Prepared in cooperation with the New Jersey Department of Environmental Protection

Implementation of MOVE.1, Censored MOVE.1, and Piecewise 

MOVE.1 Low-Flow Regressions with Applications at PartialRecord Streamgaging Stations in New Jersey

Open-File Report 2018–1089

U.S. Department of the Interior

U.S. Geological Survey
Cover. Image showing R language code written by Dave Lorenz, U.S. Geological Survey on 

December 30, 2015.
Implementation of MOVE.1, Censored 

MOVE.1, and Piecewise MOVE.1 LowFlow Regressions with Applications at 

Partial-Record Streamgaging Stations in 

New Jersey

By Susan J. Colarullo, Samantha L. Sullivan, and Amy R. McHugh

Prepared in cooperation with the New Jersey Department of Environmental 

Protection

Open-File Report 2018–1089

U.S. Department of the Interior

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

RYAN K. ZINKE, Secretary

U.S. Geological Survey

James F. Reilly II, Director

U.S. Geological Survey, Reston, Virginia: 2018

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Suggested citation:

Colarullo, S.J., Sullivan, S.L., and McHugh, A.R., 2018, Implementation of MOVE.1, censored MOVE.1, and piecewise 

MOVE.1 low-flow regressions with applications at partial-record streamgaging stations in New Jersey: U.S. Geological Survey Open-File Report 2018–1089, 20 p., https://doi.org/10.3133/ofr20181089.

ISSN 2331–1258 (online)
iii

Contents

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

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

Purpose and Scope ..............................................................................................................................3

Previous Studies ...................................................................................................................................3

Methods...........................................................................................................................................................3

Ordinary Least Squares Regression..................................................................................................4

Maintenance of Variance Regression (MOVE.1).............................................................................5

Censored MOVE.1.................................................................................................................................7

Piecewise MOVE.1..............................................................................................................................10

Application of the R Script ................................................................................................................10

Selection of Flows .....................................................................................................................10

Index Station Selection.............................................................................................................12

Derivation of Weights for Averaging Predicted Flow Statistics ........................................14

Standard Error of Prediction for the 7Q10 Flow Statistic ...........................................15

Time-Sampling Error for the 7Q10 Flow Statistic.........................................................15

Standard Error of Estimate for the 7Q10 Flow Statistic ..............................................15

Predicted Flow Statistics..........................................................................................................16

Flow Chart ...................................................................................................................................18

Summary........................................................................................................................................................18

Acknowledgments.......................................................................................................................................20

References Cited..........................................................................................................................................20
iv

Figures

1. Scatterplot showing MOVE.1 regression line for base flows measured at

partial-record station 01396190, South Branch Raritan River at Four Bridges, N.J.,

and daily flows measured at index station 01380450, Rockaway River at Main Street

at Boonton, N.J .............................................................................................................................6

2. Scatterplot showing flow residuals for partial-record station 01396190, South

Branch Raritan River at Four Bridges, N.J., from MOVE.1 regression on daily flows

at index station 01380450, Rockaway River at Main Street at Boonton, N.J......................6

3. Scatterplot showing percent flow residuals for partial-record station 01396190,

South Branch Raritan River at Four Bridges, N.J., from MOVE.1 regression on daily

flows at index station 01380450, Rockaway River at Main Street at Boonton, N.J ...........6

4. Scatterplots showing A, regular MOVE.1 regression and B, censored MOVE.1

regression for partial-record station 01397800, Neshanic River near Flemington,

N.J., and index station 01401000, Stony Brook at Princeton, N.J.........................................8

5. Scatterplots showing A, regular MOVE.1 regression and B, piecewise MOVE.1

regression for partial-record station 01381900, Passaic River at Pine Brook N.J.,

and index station 01379000, Passaic River at Millington, N.J ...............................................9

6. Scatterplots showing MOVE.1 regression lines used to evaluate the relation

between base-flow measurements at partial-record station 01396190, South

Branch Raritan River at Four Bridges, N.J., and daily flows at five index stations in

New Jersey..................................................................................................................................13

7. Flow chart showing the MOVE.1 R script ...............................................................................19

Tables

1. Tests applied to flows for inclusion in MOVE.1 regressions ...............................................11

2. Percent standard errors of estimate at index stations used to weight flow 

statistics at partial-record station 01396190, South Branch Raritan River at 

Four Bridges, N.J ........................................................................................................................14

3. Flow statistics and regression metrics output by the R script for MOVE.1 ......................16

4. Percent differences in estimated MOVE.1 low-flow statistics between the legacy

code and the new R script ........................................................................................................17

5. Minimum and maximum percent differences in regression metrics between the

legacy code for MOVE.1 and the new R script for MOVE.1.................................................17
v

Conversion Factors

 U.S. customary units to International System of Units

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

inch (in.) 25.4 millimeter (mm)

foot (ft) 0.3048 meter (m)

Area

square mile (mi2) 259.0 hectare (ha)

square mile (mi2) 2.590 square kilometer (km2) 

Volume

cubic foot (ft3) 0.02832 cubic meter (m3) 

Flow rate

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s)

Abbreviations

MOVE.1 Maintenance of Variance Extension, Type 1

SAS Statistical Analysis System

R R Foundation for Statistical Computing

NWIS USGS National Water Information System

NJDEP New Jersey Department of Environmental Protection

Implementation of MOVE.1, Censored MOVE.1, and 

Piecewise MOVE.1 Low-Flow Regressions with 

Applications at Partial-Record Streamgaging Stations in 

New Jersey

By Susan J. Colarullo, Samantha L. Sullivan, and Amy R. McHugh

Abstract

The U.S. Geological Survey (USGS) uses Maintenance 

of Variance Extension Type 1 (MOVE.1) regression to 

transfer streamflows measured at long-term continuous-record 

streamgaging stations to partial-record (PR) streamgaging 

stations where intermittent base-flow measurements are 

available. MOVE.1 regression is used widely throughout the 

hydrologic community to extend historical low flows and lowflow statistics at continuous-record streamgaging stations to 

streamgaging stations that have access to only a partial record 

of low flows. The method correlates base-flow measurements 

at PR streamgaging stations with daily mean streamflows 

measured at index stations that exhibit similar streamflow 

characteristics.

Following changes in the computing platform for storing, processing, retrieving, and publishing National Water 

Information System (NWIS) hydrologic data, legacy Statistical Analysis System (SAS) code developed by the USGS to 

implement the MOVE.1 regression was no longer suitable for 

reading and processing NWIS streamflow data. To migrate the 

MOVE.1 program so that it could continue to read streamflow 

data using the new hydrologic data platform, the SAS code 

was re-written in R, an open source programming language 

and software environment for statistical computing and graphics supported by the R Foundation for Statistical Computing. 

The work described in this report was performed in a study 

conducted by USGS in cooperation with the New Jersey 

Department of Environmental Protection.

During migration from SAS to R, graphical and tabular 

output generated by the R script was compared to output produced by the legacy SAS code to ensure that equations used 

to perform the MOVE.1 regression remained the same. An 

option to perform censored MOVE.1 regression was added to 

extend the MOVE.1 methodology to cases where one or more 

measured continuous-record or PR streamgaging station flows 

are zero valued. In addition to permitting censored regression, the new R script includes an option to perform piecewise 

MOVE.1 regression when the relation between PR station and 

index station low flows varies significantly across the range of 

index station streamflows.

Together with traditional MOVE.1 regression, censored 

and piecewise MOVE.1 regression methods implemented 

by the R script offer less biased estimates than ordinary least 

squares regression for the annual 7-day 10-year and other 

low-flow statistics at PR stations for a range of base-flow 

conditions. The R script is used to implement the MOVE.1 

regression methods across a variety of computing platforms.

Introduction

The New Jersey Department of Environmental Protection 

(NJDEP) is responsible for issuing permits to surface-water 

users in New Jersey, based in part on low-flow statistics estimated by hydrologists at the U.S. Geological Survey (USGS). 

Low-flow statistics are needed by NJDEP to check for compliance with the law controlling diversions from surface waters 

in the State of New Jersey (N.J.S.A 58:1-13 et seq.) and to 

establish waste-load allocations under the New Jersey Pollutant Discharge Elimination System program (N.J.C.A. 7:14A) 

(State of New Jersey, Department of Environmental Protection, 2011; State of New Jersey, Department of Environmental 

Protection, Division of Water Quality, 2015). Many streams 

for which permits are required do not have streamflow data 

from continuous-record streamgaging stations but are instead 

measured for flows intermittently to provide at least a partial 

record of gaged flows. For such streams, low-flow information 

can be “borrowed” from continuous-record stations located 

on hydrologically similar streams by using widely accepted 

principles of statistical inference. These continuous-record 

streamgaging stations, hereafter referred to as index stations or 

reference stations, can provide useful information about missing flows at intermittently observed gaging stations, referred to 

in the remainder of this report as partial-record (PR) stations.
2 Implementation of MOVE.1, Censored MOVE.1, and Piecewise MOVE.1 Low-Flow Regressions

The cooperative NJDEP/USGS low-flow project, which 

has been active since 1959, charges USGS with providing 

NJDEP with low-flow statistics needed to meet permitting 

requirements. Owing to cost considerations, many streams for 

which permits are required do not have a minimum of 10 years 

of daily streamflow data from index stations. One goal of 

the long-term cooperative low-flow project is to establish a 

network of partial-record stations on ungaged streams and correlate low-flow measurements at the PR stations with streamflow measurements from index stations. Unlike index stations 

where streamflow is measured continuously in time, instantaneous streamflow measurements are made under base-flow 

conditions at PR stations. These measurements are an important source of cost-effective data for computing the low-flow 

statistics required for permitting at ungaged stream locations.

Until 2016, low-flow statistics were estimated by the 

USGS using legacy Statistical Analysis System (SAS) code 

that implements the Maintenance of Variance Extension Type 

1 (MOVE.1) method to correlate instantaneous base-flow 

measurements at PR stations with continuously recorded 

daily mean flows at index stations. The legacy code estimated 

low-flow statistics for PR stations on the basis of the MOVE.1 

regression line and low-flow statistics estimated at index 

stations. Typical low-flow statistics used in the State’s point 

source and water allocation permitting process include the 

annual minimum daily flow, the 75th percentile daily mean 

flow duration, and the 1Q10, 7Q10, and 30Q10 low-flow flow 

statistics. The 1Q10, 7Q10, and 30Q10 low-flow frequency 

statistics represent the smallest average discharge over any 

consecutive 1-day, 7-day, and 30-day periods, respectively, 

that occur once every 10 years, on average. The annual 7Q10 

is of particular importance because it provides the foundation 

for defining passing-flow requirements in New Jersey and 

is used to support NJDEP surface-water permitting decisions within the State (Hoffman and Domber, 2013). Because 

low-flow frequency statistics quantify base flows of varying 

durations over an average 10-year period, at least 10 years of 

continuous daily record should ideally be available to reliably 

estimate 10-year recurrence interval flow statistics (Searcy, 

1959).

