Methods for Estimating Low-Flow Statistics for Massachusetts Streams
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Methods for Estimating Low-Flow
Statistics for Massachusetts Streams
By Kernell G. Ries, III, and Paul J. Friesz
Abstract Regression equations were developed to estimate
the natural, long-term 99-, 98-, 95-, 90-, 85-, 80-,
Methods and computer software are 75-, 70-, 60-, and 50-percent duration flows; the
described in this report for determining flow- 7-day, 2-year and the 7-day, 10-year low flows;
duration, low-flow frequency statistics, and August and the August median flow for ungaged sites in
median flows. These low-flow statistics can be Massachusetts. Streamflow statistics and basin
estimated for unregulated streams in Mass- characteristics for 87 to 133 streamgaging
achusetts using different methods depending on stations and low-flow partial-record stations
whether the location of interest is at a stream- were used to develop the equations. The stream-
gaging station, a low-flow partial-record station, or gaging stations had from 2 to 81 years of record,
an ungaged site where no data are available. Low- with a mean record length of 37 years. The
flow statistics for streamgaging stations can be low-flow partial-record stations had from 8 to
estimated using standard U.S. Geological Survey 36 streamflow measurements, with a median of
methods described in the report. 14 measurements.
The MOVE.1 mathematical method and a
All basin characteristics were determined
graphical correlation method can be used to
from digital map data. The basin characteristics
estimate low-flow statistics for low-flow partial-
record stations. The MOVE.1 method is recom- that were statistically significant in most of the
mended when the relation between measured final regression equations were drainage area, the
flows at a partial-record station and daily mean area of stratified-drift deposits per unit of stream
flows at a nearby, hydrologically similar stream- length plus 0.1, mean basin slope, and an indicator
gaging station is linear, and the graphical method variable that was 0 in the eastern region and 1 in
is recommended when the relation is curved. the western region of Massachusetts.
Equations are presented for computing the The equations were developed by use of
variance and equivalent years of record for esti- weighted-least-squares regression analyses, with
mates of low-flow statistics for low-flow partial- weights assigned proportional to the years of
record stations when either a single or multiple record and inversely proportional to the variances
index stations are used to determine the estimates. of the streamflow statistics for the stations.
The drainage-area ratio method or regres- Standard errors of prediction ranged from 70.7 to
sion equations can be used to estimate low-flow 17.5 percent for the equations to predict the 7-day,
statistics for ungaged sites where no data are 10-year low flow and 50-percent duration flow,
available. The drainage-area ratio method is respectively. The equations are not applicable for
generally as accurate as or more accurate than use in the Southeast Coastal region of the State, or
regression estimates when the drainage-area ratio where basin characteristics for the selected
for an ungaged site is between 0.3 and 1.5 times ungaged site are outside the ranges of those for the
the drainage area of the index data-collection site. stations used in the regression analyses.
Abstract 1A World Wide Web application was devel- and to provide estimates of the statistics for selected
oped that provides streamflow statistics for data- locations on ungaged streams. These studies were done
collection stations from a data base and for in cooperation with the Massachusetts Department of
ungaged sites by measuring the necessary basin Environmental Management, Office of Water
characteristics for the site and solving the regres- Resources (MOWR) and are referred to as the Basin
Yield studies. The MOWR uses the streamflow statis-
sion equations. Output provided by the Web appli-
tics to develop water-resources management plans for
cation for ungaged sites includes a map of the
each of the 27 major river basins in Massachusetts
drainage-basin boundary determined for the site, (fig. 1) and provides the streamflow statistics to other
the measured basin characteristics, the estimated State and local agencies to support their decision-
streamflow statistics, and 90-percent prediction making processes.
intervals for the estimates. Five other reports have been published as a result
An equation is provided for combining of the Basin Yield studies (Ries, 1994a, 1994b, 1997,
regression and correlation estimates to obtain 1999, 2000). The first three reports describe studies
improved estimates of the streamflow statistics done to develop regression equations for use in
for low-flow partial-record stations. An equation estimating low-flow statistics for ungaged sites. The
is also provided for combining regression and fourth report describes and provides data for a network
drainage-area ratio estimates to obtain improved of 148 low-flow partial-record (LFPR) stations that was
estimates of the streamflow statistics for ungaged established in 1988 at the beginning of the first Basin
Yield study and continued through 1996, during the
sites.
third Basin Yield study. The fifth report describes a
World Wide Web application that enables users to
INTRODUCTION select sites of interest on streams and then to obtain
estimates of streamflow statistics and basin characteris-
Low-flow statistics indicate the probable tics for the sites.
availability of water in streams during times when
conflicts between water supply and demand are most Purpose and Scope
likely to arise. Because of this, low-flow statistics are
needed by Federal, State, regional, and local agencies This report, the final report of the Basin Yield
for water-use planning, management, and regulatory study series, presents methods that can be used to
activities. These activities include (1) developing estimate low-flow statistics for streams in Massachu-
environmentally sound river-basin management plans, setts, and describes the analyses done to develop and
(2) siting and permitting new water withdrawals, evaluate the methods. Methods are presented for esti-
interbasin transfers, and effluent discharges, mating statistics for locations where various amounts
(3) determining minimum streamflow thresholds for of streamflow data are available and for locations
maintenance of aquatic biota, and (4) land-use where no data are available. Previously documented
planning and regulation. Low-flow statistics are also and generally accepted methods are presented for
needed by commercial, industrial, and hydroelectric estimating low-flow statistics for locations where
facilities to determine availability of water for water streamflow data are available. Analyses done to
supply, waste discharge, and power generation. develop and evaluate methods for estimating stream-
Low-flow statistics can be calculated from flow statistics for locations where no data are available
streamflow data collected at locations where the U.S. are described. The physical setting of Massachusetts,
Geological Survey (USGS) operates data-collection as it relates to the occurrence of low streamflows, is
stations, but it is not possible to operate stations at also briefly described.
