A Graphical Method for Estimation of 

Barometric Efficiency from Continuous Data—

Concepts and Application to a Site in the Piedmont, 

Air Force Plant 6, Marietta, Georgia

U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigation Report 2007-5111

Prepared in cooperation with the

U.S. Air Force Aeronautical Systems Center, Marietta, Georgia

1028.05

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32.50 32.60 32.70 32.80 32.90 33.00 33.10 33.20 33.30

32.50

32.60

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2 4 6 8 10 12 14 16 18 20 22 24 26

1028.05

AUGUST 2003

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Well ob202b

Well ob202b

BAROMETRIC PRESSURE, IN FEET OF WATER

BAROMETRIC PRESSURE, IN FEET OF WATER

WATER LEVEL, IN FEET ABOVE NGVD 88

WATER LEVEL, IN FEET ABOVE NGVD 88

Constant-discharge test

Observed

Average

A Graphical Method for Estimation of 

Barometric Efficiency from Continuous Data—

Concepts and Application to a Site in the Piedmont, 

Air Force Plant 6, Marietta, Georgia

By Gerard J. Gonthier

Prepared in cooperation with the

U.S. Air Force Aeronautical Systems Center, Marietta, Georgia

Scientific Investigation Report 2007-5111

U.S. Department of the Interior

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

DIRK KEMPTHORNE, Secretary

U.S. Geological Survey

Mark D. Myers, Director

U.S. Geological Survey, Reston, Virginia: 2007

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U.S. Government.

Although this report is in the public domain, permission must be secured from the individual copyright owners to 

reproduce any copyrighted materials contained within this report.

Suggested citation:

Gonthier, G.J., 2007, A Graphical Method for Estimation of Barometric Efficiency from Continuous Data—Concepts 

and Application to a Site in the Piedmont, Air Force Plant 6, Marietta, Georgia: U.S. Geological Survey Scientific 

Investigations Report 2007-5111, 29 p., Web-only publication available at http://pubs.usgs.gov/sir/2007/5111/.
iii

Contents

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

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

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

Acknowledgments................................................................................................................................3

Barometric-Efficiency Concepts.................................................................................................................3

Barometric-Pressure Effects on Water Levels................................................................................3

Barometric Efficiency...........................................................................................................................4

Factors Influencing Water-Level Change.........................................................................................4

Identifying Water-Level Change that Causes Error in Barometric-Efficiency Estimation........5

Systematic Error...........................................................................................................................6

Constant Barometric-Pressure-Independent Water-Level Change ...................................6

Previous Methods of Estimating the Barometric Efficiency...................................................................6

Average-of-Ratios Method..................................................................................................................6

Median-of-Ratios Method...................................................................................................................7

Clark Method .........................................................................................................................................7

Slope Method on Water-Level and Barometric-Pressure Change ..............................................8

Using Continuous Data to Estimate Barometric Efficiency ....................................................................9

Filtering Water-Level Change that Causes Error in the Estimation of 

Barometric Efficiency.............................................................................................................9

Long-Term Barometric-Pressure-Independent Water-Level Change ..............................10

Daily Fluctuations.......................................................................................................................10

Other Barometric-Pressure-Independent Water-Level Changes .....................................12

Time-Interval Selection......................................................................................................................12

A Graphical Method for Estimating Barometric Efficiency from 

Continuous Water-Level Data ......................................................................................................13

Theory Behind the Graphical Method.............................................................................................13

Examples of Elliptical Loops in Graphs ...........................................................................................15

Determining the Slope of the Preferred Orientation of Loops.....................................................17

Application to a Site in the Piedmont Province, Air Force Plant 6, Marietta, Georgia............20

Comparison of Methods..............................................................................................................................20

Barometric-Efficiency Estimates Compared to the Domain of the Barometric Efficiency.....22

Barometric-Efficiency Estimates of Control Wells........................................................................22

Summary........................................................................................................................................................27

References Cited..........................................................................................................................................28
iv

Figures

1. Map showing monitoring wells in the study area, U.S. Air Force Plant 6, 

Marietta, Georgia, 2003................................................................................................................2

2–10. Graphs showing

2. (A) Selected water-level values as a function of barometric pressure 

 and (B) water-level change as a function of barometric-pressure change 

 in well mwg05, Air Force Plant 6, Marietta, Georgia, 2003 ........................................9

3. Example of removing barometric-pressure-independent water-level 

 change in the form of long-term linear trend, well b4mwi, 

 Air Force Plant 6, Marietta, Georgia, 2003 ..................................................................11

4. Example of removing barometric-pressure-independent water-level 

 change in the form of long-term 2nd-order, parabolic trend, 

 well pmw7d, Air Force Plant 6, Marietta, Georgia, 2003...........................................11

5. Example of time-interval selection about two inflection points 

 during maximum rates of barometric-pressure change, well b4mwi, 

 Air Force Plant 6, Marietta, Georgia, August, 2003 ...................................................12

6. Example of (A) time series of hypothetical water level and 

 barometric pressure, and (B) hypothetical water level as a 

 function of barometric pressure....................................................................................14

7. (A) Time series of water level and barometric pressure, and (B) water 

 level as a function of barometric pressure for well ob202b, 

 August 2–October 6, 2003...............................................................................................15

8. (A) Time series of water level and barometric pressure, and (B) water 

 level as a function of barometric pressure for well mwg05, 

 August 2–October 6, 2003...............................................................................................16

9. (A) Time series of water level and barometric pressure, and (B) water 

 level as a function of barometric pressure for well ob208c, 

 August 2–October 6, 2003...............................................................................................17

10. (A) Time series of water level and barometric pressure, and (B) water 

 level as a function of barometric pressure for well mwg11, 

 August 2–October 6, 2003...............................................................................................18

11. Diagram showing screen captures of software display showing (A) spreadsheet 

and (B) selected proposed barometric efficiency (BE) values ...........................................19

12–14. Graphs showing

12. Water levels in wells rw207 and ob203c, and in a nearby pond, 

 Air Force Plant 6, Marietta, Georgia, 2003 ................................................................. 23

13. The discrepancy in the estimate of the barometric efficiency as 

 a function of the ratio of the variance of ΔW to the variance of ΔB

 (using the slope method)................................................................................................ 25

14. Water level in well rw207 as a function of surface-water level in a 

 nearby retention pond, Air Force Plant 6, Marietta Georgia, 

 August 5–October 6, 2003.............................................................................................. 26
v

Tables

1. Variance of corrected water-level data using selected values of 

correction factors for two wells, b4mwi and pmw7d, Air Force Plant 6, 

Marietta, Georgia, 2003..............................................................................................................10