To provide sufficient data to reliably extend the low-flow 

record at PR stations, 2 to 3 measurements are made at a PR 

station each year under high and low base-flow conditions for 

a period of roughly 5 to 6 years. Ten to 12 base-flow measurements generally provide an adequate amount of data to 

establish a correlation between base flows intermittently measured at a PR station and low flows measured at nearby index 

stations. Regression techniques grounded in principles of 

statistical inference are then used to infer a statistical relation 

between low flows measured at index stations and instantaneous base flows measured at PR stations. This relation is then 

used to estimate various low-flow statistics for PR stations. 

In addition to enabling the transfer of low-flow statistics from 

index to PR stations, regression can also be used to fill gaps 

in the daily flow record at index stations where gaging has 

temporarily failed.

A variety of regression methods have been used to infer 

base flows and low-flow statistics for PR stations, each with 

their own advantages and limitations. When ordinary least 

squares regression (OLS) is used to extend the low-flow 

record from index to PR stations, the variance is underestimated because PR flow estimates do not include the variability 

of flows about the regression line (Hirsch, 1982). As a result, 

the variance of estimated flows at the PR station is always 

biased low when OLS is used to extend the flow record. To 

overcome this bias, Hirsch (1982) developed the MOVE.1 

regression methodology. MOVE.1 regression preserves the 

mean and variance of flows by minimizing the sum of squared 

error about the regression line along both axes of the regression plot. This approach provides an estimate of the mean flow 

that is unbiased, as well as a flow variance estimate that is 

unbiased for large sample sizes, at the PR station.

Prior to 2016, estimation of low-flow frequency statistics 

at PR stations using the MOVE.1 regression was implemented 

by running SAS code on a UNIX platform. Changes in the 

operating system that supports the current National Water 

Information System (NWIS) database required a change to the 

program used to access streamflow data. A decision was made 

to convert the existing SAS program to R, an open-source 

interpreted programming language widely used in the physical 

sciences for statistical computing that is supported by the R 

Foundation for Statistical Computing. Initial goals of the conversion were to duplicate the way in which instantaneous base 

flow and continuous daily flow data were input and processed, 

reproduce MOVE.1 regression line intercept and slope, match 

low-flow statistics predicted using MOVE.1 regression lines, 

replicate text and graphical output, and ensure proper calculation of the percent standard error of estimate (%SEE). The 

%SEE is used to assign weights for estimating weighted averages of predicted low-flow statistics at PR stations when more 

than one reference station is available to estimate a low-flow 

or flow-frequency statistic.

The next goal of script conversion was to add new 

features that allow the user to opt for alternate regression 

strategies when traditional MOVE.1 regression is not suited 

to estimating low flows or low-flow frequency statistics. 

These new features include incorporation of MOVE.1 censored regression for correlating flows that are zero valued or 

smaller than the assumed measurement error of 0.05 ft3/s, as 

well as an option for performing piecewise MOVE.1 regression for correlating flows when flow characteristics change 

across the range of measured flows. Where one or more flows 

at PR or index stations are measured as less than or equal to 

0.05 cubic foot per second (ft3/s), censored MOVE.1 regression offers unbiased estimates of frequency statistics and 

other flow metrics at PR stations. The piecewise MOVE.1 

regression option can easily accommodate abrupt changes 

in the mean of measured PR station base flows across the 

range of measured index station flows. Such changes typically arise when a physical process disproportionately affects 

large and small base flows. An example of such a process is 

water use, which has a disproportionate effect on small base 
Methods 3

flows relative to larger base flows. Like censored MOVE.1 

regression, piecewise MOVE.1 regression provides unbiased 

estimates for low flows and frequency statistics.

The new R script duplicates the data processing, statistical analysis, and graphical output of the legacy MOVE.1 

program, using the R programming language to maintain 

functionality with the new USGS database system. The script 

is compatible with the new Aquarius database system and 

features improved portability that enhances ease of use by scientists and planners across multiple operating systems. Operations performed by the legacy code and reproduced by the new 

R script include (1) reading a PR station file listing all index 

stations that are most hydrologically similar to the PR station 

to be used in the MOVE.1 regression, (2) reading flow data for 

the PR station and each of its index stations directly from the 

NWIS website, (3) performing a MOVE.1 regression for each 

index station believed to share the same streamflow characteristics as the PR station, (4) estimating frequency statistics 

using results of the regressions, (5) weighting flow statistics at 

each index station using the %SEE of its associated MOVE.1 

regression, (6) estimating the PR flow statistic as the weighted 

sum across all index stations, and (7) reproducing all text, 

spreadsheet, and graphical output. An incremental approach 

was used to move functionality from the legacy code into R 

in the order of the sequence of operations listed above. As the 

conversion progressed, output from the legacy code and new 

R script were compared, and differences in output were used 

to identify where further revisions should be made in the R 

script. Once R script revisions duplicated all operations of the 

legacy code, the script was finalized.

Other improvements to the script include limited trapping and handling of errors triggered by improper user input, 

as well as enhanced organization, formatting, and archiving of 

output. Since the R script runs on a wide variety of platforms, 

including UNIX, Windows, and macOS operating systems, the 

revised script is expected to remain operational well into the 

future, regardless of further modifications made to software 

used to support the NWIS database.

Purpose and Scope

The purpose of this report is to document a new R script 

that replaces SAS legacy code used at the USGS to estimate 

low flows at partial-record stations. The SAS code and R script 

automate MOVE.1 regression to estimate base-flow frequency 

statistics at PR stations on the basis of daily flows measured 

at hydrologically similar index stations located throughout 

the State. The legacy code was run on a UNIX platform for 

25 years, but a decision to no longer maintain a SAS UNIX 

license, coupled with a change in database software used to 

retrieve, store, process, and publish NWIS hydrologic data, 

required that the legacy code be re-written using a more 

portable programming language that can be executed on a 

Windows operating system. The SAS program was re-written 

using R, an open source programming language and software 

environment for statistical computing and graphics that is 

widely used by environmental scientists to develop statistical 

tools. This report highlights and documents key features of the 

R script relevant to performing operations needed to estimate 

low-flow statistics, including criteria used to select index and 

PR station flows, the statistical basis for determining weights 

used to estimate average flow statistics at a PR station as the 

weighted sum of flow statistics inferred from regressions 

performed at multiple index stations, and conditions for which 

the three distinct types of MOVE.1 regression are best suited. 

This study was conducted by the USGS in cooperation with 

NJDEP.

Previous Studies

The concept of maintaining the variance of streamflow 

data was first alluded to by Riggs (1968), who recommended 

use of a graphical procedure referred to as the “structural line” 

method that, in essence, averages the effects of regressing x on 

y with the effects of regressing y on x. Gillespie and Schopp 

(1982) published the first study related to estimating low-flow 

statistics at PR stations in New Jersey using this graphical 

approach. The MOVE.1 methodology, first developed by 

Hirsch (1982), is an extension of this “structural line” method. 

Using a mathematically rigorous framework, Hirsch proved 

that the MOVE.1 method of flow-record extension produces 

flow estimates at PR stations that are unbiased in the mean 

and, for sufficiently large sample sizes, also produces unbiased 

variance. The first application of the MOVE.1 methodology to 

estimate streamflows in New Jersey was published by Watson 

and others (2005), who documented use of the SAS MOVE.1 

program to estimate low-flow statistics at 500 PR stations, 

using flow data collected at 66 reference stations with less 

than 20 years of flow record and 111 reference stations with 

a minimum of 20 years of flow record. No previous studies 

documenting use of censored MOVE.1 or piecewise MOVE.1 

regression for estimating base flows or low-flow statistics at 

PR stations in New Jersey have been published. Note that, 

although Watson and others (2005) reference the regression 

methodology as “MOVE1,” the methodology is referred to as 

“MOVE.1” in this report to maintain consistency with Hirsch 

(1982) and most USGS publications.

Methods

The R script includes a variety of MOVE.1 regression options that can be used to estimate low flows and flow 

statistics at PR stations on the basis of daily flows measured at 

index stations. MOVE.1 regression overcomes many limitations associated with traditional OLS regression, including 

low bias in estimated variance. Unlike OLS regression, which 

seeks to find the regression line that minimizes the sum of 

squared PR flow residuals, the goal of MOVE.1 regression is 

to maintain the mean and variance of PR flows. For a large 
4 Implementation of MOVE.1, Censored MOVE.1, and Piecewise MOVE.1 Low-Flow Regressions

enough sample size, MOVE.1 regression will predict PR 

flows with unbiased variance (Hirsch, 1982). In addition to 

traditional MOVE.1 regression, the R script includes options 

for performing censored and piecewise variants of MOVE.1 

regression that can be used when zero-valued flows are present in the streamflow dataset or when streamflow data exhibit 

marked changes in the relation between PR and index station 

flows.

Not all index station or PR station flow measurements are 

suitable for inclusion in the MOVE.1 regression. Only flows 

measured at index stations that are hydrologically similar to 

the PR station are used to predict flows and flow statistics at a 

PR station. Index stations that exhibit hydrologic similarity to 

a PR station share common streamflow characteristics, making 

them ideal candidates for filling gaps in the streamflow record 

for that PR station. In addition to using only measured flows 

from index stations that exhibit similar streamflow signatures 

as the PR station, only flows collected at the PR station under 

base-flow conditions are used in the regression. A series of 

base-flow tests, originally established in the legacy SAS code, 

are imposed on instantaneous PR measurements and daily 

flows at index stations. These tests are used to exclude concurrent flow measurements collected when base-flow conditions 

likely did not prevail at the PR and index stations.

Performing MOVE.1 regression on the basis of low flows 

measured at a single index station can provide reliable baseflow estimates for extension of the flow record at a PR station. 

However, when streamflows at multiple, hydrologically similar index stations are strongly correlated to flows measured 

at the PR station, it makes sense to exploit flow information 

available at all available index stations. The use of multiple 

index stations can better condition flow statistics at a PR station by transferring different facets of flow behavior to the PR 

station. Using flows measured at more than one index station also protects predicted PR flow estimates from potential 

bias introduced by index station flows that may not be truly 

representative of flows at the PR station. To incorporate flows 

measured at multiple index stations, all index station flows 

could, in theory, be pooled and used to perform the MOVE.1 

regression. However, this might obscure the unique relation 

between a PR station and each of its associated index stations. 

Instead, separate MOVE.1 regressions are performed for each 

index station, with predicted low-flow statistics at the PR 

station estimated as a weighted average of low-flow statistics 

inferred from each individual regression.

Statistics from each MOVE.1 regression are weighted 

on the basis of the %SEE of the regression, a measure of the 

accuracy in PR flow estimates made using the regression line. 

