11
7.06
7.12
      Scenario  .

Figure 16. Flow-duration curves for the twelve monthly periods of scenario 7
Concentration-flow regressions
In addition to the FDCs, plots  of concentration vs flow rate (C-Q  plots) by month in
Figure 17 and Figure 18 show that the relationships between NOs-N concentrations and
discharge vary substantially from one period (month) to the next.  The  trends in data
suggest changing watershed conditions, but causes are unknown for WES 8. The results
also suggest that collapsing data for periods longer than month-based seasons may add to
variability in regressions and that the collapsed  period may  not contain nonstationary
data.
                                                                                 27

-------
Visually,  correlations  change  from  a positive  correlation from  September  through
November (Figure  17), to no correlation from December (Figure 17) through March
(Figure 18), and then return to a positive correlation with increasing slope from April
through August. The need for quantifying the correlations with two simultaneous power
equations on some of these monthly plots is apparent by noting the curvilinear trends of
the data for a given month.  These changes occur in addition to the hydrological changes
documented in the  FDCs in Figure  16.   The  causes  for changing correlations are
unknown for WES 8  but changes in water-quality processes and anthropogenic activities
in the watershed are most likely factors.
Regression results in Table 4  quantify the correlations in Figure 17 and Figure 18.  A
smaller a parameter is apparent from July through October reflecting generally lower
flows in the FDCs (Figure 16).  However, the b parameter is larger during this time
reflecting the steeper slopes of the data trends and suggesting more availability of NOs-N
in the watershed.  Parameter b values are close to zero from  January through March,
suggesting a lack of correlation. The lack of correlation can be visually seen in Figure 17
and Figure 18.  The  lvalues are generally small and suggest that there is no correlation
for larger flows for most months.  July, August, and September are exceptions.  The
values for parameter c are greatest for  October through December. The intersection flow,
Qi,  shifts to larger  flows during  much of the  second  half of the year. The maximum
during April is due to uncertainty because  of the beginning  of a shift to  a positive
correlation from no correlation in the previous month.

Table 4.  Concentration-flow rate regression parameters for 1- and 2-equation piecewise fits
                      using the form  of equation 2 for scenario 7.

Scenario.Period**
7.01
7.02
7.03
7.04
7.05
7.06
7.07
7.08
7.09
7.10
7.11
7.12
Eqn 2 Parameters*
Smaller Flows (=QO
c, L/sec
NA
NA
NA
4.764
NA
NA
2.533
2.056
4.249
6.816
6.710
7.271
d
NA
NA
NA
0.033
NA
NA
0.198
0.253
0.127
0.021
0.052
0.006

Qi, L/sec
NA
NA
NA
158.5
NA
NA
12.6
12.6
31.6
25.1
50.1
12.6
*Parameters a and b are for the lower flows, and c and d are the equivalent parameters for larger flows.

** "Period" corresponds to month number.
                                                                                28

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      a
         ID -
     £    i -
         D.I

         100
         ID -
     O    1 -
         0.1
         100
     I
     £    1 i
          OJ01     0.1
                          1      10
                         Discharge lets)
                                         100     1000
J   10 -
1
£~    1 H
   0.1

   100

                                                            10 -
                                                        £    i •
    0.1
    100

                                                                                 J
     0.01      0.1      1       10      100      1000
                    Discharge |cfs|
Figure 17.  Concentration -flow-rate graphs for NO3-N for September through February
                                      data at WE38.
                                                                                          29