2. Barometric efficiencies of wells using the average-of-ratios, 

median-of-ratios methods, Clark, slope and graphical methods, 

Air Force Plant 6, Marietta, Georgia, 2003..............................................................................21

3. Number of estimates, by method, that were outside the domain 

of the barometric efficiency for 45 wells at Air Force Plant 6, 

Marietta, Georgia, 2003..............................................................................................................22

4. Control wells with barometric efficiencies determined to be zero 

based on surface influences, Air Force Plant 6, Marietta, Georgia, 2003.........................23

5. Estimates of barometric efficiency from different methods for wells 

determined to have barometric efficiencies of near zero based on 

surface influences, Air Force Plant 6, Marietta, Georgia, 2003. .........................................24

6. Values used to correct wells rw207 and ob203c for barometricpressure-independent water-level change in the form of 

surface-water fluctuations from a nearby retention pond, 

Air Force Plant 6, Marietta, Georgia, 2003..............................................................................25

7. Estimates of barometric efficiency from different methods for 

wells ob203c and rw207 before and after water levels were corrected 

for surface-water fluctuations in a nearby retention pond, 

Air Force Plant 6, Marietta, Georgia, 2003..............................................................................26

Datums 

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

(NAVD 88).

Horizontal coordinate information is referenced to the North American Datum of 1983 (NAD 83).
vi
A Graphical Method for Estimation of 

Barometric Efficiency from Continuous Data—

Concepts and Application to a Site in the Piedmont, 

Air Force Plant 6, Marietta, Georgia

By Gerard J. Gonthier

Abstract

A graphical method that uses continuous water-level and 

barometric-pressure data was developed to estimate barometric

efficiency. A plot of nearly continuous water level (on the 

y-axis), as a function of nearly continuous barometric pressure (on the x-axis), will plot as a line curved into a series of 

connected elliptical loops. Each loop represents a barometricpressure fluctuation. The negative of the slope of the major 

axis of an elliptical loop will be the ratio of water-level change 

to barometric-pressure change, which is the sum of the 

barometric efficiency plus the error. 

The negative of the slope of the preferred orientation 

of many elliptical loops is an estimate of the barometric 

efficiency. The slope of the preferred orientation of many 

elliptical loops is approximately the median of the slopes of 

the major axes of the elliptical loops. If water-level change 

that is not caused by barometric-pressure change does not 

correlate with barometric-pressure change, the probability 

that the error will be greater than zero will be the same as 

the probability that it will be less than zero. As a result, the 

negative of the median of the slopes for many loops will be 

close to the barometric efficiency.

The graphical method provided a rapid assessment of 

whether a well was affected by barometric-pressure change 

and also provided a rapid estimate of barometric efficiency. 

The graphical method was used to assess which wells at Air 

Force Plant 6, Marietta, Georgia, had water levels affected by 

barometric-pressure changes during a 2003 constant-discharge 

aquifer test. The graphical method was also used to estimate 

barometric efficiency. Barometric-efficiency estimates from 

the graphical method were compared to those of four other 

methods: average of ratios, median of ratios, Clark, and 

slope. The two methods (the graphical and median-of-ratios 

methods) that used the median values of water-level change 

divided by barometric-pressure change appeared to be most 

resistant to error caused by barometric-pressure-independent 

water-level change. The graphical method was particularly 

resistant to large amounts of barometric-pressure-independent 

water-level change, having an average and standard deviation 

of error for control wells that was less than one-quarter that of 

the other four methods. 

When using the graphical method, it is advisable that more 

than one person select the slope or that the same person fits the 

same data several times to minimize the effect of subjectivity. 

Also, a long study period should be used (at least 60 days) 

to ensure that loops affected by large amounts of barometricpressure-independent water-level change do not significantly 

contribute to error in the barometric-efficiency estimate.

Introduction

Barometric efficiency is the water-level change caused 

by a barometric-pressure change divided by that barometricpressure change (Clark, 1967). In confined aquifer settings, an 

increase in barometric pressure usually will cause a decrease 

in water level in an open well by an amount governed by the 

barometric efficiency (Todd, 1959; Ferris and others, 1962; 

Freeze and Cherry, 1979; Kruseman and de Ridder, 1991; 

Landmeyer, 1996; Rasmussen and Crawford, 1997; and 

Batu, 1998). Aquifer-test methods depend on accurate 

measurements of pumpage-induced ground-water-level 

fluctuations for determining values of hydraulic properties. 

The barometric efficiency is used, as a correction factor, to 

remove barometric effects on water levels in wells during an 

aquifer test (Kruseman and de Ridder, 1991), and also can 

be used to help determine the degree of confinement within 

an aquifer to which the well is opened (Landmeyer, 1996), 

estimate the storage coefficient in a confined aquifer (Jacob, 

1940), and determine bulk elastic properties of the aquifer 

(Domenico, 1983; Rojstaczer and Agnew, 1989; Batu, 1998). 

Determination of barometric efficiency is challenging because 

it is difficult to distinguish the component of water-level 

change within a well caused by barometric-pressure change 

from the total water-level change within a well.
Water-level and barometric-pressure data were collected 

at Air Force Plant 6, Marietta, Georgia, during 2003 as part of 

ground-water characterization efforts (fig. 1). An aquifer test 

was conducted during August and September 2003. Beginning about 2 days after the start of the aquifer test, barometric 

pressure rose and fell during a period of about 5 days by a 

maximum pressure of 10 millimeters of mercury (equivalent 

to about 0.43 foot of water). Several wells responded with a 

water-level decline followed by a water-level increase, respectively. The barometric effects on water level had to be removed 

to better interpret the aquifer-test data. A graphical method 

was developed to assess the wells that had water levels affected 

by barometric-pressure changes during the aquifer test and to 

estimate the barometric efficiency.