every site where the statistics are needed. Because of Equations that can be used to estimate the 99-,
this, methods are needed for estimating low-flow 98-, 95-, 90-, 85-, 80-, 75-, 70-, 60-, and 50-percent
statistics for streams for which no data are available. duration flows; the 7-day, 2-year and the 7-day, 10-year
In 1988, the USGS began the first of three low flows; and the August median flow are presented
studies to develop and evaluate methods for estimating here. An evaluation of the accuracy of the equations
low-flow statistics for ungaged Massachusetts streams and limitations for their use is also provided, along
2 Methods for Estimating Low-Flow Statistics for Massachusetts Streamswith an example application. The equations provide estimated streamflow statistics and basin characteris-
estimates of low-flow statistics for streams with tics were provided. The equations provided in the
natural flow conditions, and supersede those from second report superseded those from the first report.
earlier reports. The third Basin Yield report (Ries, 1997) pro-
vides an equation for estimating August median
streamflows. This statistic is used by the U.S. Fish and
Previous Studies
Wildlife Service (1981) and some State agencies as the
Low-flow statistics for most streamgaging minimum summertime streamflow required for mainte-
nance of habitat for aquatic biota in New England. The
stations and many LFPR stations in Massachusetts
report also provides estimates of August median
were published by the USGS in a series of gazetteers
streamflows for sites on unregulated streams in Massa-
published as Water-Resources Investigations Reports,
chusetts where the values could be determined from
in a series of Hydrologic Atlas reports (see U.S.
available data, and describes how the August median
Geological Survey, 1987, for a complete listing of both
streamflow per square mile of drainage area varies
series), and in a series of ground-water assessment
throughout the State.
reports published as Water-Resources Investigations
The LFPR network described in the fourth Basin
Reports (Olimpio and DeLima, 1984; Lapham, 1988;
Yield report (Ries, 1999a) was established to provide
Myette and Simcox, 1992; DeLima, 1991; Hanson and additional data for use in the regression analyses and to
Lapham, 1992; Persky, 1993; Bratton and Parker, 1995; provide a better understanding of the physical
Bent, 1995; Friesz, 1996; Klinger, 1996). Statistics mechanisms that cause streamflow to vary in time and
provided in this report supersede those from the space. The report provides streamflow measurements
previous reports. collected systematically at the 148 LFPR stations in the
Studies that used regression analysis to network between 1989 and 1996, and also includes any
regionalize low-flow frequency statistics in the historical streamflow measurements available for the
northeastern United States include those for stations. In addition, the report provides estimated
Connecticut (Cervione, 1982), central New England streamflow statistics, basin characteristics, location and
(Wandle and Randall, 1994), Pennsylvania and New other descriptive information for each of the stations.
York (Ku and others, 1975), Maine (Parker, 1977), The estimated streamflow statistics include the 99-,
Massachusetts, New Hampshire, Rhode Island and 98-, 97-, 95-, 93-, 90-, 85-, 80-, 75-, 70-, 65-, 60-, 55-,
Vermont (Johnson, 1970), southeastern Massachusetts and 50-percent duration flows; the 7-day, 2-year and
(Tasker, 1972), and Massachusetts (Male and Ogawa, the 7-day, 10-year low flows; and the August median
1982; Vogel and Kroll, 1990; Risley, 1994). Studies flow. Basin characteristics measured include drainage
that regionalized flow-duration statistics include those area; total stream length; mean basin slope; area of
for Connecticut (Thomas, 1966), New Hampshire surficial stratified drift; area of wetlands; area of water
(Dingman, 1978), southeastern Massachusetts (Tasker, bodies; and mean, maximum, and minimum basin
elevation. The basin characteristics were measured for
1972), and Massachusetts (Male and Ogawa, 1982;
the stations from digital maps by use of a Geographic
Fennessey and Vogel, 1990; Ries, 1994a, 1994b).
Information System (GIS).
Reports for the first two Basin Yield studies The fifth Basin Yield report (Ries and others,
(Ries, 1994a, 1994b) provided equations for estimating 2000), a fact sheet, describes a World Wide Web
the 99-, 98-, and 95-percent duration streamflows and application that includes (1) a mapping tool that allows
also provided estimates of the streamflow statistics and users to specify locations on streams where low-flow
measured basin characteristics for selected ungaged statistics are needed, (2) a database that includes
streams in eastern Massachusetts river basins. The streamflow statistics, basin characteristics, location,
equations were developed for these studies by use of and descriptive information for all data-collection
regression analyses, which statistically relate the stations in Massachusetts for which streamflow
streamflow statistics to measured basin characteristics statistics were published previously, and (3) an
for the stations used in the analyses. The studies automated GIS procedure that determines the required
differed in the methods of regression analysis used to basin characteristics and solves the regression
develop the equations, the number of stations included equations provided in this report to estimate low-flow
in the analyses (more stations were used in the second statistics for the user-selected site. The World Wide
study), and the locations of ungaged streams for which Web application is further described later in this report.
Introduction 373°00´ 72°30´ 72°00´
VERMONT
01333000
01169801
01332000 01167200
01333100 01164300 01162500
01332900 01170100
01168300 01165090
1 01169000
01163250
01331400 01168400 01168650
3
01165500
01166105 7
01359967 01331380 01163298
01169600 01165250
01169900
011197015 01162900
42°30´ 01178200 01174000
01170575 01174050
01178300 01169800
K
YOR
01197120
01179900 8 01172810
2 6 01174565
NEW
01197140 01178490
01173260
01197300 4
01180650 Western Region 01174900
01180000 01171500
01197180 01175710
01197230 01171800
01171947 01173420 01175850
01180500
01180800 01173450 01175670
01171970
01198000 01181000 01175890
01176000
01176100
01198060 5 01176780
01124390
01183210 01176200
01123161
1 01198160
01185490
01177360 01176300 01123140 10
01198200
01186300 01187400 01184282
01184200
01176415
01184277
9
01123200
CONNECTICUT
42°00´
Figure 1. Locations of streamgaging stations and low-flow partial-record stations used to develop equations for
estimating low-flow statistics for ungaged Massachusetts streams and locations of streamgaging stations outside
Massachusetts used for correlation with low-flow partial-record stations, and boundaries of the 27 major river
basins and three hydrologic regions in the State.