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          10 -
          i -
         D.I
         100
       5!
       §
          10 -
         0.1
         100
      J   10
      I
      «
      £    i ^
         0.1
j*i^^"*
                                                     10 -
                              0.1
                              100
                           -I  10
                           55
                           i
                              0.1
                              100
                                                     10
           0)0-1
                  0.1
                         1      10
                        Discharge (cfs>
                                      100
                                            1000
                               0.01     0.1      1       10     100    1000
                                            DiBcharge(cfs)
 Figure 18.  Concentration -flow-rate graphs for NO3-N for March through August data at
                                       WE38.
CDCs for monthly data
Many of the CDCs for the monthly plots (Figure 19) are flat for larger flows, reflecting
the weak correlation  apparent in Figure  17 and Figure  18 for larger flows (small
regression slopes [b and  d\ in Table 4).  The months of January and February are not
plotted due to essentially flat lines for the entire range of concentrations. The intersection
points for the regressions  are apparent by noting the abrupt change in slopes of the CDCs
for the lower concentrations - to the left at larger flows the curves are relatively flat and
to the right at smaller flows the curves slope sharply down reflecting the larger low-flow
regression slopes (Table 4).
The CDCs show that by considering seasons (months in this scenario), large differences
                                                                                  30

-------
in exceedences are apparent. For example, April concentrations exceed 4 mg/L 99.9% of
the time (see 4 mg/L line), while for August and November, 4 mg/L is exceeded 50% of
the time. Assuming for discussion that if a standard of 4 mg/L were set for the year, the
stream would usually not be in compliance and field sampling conducted in April would
almost certainly show noncompliance.   The observed  level might  be due to natural
processes that are not  controllable,  suggesting that seasonal regulated levels might be
appropriate. DCs could be used to help establish monthly/seasonal levels.  Alternatively,
if a  land-management  improvement practice was implemented  in the watershed, any
change might  be detected sooner using monthly DCs because a  subtle monthly change
would be masked in the variability of annual data.  Both purposes argue for separating
flow and concentration  data records into seasons.
Another interesting feature in the CDCs  of Figure  19  are the lines for May through
September  above the 10 mg/L drinking-water regulation level, with June reaching nearly
400 mg/L, while the maximum measured NOs-N concentration at WES8 was 17.2 mg/L.
In the case of June,  a  single equation was found to fit  the data best (Table 4),  but its
regression slope (b=0.296) was greater than the slope parameter of all  the months  for the
larger flows (d).  The slope value for August was the second largest  (d=0.253), and the
position  of the CDC  in  Figure 19  was the  next largest.     Larger and  smaller
concentrations can be  computed using C-Q  equations  because the  equations  can be
extrapolated, assuming extrapolation  is valid, however.  Obtaining  a field sample at the
largest flow rate  may  not  be likely  and extrapolated values may be the only  way to
estimate the concentration at the infrequent  flows.   This  is particularly important in
sampling programs of short duration. The use of equations with FDCs makes maximum
use of usually  small water-quality data sets to provide estimates of concentrations beyond
the measured values, an advantage of DCs.
                   0.001 0.01  0.1 0.5 1   5 10 20  40  60  80   95   99 99.8  99.99
                      0.005 0.05  0.2           30 50 70   90   98 99.5 99.9
                                     Normal Percentile
                  Scenario
                                       • 7.03
                                        7.09
• 7.04
' 7.10
• 7.05
1 7.11
• 7.06
1 7.12
  Figure 19. Concentration-duration curves for the 12 periods (months) of scenario 7 for
                                     NO3-N.
                                                                               31

-------
LDCs for monthly data
LDCs  shown in  Figure 20 correspond  to the CDCs just  presented, and  show trends
similar to the FDCs with generally larger load rates for the first half of the year. Many of
the same statements can be made for LDCs that were made in the previous section for the
corresponding CDCs.
                 0.001 0.01   0.1  0.5 1
                    0.005  0.05 0.2
    5 10  20   40  60
           30  50   70
      Normal Percentile
                  95   99  99.8    99.99
               90    98  99.5 99.9
                   Scenario
7.01
7.07
 7.02
' 7.08
• 7.03
' 7.09
7.04
7.10
• 7,05
•7.11
7.Of
7.12
 Figure 20. Load-rate-duration curves for the 12 periods (months) of scenario 7 for NO3-N.
Seasonal Duration Curves (Scenario 8)
The previous section considered characterizing hydrology and water quality strictly on a
monthly basis.  In this section, seasons are identified using available watershed data, and
DCs are developed.
Determining seasons based on hydrology and water quality data for WES 8
For WES 8, seasons were identified using average monthly water  table elevations for
seven monitoring wells within WE-38, precipitation,  and average monthly flow at the
watershed outlet (Bil Gburek (2006) personal communication). Average monthly values
were normalized  by the individual gage's overall annual range.  Plots of the normalized
values showed that all the wells behaved in a similar pattern that was distinctly different
from the stream-discharge plot.  Based on the synthesis of all the gages, and changes in
slope for the normalized plots, three  distinct hydrologic periods were identified: April
                                                                                 32