 Graphical Method for Estimation of Barometric Efficiency from Continuous Data

0 375 750 FEET

0 100 200 METERS

South Cobb Drive

280

Study area boundary Well and site name

Barometer Used in this report

Not used in this report

EXPLANATION

Base modified from U.S. Geological Survey

Marietta 1:24,000, 1992

Rottenwood Creek

b4mwh b4mwi

b4mwq

mw-00-102a

mw-00-102b

mwg11

mwg21

mwg5-5r

mwos01

mwos02

mwos03

mwos04

mwos10

mwos09

mwos12

mwos11

mw-w2

mw-w2

ob202a ob202b

ob203a

ob203b

ob203c

ob204b

ob204c

ob205b

ob208b

pmw7d

rw204

rw205

ob208c

sct1 sct4

sct6 sct7

rw201

ob208a

rw202

rw203

rw207

ob209a

ob205a

mwg10

mwg05

sct5

sct3

N

PLANT 6

U.S. AIR FORCE 

Marietta

COBB

COUNTY

GEORGIA

Figure 1. Monitoring wells in the study area, U.S. Air Force Plant 6, Marietta, Georgia, 2003.
Purpose and Scope

This report introduces a graphical method that estimates 

barometric efficiency rapidly and accurately, using continuous 

water-level and barometric-pressure data. Extensive background discussion and comparison with other methods are 

provided to support the graphical method. This report also 

provides general information on how to make the best estimates of barometric efficiency, emphasizing, but not limited 

to, continuous data. General discussion includes sources of 

error in estimating barometric efficiency, how to reduce some 

of that error prior to estimation, and how various methods 

reduce the effect of the remaining error during estimation. 

Water-level change that is not caused by barometric-pressure 

change, referred to in this report as “barometric-pressureindependent water-level change,” is identified as the source 

of error in estimates of barometric efficiency. This report 

describes two simple techniques to remove some of the 

barometric-pressure-independent water-level change prior to 

using the data to estimate barometric efficiency. A technique 

for optimally selecting time intervals from barometric-pressure 

data also is presented. This report includes the theory behind 

the graphical method and examples of barometric-efficiency 

estimates using data from 45 wells at Air Force Plant 6, 

Marietta, Georgia. The graphical method is objectively 

compared to the four other methods in two separate ways.

Some examples described within this report are for wells 

opened to fractured, highly heterogeneous rock under various 

degrees of weathering (Air Force Plant 6, Marietta, Georgia; 

fig. 1). Because some of the examples in this report are for 

a fractured-rock setting and because of the possibility of 

well-bore storage and skin effects, estimates of barometric 

efficiency are considered for individual wells and not the 

“aquifer.” Hereinafter, “water-level change” refers to waterlevel change within a well that is open to the atmosphere. 

Acknowledgments

The U.S. Geological Survey conducted the research 

for this report at Air Force Plant 6, Marietta, Georgia. The 

U.S. Air Force Aeronautical Systems Center funded all 

research activities for this report. The following hydrologists from the U.S. Geological Survey contributed colleague 

reviews: Gregory S. Cherry, Earl A. Greene, James E. Landmeyer, Dorothy F. Payne, and Lynn J. Torak. Todd C. Rasmussen, Associate Professor of Hydrology at the University 

of Georgia, also contributed to the author’s understanding of 

barometric effects on water levels in very shallow wells.

Barometric-Efficiency Concepts

Basic concepts that are used in this report to estimate 

barometric efficiency are discussed below. In this section, 

how barometric-pressure changes affect water levels in wells 

is briefly discussed, barometric efficiency is defined, the type 

of water-level change that adversely affects the estimation of 

barometric efficiency is identified, and how a method reduces 

the adverse effect of some water-level change during estimation is explained.

The term study period is defined in this report as the 

entire length of time that water-level and barometric-pressure 

data are available to determine barometric efficiency (one 

measurement, every 15 minutes for about 60 days). The term 

time interval describes a shorter length of time (from 6 hours 

to 2 days), which is usually the length of time used to determine values of water-level change (DW) and barometricpressure change (DB). The term time increment refers to a 

shorter length of time between measurements taken by continuous-recorder equipment (for example, 15 minutes) 

and when DW and DB are sometimes determined. 

Barometric-Pressure Effects on Water Levels

As early as 1663, Blaise Pascal (Pascal, 1973; Gossard 

and Hooke, 1975; Crawford, 1994) found barometric-pressure changes affected water levels in some wells. Todd (1959), 

Freeze and Cherry (1979), Batu (1998), and others provide a 

detailed description of how barometric-pressure change affects 

water levels in wells open to confined aquifers. An aquifer 

with a sturdy skeletal structure will keep some of the increase 

in barometric pressure at the surface from translating to water 

in the aquifer. Meanwhile, the full increase in barometric 

pressure will load onto the water level of a well open to the 

atmosphere. This results in an imbalance in pressure between 

water inside and outside of the well opening. The water level 

in the well changes to compensate for the imbalance by:

 D

D D

W

P P

b

at aq

w

= ( ) -

g , (1)

where

DWb is the water-level change that is caused 

by barometric-pressure change, in units 

of length; 

DPat is the pressure change at the top of water in 

the well and at the land surface, in units 

of force per area; 

DPaq is the pressure change in the aquifer (water 

outside of the well screen) as a result of the 

pressure change at the land surface (DPat);

and

gw is the specific weight of the water in the well, 

in units of force per volume. 

By convention set by literature (Clark, 1967; Batu, 1998), 

an increase in water level during a time interval is a positive 

change (greater than zero); but an increase in pressure during 

a time interval, is a negative change (less than zero).

Weeks (1979), Rasmussen and Crawford (1997), and 

Spane (1999) described the effect of barometric-pressure 

change on water level in wells open to unconfined aquifers. 

Barometric-Efficiency Concepts 
 Graphical Method for Estimation of Barometric Efficiency from Continuous Data

Within the unconfined setting, there is a time lag between an 

increase in atmospheric pressure at land surface and its translation to the water table. The key factors causing the delay are the 

pneumatic diffusivity of air through the unsaturated zone and the

thickness of the unsaturated zone. The result is that the water level 

in a well will respond to an instantaneous change in atmospheric 

pressure where the DPaq in equation 1 is significantly less than 

DPat at time zero. Through time, the increase in pressure at the 

surface migrates to the water table, DPaq will increase to the value 

of DPat in equation 1, and the water in the well will recover to its 

level prior to the pressure change. Barometric-pressure change 

will not affect a well that is directly connected to a surface-water 

feature (the effective thickness of the unsaturated zone is zero) 

or when the top of the well screen is above the water table.

Other factors that affect barometric efficiency of a well 

include well-skin and well-bore storage effects. A well with 

a clogged well screen or relatively impermeable material 

surrounding the well screen (well skin) will have a delayed 

response to barometric-pressure change (Rasmussen and 

Crawford, 1997; and Spane, 1999) and may cause the barometric efficiency (see definition below) to change with time. 

In this report, discussion of how methods estimate the barometric efficiency focuses on confined settings, and assumes 

that barometric efficiency does not change with time.

Barometric Efficiency

The definition of barometric efficiency equates to 

the following:

α = W

B

Δ b

Δ , (2)

where 

DWb is that part of the water-level change caused 

by a barometric-pressure change DB

and 

α is the barometric efficiency. 