4 Methods for Estimating Low-Flow Statistics for Massachusetts Streams71°00´
01073860
71°30´ 01100700
01101000
AT
NEW HAMPSHIRE 01101100
13 16
LA
01095928 01096505
01096515
NT
01089400
01096000 01096504 01100608
17
IC
01094396 15
OC
01095915
01094340 01102053
01097300
18
EA
11 01097280
01096910
01102490
19a
N
01096935
01095220
01094760
01103015 Massachusetts
01095380 14a 01096855 14b
BOSTON Bay
01096805
01103440
Eastern Region 01103435
01105630
01103445 01105582
01109460
20 01104960 01105575 19c 01105830
01105670
01104980 01105600
01111142 01104840
12 01103253
19b 01105568 01105820
01111200 01105270
01103330
01112190 21a 70°00´
01107000
01111225
9 01108600 01106460 01105861
01111300
RHODE ISLAND
27 25
01108140
01108180
01107400
21b
01109200 70°30´
01109087
01109225 011058839
01109090 011059106 Southeast
24 22
26
01105937 01105930
Coastal
01105947
01106000
01105935 Region
41°30´ Nantucket
Sound
23
0 50 MILES
0 50 KILOMETERS
23
Figure 1. Locations of streamgaging stations and low-flow partial-record stations used to develop equations for
estimating low-flow statistics for ungaged Massachusetts streams, locations of streamgaging stations outside
Massachusetts used for correlation with low-flow partial-record stations, and boundaries of the 27 major river
basins and three hydrologic regions in the State—Continued.
Introduction 5Physical Setting surrounded by upland areas underlain by till with
exposed bedrock outcrops. Till is an unsorted glacial
Massachusetts encompasses an area of 8,093 mi2 deposit that consists of material ranging in size from
in the northeastern United States. State environmental clay to large boulders. Till yields little water to adjacent
agencies divide Massachusetts into 27 major river streams in comparison to yields from coarse-grained
basins for planning and regulatory purposes (fig. 1). stratified drift. As a result, during summertime, streams
Some of these designated river basins are actually part in till areas tend to have less flow per unit of drainage
of larger river basins that extend into neighboring area than streams in areas of coarse-grain stratified
states. The Millers, Deerfield, Chicopee, and Westfield drift, and some small streams in till areas may go dry
River Basins are part of the Connecticut River Basin. (Wandle and Randall, 1994).
The Nashua, Concord, and Shawsheen River Basins are Ries (1997) defined three hydrologic regions in
part of the Merrimack River Basin. Several designated Massachusetts based on differences in August median
basins in coastal areas of Massachusetts were streamflow per square mile of drainage area (fig. 1).
comprised by grouping land areas drained by multiple These regions were the Western, the Eastern, and the
streams that discharge to the same receiving body of Southeast Coastal regions. The Western region was
salt water, such as Boston Harbor and Buzzards Bay. defined by all major basins that drain to the
The climate of Massachusetts is humid. Connecticut River plus those west of the Connecticut
Precipitation is distributed fairly evenly throughout the River Basin (basins 1 through 8 on fig. 1). The Eastern
State and throughout the year, and averages about region was defined as all basins east of the Western
45 in. annually. Average annual temperatures range region except Cape Cod, the Islands, the southern part
from 50˚F in coastal areas to 45˚F in the western of the South Coastal Basin, and the eastern part of the
mountains. Average monthly temperatures range from Buzzards Bay Basin, which define the Southeast
about 30˚F in February to about 71˚F in July in coastal Coastal region. August median flows per square mile
areas, and from about 20˚F in January to about 68˚F in were significantly higher, on average, in the Western
July in the western parts of the State (U.S. Commerce region than in the Eastern region.
Department, National Oceanic and Atmospheric Differences in August median streamflow per
Administration, 1989). Average evapotranspiration unit area between the Western and Eastern regions
ranges from 19 in. in southeastern Massachusetts to appeared to be more a function of climate and
22 in. in the western Mountains (Randall, 1996). physiography than surficial geology. Percentages of
Several physical characteristics vary from east to stratified-drift deposits were generally lower in the
west in Massachusetts. Elevations range from sea level Western region than in the Eastern region, but August
along the coast in eastern Massachusetts to almost median streamflows were higher in the Western region
3,500 ft in the western mountains. Basin relief and than in the Eastern region. The higher low flows per
mean basin slope, which are highly related, also tend to unit area in the Western region than in the Eastern
increase from east to west in Massachusetts. The extent region is likely explained by the combination of lower
of lakes, ponds, and wetlands, as a proportion of total mean annual temperatures, higher mean elevations,
basin area, generally decreases from east to west in higher relief, higher precipitation, lower evapo-
Massachusetts. The extent of coarse-grained stratified transpiration, and lower areal percentages of wetlands
drift, as a proportion of total basin area, also generally and water bodies in western Massachusetts than in
decreases from east to west. eastern Massachusetts.
Except during and for a short time after storms, The Southeast Coastal region is underlain
summertime flow in Massachusetts streams comes entirely by stratified-drift deposits, which are mostly
from ground water discharged by aquifers in coarse grained. Surface-water drainage boundaries in
unconsolidated deposits adjacent to the streams. This this region often do not coincide with contributing
discharge is termed base flow. High-yielding aquifers areas of ground water for streams in the area. In
usually are in stratified drift, sand and gravel deposits addition, dam regulations, diversions, or controls by
that are located primarily along the valley floors of cranberry bogs affect most streams in the region. As a
inland river basins and in coastal areas of southeastern result, insufficient data were available to define the
Massachusetts. The stratified-drift deposits usually are natural flow characteristics of streams in this region.
6 Methods for Estimating Low-Flow Statistics for Massachusetts StreamsAcknowledgments as part of the three Basin Yield projects. Data for many
of the network stations are used in the analyses
The authors would like to thank the MOWR described here.
for its long-term support of this work. Thanks also Miscellaneous-measurement stations usually are
to Aleda Freeman, Christian Jacqz, and the rest of established to obtain streamflow data for hydrologic
the staff of MassGIS, to John Rader, formerly of studies of limited regional extent and short duration.