Since regressions characterized by high standard errors are 

more uncertain than those associated with low standard errors, 

the reciprocal of the %SEE, estimated for each index station 

regression, is used to weight predictions of low-flow statistics 

before summing them across all index stations. Using the 

reciprocal of the %SEE, more certain PR flow estimates are 

weighted more heavily so that they contribute a disproportionately larger effect on predicted PR flows than those produced 

by regressions that have a high degree of uncertainty attached 

to them.

Ordinary Least Squares Regression

OLS regression has been used for several decades to 

extend instantaneous flows at a PR station ( y ( )i i = ...1 N1  ) 

on the basis of continuously measured flows at an index station (x i( ) i N = ...1 1 2 + N ) where N1 denotes the number of 

PR base flows measured on dates shared by the PR and index 

stations, N2 is the number of low flows measured at the index 

station on days when flow at the PR station is missing, and 

N1 2 + N is the total number of measured low flows at the index 

station. Flows are typically transformed by computing the 

logarithm to force the probability of flows to be more normal 

and to produce a more linear relation between y ( )i and x i . 

Values of missing log-transformed flows at the PR station 

( y ( )i i = + N N 1 1 ... + N2  1 ) have traditionally been estimated 

using the OLS regression equation

y

( )i a = + bx ( )i . (1)

The values of intercept a and slope b in equation 1 are 

estimated by minimizing the sum of squared residuals for the 

N1 log-transformed base-flow measurements available for 

the PR station that have concurrent log-transformed measured 

daily flows at the index station:

i

N

y i y i ∑ =  ( ) − ( )   1 

2 1  (2)

where y

( )i and y i  are predicted and measured log-transformed base flows at the PR station. Solving equation 1 for 

values of a and b that yield the minimum sum of squared 

residuals produces the solution

y i m y r

S

S

x i m x y

x

 ( ) = ( ) +  ( ) − ( )   1 1  1

1

(3)

where Sx1 and Sy1 are the standard deviations of the N1 concurrently measured log-transformed index and PR flows, r

represents the correlation coefficient describing the strength of 

the linear relation between the N1 concurrent log-transformed 

index and PR base flows, and m x  1 and m y  1 are the 

means of the N1 concurrently measured log-transformed index 

and PR flows (Hirsch, 1982).

Use of OLS regression provided by equation 3 to 

estimate base flows or low-flow statistics at PR stations on 

the basis of measured low flows at index stations has been 

shown by Matalas and Jacobs (1964) to produce variances in 

predicted PR low flows that are biased low by a factor of r 2, 

the squared correlation coefficient for concurrent index station and PR station base flows. As a consequence of this low 

bias in flow variance, variability in PR flows estimated using 

OLS regression is underestimated, causing the frequency of 
Methods 5

estimated PR flows in both tails of the flow distribution to 

likewise be underestimated. This tendency for the frequency 

of predicted extreme low-flow events to be underestimated by 

OLS regression generally produces overestimated low flows 

and underestimated high flows, a known drawback of traditional OLS regression (Helsel and Hirsch, 2002). To better 

reproduce the tails of the underlying PR flow distribution and 

obtain more reliable estimates of low flows at PR stations, an 

alternative regression strategy is needed to transfer low flows 

and low-flow statistics from index stations.

Maintenance of Variance Regression (MOVE.1)

To produce unbiased estimates of the mean and variance at PR stations, Hirsch (1982) took a different regression 

approach to estimating low flows. Rather than minimize the 

sum of squared error, a new regression equation was derived 

to preserve the mean and variance of low-flow predictions. 

Hirsch (1982) developed the MOVE.1 regression by imposing 

the constraints that the mean and variance of the N1 measured 

and predicted concurrent flows at a PR station be equal:

i

N

y i y i ∑ ∑ = ( ) = ( ) 1 i=1

1 N1

 (4a)

i

N

y i m y y i m y ∑ =  ( ) − ( )  

 =  ( ) − ( )   1 1 

2

1

2 1

i

N

∑ =1

1

 . (4b)

When the ordinary least squares regression equation 3 is constrained by equations 4a and 4b rather than by minimizing the 

sum of squared residuals given by expression 2, the following 

MOVE.1 regression equation is derived:

y i m y

S

S

x i m x y

x

 ( ) = ( ) +  ( ) − ( )   1 1  1

1

(5)

where m y  1 and m x  1 are sample means for the N1 concurrent low flows at the PR and index stations, respectively. 

Although MOVE.1 regression equation 5 produces an unbiased sample mean of PR flows (Stedinger and Thomas, 1985), 

the sample variance for PR flows, Sy

2 , is an asymptotically 

unbiased estimate of the true variance that will be biased high 

for a finite sample size (Hirsch, 1982).

Note that, in comparing the second term in OLS regression equation 3 to the second term in MOVE.1 regression 

equation 5, it is evident that the variance of the PR low-flow 

estimate for the OLS regression is equal to the variance of the 

MOVE.1 estimate, scaled by a factor of r 2 (Helsel and Hirsch, 

2002). For the general case where r is less than 1, PR flows < 1

estimated by the OLS regression will be characterized by a 

smaller variance than PR flows estimated using the MOVE.1 

regression. Because the MOVE.1 regression, by definition, 

preserves the variance of observed flows, the OLS regression 

will generally underestimate the variance of PR station flows. 

The OLS flow estimate will have the same variance as the 

MOVE.1 flow estimate only when flows at the PR station and 

index station are perfectly correlated, with r = 1.

One important property of the MOVE.1 regression is that 

it minimizes the sum of squared deviations of measured PR 

and index station flows from the regression line in both the x

and y dimensions to define the regression line, also known as 

the line of organic correlation (LOC). The LOC lies between 

traditional OLS regression lines obtained by regressing y on 

x and x on y (Kritskiy and Menkel, 1968). MOVE.1 produces 

the same unique LOC, regardless of whether x or y is used 

as the dependent variable. Halfon (1985) refers to MOVE.1 

as the geometric mean regression because the slope of the 

MOVE.1 regression line is equal to the geometric mean of the 

slopes associated with the traditional OLS regression line for 

x regressed on yy ( ) bxy and for y regressed on x byx x ( ):

b bxy yx . (6)

Yet another way of viewing the MOVE.1 regression is 

that it is equivalent to minimizing the sum of the areas of 

right triangles, each defined by vertical and horizontal lines 

extending from measured flows to the regression line, indicating that the LOC minimizes errors along x and y axes (Hirsch 

and Gilroy, 1984). Another characteristic of MOVE.1 that 

makes it preferable to OLS for flow extension is the fact that 

the distributional properties of estimated PR flows produced 

by a MOVE.1 regression are similar to those of measured PR 

flows. Preservation of distributional properties is important 

when estimates of flow percentiles at PR stations, not just 

mean flow or a single flow estimate, are of interest (Helsel and 

Hirsch, 2002).

The MOVE.1 regression line estimated using low-flow 

data from PR station South Branch Raritan River at Four 

Bridges, N.J., (01396190) and index station Rockaway River 

at Main Street at Boonton, N.J., (01380450) are shown in 

figure 1. Note that the station at Four Bridges was operated as 

an index station from water years1 1999 through 2012. Despite 

the fact that the station has 13 years of continuous flow record, 

for purposes of this study, it is considered a PR station because 

it has less than 20 years of daily streamflow record. The points 

plotted in figure 1 represent concurrent log-transformed flows 

measured at the two stations on the same days. As one would 

expect, because both stations are affected by similar rainfall 

events and other physical processes, smaller PR station flows 

correspond to smaller flows measured at the index station. The 

points generally fall close to the regression line, indicating that 

daily flows measured at the index station explain much of the 

variation in daily flows at the PR station. The strong degree of 

correlation between PR station flows and index station flows 

is further supported by the high value of 0.96 estimated for R2, 

the coefficient of determination for the MOVE.1 regression. 

On the basis of this value of R2, the MOVE.1 regression line 

explains 96 percent of the variability of PR station flows about 

1

A water year is the 12-month period beginning October 1 and ending September 30 of the following year. It is designated by the year in which it ends.
6 Implementation of MOVE.1, Censored MOVE.1, and Piecewise MOVE.1 Low-Flow Regressions

their mean. Note that the coefficient of determination, R2, is a 

measure of how much variance is explained by the regression, 

whereas the correlation coefficient r describes the degree of 

linear correlation between index and PR flows.

Also shown plotted in figure 1 and other regression plots 

are values of mean annual flow (QAvg) and the 7Q10 lowflow statistic. Mean annual flow can be considered an upper 

bound on base-flow statistics computed by the script, with the 

7Q10 and other low-flow frequency statistics equal to QAvg 

only when streamflow measured at the index station is constant in time. The difference between 7Q10 and QAvg can be 

considered a measure of the degree of daily flow variability at 

the index station used for the regression.

EXPLANATION

MOVE.1

OLS

7Q10

QAvg

0.5

1

1

5

5

10

10

50

50

100

100

500

500

1,000

1,000 5,000

Index station daily flow, in cubic feet per second

Partial-record station base flow, 

in cubic feet per second

Figure 1. MOVE.1 regression line for base flows measured at 

partial-record station 01396190, South Branch Raritan River at 

Four Bridges, N.J., and daily flows measured at index station 

01380450, Rockaway River at Main Street at Boonton, N.J.

(MOVE.1, Maintenance of Variance Extension, Type 1; OLS, ordinary least 

squares regression; QAvg, average daily flow; 7Q10, 7-day 10-year low flow)

The plot of PR flow residuals versus index flows output 

by the R script is shown in figure 2. These residuals represent 

deviations of measured PR flows from the MOVE.1 regression line presented in figure 1. Examination of the residual 

plot reveals that larger deviations from the MOVE.1 regression line tend to occur for larger index-station daily flows, 

indicating an increase in variance with increasing flows. This 

increase in flow variance with increasing values of flow, often 

referred to as the proportional effect, is typical for variables 

that follow a lognormal distribution (Journel and Huijbregts, 

1978). To eliminate this effect, the R script normalizes all 

residuals by index-station flows to produce the plot of percent residual versus index flow shown in figure 3. The diffuse 

cloud of points in figure 3, free of obvious trends or discontinuities, indicates the absence of significant structural error 

in the MOVE.1 regression model for PR station 01396190 

flows regressed on daily mean flows measured at index station 

−20

−10

0

10

10 20 50 100 200 500

20

Residual, in cubic feet per second

Index station daily flow, in cubic feet per second

Figure 2. Flow residuals for partial-record station 01396190, 

South Branch Raritan River at Four Bridges, N.J., from MOVE.1 

regression on daily flows at index station 01380450, Rockaway 

River at Main Street at Boonton, N.J.

−100

10 20 50 100 200 500

−80

−60

−40

−20

0

20

40

50

Residual, in percent

Index station daily flow, in cubic feet per second

Figure 3. Percent flow residuals for partial-record station 

01396190, South Branch Raritan River at Four Bridges, N.J., from 

MOVE.1 regression on daily flows at index station 01380450, 

Rockaway River at Main Street at Boonton, N.J.