The barometric-pressure change (DB) equates to DPat /γw, 

and is measured in units of length of water. 

As with the previously mentioned pressure change, an 

increase in barometric pressure during a time interval is a 

negative change (Clark, 1967; Batu, 1998). The barometric 

efficiency is dimensionless and ranges from zero to one. 

The water level, in time series, then can be corrected 

for barometric effects by:

W W B B ( )t corr t uncorr ( ) ( )t = − − α ( ) 0 , (3)

where 

W(t)corr is the water level, at time t, corrected for 

barometric pressure; 

W(t)uncorr is the uncorrected water level, at time t ; 

α is the barometric efficiency; 

and 

(B0–B(t)) is the barometric pressure B(t), at time t, 

referenced to a barometric-pressure 

datum B0

.

Factors Influencing Water-Level Change

Measurable water-level change in a well (DW) is the 

water level in a well at time (t+1), minus the water level in a 

well at time (t):

DW W W = - ( ) ( ) t+1 t . (4)

Barometric-pressure change (DB) that causes the water-level 

change (DWb) is measurable as the barometric pressure at time 

(t) minus the barometric pressure at time (t+1):

DB B B = -( ) ( )1 t t+ . (5)

The order of time (t) and time (t+1) in equation 4 is inverted 

with respect to equation 5 in order to follow the convention 

that DB is negative for an increase in barometric pressure 

during a time interval. Throughout this report, plots of DW

(on the y-axis) and DB (on the x-axis) for wells that are 

affected by barometric-pressure change have positive slopes 

whereas plots of W (on the y-axis) and B (on the x-axis) for 

the same data sets have negative slopes.

The measurable water-level change (DW) in a well is 

caused by one or more of several possible causes: 

DW=DWb+DWr+DWl+DWp+DWg+DWm

+DWe+DWs+DWo, (6)

where 

DW is the water-level change in a well 

during a time interval; 

DWb is the water-level change caused 

by barometric-pressure change; 

DWr is the water-level change caused 

by recharge; 

DWl is the water-level change caused 

by seasonal or long-term trends; 

DWp is the water-level change caused 

by local or regional pumping; 

DWg is the water-level change caused 

by earth tides; 

DWm is the water-level change caused 

by ocean tides; 

DWe is the water-level change caused 

by evapotranspiration; 

DWs is the water-level change caused 

by surface-water fluctuations; 

and 

DWo is the water-level change caused 

by other influences, during a 

specific time interval.

Water-level change that is caused by recharge (DWr ) can 

be from rain events or snow melt. Examples of recharge used 

within this report are rain events. The response to a rain event 

is characterized by a sharp increase in water level, followed 

by a recession curve with the rate of water-level change 

decreasing with time after the rain event. 
Seasonal or long-term trends (DWl ) involve waterlevel changes during lengths of time greater than barometricpressure fluctuations. If one season is wetter than another, a 

long-term climate pattern will be reflected in water-level data 

during a study period. This long-term trend may be represented through time as a linear (constant change) or higherorder curved line.

Water-level change caused by pumping (DWp ) includes 

aquifer tests and other local pumping, or irrigation and other 

regional pumping. Regional pumping often occurs on a seasonal time scale, but the effects may be more pronounced during a shorter period of time. Study periods used to determine 

barometric efficiency and other correction factors should be 

during a time outside of an aquifer test (Halford, 2006).

Gravitational influences of the sun and moon along with 

movements of the moon and earth create changes in gravitational force on the earth (Rinehart, 1975). These forces cause 

earth tides. Earth tides are slight changes in the shape of the 

geoid that, in turn, lead to compression or tension within the 

earth’s crust. For example, when the vertical component of the 

force of gravity is relatively high, the fractured-crystalline rock 

at Air Force Plant 6 usually is compressed, fractures constrict, 

storage is temporarily decreased, and the water in the fractures 

is squeezed to higher values of head. Conversely, when the 

force of gravity is relatively low, the fractured-crystalline rock 

is stretched, fractures dilate, storage is temporarily increased, 

and values of head decline. The result is the occurrence of 

sinusoidal fluctuations in water level (DWg

, two peaks and two 

troughs during about 25 hours) and a change in amplitudes, 

which cycles every 2 weeks (Marine, 1975; Domenico, 1983; 

and Galloway and Rojstaczer, 1988). Earth-tide effects are 

not to be confused with ocean-tide effects, which affect some 

wells that are close to the coast.

Ocean tides cause water-level change in wells (DWm ) by 

compression of a confined aquifer due to the increased weight 

of ocean water during high tide (Robinson and Bell, 1971; 

Batu, 1998). Under a confined aquifer setting, the ocean-tidal 

efficiency plus the barometric efficiency should, theoretically, 

equal one. Ocean tides cause water-level change in wells open 

to unconfined aquifers due to the head-gradient-induced flow 

of water into the unconfined aquifer.

Evapotranspiration from vegetation during daylight 

hours can act as local pumping in shallow aquifers. The water 

level in the well reaches a peak during sunrise and reaches a 

minimum just after sunset. There is a peak and trough each 

day with a wavelength of 1 day. Effects of evapotranspiration 

on ground-water level (DWe ) are greatest during the growing 

season and normally do not affect ground-water levels during 

winter when the temperature stays at or below freezing.

Wells that are in close proximity or have fractures 

directly connected to surface-water bodies may respond 

directly to surface-water fluctuations (DWs ). Surface-water 

bodies include ponds and streams. Wells near streams may 

exhibit water-level fluctuations that look like responses to 

rain events but are actually responding to a combination of 

rain events and stream-flood events.

Identifying Water-Level Change that Causes 

Error in Barometric-Efficiency Estimation

Water-level change that is caused by barometric-pressure 

change (DWb ) cannot be directly measured. Equation 6 can 

be simplified by grouping all of the water-level fluctuations 

that are independent of barometric-pressure change into one 

term (DWi ):

 D D D W W W = +b i . (7)

Equation 7 separates the measurable water-level change in 

a well into two separate components based on their cause. 

While the term DWb is the water-level change that is caused 

by barometric-pressure change (related to the “α B B 0 t ( ) − ( )” 

in equation 3), the term DWi is the water-level change that is 

not caused by barometric-pressure change (related to W(t)corr in 

equation 3). In this report, the terms Wi

, W, and B are values 

relative to an implied reference datum.