-------
through August, September through November, and December through March (identified
as scenarios 8.01, 8.02, and 8.03, respectively; Figure 21).  A notable feature in Figure 21
is the 1-month lag of well levels following stream flow.
            1234567
                                9  10 11  12  1  2  3
                                       Month
                                                    5  6
                                                              9  10  11 12
Figure 21. Normalized plots of monitoring well levels and stream flow data for WE38; used
      to identify seasons for scenario 8 (Bil Gburek (2006) personal communication).
FDCs for seasonal data
The  data were collapsed across the entire period of record for the months comprising
each season, and FDCs developed for the three seasons (Figure 22).  Season 3 (Dec-Mar)
had the largest flows and season 2 (Sept-Nov) had the smallest flows.  As for scenario 7,
the visual difference between seasonal discharges is apparent.
                                                                               33

-------
         n
        Pi
        _2  10.00:
                   0.001 0.01  0.1 0.5 1
                      0.005 0.05 0.2
                          Scenario
 5  10  20   40  60
        30   50  70
 Normal Percentile
- 8.01      - 8.02
  95    99  99.8  99.99
90    98  99.5 99.9
                                                            8.03
       Figure 22.  Flow-duration curve for the three periods (seasons) of scenario 8.
C-Q regressions for seasonal data
The seasons identified by examining the variety of hydrological data available for WES 8
showed visually different C-Q relationships. The two piecewise regressions for scenario
8.01 and 8.02 in Figure 23 had identical intersection flow values (Q;=12.6 L/sec;  Table
5), while Q;=39.8 L/sec fit the data better for  scenario 8.03. The trend of the points for
8.03 was noticeably flatter for the range of sampled flows.  This is consistent with winter
data for scenario 7. The concentrations for 8.02 were generally larger than 8.01 for the
same flow rates,  resulting  in  larger coefficients  (a and c).   The  slopes of the
corresponding 8.01 and 8.02 piecewise lines were similar for both small and large flows.
These parameter combinations resulted in scenarios 8.01 and 8.02 being visually parallel
to one another, with 8.02 having higher concentrations.
                                                                                   34

-------
                    Scenario
 100           1000
    Flow Rate, L/sec

™~ 8.01B  0 0 O 802A
                                                     1 8.02B  • • • 8.03A
 Figure 23. Piecewise-linear concentration-flow rate regressions for scenario 8. A refers to
       the graph of empirical data and B refers to the regression for each scenario.
Table 5.  Concentration-flow rate regression parameters for a 2-equation piecewise fit using
                         the form of equation 2 for scenario 8.

Scenario.Period
8.01
8.02
8.03
Eqn 2 Parameters*
Smaller Flows (=QD
c, L/sec
2.732
3.309
6.472
d
0.157
0.182
0.003

Qi, L/sec
12.6
12.6
39.8
*Parameters a and b are for the lower flows, and c and d are the equivalent parameters for larger flows.
CDCs for seasonal data
The combination of FDCs and C-Q regressions between periods 8.01 and 8.02 made the
CDCs similar for  lower discharges (Figure 24).  At larger discharges 8.02 had larger
concentrations.  This is in contrast to the FDC curves, which showed that 8.01  and 8.02
had similar larger discharges (Figure 22). The larger concentrations for 8.02 for the same
discharges resulted in the higher position of the 8.02 curve. The C-Q regression for 8.03
was generally high and flat compared with the other two seasons (Figure 23), and was the
reason for the flat CDC in  Figure 24.  The  graph  shows, for example,  that NOs-N
concentrations exceed 10 mg/L  about 3% of the time for season 2, but only exceed  10
mg/L about  0.03% for season  1.   The  8.01  and 8.02  comparison  again shows the
                                                                                35