01380450. Absence of structural error implies that most physical processes affecting the relation between PR station flows 

and index station flows have been integrated into the regression and indicates that processes common to both stations 

exert roughly the same effect on both sets of flows.
Methods 7

Censored MOVE.1

Under some circumstances, low-flow data cannot support 

the use of a MOVE.1 regression to predict the flow statistics. 

When zero-valued flows are part of the flow record at either 

the PR station or the index stations, a censored version of 

MOVE.1 ideally is implemented to avoid low bias in estimated PR flows. The new R script includes an option for performing left-censored MOVE.1 regression when flows equal 

to or less than 0.05 ft3/s occur in the concurrent flow record. 

The censoring threshold of 0.05 ft3/s was chosen because, for 

this study, estimated flows less than 0.05 ft3/s were rounded to 

0.0 ft3/s to account for measurement errors that typically occur 

when collecting streamflow data.

Left-censored MOVE.1 regression was not included in 

the legacy code. Instead, a constant of 0.01 ft3/s was added 

to all zero-valued flows prior to log transformation and, to 

prevent structural error in the regression, to all non-zero flows 

as well. Although this constant was subtracted from estimated 

flows, it introduced bias into the MOVE.1 regression because 

adding a value of 0.01 ft3/s to small flows has a disproportionate effect on the regression compared with the addition of the 

same value to larger flows. With publication of the censored 

MOVE.1 regression R function censMove.1 developed by 

David Lorenz of USGS in 2014 (https://github.com/USGS-R/

smwrQW/blob/master/R/censMove.1.R), it became possible 

to avoid this pitfall. The censMove.1 function retains all the 

advantages of the regular MOVE.1 regression methodology 

but allows for input of zero-valued concurrent flows measured 

at index stations, PR stations, or both. During implementation of the censored MOVE.1 regression, index and PR 

station flows are assumed to be characterized by a bivariate 

log-normal distribution. If all measured index station and PR 

station flows are uncensored, the actual measured flow values 

are input to the regular MOVE.1 regression. If a measured 

PR flow value is censored and its concurrent index flow is 

not censored, then the censored PR flow is set equal to the 

expected value of PR flows, given the censored PR flow value 

and the uncensored value of its concurrent measured index station flow. Likewise, if a measured index flow value is censored 

and its concurrent PR flow is not, then the censored index flow 

is assumed equal to the expected value of index flows, given 

the censored index flow value and the uncensored value of its 

concurrent measured PR station flow. Finally, if both measured PR and index flows are censored, then their values are 

set equal to their corresponding expected values, given their 

censored values, without regard for the censored value of the 

concurrent flow variable. 

 During implementation of the censored MOVE.1 

regression, index and PR station flows are assumed to be 

characterized by a bivariate log-normal distribution. If all 

measured index station and PR station flows are uncensored, 

the actual measured flow values are input to the regular 

MOVE.1 regression. If a measured PR flow value is censored 

and its concurrent index flow is not censored, then the censored PR flow is set equal to the expected value of PR flows, 

given the censored PR flow value and the uncensored value of 

its concurrent measured index station flow. Likewise, if a measured index flow value is censored and its concurrent PR flow 

is not, then the censored index flow is assumed equal to the 

expected value of index flows, given the censored index flow 

value and the uncensored value of its concurrent measured PR 

station flow. Finally, if both measured PR and index flows are 

censored, then their values are set equal to their corresponding 

expected values, given their censored values, without regard 

for the censored value of the concurrent flow variable.

To illustrate, if index flow is less than 0.05 ft3/s but flow 

at the PR station is not, then index station flow is estimated as 

the mean of the conditional distribution of index station flows, 

given an index station flow of 0.05 ft3/s and the uncensored 

value of PR station flow. Likewise, for the case of censored 

index flow and uncensored PR flow, index flow is estimated 

as the expected value of index flow, given its censored value 

and the value of the concurrent PR flow. If index and PR station flows are censored, their values are estimated as expected 

values of distributions conditioned on their censored values, 

without regard to the censored value of the concurrent flow.

Results of regressions for PR station Neshanic River 

near Flemington, N.J., (01397800) and index station Stony 

Brook at Princeton, N.J., (01401000) using regular and 

censored MOVE.1 regression methodologies are shown in 

figure 4. When 0.01 ft3/s was added to all flows so that the 

three zero-valued PR flows at station 01397800 were no longer equal to zero, the regular MOVE.1 regression produced a 

line with a smaller intercept and larger slope than the censored 

MOVE.1 regression with unmodified flows. It is evident from 

examination of the two regression plots that predicted PR flow 

statistics, based on the regular MOVE.1 line in figure 4A, will 

be biased high for large index station flows and biased low for 

small index station flows. Because all streamflows input to the 

regular MOVE.1 regression are modified by adding 0.01 ft3/s 

and streamflows input to the censored MOVE.1 regression 

represent unaltered flows, the two types of regression operate on different sets of flows. Differences between flows in 

the two datasets are especially pronounced in lower ranges of 

flow, where the addition of 0.01 ft3/s disproportionately affects 

flows relative to larger flows, and can dramatically change the 

estimate of %SEE. As a consequence of using fundamentally 

different streamflow datasets for the regular MOVE.1 and censored MOVE.1 regressions shown in figure 4, %SEE values 

from the two types of MOVE.1 regression cannot be used to 

objectively assess their relative goodness of fit.
8 Implementation of MOVE.1, Censored MOVE.1, and Piecewise MOVE.1 Low-Flow Regressions

EXPLANATION

MOVE.1

OLS

QAvg

0.001

0.1

10

Partial-record station base flow, in cubic feet per second

0.1 0.5 1 5 10 50 100

Index station daily flow, in cubic feet per second

A. Regular regression of partial-record station 01397800 on index station 01401000

Partial-record station: Neshanic River near Flemington, N.J. 

Index station: Stony Brook at Princeton, N.J.

7Q10=0

EXPLANATION

MOVE.1

OLS

7Q10

QAvg

0.001

0.1

10

Partial-record station base flow, in cubic feet per second

0.1 0.5 1 5 10 50 100

Index station daily flow, in cubic feet per second

B. Censored regression of partial-record station 01397800 on index station 01401000

Partial-record station: Neshanic River near Flemington, N.J. 

Index station: Stony Brook at Princeton, N.J.

Figure 4. A, Regular MOVE.1 regression and B, censored MOVE.1 regression for partial-record station 01397800, Neshanic River near 

Flemington, N.J., and index station 01401000, Stony Brook at Princeton, N.J.

(MOVE.1, Maintenance of Variance Extension, Type 1; OLS ordinary least squares regression; QAvg, average daily flow; 7Q10, 7-day 10-year low flow)
Methods 9

EXPLANATION

MOVE.1

OLS

7Q10

QAvg

0.5 1 5 10 50 100 500 1,000 5,000

10

50

100

500

1,000

5,000

10,000

Partial-record station base flow, in cubic feet per second

Index station daily flow, in cubic feet per second

A. Regular regression of partial-record station 01381900 on index station 01379000

Partial-record station: Passaic River at Pine Brook, N.J. 

Index station: Rassaic River near Millington, N.J.

EXPLANATION

MOVE.1A

MOVE.1B

OLS

7Q10

QAvg

Partial-record station base flow, in cubic feet per second

Index station daily flow, in cubic feet per second

B. Piecewise regression of partial-record station 01381900 on index station 01379000

Partial-record station: Passaic River at Pine Brook, N.J. 

Index station: Passaic River near Millington, N.J.

0.5 1 5 10 50 100 500 1,000 5,000

10

50

100

500

1,000

5,000

Figure 5. A, Regular MOVE.1 regression and B, piecewise MOVE.1 regression for partial-record station 01381900, Passaic River at Pine 

Brook N.J., and index station 01379000, Passaic River at Millington, N.J.

(MOVE.1, Maintenance of Variance Extension, Type 1; MOVE.1A, regular; MOVE.1B, piecewise; OLS, ordinary least squares regression; QAvg, average daily 

flow; 7Q10, 7-day 10-year low flow)
10 Implementation of MOVE.1, Censored MOVE.1, and Piecewise MOVE.1 Low-Flow Regressions

Piecewise MOVE.1

In some cases, PR flows may exhibit a distinctly 

nonlinear relation with index flows, indicating that different 

processes affecting streamflow are in play across different 

ranges of index flows. Reasons for such statistically nonhomogeneous behavior can be attributed to a number of 

causes, including water use that disproportionately reduces 

streamflows and causes the type of plot shown in figure 5A. 

Under such circumstances, a piecewise approach to MOVE.1 

regression can help resolve the statistically non-stationary 

behavior by fitting two or more regression lines to the flow 

data.

At the time of publication of this report, an R function 

that performs piecewise MOVE.1 regression had not yet 

been developed. In place of a formal MOVE.1 piecewise 

regression, an iterative approach first described by Crawley 

(2013) was included as a regression option in the R script. 

The piecewise regression option allows the user to implement 

regular MOVE.1 regression by defining two separate regression lines that intersect at a breakpoint where a marked change 

in the relation between measured flows at the PR station and 

the index station occurs. To perform the piecewise regression, 

the user is prompted to enter upper and lower bounds on the 

index flow interval within which an obvious discontinuity in 

the relation between PR and index flows occurs. Only one 

breakpoint defining the shared endpoint for the two regression 

segments is allowed to occur within this interval. Additional 

R code was written to iteratively vary the location of this 

single breakpoint in fixed increments across the user-defined 

index flow interval and estimate standard error at each trial 

breakpoint. For an unbiased estimator, the mean squared error 

and the standard error of estimate (SEE) are equal. The trial 

breakpoint that produces the smallest SEE defines where the 

optimal breakpoint occurs along the index flow axis. Independent MOVE.1 regressions are then performed for flows greater 

than and less than the shared breakpoint, each characterized 

by a unique slope and intercept. Future improvements in the 

R script may include a piecewise regression methodology that 

doesn’t require the user to provide index flow interval bounds 

that define the upper and lower limits constraining where the 

true breakpoint is likely to be located.

Results of regular MOVE.1 and piecewise MOVE.1 

regressions performed for PR station Passaic River at Pine 

Brook, N.J., (01381900) and index station Passaic River at 

Millington, N.J., (01379000) are shown in figures 5A and 5B. 

Although the regular MOVE.1 regression line does a fairly 

good job of minimizing error globally across the full range 

of index flows, it fails to locally minimize error in the lower 

ranges of index flow. Piecewise MOVE.1 regression does not 

provide a meaningful %SEE for both regression line segments that can be compared to the %SEE for regular MOVE.1 

regression. However, the piecewise regression presented in 

figure 5B provides a smaller %SEE for index flows less than 

the 35 ft3/s breakpoint than regular MOVE.1 regression does 

for the same range of smaller flows, indicating that piecewise 

regression explains more of the variability in measured PR 

flows than regular MOVE.1 regression for index flows smaller 

than 35 ft3/s.