From equations 2 and 7, it can be seen that the ratio of 

the total water-level change to the barometric-pressure change 

(DW/DB) is an estimate of barometric efficiency (αˆ ) and can 

be defined as:

 ˆ ( )

a = +D D

D

W W

B

b i ,

or (8)

 aˆ = +

D

D

D

D

W

B

W

B

b i ,

where 

DWb+DWi is the measurable water-level change (DW)

and 

 aˆ may or may not be close to the actual 

value of the barometric efficiency. 

Rearranging equation 8 and substituting α for DWb /DB yields:

a a e ˆ - = =sr

Wi

B

D

D . (9)

It can be seen that the error ( esr ) in the estimate of barometric 

efficiency determined from a single ratio of water-level change 

to barometric-pressure change (DW/DB) is the ratio of barometric-pressure-independent water-level change to barometricpressure change (DWi

 /DB).

The key to an accurate estimation of barometric efficiency is to reduce the error-causing effect of barometricpressure-independent water-level change (represented by 

DWi ) from the estimation (Clark, 1967; and Davis and Rasmussen, 1993). Some barometric-pressure-independent waterlevel change can be removed prior to estimating barometric 

efficiency because it can be identified and avoided or quantified and removed. Examples of avoidable DWi

 include features 

that can be seen on hydrographs, such as spikes in water level 

associated with rain events or sudden decreases in water level 

Barometric-Efficiency Concepts 
Graphical Method for Estimation of Barometric Efficiency from Continuous Data

caused by the start of pumping. Other forms of DWi that 

can be removed from the data include long-term trends and 

daily fluctuations. Techniques for removing such barometric-pressure-independent water-level change are reviewed in 

the Filtering Water-Level Change that Causes Error in the 

Estimation of the Barometric Efficiency section of this report. 

After removing as much barometric-pressure-independent 

water-level change as possible, some barometric-pressureindependent water-level change will remain in the data.

Methods that estimate barometric efficiency do so in 

the presence of remaining barometric-pressure-independent 

water-level change. The best methods for estimating barometric efficiency use many measurements of barometric-pressure 

change and corresponding water-level change from many 

time intervals and rely on barometric-pressure-independent 

water-level change to not be correlated to barometric-pressure 

change. With many measurements of water-level and barometric-pressure change, the constant relation between DWb

and DB becomes more apparent while the lack of correlation 

between DWi and DB becomes less apparent.

Systematic Error

Correlation between DWi and DB tends to produce a 

systematic error and, thus, the error (DWi /DB) may be difficult to distinguish from DWb /DB. Such correlation can occur 

when both the barometric-pressure-independent water-level 

change and the barometric-pressure change share a common 

source. Two possible situations that may create correlations 

between DWi and DB are repetitive weather patterns and 

earth tides. 

During repetitive synoptic weather conditions, rain 

events may occur at similar times with respect to barometric-pressure change. For example, a cold front approaches, 

barometric pressure decreases, and a rain event causes water 

levels to rise in the well (DWr ); then a high-pressure area 

moves over the well, and the water level in the well recesses 

from the rain event. If similar rain events repeatedly occur 

at about the same time with respect to barometric-pressure 

fluctuations, the effect of the rain event on DW may be 

difficult to distinguish from the barometric-pressure effect 

because of the correlation between the rain event and the 

barometric-pressure change. 

Earth tides are nearly semidiurnal (two peaks and 

two troughs in about 25 hours); atmospheric tides are semidiurnal fluctuations in barometric pressure (two peaks and 

two troughs in 24 hours). At Air Force Plant 6, diurnal 

barometric-pressure fluctuations included two daily minima 

at 3:20 a.m. and 5:20 p.m. and two daily maxima at about 

10:30 a.m. and 10:45 p.m. Actual times of the minima and 

maxima occurred plus/minus an hour. Due to the similar 

wavelength between atmospheric tides and earth tides, 

barometric-pressure changes may sometimes correlate 

to earth-tide fluctuations (DWg) within wells.

Constant Barometric-Pressure-Independent 

Water-Level Change

A constant barometric-pressure-independent water-level 

change with time will cause a random error in the estimate 

of barometric efficiency. For example, long-term water-level 

trend was a common form of constant barometric-pressureindependent water-level change at Air Force Plant 6. A 

constant DWi will not correlate to DB as long as there is no 

long-term barometric-pressure trend.

Previous Methods of Estimating the 

Barometric Efficiency

In this report, four methods of estimating barometric 

efficiency are compared with the graphical method. This 

section briefly describes each of these methods, and discusses 

how they reduce the effect of remaining barometric-pressureindependent water-level change (DWi

) during the estimation 

of barometric efficiency. Discussion emphasizes how the 

estimation process of each method affects DWi, DWb, and 

DB, and how DWi creates error. A special case of DWi being 

constant for each time interval (representing long-term trends) 

also is discussed. 

Average-of-Ratios Method

The average of ratios of water-level change to barometricpressure change (DW/DB) for many time intervals can be used 

as an estimate of barometric efficiency. This method is mathematically simple and also can be used to demonstrate how the 

effects of barometric-pressure-independent water-level change 

are reduced during method estimation. Multiplying both sides 

of equation 7 by 1/DB or modifying equation 8 yields:

D

D

D

D

D

D

W

B

W

B

W

B

= +b i . (10)

Expressing equation 10 as the sum of many (n) time 

intervals yields:

D

D

D

D

W

B n WB

j

j j

n ij

j j

n

= =

å å = + 1 1

a , (11)

where 

 nα is D D W B bj j

j

n

=

å1

, where each DWbj /DBj is 

 a constant value for every 

 time interval j. 
If DWi

 does not correlate with DB, then values of DWi /DB

(error) will be scattered about zero and the sum of many values of DWi /DB will not be far from zero. Dividing both sides 

of equation 11 by the number of time intervals (n) yields the 

“average of ratios” of DW/DB:

D

D

D

D

W

B

n

W

B

n

j

j j

n ij

j j

n

= =

å å

= + 1 1a . (12)

From equation 12, the average value of DW/DB equals 

the barometric efficiency plus the average value of DWi /DB. 

As the number of time intervals increases, the average value of 

DWi /DB will approach zero and the average value of DW/DB 

will be close to the barometric efficiency. This concept is 

roughly similar to a noise-reduction process that is used in 

reflection seismics (Yilmaz and Doherty, 1987).

The average of many values of DW/DB will be a reliable 

estimate of barometric efficiency if there are no large outliers. 

Outliers of DW/DB will occur within time intervals or increments when (1) DB is close to zero or (2) DWi

 is large. Values 

of DB will be close to zero when a time interval or increment 

used to determine a value of DW/DB straddles a maximum or 

minimum of barometric pressure. Values of DWi will be large 

when a time interval or increment straddles a sudden change 

in barometric-pressure-independent water level, such as during 

the beginning of a rain event or aquifer test. 