MassGIS and USGS, and to Peter Steeves of USGS. The number and streamflow range of measurements
These people worked together to prepare numerous made at these stations varies depending on the
digital data layers needed for measuring the basin objectives of the study. High-flow as well as low-flow
characteristics used in the regression analyses, and to measurements commonly are made at miscellaneous-
develop methods for automating measurements of the measurement stations. Low-flow statistics can be
basin characteristics. The authors would also like to estimated for miscellaneous-measurement stations
express their appreciation to the many other USGS when the number and range of low-flow measurements
employees who assisted with collection and analysis collected at the stations approximates the requirements
of streamflow data, measurements of basin for measurements at a LFPR station.
characteristics, and preparation of this report. Many stations in Massachusetts have been
operated at different times as both LFPR stations and
miscellaneous-measurement stations. Methods used in
ESTIMATING METHODS FOR this study to estimate low-flow statistics for LFPR
DATA-COLLECTION STATIONS stations and miscellaneous-measurement stations were
the same and are described in the section “Low-flow
statistics for low-flow partial-record stations.” Because
The USGS operates three types of data-
the data and analysis methods were the same, both
collection stations for which low-flow statistics can
station types are referred to as LFPR stations for the
be estimated. These include (1) streamgaging stations,
remainder of this report.
(2) low-flow partial record (or LFPR) stations, and
(3) miscellaneous-measurement stations. Methods
used to estimate streamflow statistics at data-
Low-Flow Statistics for
collection stations differ depending on the type of
Streamgaging Stations
statistic and on the type of station. Continuous records
of streamflow are obtained at streamgaging stations.
Daily mean flows for all complete climatic
Streamflow statistics generally are determined directly
years of record are used to determine flow-duration
from the records for these stations using methods
and low-flow frequency statistics for streamgaging
described in the section “Low-flow statistics for stations. A climatic year begins on April 1 of the year
streamgaging stations.” noted and ends on March 31 of the following year.
Low-flow partial-record and miscellaneous- Daily mean flows for all complete Augusts for the
measurement stations are often established where period of record are used to determine August median
streamflow information is needed, but either (1) it is flows. Daily mean flows for USGS streamgaging
not physically or economically feasible to continuously stations in Massachusetts can be obtained by
monitor streamflows at the location, or (2) the amount downloading them from the World Wide Web address:
or accuracy of the streamflow information needed does http://waterdata.usgs.gov/nwis-w/MA/, or by
not require continuous monitoring at the location. At contacting the Massachusetts–Rhode Island District
LFPR stations, a series of streamflow measurements information officer at the address provided on the
are made during independent low-flow periods when back of the title page of this report.
all or nearly all streamflow is from ground-water The USGS has established standard methods
discharge. Usually about 10 low-flow measurements for estimating flow-duration (Searcy, 1959) and
are obtained systematically over a period of years. Ries low-flow frequency statistics (Riggs, 1972) for
(1999) summarized a network of LFPR stations streamgaging stations. The computer software
operated in Massachusetts during 1989 through 1996 programs IOWDM, ANNIE, and SWSTAT can be used
Estimating Methods for Data-Collection Stations 7to format input data, manage and display data, and statistics and probabilities of recurrence (recurrence
complete the statistical analyses, respectively, required intervals) of individual annual values can be analyzed.
to determine flow-duration and low-flow frequency A disadvantage of the approach is that generally at
statistics for streamgaging stations (Lumb and others, least 10 years of record are needed to determine the
1990; Flynn and others, 1995). These programs can be statistics with reasonable confidence.
downloaded from the World Wide Web address:
http://water.usgs.gov/software/surface_water.html.
Low-Flow Frequency Statistics
Flow-Duration Statistics Low-flow frequency statistics are determined for
A flow-duration curve is a graphical streamgaging stations from series of annual minimum
representation of the percentage of time streamflows mean flows for a given number of days. The statistics
for a given time step (usually daily) are equaled or can be computed for any combination of days of
exceeded over a specified period (usually the minimum mean flow and years of recurrence. For
complete period of record) at a stream site. Flow- example, the 7-day, 10-year low flow is determined
duration curves usually are constructed by first ranking from the annual series of minimum 7-day mean flows
all of the daily mean discharges for the period of record at a station. The mean flow for each consecutive 7-day
at a gaging station from largest to smallest, next period is computed from the daily records, and the
computing the probability for each value of being lowest mean value for each year represents that year in
equaled or exceeded, then plotting the discharges the annual series. The 7-day minimum mean flows are
against their associated exceedance probabilities usually fit to a log-Pearson Type III distribution to
(Loaiciga, 1989, p. 82). The daily mean discharges
determine the recurrence interval for an individual
are not fit to an assumed distribution. Flow-duration
7-day minimum mean flow (Riggs, 1972), although
analysis can be done by use of the USGS software
described above or by use of most commercially other researchers sometimes have used other
available statistical software. distributions (Vogel and Kroll, 1989). The value that
recurs, on average, once in 10 years is the 7-day,
Flow-duration statistics are points along a flow-
10-year low flow. The 7-day, 10-year low flow is used
duration curve. For example, the 99-percent duration
streamflow is equaled or exceeded 99 percent of the by the U.S. Environmental Protection Agency and by
time, whereas the 50-percent duration streamflow is many state and local agencies to regulate waste-water
equaled or exceeded 50 percent of the time. Strictly discharges into surface waters.
interpreted, flow-duration statistics reflect only the The USGS has, to a large extent, automated the
period for which they are calculated; however, when process of determining low-flow frequency statistics
the period of record used to compute the statistics is for streamgaging stations. The computer program
sufficiently long, the statistics often are used as an SWSTAT (Lumb and others, 1990, p. 141) determines
indicator of probable future conditions (Searcy, 1959).
the annual series of minimum mean flows, ranks them,
Vogel and Fennessey (1994) presented an fits them to a log-Pearson type III distribution, and
alternative method for determining flow-duration plots the resulting line of fit through the annual values.
statistics that indicate future conditions. This method How well the data fit the distribution, and the ultimate
requires determining flow-duration statistics for each
low-flow frequency values to be used, are left to the
individual year of record at a gaging station, then using
judgment of the individual hydrologist. Usually at least
the median of the annual values to represent the long-
term flow-duration statistics. Median annual flow- 10 years of record are needed to determine the statistics
duration statistics determined by use of this alternative with reasonable confidence. The annual series should
method tend to be higher than those calculated from the be checked for trends, and corrected if necessary,
entire period of record by use of the traditional before the log-Pearson analysis is done. The output
approach. The advantages of using the alternative from the analysis should be checked for outliers, and
method over the traditional approach are that corrected if necessary, before the frequency curve is
confidence intervals can easily be attached to the finalized.