-------
importance of considering seasons. The concentration of NOs-N never exceedslO mg/L
for season 3 (8.03).
  100.00:
   10.00^
   1.00:
   0.10:
   0.01 i
           0.001  0.01   0.1  0.5 1
               0.005  0.05  0.2
                         Scenario
5  10  20   40  60   80     95
         30    50   70     90
   Normal Percentile
99  99.8   99.99
  99.5 99.9
                                     8.01
                                                         8.03
   Figure 24. Concentration-duration curve for the three periods (seasons) of scenario 8
                         computed from regression equations.
Raw  data  were superimposed on the computed CDC lines  to  show that the  CDCs
developed from regressions are representative of trend of points if only the raw data were
used (Figure 25).  The advantage of extrapolation using the equation at the extremes is
also apparent where there are no data for these infrequent wet and  dry periods.  The
points in Figure 25 were plotted at the same  exceedance level as the associated sampled
discharge on the FDC.
                                                                                 36

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                 0.001  0.01  0.1  0.5 1
                    0.005 0.05  0.2


                Scenario  •••B.OID  —
  5 10  20   40  60   80    95
         30  50  70   90    <
   Normal Percentile
1 8.01F  A 4 A 8.02D     8.02F ° ° ° 8.03D
 99  99.8   99.99
i  99.5 99.9
  Figure 25. Concentration-duration curves and superimposed data for the three periods
                               (seasons) of scenario 8.
LDCs for seasonal data
LDCs developed from the seasonal regressions and FDCs (Figure 26) show that load rates
are similar for large flows for all three seasons, but 8.03 has larger load rates compared
with the other two seasons for smaller load rates. The other two seasons are very similar
at smaller load rates.
The LDCs with data superimposed (Figure 27)  show that the LDCs computed with the
FDC and regressions are representative of the central trend of the raw data.  Note that the
curve for  8.03 follows the nonlinear trend of the data at the small load rates.  Assuming
extrapolation is  valid, the LDCs will provide an estimate  of load rates at the extremes
where  sample data were not collected but where there is a more extensive discharge
record.
                                                                                 37

-------
1.00E+03'
1.00E+02'
1.00E+01'
o
^ 1.00E+00'
g 1.00E-01-
C3
P^ 1.00E-02'
1.00E-03'
1.00E-04'
1.00E-05'
1.00E-06'
1.00E-07'
	 ,






>»














<





"«*
*v





s





<.





^





0.001 0.01 0.1 0.5
0.005 0.05 0.2
V





j^.











>•





^^






"^7





X




-^.
v











$





i*





X
T-u





L






^v





\




•s







\
\






•
5 10 20 40 60 80 95 99 99.8 99.99
30 50 70 90 98 99.5 99.9
Normal Percentile
                                    Scenario
 Figure 26.  Load-rate-duration curve for the three periods (seasons) of scenario 8.
  1.00E+02-
  1.00E+00
  1.00E-01 '
  1 .OOE-03'
  1 .OOE-05
  1 .OOE-06 -
              0.001  0.01 0.051.1   0.5 1
                 0.005      0.2

                  Scenario   • • • 3.01 D
5  10  20   40   60
          30   50  70
       Normal Percentile
                                                                   95
90
         99   99.8    99.99
           99.5  99.9
                                        8.01 F
                                                 A 8.02D
                                                            8.02F  ° a ° 8.03D
Figure 27. Load-rate-duration curves and superimposed data for the three periods
                                (seasons) of scenario 8.
                                                                                           38