Application of the R Script

The R script begins execution by installing all libraries 

that support R functions needed to read NWIS flows, perform 

MOVE.1 regressions, and estimate flow statistics at the index 

stations. The user is prompted to select the location of the 

output directory using a folder selection dialog. The R script 

then warns the user that certain function files must be present 

for the script to run correctly. The script then asks the user to 

select the location where the text input file containing station 

identifiers for the PR stations and their index stations resides. 

The user is then offered the option of appending all output to 

existing output files. An interface with three radio buttons, 

each corresponding to one of the three MOVE.1 regression 

variants, allows the user to select the type of regression to 

be performed. The final interface allows the user to uncheck 

boxes corresponding to index stations that the user does not 

want to include in the regression analysis.

Once the user specifies the location of the station identifier input file and the type of MOVE.1 regression, the R script 

reads all station identifiers, extracts index station and PR 

station flows from the NWIS database, and loads historical 

low-flow statistics previously estimated for all New Jersey 

index stations on the basis of daily flows measured until water 

year 2001. The R script then loops through each index station 

associated with the PR station, performing a MOVE.1 regression using all index and PR station flows that meet a series of 

base-flow criteria, and computes the SEE for the regression 

corresponding to each index/PR station pair. Once all index 

station regressions have been performed for the PR station, the 

MOVE.1 regression line for each regression is used to predict 

flow statistics at the index station. The reciprocal of %SEE

for each regression is then used to weight the flow statistic at 

each index station, and the weighted index station statistics are 

summed across all index stations to estimate the flow statistic 

at the PR station.

Selection of Flows

As part of an ongoing agreement between the U.S. 

Geological Survey Water Science Center in New Jersey 

and NJDEP, instantaneous streamflow measurements at PR 

stations are made under base-flow conditions 2 to 3 times a 

year at 80 designated low-flow PR stations throughout New 

Jersey. In total, there are more than 600 PR stations in New 

Jersey with enough instantaneous streamflow measurements 

to potentially be used to estimate low-flow statistics. However, 

because not all measurements are actually made under baseflow conditions, some of these flows must be excluded from 

the regression to obtain reliable estimates of base flows at the 

PR station. 
Methods 11

To determine whether base-flow conditions prevailed 

at the time of measurement at the PR and index station, each 

flow record is subjected to a sequence of screening tests. If 

a given flow record satisfies all base-flow test criteria listed 

in table 1, it is included in the MOVE.1 regression. Note 

that, owing to the discrete nature of instantaneous base flows 

measured at PR stations, they were subject to a different set 

of tests than index daily flows measured at index stations. 

Although base-flow tests at PR stations rely solely on flags 

contained in the NWIS dataset, base-flow testing at index stations depends on changes in daily continuous flow over one or 

more consecutive days.

As shown in table 1, PR measurements flagged as not 

measured under base-flow conditions, with a BASEFLOW 

attribute NBAS indicating “not base flow” in the instantaneous measurement file, are assumed to have been measured 

during a time when base-flow conditions were known not 

to have occurred. These measurements were automatically 

excluded from the MOVE.1 regression without further consideration. Instantaneous PR flow measurements flagged with 

a BASEFLOW attribute of unspecified conditions (UNSP) 

or base-flow conditions (BASE), indicating that it was either 

unknown whether the flow measurement was made under 

base-flow conditions or that base-flow conditions were known 

to exist at the time of measurement, were included in the 

MOVE.1 regression. After filtering for base-flow conditions 

at the time of measurement, instantaneous flow measurements 

made at PR stations were then subject to a gage-height test 

to eliminate instantaneous flows made at times when stream 

stage was observed to be changing rapidly. For an instantaneous measurement at a PR station to be considered an 

accurate base-flow value, stream stage could not change by 

more than 0.02 foot (ft) during the time it took to make the 

flow measurement, unless the measurement was attributed as 

being made during tidal conditions, and the appropriate tidal 

adjustment was made to the measured flow. 

Flows measured at index stations were subject to an 

entirely different sequence of base-flow tests than PR station 

flows. A number of flow criteria at index stations must be 

passed if the concurrent flow at the PR station has an attribute 

of UNSP or BASE. These criteria require comparison of daily 

flows on consecutive days, data that are typically unavailable 

at PR stations where instantaneous flow measurements are 

made. Index station daily flows are considered to be measured 

under valid base-flow conditions only when they drop by less 

than 30 percent and rise by less than 10 percent, relative to 

daily flow measured on the previous day.

In certain cases, instantaneous PR measurements indicate that base-flow conditions are present at the partial-record 

station despite a concurrent increase in measured daily flow at 

an index station relative to flow observed at the index station 

on the previous day. These conditions are generally encountered following late-day convective rainfall events that prevail 

during summer months. Because convective rainfall events 

are generally localized in spatial extent, they can increase 

flow at an index station without significantly influencing 

flow at the associated PR station. Under such circumstances, 

flow at the index station is estimated when the base-flow flag 

BASEFLOW is equal to BASE for the concurrent PR flow. 

To provide this estimate of index site flow, an additional test 

is subsequently performed to determine whether daily flow 

measured at the index station 2 days previously was greater 

than flow measured 1 day prior to the daily flow measurement 

at the index station. If true, index station flow is estimated to 

be equal to flow from the previous day, corrected by the rate of 

recession observed between 1 and 2 days prior. If not true, the 

index station measurement is excluded from the regression.

Table 1. Tests applied to flows for inclusion in MOVE.1 regressions.

[PR, partial-record]

Name of 

base-flow test

Type of 

station tested Description Exception

Base-flow flag check PR station Measurements flagged as 

BASEFLOW=NBAS in 

the measurement file are 

excluded.

None

Gage-height test PR station Gage-height change during the flow measurement 

(GHCHGF) must be less 

than or equal to 0.02 foot.

If a measurement is flagged as being adjusted for tide effect 

(BASEFLOW=‘TADJ’) and appropriate flow adjustment 

was made.

Base-flow test Index station Daily flow must decrease by 

less than 30 percent or rise 

by less than 10 percent, 

relative to flow measured 

on the previous day.

1. If BASEFLOW=‘BASE’ for concurrent PR flows, then index 

station daily flow is checked to determine whether it has been 

receding over the previous 2 days. If true, daily flow at the 

index station on the day of the measurement is estimated by 

linear interpolation.

2. If the absolute value of the change in daily flow is less than 0.5 

cubic foot per second.
12 Implementation of MOVE.1, Censored MOVE.1, and Piecewise MOVE.1 Low-Flow Regressions

In a final base-flow test of index station flow, an exception is made for index stations with very low daily flows that 

do not pass criteria for percent change in flow from the previous day. If the absolute value of change in daily flow is less 

than 0.5 ft3/s from the previous day, the index station flow is 

included in the regression. Owing to insensitivity of the stagedischarge rating for very low flows at some index stations, a 

change of only 0.01 ft in stage height can produce a change 

in index flow of 20 percent or more. Under these conditions, 

small fluctuations in continuous gage height record unrelated 

to rainfall events can produce relatively large percent differences at low flows computed from a rating curve. These index 

flows are included in the regression, as it is assumed that baseflow conditions persist.

The final set of concurrent PR and index station base-flow 

measurements included in the MOVE.1 regression excludes 

all PR station measurements with a BASEFLOW attribute of 

NBAS, all PR measurements that fail the gage-height test, and 

all index-station flows that fail the consecutive-day base-flow 

tests. Failure of any base-flow test performed on a PR station 

measurement automatically excludes the concurrent measurement made at the index station and vice versa. Exclusion of 

concurrent measurement pairs when a base-flow test failed 

for either station ensured that the MOVE.1 regression was 

performed on pairs of flows measured under similar base-flow 

conditions, helping to prevent a high bias for predictions of the 

annual 7Q10 and other low-flow statistics computed using the 

MOVE.1 regression line.

Index Station Selection

New Jersey is fortunate to have over 120 index stations 

with more than 20 years of daily flows. To increase confidence 

in low-flow frequency statistics estimated at PR stations, 

only index stations with a minimum of 20 years of continuous record were used for this study. The MOVE.1 regression 

provides a means of transferring flow information from an 

index station for which continuous daily streamflow records 

are available to a PR station for which only instantaneous 

measurements of base flow are available or, in some instances, 

for which the continuous daily mean flow record is less than 

20 years. Selection of index stations suitable for inferring flow 

estimates at a given PR station can be somewhat subjective. 

Only index stations that are not strongly affected by diversions, exhibit no significant trends in flow over time, drain 

basins characterized by similar physical properties affecting 

streamflow, and share the same physiographic province as the 

PR station are considered as candidates for inclusion in the 

MOVE.1 regression model. Such index stations tend to be 

hydrologically similar to the PR station and characterized by 

streamflow hydrographs with similar types of flow characteristics as the PR station.

In New Jersey, there are often multiple index stations 

that are appropriate for use in transferring flow information 

to a single PR station. Use of daily flows measured at multiple index stations provides more reliable estimates of flow 

statistics at PR stations than use of daily flows at a single 

index station. Rather than pooling daily flows measured at the 

five index stations and performing a single MOVE.1 regression, separate regressions were performed for each of the five 

index stations, and estimated flows and flow statistics were 

weighted according to the SEE for each regression. This pairwise approach to regression preserves unique flow information 

shared between the PR station and each of its index stations. 

Because index station selection can be subjective even when 

adhering to established guidelines set forth to include only 

stations with similar hydrologic characteristics, incorporating flow information from multiple index sites based on the 

strength of their MOVE.1 regressions helps eliminate some of 

that subjectivity.

To illustrate how multiple index stations are used to 

infer flow statistics at a PR station, separate regressions were 

performed for each of the five hydrologically similar index 

stations associated with PR station 01396190. The five separate MOVE.1 regressions shown in figure 6 exhibit varying 

degrees of scatter about their regression lines. Differences in 

the amount of scatter of flow measurements about a regression 

line need to be accounted for when the line is used to predict 

a low-flow statistic for the PR station. Clearly, a flow statistic predicted using a MOVE.1 regression at an index station 

characterized by a large amount of flow variation about the 

regression line should have less effect on the estimate of the 

statistic at the PR station, and be assigned a smaller weight, 

than a statistic estimated using an index-station regression that 

shows a tighter degree of scatter about the line. To provide an 

example of how a low-flow statistic is estimated for a PR station as a weighted average of flow statistics predicted at each 

of the five index stations, the annual 7Q10 at the PR station is 

estimated using the equation

7Q10 7Q10 1 PR j

L = j j ∑ = ω * (7a)

j

L

j ∑ =1

ω = 1 (7b)

where L is the number of index stations used to estimate the 

low-flow statistic at the PR station, ωj denotes the weight at 

the jth index station, 7 1 Q 0 j is the estimated annual 7Q10 at 

the jth index station, and 7 1 Q 0PR is the annual 7Q10 at the PR 

station, estimated as the weighted average of the annual 7Q10 

values across the L index stations, with all weights required to 

sum to 1.
Methods 13

EXPLANATION

MOVE.1

OLS

7Q10

QAvg

1 5 10 50 100 500 1,000 5,000

Index station daily flow, in cubic feet per second

0.5

1

5

10

50

100

500

1,000

Partial-record station base flow, in cubic feet per second

A. MOVE.1 regression of partial-record station 01396190 on index station 01380450

Partial-record station: South Branch Raritan River at Four Bridges, N.J.