8 Methods for Estimating Low-Flow Statistics for Massachusetts StreamsAugust Median Flows to extend the records for short-term streamgaging
stations to estimate flow-duration statistics that reflect
August median flows at streamgaging stations
long-term conditions for the stations. These methods
can be determined by two methods. The U.S. Fish and
are similar to those described below for estimating
Wildlife Service (USFWS) (1981) recommends
low-flow statistics for LFPR stations.
calculating August median streamflows as the median
value of the annual series of August monthly mean
streamflows for the period of record at a gaging station. Low-Flow Statistics for
The USFWS uses the August median flow calculated in
Low-Flow Partial-Record Stations
this manner as the minimum streamflow required for
summertime maintenance of habitat for biota in New Streamflow statistics for LFPR stations are
England streams. estimated by relating the low streamflow measurements
Charles Ritzi and Associates (1987) suggested made at the stations to daily mean discharges on the
calculating August median flows as the median of the same days at nearby, hydrologically similar
daily mean flows for all complete Augusts during the streamgaging stations. Lines or curves of correlation
period of record at a streamgaging station. Kulik are developed between the same-day discharges at
(1990) and Ries (1997) also used this method for the LFPR stations and the selected streamgaging
calculating August median flows. This method stations, and then the streamflow statistics for the
typically results in values of August median flows that gaging stations are entered into the relations to
are somewhat lower than those determined by use of determine the corresponding streamflow statistics for
the method suggested by the USFWS. The monthly the LFPR stations. A mathematical correlation method
mean values used by the USFWS to calculate August described by Hirsch (1982) is used when the relations
median flows tend to be skewed by infrequent storm are linear. A graphical correlation method described
events that cause the monthly means to be larger than by Riggs (1972) and Searcy (1959) is used when
the medians, thus “the median is a more useful statistic the relations are nonlinear. These methods were
than the mean for describing the central tendency” of recommended for use by the USGS Office of Surface
the daily data (Kulik, 1990). Water in Technical Memorandum No. 86.02, Low-
Flow Frequency Estimation at Partial-Record Sites,
Streamflow Statistics for issued December 16, 1985. Both methods assume that
Streamgaging Stations with Short Records the relation between the discharges at the LFPR station
and the streamgaging station remains constant with
Streamflow statistics are often needed for
time, thus the relation between the same-day flows can
streamgaging stations with short records that may not
be used to estimate streamflow statistics that represent
reflect long-term conditions, and thus may not be
long-term conditions.
useful as indicators of future conditions. Streamflow
Medium- to high-range streamflow measure-
record extension or augmentation can be used to adjust
ments made at some LFPR stations can be useful for
the records for these stations to reflect a longer period.
estimating flow statistics near the median flow.
This is usually done by developing a relation between
Commonly, however, measurements made in these
the daily mean streamflows or the streamflow statistics
ranges need to be excluded from the analyses because
at the short-term station and the daily mean
the measurements were made at times when flow was
streamflows or the streamflow statistics for the same
rapidly changing, thus the measurements correlate
period at a nearby and hydrologically similar gaging
poorly with same-day mean flows at gaging stations.
station with a long record.
Vogel and Kroll (1991) demonstrated the value
Mathematical Method
of augmenting streamflow records to obtain improved
estimates of low- and peak-flow frequency statistics A mathematical record-extension method known
for streamgaging stations. They also described methods as the Maintenance Of Variance Extension, Type 1
that can be used for augmenting records to estimate (MOVE.1) method (Hirsch, 1982) can be used to
these statistics. Searcy (1959, p.12–14) and Ries estimate streamflow statistics for LFPR stations when
(1994a, p. 21–22) described methods that can be used the relation between the logarithms of the same-day
Estimating Methods for Data-Collection Stations 9sy
discharges at the LFPR station and a nearby gaging Yˆ i = Y + ---- ( X i – X ) , (1)
sx
station is linear. The method is applied by first
calculating logarithms-base 10 of the same-day flows
for the LFPR and gaging stations and graphing the and then retransforming the estimates by
values to ascertain the linearity of the relation. The exponentiating the values ( 10 Ŷ i ) to convert the
correlation coefficient is also computed as an indicator estimates into their original units of measurement.
of linearity. If the relation appears linear, the MOVE.1
The MOVE.1 relation between an LFPR station,
method is used; if not, a graphical method is used, as
explained below. Hemlock Brook near Williamstown, Mass., and a
streamgaging station, Green River at Williamstown,
When the graph of the data appears linear, the
Mass., is shown as an example in figure 2. The line
means (Y and X) and standard deviations (sy and sx) of
through the data points was determined by inserting the
the logarithms-base 10 of the same-day flows for the
LFPR and gaging stations and the logarithms-base 10 same-day flows for the gaging station into the MOVE.1
of the streamflow statistics (Xi) for the gaging station equation as the Xi values to obtain estimated same-day
are calculated. Estimates of the streamflow statistics flows for the LFPR station, then connecting the points
( Ŷ i ) for the LFPR station are obtained by inserting the to illustrate how the MOVE.1 estimates fit the original
calculated values into the MOVE.1 equation: data.
10
CORRELATION COEFFICIENT = 0.974
HEMLOCK BROOK AT WILLIAMSTOWN, MASS.
DISCHARGE, IN CUBIC FEET PER SECOND
1
SAME-DAY DISCHARGES
MOVE.1 RELATION
0.1
1 10 100
GREEN RIVER NEAR WILLIAMSTOWN, MASS. DISCHARGE, IN CUBIC FEET PER SECOND
Figure 2. Example MOVE.1 relation between a low-flow partial-record station, Hemlock Brook near
Williamstown, Mass., and a streamgaging station, Green River at Williamstown, Mass.