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Apparent Changes in Annual Concentration-Discharge Records (Scenario 9)
The previous two sections used duration curve concepts to investigate the separation of
watershed water  quality and  hydrology  into  seasons,  either arbitrarily  assigned or
determined from data.  In this  section, DCs are developed after inspection of the raw
water-quality  data to identify  periods when there  are  apparent  changes in watershed
response to precipitation.  Seasons are not considered.  The cause of the differences is
unknown,  but the different data sets show how a watershed  could  respond when, for
example, a best-management practice(s) is implemented in the watershed.
                     100
                  I
                     10 ---•     -^TfS
                     0.1
                                                     o  1984-1986
                                                     o  1987-1989
                                                     •  1990-1992
                                                       -r-
                                               10
                                                       100
                  6"
                  j--


                  o"
                                                               1000
                     10 --
                      1 •
                     0.1
                                       1        10

                                       Discharge (cfs)
 Figure 28.  Concentration-flow-rate graphs for NO3-N for each year to identify periods of
  similar relationships for scenario 9. Open circles depict the remainder of the data set in
                         each graph for comparison purposes.
                                                                                 39

-------
        o   1000.00:
        U
        Pi

        J2    10.001
        PH
             0.101
                   0.001 0.01   0.1 0.5 1
                      0.005 0.05 0.2
5  10  20  40  60   80    95
       30  50  70    90
Normal Percentile
 99  99.8  99.99
i  99.5 99.9
                          Scenario
                                       9.01
                                                9.02
                                                         9.03
            Figure 29. Flow-duration curve for the three periods of scenario 9.

Figure 28 shows plots of C-Q data for WES8 for different years. Years are grouped by
visual inspection of similarity of plots, and are subsequently used in analyses.  The FDCs
for the three periods (Figure 29) show that periods 1 and 2 are similar, while period 3 has
noticeably lower flow rates.
C-Q regressions for annual periods
Piecewise regressions for the  three periods  are  presented  in Figure 30  and Table  6.
Generally, period 3  shows  larger concentrations for  similar flow  rates, particularly  at
smaller flow  rates.   Periods  9.01  and  9.02 show  mixed  results -  9.02   has  larger
concentrations at smaller discharges, while  9.01  has  larger concentrations at  larger
discharges.  The intersecting discharge is  similar for 9.01 and 9.02 but much smaller for
9.03 (Table 6). Regression parameters c and d are similar for 9.01 and 9.03 for the larger
flows.
                                                                                   40

-------
               Scenario + + + 9.0
     Flow Rate, L/sec
•9.01B  °°°9.02A 	 9.02B  • • • 9.03A 	 9.03B
  Figure 30. Piecewise-linear concentration-flow rate regressions for the three periods for
                                scenario 9 for NO3-N.

Table 6. Concentration-flow rate regression parameters for a 2-equation piecewise fit using
                        the form of equation 2 for scenario 9.

Scenario.Period
9.01
9.02
9.03
Eqn 2 Parameters*
Smaller Flows (=QO
c, L/sec
3.78
4.38
3.83
d
0.12
0.05
0.14

Qi, L/sec
22.4
25.1
6.3
*Parameters a and b are for the lower flows, and c and d are the equivalent parameters for larger flows.
CDCs for annual periods
Unlike the FDCs for the three periods, the CDCs for periods 9.01 and 9.03 have similar
concentrations at the larger  flow rates,  while period 9.02 has smaller concentrations
(Figure 31). A comparison between relative positions of the FDCs and CDCs shows that
DCs can separately account  for flow regime and concentration-flow relationships, and
that both of these DCs  can affect the quantification of durations of concentrations in a
watershed.  At the low flows, 9.03 has noticeably smaller concentrations due to the lower
small flows in Figure 29.   The CDC for  9.02 is larger than that for 9.01 because of the
relative position of the FDCs and  C-Q regressions. Figure 32  shows the raw  data
superimposed  on the CDCs, and  again  shows the  representativeness of the computed
CDCs.
                                                                                41