Index station: Rockaway River at Main Street at Boonton, N.J.

%SEE=26.9

EXPLANATION

MOVE.1

OLS

7Q10

QAvg

0.5 1 5 10 50 100 500 1,000

Index station daily flow, in cubic feet per second

0.5

1

5

10

50

100

500

1,000

Partial-record station base flow, in cubic feet per second

B. MOVE.1 regression of partial-record station 01396190 on index station 01398500

Partial-record station: South Branch Raritan River at Four Bridges, N.J.

Index station: North Branch Raritan River near Far Hills, N.J.

%SEE=35.8

EXPLANATION

MOVE.1

OLS

7Q10

QAvg

0.5 1 5 10 50 100 500 1,000 5,000

Index station daily flow, in cubic feet per second

0.5

1

5

10

50

100

500

1,000

Partial-record station base flow, in cubic feet per second

C. MOVE.1 regression of partial-record station 01396190 on index station 01440000

Partial-record station: South Branch Raritan River at Four Bridges, N.J.

Index station: Flat Brook near Flatbrookville, N.J.

%SEE=35.5

EXPLANATION

MOVE.1

OLS

7Q10

QAvg

1 5 10 50 100 500 1,000 5,000

Index station daily flow, in cubic feet per second

0.5

1

5

10

50

100

500

1,000

Partial-record station base flow, in cubic feet per second

D. MOVE.1 regression of partial-record station 01396190 on index station 01443500

Partial-record station: South Branch Raritan River at Four Bridges, N.J.

Index station: Paulins Kill at Blairstown, N.J.

%SEE=33.5

EXPLANATION

MOVE.1

OLS

7Q10

QAvg

2 5 10 20 50 100 200 500 1,000 2,000

Index station daily flow, in cubic feet per second

0.5

1

5

10

50

100

500

1,000

Partial-record station base flow, in cubic feet per second

E. MOVE.1 regression of partial-record station 01396190 on index station 01445500

Partial-record station: South Branch Raritan River at Four Bridges, N.J.

Index station: Pequest River at Pequest, N.J.

%SEE=28.2

Figure 6. MOVE.1 regression lines used to evaluate the relation between base-flow measurements at partial-record station 01396190, 

South Branch Raritan River at Four Bridges, N.J., and daily flows at five index stations in New Jersey.

(%SEE, percent standard error of estimate; MOVE.1, Maintenance of Variance Extension, Type 1; OLS, ordinary least squares regression; 7Q10, 7-day 10-year 

low flow)
14 Implementation of MOVE.1, Censored MOVE.1, and Piecewise MOVE.1 Low-Flow Regressions

Derivation of Weights for Averaging Predicted 

Flow Statistics

The SEE, which quantifies the degree of scatter in PR 

flows about the regression line, offers a useful metric for 

assigning meaningful weights when combining flow statistics 

predicted from separate MOVE.1 regressions performed at 

L index stations, where L varies with the PR station. In the 

context of regression, the SEE is a measure of the accuracy 

of predictions made using the regression line. A larger value 

of SEE indicates a greater amount of unexplained variance in 

measured PR station flows when regressed on flows observed 

at an index station. Large values of SEE indicate that flow 

statistics estimated for the PR station are highly uncertain and 

should be assigned less weight than flow statistics estimated 

from regressions performed at index stations with smaller 

values of SEE that account for more variation in observed PR 

station flows.

The percent standard error of estimate, or %SEE, is the 

percent of unexplained variance relative to the value of the 

estimate. %SEE is a better measure of uncertainty than SEE

because it accounts for the proportional effect encountered 

in natural streamflow processes, where greater variances are 

observed for larger flows. To lessen the weight of predictions made using MOVE.1 regressions characterized by a 

high degree of unexplained variability in flows and increase 

the weight of predictions made using regressions that exhibit 

a small amount of unexplained flow variability, weights 

ωj , j=1, 2, ..., L are defined as the reciprocal of %SEE for the 

jth index station regression:

j

j SEE  1

% j=1, 2, ..., L (8)

where %SEEj is the percent standard error of estimate at the 

jth index station. By assigning weights as the reciprocal of 

the %SEE, flow statistics at index stations that do a poor 

job of explaining measured PR flows will be provided less 

weight in the calculation of PR station flow statistics than 

index stations with flows that explain more of the variation in 

measured PR flows. Use of the reciprocal of the %SEE penalizes highly uncertain estimates characterized by large standard 

errors while assigning greater weight to estimates associated 

with less uncertainty, without disproportionately penalizing 

estimates of PR low-flow statistics made using regressions 

performed at index stations characterized by larger flows. 

Weighting of regression estimates by the reciprocal of their 

percent standard errors of estimate is typically used in statistical analysis to combine independent estimates (Hartung and 

others, 2008).

To illustrate how %SEEs from MOVE.1 regressions 

are used to weight index-station flow statistics, consider the 

effect of low-flow statistics predicted on the basis of MOVE.1 

regressions for base-flow measurements made at the PR 

station at South Branch Raritan River at Four Bridges, N.J., 

(01396190) on the basis of daily flows at L=5 index stations. 

The regression line for index station 01380450 (fig. 6A) 

explains more of the variance in measured flows at the PR 

station, showing a smaller amount of scatter in measurements about the regression line and, as indicated in table 2, 

the smallest %SEE of all L index stations. As a result of this 

small %SEE, flow statistics estimated by regressing on mean 

daily flows at index station 01380450 are expected to be more 

accurate than statistics predicted using the regression on mean 

daily flows at the other index stations. Because a weight equal 

to the reciprocal of this smaller %SEE will be larger than the 

weight associated with the %SEE values for the other index 

stations, the weighted average will favor estimates of lowflow statistics made using the regression for index station 

01380450.

The %SEE for a t-year flow statistic is calculated using 

its standard error of prediction SEPt )( , along with its timesampling error SEt ( ). For ease of computation, the R script 

calculates SEP10 and SE10 , the standard error of prediction 

and time-sampling error corresponding to the 10-year recurrence interval, rather than computing them on the basis of the 

recurrence interval of the particular flow statistic of interest. 

The rationale for using values of SEP10 and SE10 to calculate the %SEE for weighting all flow statistics at PR stations 

can be traced to the widespread use of the annual 7Q10 as a 

common statistic for computing waste load allocations and 

for defining minimum passing flows required to preserve 

downstream water rights and maintain the health of aquatic 

ecosystems in the State of New Jersey (Hoffman and Domber, 

2013). Equations used to compute SEP10 and SE10 for each of 

the L MOVE.1 regressions, along with the L values of %SEE

needed to estimate the L weights, are provided in the following sections.

Table 2. Percent standard errors of estimate at index stations 

used to weight flow statistics at partial-record station 01396190, 

South Branch Raritan River at Four Bridges, N.J.

[ID, identifier; %SEE, percent standard error of estimate]

Index Station ID %SEE

01380450 26.9

01398500 35.8

01440000 35.5

01443500 33.5

01445500 28.2
Methods 15

Standard Error of Prediction for the 7Q10 Flow Statistic

The %SEE depends, in part, on the standard error of prediction (SEP), a measure of the variability in measured flows about 

the MOVE.1 regression line not caused by systematic error or bias. The SEP depends on the number of measured flows used in 

the regression, how far the regression relation is extended beyond measured flows to make predictions of flows or flow statistics at the PR station, and the standard error of index station statistics (Telis, 1991). The SEP for the 10-year recurrence interval 

associated with the annual 7Q10, or SEP10 , is determined using the equation

SEP SEE

N

X m x

x i m x ols

i

10 N

2

1

10 1

2

1 1

2 1 1

1

= + + ( ) − ( )

( ) − ( )

















+

∑ = (

.

S

S

r x i m x y

x

2

2

1

2

0 5

( ) 1− ( ) ( ) − ( )















 (9)

where SEP10 is the standard error of prediction in log units using the MOVE.1 regression for flows with 10-year recurrence 

interval (Telis, 1991), SEEols is the standard error of estimate in log units estimated from ordinary least-squares regression, N1 is the number of concurrent discharge measurements at the PR and index stations used for the regression, X10 is the 

logarithm of the 10-year low-flow statistic at the index station, m x  1 is the mean of the logarithms of the N1 concurrent flows 

for the index station, x i  is the logarithm of the ith discharge for the index station, Sy is the standard deviation of log-transformed measured flows at the PR station, Sx is the standard deviation of log-transformed measured flows at the index station, and 

r is the correlation coefficient for PR flows regressed on index station flows using OLS regression.

Time-Sampling Error for the 7Q10 Flow Statistic

In addition to being dependent on the 7Q10 SEP, the %SEE for the annual 7Q10 is a function of time-sampling errors 

attributed to the use of instantaneous flow measurements to derive flows of finite duration. Time-sampling errors, sometimes 

known as gage errors, are errors that arise when flow statistics for a given frequency and duration are estimated on the basis 

of computed mean daily flows. The accuracy of mean daily flows is a function of the accuracy of streamflow measurements, 

rating curves, and computation methods. Because the annual 7Q10 metric and flow-duration statistics predicted by the MOVE.1 

regression line are derived using daily flows averaged over long periods of time, errors associated with such flow statistics will 

tend to be smaller than errors associated with daily flow measurements.

The time-sampling error for the annual 7Q10 is estimated using the equation developed by Kite (1988):

SE S

N 10 = δ (10)

where SE10 is the time-sampling error in logarithmic units, S is the standard deviation of log-transformed annual 7-day low 

flows, N is the number of years in the annual 7-day low-flow record, and δ is a function of the skew of the distribution of annual 

7-day flows. Greater variability in 7Q10 estimates, a shorter period of record, and a highly skewed distribution of 7-day flows 

will produce large time-sampling errors. The time-sampling error can be converted to percent sampling error using the relation

% .SE SE

10

2 3 0 5

100 10 1 10

2

= − 

  . (11)

Standard Error of Estimate for the 7Q10 Flow Statistic

The SEE for flows with a 10-year recurrence interval is computed using the values of SEP10 and the SE10 , based on the 

equation for SEE10 provided by Telis (1991):

SEE10 = +





 



















 SEP

S

S

SE y

x

10

2

2

10

2

0 5.