10 Methods for Estimating Low-Flow Statistics for Massachusetts StreamsGraphical Method Combining Estimates Determined
from Multiple Index Sites
The graphical method (Searcy, 1959; Riggs,
1972) is used when curvature is apparent in the plot of Selection of individual gaging stations for
logarithms-base 10 of the same-day flows. The method relation to a LFPR station is based on distance between
is applied by first plotting the original (non-log) values the stations and similarity of basin characteristics
of the same-day flows on log-log paper and drawing a between the stations. In Massachusetts, the measured
smooth curve through the plotted points that appears to streamflows at a LFPR station usually will correlate
best fit the data. Next, the calculated streamflow well with more than one gaging station. When this
statistics for the gaging station are entered into the
happens, MOVE.1 or graphical relations between a
curve of relation and corresponding values for the
given LFPR station and each of several gaging stations
LFPR station are read from the graph. Log-log plots
can be developed to estimate the streamflow statistics
sometimes have extreme curvature in the very low
for the LFPR station. This process results in multiple
end of the relation. Because of this, it is sometimes
estimates of the streamflow statistics for a single LFPR
necessary to replot the data on arithmetic paper to
station, when only a single best estimate is desired.
adequately define the relation in this range and to avoid
long downward extrapolations that would otherwise be Tasker (1975) stated that when independent
necessary with log-log plots. multiple estimates of streamflow statistics are available
The graphical relation between an LFPR station, for a single station, the best estimate can be obtained
Hopping Brook near West Medway, Mass., and a by weighting each individual estimate by its variance
streamgaging station, West River near Uxbridge, and averaging the weighted estimates. This final
Mass., is shown as an example in figure 3. The curve weighted estimate is best because its variance is less
was fit through the data visually to minimize overall than or equal to the variances of each of the
differences between the observed and fit values. individual estimates.
100
HOPPING BROOK NEAR WEST MEDWAY, MASS.
DISCHARGE, IN CUBIC FEET PER SECOND
10
1
0.1
SAME-DAY DISCHARGES
GRAPHICAL RELATION
0.10
1 10 100
GREEN RIVER NEAR WILLIAMSTOWN, MASS. DISCHARGE, IN CUBIC FEET PER SECOND
Figure 3. Example graphical relation between a low-flow partial-record station, Hopping Brook near West
Medway, Mass., and a streamgaging station, West River near Uxbridge, Mass.
Estimating Methods for Data-Collection Stations 11Calculated variances for each individual hydrologically similar gaging station. Modifications
estimate of the streamflow statistics for each LFPR to the Hardison and Moss equation were needed to
station were needed to obtain the final best estimates generalize its use for other streamflow statistics and to
for the stations. Variances were calculated by use of allow for the MOVE.1 or graphical methods of line
the equation fitting to be used rather than the ordinary-least-squares
method of line fitting. Assumptions for use of equation
VR 2 SE S, G 2 M 2 are generalized from Hardison and Moss (1972):
V S, U = ------ 1 + -------------- + -------------- + -------------- --------------
1 z M
M M – 3 M – 3 s B, G M – 3 1. The true relation between the logarithms of the
2 base-flow measurements at the LFPR station
+ b V S, G , (2) and the same-day mean streamflows at the
gaging station is the same as the true relation
where: between the logarithms of the data from which
VS,U is the sample variance of the streamflow the low streamflow statistics are calculated. In
statistic at the LFPR station, in log units; the case of the 7-day low-flow statistics, the
VS,G is the sample variance of the streamflow data are calculated from an annual series of
statistic at the gaging station, in log units; minimum 7-day mean flows. In the case of the
VR is the variance about the MOVE.1 or graphical flow-duration and August median statistics, the
line of relation; data are calculated from the daily mean flows.
M is the number of base-flow measurements; 2. The relation between the logarithms of the data
SES,G is the standard error of the streamflow statistic from which the low-flow statistics are
at the gaging station, which equals the square calculated is the same as the relation between
root of VS,G; the flow statistics for the stations.
b is computed as r(sB,U/sB,G), where r is the
3. The time-sampling errors in the streamflow
correlation coefficient between the low
statistics that are used to enter the regression
streamflow measurements made at the LFPR
equation are independent of the variation in the
station and the same-day mean discharges at
base-flow measurements used to define the
the gaging station (the value of r can be set to
equation.
1 when MOVE.1 is used to obtain the
estimate), and sB,U is the standard deviation 4. The logarithms of the measured streamflows at
of the logarithms-base 10 of the low the LFPR station and the same-day mean
streamflow measurements made at the LFPR streamflows at the gaging station follow a
station; bivariate normal distribution.
sB,G is the standard deviation of the logarithms-base 5. The M measurements made at the LFPR station
10 of the mean discharges at the gaging are statistically independent estimates of the
station on the same days the low-flow base-flow relation.
measurements were made at the ungaged Hardison and Moss noted that the first four
site; and assumptions appeared to be reasonable under the
z is the number of standard deviation units conditions in which application of the original equation
between the mean of the logarithms-base 10 2 was intended. These assumptions are reasonable for
of the same-day mean discharges at the the modified equation 2 as well. Hardison and Moss
gaging station and the logarithm-base 10 of also noted that assumption 5 could be satisfied by
the streamflow statistic at the gaging station. applying criteria for using only those measurements
Equation 2 is modified from an equation that can be reasonably assumed independent to define
developed by Hardison and Moss (1972) to determine the relation. The criterion usually applied is that the
the variance of estimates of 7-day, T-year low flows base-flow measurements used in the relation should be
obtained from an ordinary-least-squares (OLS) separated by significant storm events (Stedinger and
regression of the logarithms-base 10 of base-flow Thomas, 1985). Collection of low streamflow
measurements at a LFPR station to the logarithms-base measurements at LFPR stations in Massachusetts has
10 of same-day mean discharges at a nearby, generally followed that criterion.