-------
                      0.001 0.01   0.1  0.51    5  10   20  40  60   80    95   99  99.8   99.99
                        0.005 0.05 0.2              30  50 70    90    98  99.5 99.9
                                        Normal Percentile
                                 Scenario  	9.01   	9.02
                                                                  ' 9.03
       Figure 31. Concentration-duration curve for the three periods of scenario 9.
                     0.001 0.01  0.1  0.5 1    5  10  20   40  60  80    95  99  99.8   99.99
                        0.005 0.05 0.2              30   50  70    90    98  99.5  99.9
                                          Normal Percentile
                    Scenario •••9.010  ^—9.01F  &&& 9.02D  ^^ 9.02F  OOOgoSD  ^^9.03F


Figure 32. Concentration-duration curves and superimposed data for the three periods of
                                          scenario 9.
                                                                                              42

-------
       u
       cd
                                   D '-
                                          B
                                                  ^oid ra ereiuctior
                                       5 10  20   40  60   80    95   99  99.8   99.99
                                              30   50  70   90    98  99.5  99.9
                         Scenario
 Normal Percentile
' 9.01    	 9.02
                                                                  9.03
       Figure 33. Load-rate-duration curve for the three periods of scenario 9.
                       0.0010.01  0.1 0.51   5  10  20 40  60  80   95  99 99.8  99.99
                         0.005 0.05 0.2            30  50 70    90   98  99.5  99.9
                                         Normal Percentile
                 Scenario  •••9.010     9.OIF  *i! * 9.020       9.02F  °°° 9.030     9.03F


Figure 34. Load-rate-duration curves and superimposed data for the three periods of
                                        scenario  9.
                                                                                            43

-------
LDCs and an illustration of quantifying changes in watershed conditions

The LDCs for annual periods (Figure 33 and Figure 34) show the same relative positions
as the FDCs, with period 9.03 having lower load rates.  Figure 33  illustrates how, by
using WE38 data, DCs can be used to quantify improvements in water quality by either a
reduction in exceedance (ER) or a reduction in load rate (LRR) if a management change
is implemented on a watershed, and follows the presentation of the concept in Figure 8.
Assume that a management change was implemented during period  9.03 from baseline
conditions in period 9.02.  At point B  during the baseline condition,  the exceedance of
10% at a load rate of 1.9 g/sec was reduced in half to point A at 5% due to the assumed
management change from 9.02 to 9.03. Similarly, at an exceedance level of 1% (point
C), the load-rate was reduced in half from 10 g/sec to 5 g/sec at point D due the assumed
management change.  Assuming that there was a management change from 9.01 to 9.02,
the ER and LRR for this change in management was 0 for both variables at the same
levels just discussed.  This demonstrates how DCs can quantify the improvement in risk
reduction and load rates.  This is in contrast to computing a change in average for an
entire flow range which may mask subtle changes for which a watershed manager could
receive credit if a management change had been implemented. DCs provide a method to
quantify changes in different parts of the flow regime.

The same concept applies to CDCs (Figure 32). However, concentrations are what are
measured in the stream channel, but the regulated quantity is the load rate.  DCs provide a
simple method for converting concentrations to loads.

Case Study Conclusions

The following conclusions can be made from the case study using WE38  hydrology and
water-quality data:

•  Concentration-discharge (C-Q) correlations and regression forms of NOs-N vary with
   season.
•  Subdividing annual runoff and  water  quality data into seasons is important  for
   identifying  and  quantifying  natural  processes   affecting  these   variables,  for
   constructing DCs,  for detecting changes in watershed response to precipitation,
   following land-management changes, and for regulatory purposes.
•  CDCs and LDCs developed from regressions and FDCs follow the central trend of
   measured data in  which  there  can  be much  variability,  and are  useful  for
   characterizing the constituent response over the entire range of measured flows.
•  C-Q regressions in conjunction with FDCs  allow  extrapolation of limited data to
   extremes of measured flow rates (low and high flows). Obtaining a field sample at the
   largest flow rate may not be likely and extrapolated values may be the only way to
   estimate the concentration at the infrequent flows, particularly important in sampling
   programs of short duration.
•  DCs can be used to identify seasons of year during intervals when flow and water-
   quality conditions are similar.
•  DCs separately account for changes in watershed hydrology through the FDC and
                                                                             44