The error in the estimate of barometric efficiency using 

the average-of-ratios method ( εaor ) will be the second term 

on the right-hand side of equation 12:

 eaor

ij

j j

n W

B

n = =

å D

1 D . (13)

From equation 13, it can be seen that one or more time 

intervals with a DB very close to zero will cause a very large 

error in the estimate of barometric efficiency. Therefore, the 

average-of-ratios method is vulnerable to any time interval 

having a small value of DB. Confidence intervals of barometric efficiency can be estimated from the spread of values 

of DW/DB using basic statistical techniques.

In the special case when DWi is constant (const Wi ) for 

all time intervals, error in the estimate of barometric efficiency 

using the average-of-ratios method will be:

eaor

j

n

i

B

n = ( ) const W =

å 1

1 D D , (14)

indicating that if the sum of the reciprocals of values of DB for 

all time intervals is zero, then the average-of-ratios estimate of 

barometric efficiency will not include error from a constant DWi.

Median-of-Ratios Method

The median-of-ratios method is similar to the averageof-ratios method, except that it uses the median instead of the 

average as a measure of location. In the case of outliers for 

DW/DB, as previously discussed, a more robust measure of 

location—such as the median—can provide an estimate that 

is closer to the barometric efficiency than provided by the 

average. If DWi does not correlate to DB, the probability that 

DWi /DB (the error) will be greater than zero will be the same 

as the probability that the error will be less than zero. As a 

result, the median for many values of DWi /DB will be close to 

zero. Thus, the median value of DW/DB for many time intervals will be close to the value of the barometric efficiency.

Clark Method

The Clark method (Clark, 1967) is presented as a method 

that can estimate barometric efficiency in the presence of a 

constant DWi; time intervals are of a constant length. This 

method has become a standard for comparing other methods or 

data-processing techniques. Rasmussen and Crawford (1997) 

compared a version of the Clark method, modified by Davis 

and Rasmussen (1993), to regression techniques.

The Clark method performs a sign test on DW compared 

to the sign of DB for each time interval:

SD SD W W j = + j j -1 w , (15)

where 

SDWj is the ongoing sum after the j

th time interval

and 

SDWj–1 is the ongoing sum after the (j–1) th time interval. 

The term wj is the absolute value of DWj , if the sign of DWj

agrees with the sign of DBj (during a specific time interval, 

water level rises while barometric pressure falls or water level 

falls while barometric pressure rises); the term wj will be 

the negative of the absolute value of DWj if the sign of DWj

disagrees with the sign of DBj (both water level and barometric pressure rise or fall). 

If DBj is zero, then DWj is ignored so that changes in 

water level that occur when there is no change in barometric 

pressure are eliminated from the estimation of barometric 

efficiency. With each time interval, the absolute value of DB

is added to the ongoing sum SDB:

SD SD D B B B j = + j-1 j . (16)

The estimate of barometric efficiency is the slope of a line 

that fits SDWj plotted on the y-axis and SDB plotted on the 

x-axis.

Previous Methods of Estimating the Barometric Efficiency 
 Graphical Method for Estimation of Barometric Efficiency from Continuous Data

The sign test of DW compared to the sign of DB from 

Clark (1967) is a sign test of DWi compared to the sign of DB. 

When the sign of DWi agrees with the sign of DB:

wj = +D D W W bj ij , (17)

when the sign of DWi disagrees with the sign of DB:

wj = -D D W W bj ij . (18)

In the case when the sign of DW agrees with the sign of DB, 

(w is greater than zero) two possible cases exist: (1) the sign of 

DWi agrees with the sign of DB (equation 17) and (2) the sign 

of DWi disagrees with the sign of DB and the absolute value 

of DWi is less than the absolute value of DWb (equation 18). 

In the case when the sign of DW disagrees with the sign of 

DB and w is less than zero, the sign of DWi disagrees with the 

sign of DB and the absolute value of DWi is greater than the 

absolute value DWb (equation 18). It can be seen from equations 15, 17, and 18 that the absolute value of DWb

 is always 

added to SDW.

The value of SDW/SDB for many (n) time intervals 

is an estimate of barometric efficiency (also see Davis and 

Rasmussen, 1993):

aˆ n

n

n

W

B = SDSD . (19)

By separating DWi and DWb from DW:

SD

SD

SD

SD

SD

SD

W

B

W

B

W

B

n

n

bn

n

in

n

= + , (20)

where 

SDWbn is the sum of the absolute values of DWb

after n time intervals, 

and 

SDWin is the sum of the values of DWi of different 

signs after n time intervals, with the 

absolute value of DWi being added when 

the sign of DWi agrees with the sign DB

and being subtracted when the sign of 

DWi disagrees with the sign of DB. 

If DWi does not correlate to DB then about as many time 

intervals will have the sign of DWi agree with the sign of 

DB (DWi will be added to SDWi ) as will have the sign

of DWi disagree with the sign of DB (DWi will be

subtracted from SDWi ). After many time intervals,

SDWi will not be far from zero and SDWi /SDB will 

be insignificant compared to SDWb /SDB. The Clark method

need not be restricted to time intervals of constant length

and need not be performed on a plot of values of SDWj 

on the y-axis and SDBj on the x-axis.

The error in the estimate of barometric efficiency using 

the Clark method will be:

eClark

in

n

W

B = SDSD . (21)

The estimates of the barometric efficiency converge to a 

solution when there is little difference in the value of

SDW/SDB with successive time intervals 

( SD SD SD SD W B W B ( j j)-( j-1 j-1 ) is close to zero).

In the special case when DWi is constant, error in 

the estimate of barometric efficiency using the Clark 

method will be:

eClark

i

n

a d const W

B = ( ) - DSD

, (22)

where

a is the number of time intervals 

when DB is greater than zero.

and 

d is the number of time intervals 

when DB is less than zero. 

The sum of a and d equals n, the total number of time intervals. The estimates of barometric efficiency using the Clark 

method are resistant to a constant DWi when the number of 

time intervals with a DB of less than zero equals the number 

of time intervals with a DB greater than zero (Davis and 

Rasmussen, 1993).