12 Methods for Estimating Low-Flow Statistics for Massachusetts Streams2
2 s B, U
N U = R S I S, G k ----------
2 2
When estimates are obtained for LFPR stations
s B, G
from relations with more than one streamflow-gaging 2 2 2
station, the individual estimates, Q S, U i , (where i = 1, V R 1 + z 2 b RS I S, G
⁄ ------- -------------- + ---------------------- , (6)
..., n, and n is the number of individual estimates of M K NG
statistic S for LFPR station U) can be weighted by the
reciprocals of their variances, determined from where all variables are as previously defined, and:
equation 2, to obtain minimum-variance estimates, NU is the equivalent years of record at the partial-
Q S, U , for each of the statistics from the equation record station;
w n
∑ ( Q S, U ⁄ V S, U )
i i
NG is the years of record at the streamflow-gaging
station used in the relation;
i=1
Q S, U = -----------------------------------------
n - . (3) IS,G is the standard deviation of the logarithms-base
∑
w
( 1 ⁄ V S, U ) 10 of the observed flows (annual series for
i
i=1
frequency statistics or daily flows for
Weighted variances, V S, U , can be determined for the duration statistics) at the streamflow-gaging
w
weighted estimates from the equation station
k is from equation 9 of Hardison and Moss
n (1972),
V S, U
w
= 1⁄ ∑ ( 1 ⁄ V S, U i ) . (4)
r + -------------- ( 1 – r ) ; and
i=1 2 M–4 2
k =
M – 2
(7)
Standard errors, SE S, U , in percent, for the weighted
w
estimates can be obtained from the equation (Stedinger K is from equation 3 of Hardison and Moss
and Thomas, 1985, p. 18) (1972),
2
SE S, U = 100 exp ( 5.3018V w ) – 1 . (5) K = (1 + z )
w
2
z M S ,G M
2 SE
1
Equation 2 does not account for errors inherent in the ⁄ 1 + -------------- + -------------- + -------------
- -------------- ; (8)
M – 3 M – 3 sB ,G M – 3
discharge measurements made at the LFPR station or
in the mean daily discharges determined for the gaging RS is a correction factor that depends on the
stations. In addition, the estimates obtained for an streamflow statistic being estimated, and is
LFPR station by use of the MOVE.1 or graphical determined by combining the equation that
method with multiple gaging stations are not truly appears in table 1 of Hardison and Moss
independent because of cross correlation of the (1972),
streamflow records at the gaging stations. As a result,
the final estimates obtained using equations 2 and 3
may not truly be the best possible, and the true R S = ( SE S, G N ) ⁄ I S ,G , (9)
variances and standard errors are somewhat larger than
those obtained using equations 4 and 5. with the equation
2
The equivalent years of record also can be 1 + kS ⁄ 2
computed for estimates of streamflow statistics for the SE S, G = I S ,G --------------------- (10)
N
LFPR stations. The equivalent years of record is the
length of time that a streamgaging station would need from Hardison (1969, p. D212) to obtain
to be operated at the location of the LFPR station to
2 2
obtain an estimate of the streamflow statistic with equal RS = 1 + k S ⁄ 2 . (11)
accuracy. The equivalent years of record for LFPR
stations is computed from an equation developed by Subscripts have been changed from their original
combining equations 7, 8, and 9 in Hardison and Moss appearance in equations 6 to 11 to generalize from
(1972) and solving for the number of years of record. T-year statistics to other streamflow statistics. In
The resulting equation is: equations 10 and 11 above, kS is the number of
Estimating Methods for Data-Collection Stations 13standard deviation units between the streamflow topographic maps. Streamflow statistics are computed
statistic and the mean of the data from which it is for the index station, then the statistics (numerical
calculated. From assumption 4 above, the annual series values) are divided by the drainage area to determine
of 7-day low flows and the daily mean streamflows streamflows per unit area at the index station. These
from which the flow-duration statistics and the August values are multiplied by the drainage area at the
median streamflows are calculated are distributed log- ungaged site to obtain estimated statistics for the site.
normally, and thus kS can be obtained from a table of This method is most commonly applied when the index
standard normal deviates as appears in most statistical gaging station is on the same stream as the ungaged site
textbooks. Values for the 99-, 98-, 95-, 90-, 85-, 80-, because the accuracy of the method depends on the
75-, 70-, 60-, and 50-percent duration streamflows, the proximity of the two, on similarities in drainage area
August median streamflow, and the 7-day, 10- and and on other physical and climatic characteristics of
2-year streamflows are 2.3263, 2.0537, 1.6449, 1.2816, their drainage basins.
1.0364, 0.8416, 0.6745, 0.5244, 0.2533, 0.0, 0.0, Several researchers have provided guidelines as
1.2816, and 0.0, respectively (Iman and Conover, 1983, to how large the difference in drainage areas can be
p. 434–435). When estimates for LFPR stations are before use of regression equations is preferred over use
obtained from relations with more than one of the drainage-area ratio method. Guidelines have
streamgaging station, the individual calculations of been provided for estimating peak-flow statistics, and
equivalent years of record can be weighted by the usually the recommendation has been that the drainage
reciprocals of the variances of the estimated streamflow area for the ungaged site should be within 0.5 and 1.5
statistics, determined from equation 2, then the times the drainage area of the index station (Choquette,
individual weighted equivalent years of record can be 1988, p. 41; Koltun and Roberts, 1990, p. 6; Lumia,
averaged to obtain the final weighted equivalent years 1991, p. 34; Bisese, 1995, p. 13). One report (Koltun
of record for the LFPR station by substituting the and Schwartz, 1986, p.32) recommended a range of
equivalent years of record estimates for the discharge 0.85 to 1.15 times the drainage area of the index station
estimates in equation 3 above. for estimating low flows at ungaged sites in Ohio. None
of these researchers provided any scientific basis for
use of these guidelines. R.E. Thompson, Jr. (U.S.
ESTIMATING METHODS FOR Geological Survey, written commun., 1999), however,
UNGAGED STREAM SITES recently completed a study that provides evidence
supporting use of ratios between 0.33 and 3.0 for
Estimates of streamflow statistics often are
streams in Pennsylvania.
needed for sites on streams where no data are available.