-------
   water quality through C-Q regressions. This result suggests opportunities to estimate
   constituent transport and  concentrations, and  hydrology separately for ungauged
   basins.
•  The simple  duration-curve  methodology  can  be used  to quantify  the  load-rate
   reduction (decrease in load rates [and concentrations]) and exceedance reductions (the
   reduction in the percent of time a given  concentration is  exceeded, or reduced risk)
   due to a land-management change.
•  LDCs developed from instantaneous data must be integrated over time (e.g., daily) to
   aide in TMDLs assessments.  This can be accomplished by using the finding in this
   report that FDCs developed from average daily flows and instantaneous flows are
   nearly identical for mid- to low flows.  However, there is a disparity at higher flows.

Report Conclusion and Summary:  The  Utility  of Duration
Curve-Based Methods
Watershed managers are faced with significant challenges when it comes to assessing the
condition of water resources with respect to  hydrology, ecology and water quality. The
challenge becomes acute when  managers are asked to  quantify or predict changes in
condition, and to link change to specific management actions.  Simple tools that can
clearly depict  the condition of water resources and quantify  change are  extremely
valuable.  Duration-curve based  methods fit this description and,  along with other
methods  such as  watershed models, GIS-based analysis,  indices  and  multivariate
statistical methods, add to the set of analytical tools available to watershed managers and
other decision-makers.   Duration  curves can help to maximize the information  in
available data  and provide a quantitative measure of watershed hydrology and water
quality. The DC method can provide a representation of the current stream or watershed
condition and,  by using expected reductions in concentration or a desired water  quality
standard, can depict future watershed land-management scenarios.  The DC method has
the potential to quantify the magnitude of change in stream-water-quality characteristics
after a land-management change in terms of load reduction and the reduction in the risk
of exceeding selected water-quality levels. Duration-curve based methods require  further
investigation of how regression parameters for concentration-flow regressions change for
changing land  use (Bonta and Dick 2003, Bonta 2005).  The DC method has  untapped
potential for: 1) allocating the sources of flow and chemical constituents in the watershed
through the concept of mixed distributions; 2) quantifying the frequency distributions of
individual durations  of flows,  concentrations,  and loads  that will  facilitate  setting
regulated concentrations and  loads for selected biological  species; 3)  evaluating  the
outputs of watershed models; and 4) understanding and quantifying the physical, climatic
and hydrologic variables that contribute to watershed flow condition and water quality.
The  investigations pursued in this report represent a continuing evolution of duration-
curve based methods. They include the following:

•  The use of regression relationships with CDCs and LDCs, and the minimum number
   of samples  needed to construct stable curves can be estimated, but needs further study
   for more and  different types of water quality constituents.

•  Flow-averaging and the time step used in constructing FDCs can affect the structure
   of DCs, especially for extreme low and high-flow conditions.
                                                                             45

-------
•  The construction of, and comparison between, curves based on monthly and seasonal
   data can add value to an analysis because of changes in the underlying relationship
   between flow and concentration for a given constituent.  Regression relationships
   may, for example, be stronger during certain times of the year (e.g., relating  to
   fertilizer application).  Stronger,  more significant relationships between flow and
   concentration yield less uncertainty and a greater likelihood of detecting changes  in
   condition due to management practices or  other change in watershed condition.

•  Examination of changes in annual  data for consecutive time periods using duration-
   curve based methods can be used to quantify changes in concentration and load rate,
   but questions of uncertainty still need to be more fully addressed.
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