Slope Method on Water-Level and 

Barometric-Pressure Change

Ferris and others (1962) first presented the slope method 

on DW and DB from several time intervals. Measures of DW

are plotted on the y-axis, and those of DB are plotted on the 

x-axis. A line is fitted to the plotted points. The slope of the 

fitted line is the estimate of barometric efficiency. Using ordinary least squares to obtain an objective best fit of the plotted 

points, the estimate of barometric efficiency (αˆ ) will be the 

slope from the ordinary least squares best fit (modified from 

Ott, 1988):

aˆ = D D

D D

W B

B B

S

S , (23)

where

 D D D D D D W B j j

j

n

S = - ( ) W W B B ( ) - =

å1

(24)

and

D D D D B B j

j

n

S = - ( ) B B =

å1

2

, (25)

where 

j designates the j

th time interval; 

 DW is the average water-level change for 

all time intervals; 
and 

DB is the average barometric-pressure change 

for all time intervals. 

The r-squared value will provide information about the scatter of the plotted points and the reliability of the estimate of 

barometric efficiency.

As with other methods, replacing DW with DWi will 

yield the error in the estimate of barometric efficiency:

eslope

W B

B B

S i

S = D D

D D

, (26)

where

D D D D D D W B ij i

j

n

j S i

= - ( ) W W B B ( ) - =

å1

. (27)

The error in the estimate of barometric efficiency is then the 

slope of DWi as a function of DB. The larger the sample size, the 

lower the probability that DWi (not correlated to DB) will plot 

against DB as a slope that is significantly different from zero.

In the special case that DWi is constant, the slope of DWi

as a function of DB will be zero. Therefore, a constant DWi

will not cause error in the estimate of barometric efficiency 

using the slope method. The constant DWi will be expressed 

as the y-intercept of the ordinary least squares calculation.

Some estimates of barometric efficiency use the slope 

method with water level (W) on the y-axis and barometric 

pressure (B) on the x-axis (Hare and Morse, 1999). The negative of the slope of the best-fit line through points of water 

level (W) plotted against barometric pressure (B) is the estimate of barometric efficiency. It is advisable to use DW and 

DB in the slope method rather than W and B because DW and 

DB are more mathematically accurate (barometric efficiency 

is defined as the ratio of water-level change to barometricpressure change, not water level to barometric pressure). 

Plots of DW and DB, also, show less scatter, compared to 

plots of W and B (see fig. 2 for an example).

Using Continuous Data to Estimate 

Barometric Efficiency

With advances in data-collection equipment, continuous data are more readily available. A graphical method was 

developed that uses continuous water-level and barometricpressure data to estimate barometric efficiency. Continuous 

data provide more information than discrete data to estimate 

the barometric efficiency. Continuous data in this report are 

actually “nearly continuous” data where the parameter resolution and sample frequency are great enough to define the shape 

of barometric-pressure and water-level fluctuations through 

time. This section of the report (1) describes simple techniques 

for removing some of the barometric-pressure-independent 

water-level change from the water-level data prior to using a 

method to estimate the barometric efficiency, and (2) discusses 

the optimal selection of time intervals from time-series data.

Filtering Water-Level Change that Causes Error 

in the Estimation of Barometric Efficiency

Simple techniques exist for removing barometricpressure-independent water-level change related to time—

long-term water-level change and daily fluctuations. While 

discussion in this report is limited to these two forms of 

barometric-pressure-independent water-level change, other 

forms that can be quantified should be removed. 

Decreasing the variance of water-level data by correcting the data with an independent variable that is not correlated to barometric pressure will decrease the amount of 

barometric-pressure-independent water-level change in the 

water-level data. The reduction in barometric-pressureindependent water-level change from the water-level data can 

be monitored by comparing the variance in corrected waterlevel data to the variance in uncorrected water-level data.

Using Continuous Data to Estimate Barometric Efficiency 

–0.20

–0.15

–0.10

–0.05

0.00

0.05

0.10

0.15

0.20

–0.4 –0.3 –0.2 –0.1 0.1 0.2 0.3 0.4

B.

A.

BAROMETRIC PRESSURE RELATIVE TO ARBITRARY DATUM, IN FEET OF WATER

BAROMETRIC-PRESSURE CHANGE CALCULATED FROM

BAROMETRIC PRESSURES IN GRAPH A, IN FEET OF WATER

SELECTED RELATIVE WATER LEVEL ABOVE

OR BELOW (–) INITIAL MEASUREMENT, IN FEET

WATER-LEVEL CHANGE CALCULATED FROM 

WATER LEVELS IN GRAPH A, IN FEET

0

–0.1

–0.08

–0.06

–0.04

–0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

–0.3 –0.2 –0.1 0 0.1 0.2 0.3

y = –0.264 x + 0.023, r 2= 0.631

OLS (ordinary least squares) estimate 

 of BE (barometric efficiency) is 0.264

OLS estimate of BE is 0.335

y = 0.335 x + 0.002, r 2= 0.971

Figure 2. (A) Selected water-level values as a 

function of barometric pressure and (B) water-level 

change as a function of barometric-pressure change 

in well mwg05, Air Force Plant 6, Marietta, Georgia, 

2003. See figure 1 for well location.
10 Graphical Method for Estimation of Barometric Efficiency from Continuous Data

Long-Term Barometric-Pressure-Independent 

Water-Level Change

Long-term barometric-pressure-independent water-level 

change (DWi ) can be identified from a water-level hydrograph 

if the following criteria are met: (1) the time length of the trend 

is much longer than the time length of barometric-pressure 

fluctuations, (2) barometric pressure lacks a long-term trend, 

and (3) the water-level trend is easily identifiable and can be 

represented by a simple mathematical expression. Barometricpressure fluctuations during the aquifer-test period at Air Force 

Plant 6 ranged from 1 to 7 days. Long-term water-level trends 

extended for the length of the study period, which was at least 

60 days. At Air Force Plant 6, barometric pressure fluctuated 

about an average value, exhibiting no long-term trend. 

Examples of correcting for a linear change (constant 

rate of change), and a second-order, parabolic trend (increase 

to a maximum water level followed by a decrease) are 

described below. Data corrections decrease some barometricpressure-independent water-level change using time as the 

independent variable. Conceptually, a simple shape (line or 

parabola) that represents the long-term trend is fitted and 

subtracted from the water-level data. The resulting residual 

of the trend is corrected data with a variance in water level 

that is smaller than the variance in water level from the uncorrected data. Optimal correction factors provide the minimum 

variance in corrected water level. The smaller the variance of 

water level, the more horizontal the curve of water level as a 

function of time. 

In the linear trend, water-level change is constant with 

time. The correction formula is:

W W t t ( )t corr t uncorr = - - ( ) t ( )0 , (28)

where

W(t)corr is the water level at time t corrected for a 

long-term linear trend; 

W(t)uncorr is the uncorrected water level at time t; 

and

t is the rate of water-level change per unit of 

time determined by ordinary least squares. 