The two methods used most commonly to estimate Because of uncertainty in an appropriate range
statistics for ungaged sites are the drainage-area ratio for use of the drainage-area ratio method for streams in
method and regression equations. The drainage-area Massachusetts, an experiment was designed to
ratio method is most appropriate for use when the determine the ratio range in which the method is likely
ungaged site is near a streamgaging station on the same to provide better estimates of low streamflow statistics
stream (nested). Regression equations can be used to than use of regression equations. Five river basins with
obtain estimates for most ungaged sites. Additional one or more continuous gaging stations in each basin
details on application of these methods is provided were chosen for the experiment to represent the varied
below. topography, geology, and precipitation of
Massachusetts. Two basins, the Green and the West
Branch Westfield, are in the mountainous western part
Drainage-Area Ratio Method of the State; two basins, the Quaboag and the
Squannacook, are in the foothills of the central part of
The drainage-area ratio method assumes that the the State; and one basin, the Wading, is in the flat,
streamflow at an ungaged site is the same per unit area low-lying landscape typical of eastern Massachusetts.
as that at a nearby, hydrologically similar streamgaging A total of 25 LFPR stations were established
station used as an index. Drainage areas for the upstream and downstream from 8 streamgaging
ungaged site and the index station are determined from stations in the 5 basins. Most of the LFPR stations have
14 Methods for Estimating Low-Flow Statistics for Massachusetts Streamssmaller drainage areas than those for the streamgaging and one discontinued streamgaging station) at which
stations because historically most streamgaging streamflow measurements were made in the Wading
stations in Massachusetts have been established near River Basin, only three of the stations (01108490,
the downstream ends of rivers. Locations and drainage 01108600, and 01108700) were used to compare
boundaries for the streamgaging stations and LFPR results of the different estimation methods. Discharges
stations are shown for each basin in figures 4A to 4E. and basin characteristics from stations 01108440 and
Station descriptions for the stations are in table 1. 01108470 were subtracted from station 01108490, and
Seven to ten discharge measurements were made station 01108500 was subtracted from 01108700 to
at each of the LFPR stations during 1994 and 1995. determine discharges and basin characteristics
The measurements were published in the Mass. annual representative of the naturally flowing areas above
data reports for those years (Gadoury and others, 1995; those stations. The adjusted discharges and basin
Socolow and others, 1996, 1997). The measurements, characteristics were used to estimate unregulated
along with historical measurements available at three streamflow statistics for the stations. Station 01108600
stations, were used to estimate streamflows at the 99-, was not affected by regulation or diversions.
98-, and 95-percent durations and August median flows
for the stations using the methods described above for The drainage area for the Wading River below
LFPR stations. Estimates of the flow-duration statistics the West Mansfield streamgaging station (station
were also derived for the stations using the drainage- 01108500) and above the Norton streamgaging station
area ratio method and the regression equations (station 01109000) is not affected by regulation or
developed by Ries (1994b, 1997). The regression diversions, whereas the drainage area above the West
equations presented later in this report were not used Mansfield station is affected by regulation and
because they were not yet available at the time of the diversions. Streamflow statistics for the West Mansfield
analysis. station were subtracted from those for the Norton
Two gaging stations were available in some station to obtain the streamflow statistics for the
basins for the analysis (table 1). To increase the sample naturally flowing part of the drainage area above the
size for the analysis of the drainage-area ratio method, Norton station.
drainage-area ratio estimates and regression-equation The four streamflow statistics (99-, 98-, and
estimates were determined for the streamgaging
95-percent duration and August median streamflows)
stations in addition to the estimates determined from
estimated by the three different methods (correlation,
the records for the stations. The drainage-area ratio
drainage-area ratio, and regression equation) for each
estimates were determined for each streamgaging
of the LFPR and streamgaging stations used in the
station by applying the flow per unit area for one
streamgaging station to the drainage area for the other analysis are presented in table 7 (back of the report).
streamgaging station. The longest common period of The estimates derived by correlation, shown in the
record available for the streamgaging stations in each column labeled “Correlation method estimate or
basin was used to compute the streamflow statistics for computed,” were considered the best estimates
the analysis to avoid differences in the statistics due to available for the LFPR stations for the analysis,
differences in record length. and they were compared to the estimates derived by
The Wading River Basin, unlike the other four the other methods. The correlation estimates were
basins used in the experiment, has water withdrawals considered the best estimates because they were
and regulated streamflows in parts of the basin (see derived from actual streamflow data for the stations,
table 1, remarks). It was chosen for use in the whereas the drainage-area ratio and regression
experiment because the unregulated part of the basin is estimates were derived indirectly based on an
the largest unregulated area in southeastern assumed or statistical relation between the basin
Massachusetts. Discharges, drainage areas, and other characteristics for the LFPR stations and stream-
basin characteristics used to solve the regression gaging stations. Statistics shown for streamgaging
equations were adjusted for stations downstream from stations in the column labeled “Correlation method
the diversions and regulation to correct for these estimate or computed” were computed from daily-
activities. Of the seven stations (including one active flow records.
Estimating Methods for Ungaged Stream Sites 1571o00'
A. SQUANNACOOK 73o00' 72o00'
RIVER BASIN 42o30'
70o00'
0 50 MILES 42o00'
0 50 KILOMETERS 41o30'
71o50' 71o40'
42o45'
01095977
01095930
01095928
01095990
01096000
01096035
42o35' EXPLANATION
BASIN BOUNDARY
01096000 STREAMFLOW GAGING
STATION AND NUMBER
0 2.5 5 MILES
01095930 LOW-FLOW PARTIAL-
RECORD STATION
AND NUMBER
0 2.5 5 KILOMETERS
Figure 4. Locations and drainage boundaries of low-flow partial-record stations and gaging stations in the (A) Squannacook,
(B) Wading, (C) Quaboag, (D) Green, and (E) West Branch Westfield River Basins, Massachusetts.
16 Streamflow Statistics and Basin Characteristics for Low-Flow Stations in Massachusetts71o00'
73o00'
B. WADING 72o00'
42o30'
RIVER BASIN
70o00'
0 50 MILES 42o00'
0 50 KILOMETERS 41o30'
71o20' 71o10'
42o05'
01108440
01108470
01108490
01108500
SHADED AREAS ARE AFFECTED 01108600
BY REGULATION, DIVERSIONS, OR BOTH
01108700
01109000
EXPLANATION
41o55' BASIN BOUNDARY
01109000 STREAMFLOW GAGING
STATION AND NUMBER
01108600 LOW-FLOW PARTIAL-
RECORD STATION 0 1.0 2 MILES
AND NUMBER
0 1.0 2 KILOMETERS
Figure 4. Locations and drainage boundaries of low-flow partial-record stations and gaging stations in the (A) Squannacook,
(B) Wading, (C) Quaboag, (D) Green, and (E) West Branch Westfield River Basins, Massachusetts—Continued.
Estimating Methods for Ungaged Stream Sites 17You can also read