 . (12)
16 Implementation of MOVE.1, Censored MOVE.1, and Piecewise MOVE.1 Low-Flow Regressions

Use of the reciprocal of SEE10 to define index station 

weights for the weighted average of a flow statistic at the PR 

station disproportionately penalizes flow information at index 

stations that have larger flows because these larger flows generally have larger standard errors of estimate. To account for 

the fact that regressions involving greater flows tend to have 

larger values of SEE10 , the percent standard errors of estimate 

used to define values of ωj in equation 8 were estimated as

% .SEE SEE

10

2 3 0 5

100 10 1 10

2

= − 

  . (13)

%SEE10 values were used to define weights ωj

 , j = 1, 

2, . . ., L needed to calculate the average of flow statistics 

predicted across all L MOVE.1 regressions, regardless of the 

actual flow statistic being predicted. Values of ωj for each 

PR station were standardized by summing them across all 

L index-station regressions, and dividing each weight by the 

sum before using them to weight each of the MOVE.1 predictions associated with the PR station.

Use of %SEE10 to define weights for arbitrary flow 

statistics predicted at PR stations can be justified if the %SEE

associated with the flow statistic, normalized to sum to 1, is 

approximately equal to the percent standard error for prediction of the annual 7Q10. Under conditions of small variance in 

base flows, low-flow statistics derived for different durations 

and recurrence intervals will tend to become more similar, 

with percent standard errors of estimate that approach %SEE10

and index-station weights that are approximately equal to 

those estimated for the annual 7Q10. Future improvements 

in the R script may include explicit derivation of percent 

standard errors of estimate for low-flow statistics of arbitrary 

duration and recurrence, with index-station weights customized to the particular flow statistic of interest.

Predicted Flow Statistics

The MOVE.1 R script computes the same low-flow 

frequency statistics as the legacy SAS code. Flow statistics 

and regression parameters requested by NJDEP for permitting purposes and output by the R script include low-flow 

frequency statistics, arithmetic and harmonic means of daily 

flows, the standard error of the regression, and the Kendall tau 

trend statistic (table 3). Values of these statistics for all index 

stations associated with the PR station are read from a static 

file of flow statistics for New Jersey PR stations published 

in Watson and others (2005). These published flow statistics, 

compiled using daily streamflows through water year 2001, 

are used to predict equivalent statistics at the PR station on the 

basis of the L MOVE.1 regressions, weighted by normalized 

reciprocals of %SEE10 and summed to obtain a weighted average for the flow statistic at the PR station.

Low-flow frequency statistics estimated at the PR 

station include the 1-day, 7-day, and 30-day low-flow frequencies; 25-percent, 50-percent, and 75-percent flow durations; 

Table 3. Flow statistics and regression metrics output by the 

R script for MOVE.1.

[%, percent; ft3/s, cubic foot per second; mi2, square mile; yr, year]

Variable Description

Q1.10 Annual 1-day 10-yr (1Q10) low flow (ft3

/s)

Q7.10 Annual 7-day 10-yr (7Q10) low flow (ft3

/s)

Q7.10.DA Annual 7-day 10-yr (7Q10) low flow divided 

by drainage area (ft3

/s/mi2)

WIN.Q7.10 Winter 7-day 10-yr (7Q10) low flow (ft3

/s)

Q30.5 Annual 30-day 5-yr (30Q5) low flow at the 

index station (ft3

/s)

Q30.10 Annual 30-day 10-yr (30Q10) low flow at the 

index station (ft3

/s)

WIN.Q30.5 Winter 30-day 5-yr (30Q5) low flow at the 

index station (ft3

/s)

WIN.Q30.10 Winter 30-day 10-yr (30Q10) low flow at the 

index station (ft3

/s)

QAVG Arithmetic daily mean flows (ft3

/s)

HARMEAN Harmonic mean of daily flows (ft3

/s)

DURA.25 Daily flow exceeded 25% of the time (ft3

/s)

DURA.50 Daily flow exceeded 50% of the time (ft3

/s)

DURA.75 Daily flow exceeded 75% of the time (ft3

/s)

PERCENT.PRED Standard error of MOVE.1 prediction (%)

PERCENT.GAGE Time-sampling, or gage, error (%)

PERCENT.EST Standard error of MOVE.1 estimate (%)

CORRELATION MOVE.1 correlation coefficient

POWER Slope of MOVE.1 regression line

CONSTANT Intercept of MOVE.1 regression line

KENDALLTAU Kendall trend test statistic for MOVE.1 

residuals

KENDALLPROB Kendall trend test significance level for 

MOVE.1 residuals
Methods 17

Table 4. Percent differences in estimated MOVE.1 low-flow statistics between the legacy code and the new R script.

[Variables are defined in table 3]

Variable Q1.10 Q7.10 WIN.Q7.10 Q30.5 Q30.10 WIN.Q30.5 WIN.Q30.10 HARMEAN DURA.25 DURA.50 DURA.75

Minimum percent 

difference −3.67 −3.61 −1.27 −1.57 −2.35 −3.99 −1.79 −1.52 −2.79 −1.25 −1.53

Maximum percent 

difference 2.65 3.70 1.56 2.08 2.37 0.55 1.26 0.47 0.93 0.54 0.89

Table 5. Minimum and maximum percent differences in regression metrics between the legacy code for MOVE.1 and the new R script 

for MOVE.1.

[Variables are defined in table 3]

Variable PERCENT.PRED PERCENT.GAGE PERCENT.EST CORRELATION POWER CONSTANT KENDALLTAU KENDALLPROB

Minimum 

percent 

difference

−16.76 −0.47 −16.78 −0.41 −0.26 −4.30 −7.57 −12.14

Maximum 

percent 

difference

0.37 1.12 0.44 0.35 0.63 18.81 9.25 13.26

25-percent/75-percent flow duration ratio; annual mean flow; 

and harmonic mean flow. The 7-day and 30-day low-flow 

frequencies at the PR station are computed for annual and 

winter periods, with the winter period extending from November through March. The Kendall Tau statistic is of particular 

importance because it is used to determine the likelihood that 

significant trends in flow residuals occur over time, a situation 

that would give rise to non-stationary conditions and potential 

bias in predictions of flow statistics at PR stations. All flow 

statistics and regression metrics are output to a .csv file to 

facilitate further analysis. 

In comparing the output of the R script to the legacy 

code, some minor differences were observed. As seen in 

table 4, computed differences in estimated flow statistics 

are less than 4 percent. Differences in output generally stem 

from the dissimilar ways that the legacy code and the R script 

handle data, including numerical rounding and differences in 

the streamflow values selected for the regression. These differences can alter results of the MOVE.1 regression enough 

to produce large percent differences in estimated flow statistics, particularly in lower ranges of flow. The table does not 

include differences for estimated flow statistics of less than 

0.1 ft3

/s because the percent difference between estimates 

output by the legacy code and the R script can become large 

at such small values when estimated flow statistics made 

using log-transformed flows are back-transformed. Another 

reason for not including differences for estimates less than the 

0.1 ft3/s threshold is that estimates below 1 ft3/s are typically 

rounded to the nearest 0.1 ft3

/s. Because this practice alters 

the difference between estimated flow statistics output by the 

MOVE.1 legacy program and statistics output by the MOVE.1 

R script, it makes little sense to consider differences less than 

the 0.1 ft3/s threshold.

Several factors contribute to the slight differences in 

flow estimates produced by the MOVE.1 legacy code and 

the new MOVE.1 R script. The principal factor contributing to differences in estimated flows can be largely attributed 

to differences in rounding that occur during input of flows. 

The legacy code retrieves mean daily flows directly from the 

NWIS database as unrounded values with a floating decimal, 

whereas the R script reads mean daily flows from NWISweb 

and is limited to three significant figures. A secondary factor 

contributing to differences in output is associated with errors 

discovered in the legacy code. In the process of converting 

from the legacy code to R, it was determined that the legacy 

code was not properly screening data when more than one 

measurement was made at a PR station during a single day. 

The legacy code failed to apply base-flow tests to subsequent 

measurements made on the same day and automatically 

included all daily measurements in the regression, whereas 

the new R script for MOVE.1 correctly imposes base-flow 

tests on each measurement made on the same day. Lastly, the 

new Aquarius database system has the ability to recognize 

data entry errors and, as a safeguard, does not include those 

measurements in the output file read by the new R script. For 

example, if any part of a flow record is considered to be in 

error as determined by the business rules of the new database, 

that measurement record is not included in the NWIS output 

file, and is therefore not read by the R script. To illustrate, if 

the start time of the flow measurement is incorrectly entered 

as beginning before the time of the site visit, that measurement 

is filtered from the output file of the database. Because no 

such safeguards were in place when the legacy code was run, 

any and all data input to the NWIS database were included in 

the MOVE.1 regression, regardless of whether the data were 

incorrectly input to the database.
18 Implementation of MOVE.1, Censored MOVE.1, and Piecewise MOVE.1 Low-Flow Regressions

Percent differences between regression metrics from 

the R script and the legacy code were as high as nearly 

19 percent, as shown in table 5. Errors in the percent predicted 

(PERCENT.PRED) and percent estimated (PERCENT.EST) 

calculations are consistently high. These differences are a 

reflection of the differences in estimated flows caused by one 

or more of the three factors mentioned above. Large differences in the regression intercept (CONSTANT) and Kendall 

probability (KENDALL.PROB) were caused by the very small 

values used to compute them. Overall, the R script reproduced 

low-flow estimates output by the legacy code, and when it 

didn’t, differences could generally be traced to the three principal factors mentioned above. 

Flow Chart

The flow chart in figure 7 graphically illustrates the order 

of operations used by the R script to read PR station identifiers 

and associated index station identifiers; read base flows and 

daily flows; select flow records according to the sequence of 

base-flow tests; perform OLS regression and MOVE.1, censored MOVE.1, or piecewise MOVE.1 regressions; compute 

standard errors of estimate at each index station; calculate the 

weighted average of flow statistics at the PR station; make 

predictions of flow statistics at PR stations on the basis of 

regression results; and generate pdf files containing plots of regression lines and residuals for each index station. Finally, the 

R script outputs weighted averages of flow statistics predicted 

at the PR station to a comma delimited .csv file.

Summary

To accommodate changes in software used to maintain 

and serve data from the U.S. Geological Survey (USGS) 

National Water Information System (NWIS) database, the 

USGS successfully migrated SAS legacy code to estimate 

flows at partial-record (PR) stations to an R script. The 

R script performs MOVE.1 regressions to predict low-flow 

statistics at a PR station, based on daily flows measured at 

index stations believed to share similar streamflow characteristics. Low-flow statistics at the PR stations are used in support 

of the New Jersey Department of Environmental Protection 

(NJDEP) surface-water permitting process. MOVE.1 regression extends the record at a PR station by preserving the 

mean and variance of measured and predicted low flows. The 

R script can be executed on various computing platforms, 

including those with UNIX, Windows, and macOS operating 

systems. 

The R script begins execution by reading a file containing 

the station identifiers of the PR station and all index stations 

considered to be hydrologically similar to the PR station. The 

R script then reads instantaneous base flows at the PR station and daily flows at one or more index stations, screens 

flows to ensure that all flow records meet a series of base-flow 

criteria, transfers flow information from index stations to the 

PR station by performing MOVE.1 regressions on concurrent 

flows, predicts low-flow statistics at the index stations on the 

basis of their regression lines, computes index station weights 

as reciprocals of the percent standard errors of estimate for 

index station regressions, estimates low-flow statistics at the 

PR station as the weighted sum of low-flow statistics estimated at multiple index stations, generates pdf files containing 

regression and residual plots, and outputs predicted low-flow 

statistics and their weighted averages to a .csv text file that can 

easily be imported to a spreadsheet. Low-flow statistics output 

by R script include the annual 7Q10, defined as the lowest 

annual 7-consecutive-day flow that occurs, on average, during 

a 10-year period. The annual 7Q10 provides the foundation for 

defining passing-flow requirements in New Jersey and is used 

to support NJDEP surface-water permitting decisions within 

the State.

The R script replicates MOVE.1 output generated 

by the legacy code and adds functionality by introducing new regression options for the user. These new options 

include censored MOVE.1 regression, which calls the 

USGS-developed censMove.1 R function to perform regression when zero-valued flows occur at index or PR stations. 

A piecewise MOVE.1 regression option was also added to 

be used where measured base flows at the PR station fail to 

exhibit statistically homogeneous behavior across the range of 

measured index station flows. The piecewise regression option 

permits two regression lines to be identified by iteratively 

changing the slopes and intercepts of the MOVE.1 regression 

line segments until the sum of squared error is minimized.

Equations used to derive the standard error of prediction, 

the time-sampling error, and the percent standard errors of 

estimate in the legacy code were reproduced in the R script. 

The standard errors of estimate provide measures of the degree 

of uncertainty in estimated low-flow statistics made at each 

index station using its associated regression line. Normalized 

reciprocals of the percent standard errors of estimate are used 

to weight terms in the average of low-flow statistics predicted 

at each of the index stations. The MOVE.1 regression methodology is used by the NJDEP to predict the annual 7Q10 at 

PR stations, using index station weights that are based on the 

7Q10 low-flow statistic. Future improvements to the R script 

may include development of a formal framework for estimating other low-flow statistics at PR stations that explicitly 

calculates weights for the particular t-year recurrence statistic 

of interest, as well as development of a piecewise regression 

approach that doesn’t require the user to enter an index flow 

interval where the breakpoint appears likely to be located.
Summary 19

Start

Install and load R libraries

and packages

Download and unzip the base

flow measurement files from

the Aquarius system

Prompt the user to select the

location for the “Output” subdirectory, create it, and tell the user 

where it is

Prompt the user to select the

station id file for the partial-record 

site of interest (or, alternatively, a 

“stacked” station id file containing 

multiple PR stations)

Ask the user if predicted flow

statistics should be appended to 

flow statistics predicted from

previous regressions

Ask the user to select which type

of regression to use by clicking on 

a radio button

From the station id file or the

“stacked” station id file, read

the station id for the next 

partial-record station and all

index stations associated with it

Create a new subdirectory in the

“Output”directory for the current PR site,

using the PR station id and appending a

sequence number and the regression

type to the subdirectory name

From the station id file, read the

beginning and end dates for the

period of record, the upper and

lower index station recession

bounds used in a base-flow 

test, and the index station ids

Populate checkboxes with

index station ids associated

with the PR site and allow the

user to select which index sites 

to include in the regression

Read the base-flow

measurement file

downloaded from Aquarius

Regular Piecewise Censored

Yes No

Read the skew and time sampling 

error parameter vectors that 

contain entries in the Kite (1988) 

table needed to estimate the 

timesampling error

Read the stat2001 file, which 

contains low-flow statistics 

for all index stations, 

estimated from daily flows 

observed since 2001

Loop through all

index stations

selected by user

Create a .txt file and a .pdf file to

store predicted flows, residuals and

regression plots in the current PR

subdirectory for the current station

Read daily flows for 

the current index 

station from the 

NWIS database

Report duplicate

dates to user

Lag daily index flows by 

1 and 2 days for use in 

baseflow testing

Join index station flows to

PR station flows on

concurrent dates

Remove all flow records

that do not pass all 

baseflow tests

Add 0.01 ft3/s to all flows

Type of MOVE.1

Regression Selected?

Perform MOVE.1

regression using R

function move.1

Subtract 0.01 ft3/s from all

flows

Use the regression line to predict PR station

flow statistics from index station flow statistics

at the current index station

Prompt user to select

upper and lower 

bounds on breakpoint 

interval

Subtract 0.01 ft3/s

from all flows

Perform MOVE.1

regression using R

Loop through all function censMove.1

breakpoints in the

breakpoint interval

Perform MOVE.1 regression

using R function move.1 for

segment 1

Perform MOVE.1 regression

using R function move.1 for

segment 2

Estimate the mean-squared

error for the regression

End

End

Final piecewise regression is

the regression with the

breakpoint providing the

smallest mean-squared error

Calculate the weight for the current index station

Plot the MOVE.1 regression line and

residual plots to a .pdf file, write

predicted flow statistics at the PR

station for the current index station to

a .csv file, and write summary to a .txt

Find the weighted average for each flow statistic

at the PR station by adding weighted predicted

statistics at all its index stations

Write all predicted flow statistics and their

weighted average to a .csv file

Stacked station id file

with another partial

record station?

Stop

Figure 7. Flow chart for the MOVE.1 R script.

(PR, partial-record; ft3/s, cubic feet per second)
20 Implementation of MOVE.1, Censored MOVE.1, and Piecewise MOVE.1 Low-Flow Regressions

Acknowledgments

Special thanks go to Dave Lorenz of the USGS for 

developing the move.1, censMove.1 , and predict.move.1 R 

functions required to perform MOVE.1, censored MOVE.1, 

and piecewise MOVE.1 regressions invoked by the R script 

and to estimate flow statistics based on results of the regressions (https://github.com/USGS-R/smwrStats/blob/master/R/

move.1.R, https://github.com/USGS-R/smwrQW/blob/

master/R/censMove.1.R). We appreciate insights offered by R. 

Ward Hickman (USGS) relating to the workings of the legacy 

code that provided the basis for the R script. Thanks are also 

extended to Will Farmer (USGS), who verified the equation 

used to estimate the percent standard error of estimate needed 

to calculate weights for the weighted average of low-flow 

statistics at index stations.

References Cited

Crawley, M.J., 2013, The R book: Chichester, UK, John Wiley 

and Sons, Ltd., 1076 p.

Gillespie, B.D., and Schopp, R.D, 1982, Low-flow characteristics and flow duration of New Jersey streams: U.S 

Geological Survey Open-File Report 81–1110, 164 p. [Also 

available at https://pubs.usgs.gov/of/1981/1110/report.pdf.]

Halfon, E., 1985, Regression method in ecotoxicology: a better formulation using the geometric mean functional regression: Environmental Science and Technology, v. 19, iss. 8, 

p. 747–749.

Hartung, J., Knapp, G., and Sinha, B.K., 2008, Statistical 

meta-analysis with applications: Hoboken, N.J., John Wiley 

and Sons, Ltd., 248 p.

Helsel, D.R, and Hirsch, R.M., 2002, Statistical methods in 

water resources: U.S. Geological Survey Techniques of 

Water-Resources Investigations, book 4, chap. A3, 510 p.

Hirsch, R.M., 1982, A comparison of four streamflow regression techniques: Water Resources Research, v. 18, no. 4, 

p. 1081–1088.

Hirsch, R.M., and Gilroy, E.J., 1984, Methods of fitting a 

straight line to data: examples in water resources: Journal of the American Water Resources Association, v. 20, 

p. 705–711.

Hoffman, J.L., and Domber, S.E., 2013, History of passing 

flows in New Jersey, with contemporary and future applications: New Jersey Geological and Water Survey Open-File 

Report OFR 13–1, 62 p.

Journel, A.G., and Huijbregts, C.J., 1978, Mining geostatistics: 

San Diego, Calif., Academic Press, 600 p.

Kite, G.W., 1988, Frequency and risk analyses in hydrology: 

Littleton, Colo., Water Resources Publications, 257 p.

Kritskiy, S.N., and Menkel, M.F., 1968, Some statistical methods in the analysis of hydrologic data: Soviet Hydrology 

Selected Papers 1, p. 80–88.

Matalas, N.C., and Jacobs, B., 1964, A correlation procedure 

for augmenting hydrologic data: U.S. Geological Survey 

Professional Paper 434–E, 7 p.

Riggs, H.C., 1968, Some statistical tools in hydrology: U.S. 

Geological Survey Techniques of Water-Resources Investigations, book 4, chap. A1, 39 p.

Searcy, J.K., 1959, Flow-duration curves, Manual of hydrology—Part 2. Low-flow techniques: U.S. Geological Survey 

Water-Supply Paper 1542–A, 33 p.

State of New Jersey, Department of Environmental Protection, 2011, Surface water quality standards: New Jersey 

Administrative Code, Title 7, Chapter 9B, 113 p., accessed 

December 22, 2017, at http://www.nj.gov/dep/rules/rules/

njac7_9b.pdf.

State of New Jersey, Department of Environmental Protection, 

Division of Water Quality, 2015, NJPDS Rules N.J.A.C. 

7:14A, accessed December 22, 2017, at http://www.state.

nj.us/dep/dwq/714a.htm.

Stedinger, J.R., and Thomas, W.O., Jr., 1985, Low-flow 

frequency estimation using base-flow measurements: U.S. 

Geological Survey Open-File Report 85–95, 22 p.

Telis, P.A., 1991, Low-flow and flow-duration characteristics 

of Mississippi streams: U.S Geological Survey WaterResources Investigations Report 90–4087, 214 p.

Watson, K.M., Reiser, R.G., and Schopp, R.D., 2005, Streamflow characteristics and trends in New Jersey, water years 

1897–2003: U. S. Geological Survey Scientific Investigations Report 2005–5105, 131 p.
For additional information, contact:

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U.S. Geological Survey

3450 Princeton Pike, Suite 110

Lawrenceville, NJ 08648

or visit our website at:

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Colarullo and others—Implementation of MOVE.1, Censored MOVE.1, and Piecewise MOVE.1 Low-Flow Regressions—Open-File Report 2018–1089

ISSN 2331–1258 (online)

https://doi.org/10.3133/ofr20181089