Figure 3 shows an example of correcting a well hydrograph for 

a linear, long-term trend (well b4mwi). Table 1 lists variances 

of the 6,411 uncorrected and corrected water-level values. 

Correcting the water-level data with a value for t of –0.0153, 

the water-level variance is decreased from 0.08954 to 0.00290, 

reducing more than 95 percent of the variance. The resulting 

hydrograph of corrected water level shows little to no longterm trend (is horizontal) yet retains other water-level features 

within the graph (fig. 3).

In the second-order, parabolic trend, the correction 

formula is:

W W

t t

p t corr t uncorr

v

( ) ( ) = + ( ) − 2

4 , (29)

where

t

v is the time when the long-term water level 

reaches a maximum or minimum value 

(the maximum value in the example in 

this report)

and 

p is a parameter that is adjusted to remove 

the trend. 

Figure 4 shows an example of correcting a well hydrograph for 

a second-order, long-term trend (well pmw7d). Table 1 lists 

variances of the 6,412 15-minute uncorrected and corrected 

water-level values. Correcting the water-level data with a value 

for p of 1,950, the water-level variance is decreased from 

0.00425 to 0.00138, reducing more than 65 percent of the variance. The resulting hydrograph of corrected water level shows 

no long-term trend (is horizontal) yet retains other water-level 

features within the graph (fig. 4).

Daily Fluctuations

Some short-term fluctuations can be removed using a 

frequency filter or by using a centered daily-moving arithmetic 

mean (hereinafter, the arithmetic mean is referred to as the 

average). Geldon and others (1997) used a low-pass Butterworth filter to remove water-level fluctuations with frequencies 

of greater than 0.8 cycles per day. Centered daily-moving averages of both the barometric pressure and water level also can 

remove daily fluctuations that interfere with estimations of barometric efficiency. Centered daily-moving average at each time 

increment (for example, 15 minutes) is the average water level 

or barometric pressure for all time increments during a 24-hour 

Table 1. Variance of corrected water-level data using selected 

values of correction factors for two wells, b4mwi and pmw7d, 

Air Force Plant 6, Marietta, Georgia, 2003. 

[n, sample size or number of 15-minute measurements of water level, amounting to a study period of about 67 days; τ, correction factor used in correcting 

for a linear, long-term trend; –, negative; NA, not applicable; p, correction 

factor used in correcting for a second-order, parabolic, long-term trend]

Variable name 

or description

Well name

b4mwi pmw7d

Sample size (n) 6,411 6,412

Variance of uncorrected 

water levels

0.08954 0.00425

Trend Linear Second-order 

parabolic

Selected value of τ –0.0153 NA

Selected value of p NA 1,950

Variance of corrected 

water levels

0.00290 0.00138
period with the time increment in the center of the period. As 

an example, if continuous recorders are measuring barometric 

pressure and water level every 15 minutes, the daily-movingaverage value for the time March 5 at 10:15 a.m. will be the 

average barometric pressure and the average water level for all 

97 15-minute intervals starting March 4 at 10:15 p.m. through 

March 5 at 10:15 p.m. The daily-moving-average values for the 

time March 5 at 10:30 a.m. will be the average barometric pressure and the average water level for all 97 15-minute intervals 

starting March 4 at 10:30 p.m. through March 5 at 10:30 p.m. 

Daily-moving averages are very easy to apply to time-series 

data and are effective in removing daily water-level fluctuations. As long as daily-moving averages are applied to both the 

barometric-pressure and water-level values, there appears to be 

no adverse effect on the estimation of barometric efficiency for 

long-term barometric-pressure fluctuations on the scale of days. 

Because earth-tide fluctuations (DWg ) cycle slightly longer 

than 24 hours, it may be advisable to extend the time used for 

daily-moving averages to slightly longer than 24 hours. 

It should be confirmed that no time lag in barometric 

effects are present, prior to using filters or daily-moving averages to reduce short-term fluctuations in barometric pressure 

or water level. If a time lag is present, depending on the scope 

of the study, barometric efficiency may need to be estimated 

at more than one time scale (Toll and Rasmussen, 2007). 

While the centered daily-moving average water level is used 

to estimate barometric efficiency, it is not used to calculate 

drawdown in a well for an aquifer test.

Using Continuous Data to Estimate Barometric Efficiency 11

July 28 Aug 7 Aug 17 Aug 27 Sept 6 Sept 16 Sept 26 Oct 6 Oct 16

2003

WATER LEVEL, IN FEET ABOVE NAVD 88

1,041.0

1,041.2

1,041.4

1,041.6

1,041.8

1,042.0

1,042.2

1,042.4

Uncorrected water level

Corrected water level

Figure 3. Example of removing barometric-pressure-independent water-level change in the form of 

long-term linear trend, well b4mwi, Air Force Plant 6, Marietta, Georgia, 2003.

July 28 Aug 7 Aug 17 Aug 27 Sept 6 Sept 16 Sept 26 Oct 6 Oct 16

2003

WATER LEVEL, IN FEET ABOVE NAVD 88

Uncorrected water level

Corrected water level

1,067.80

1,067.85

1,067.90

1,067.95

1,068.00

1,068.05

1,068.10

1,068.15

1,068.20

1,068.25

Figure 4. Example of removing barometric-pressure-independent water-level change in the form of 

long-term 2nd-order, parabolic trend, well pmw7d, Air Force Plant 6, Marietta, Georgia, 2003.
Other Barometric-Pressure-Independent 

Water-Level Changes

As long as an identified trend does not correlate to barometric pressure, and it can be feasibly quantified, it should be 

removed prior to barometric-efficiency estimation. Recession 

curves after rain events (DWr ), earth tides (DWg ), and ocean 

tides (DWm ) are examples of other barometric-pressureindependent water-level change that might be removable from 

data prior to estimating barometric efficiency. Methods discussed in this report can be applied to earth-tide fluctuations 

to determine tidal efficiency by using one of the directional 

components of microgravity as the independent variable. Tidal 

efficiency then can be used to correct water-level data for earth 

tides prior to determining barometric efficiency.

Time-Interval Selection

Methods described in the previous section require that 

DW and DB be measured for multiple time intervals. Time 

intervals should be selected such that the value of DWi is as 

small as possible compared to the value of DB, as is reflected 

in equation 9. 

One way to optimize the quality of data in the time intervals is to focus on the inflection points of barometric-pressure 

fluctuations (fig. 5). Such inflection points are those times when 

the barometric pressure is changing most rapidly. If DWi does 

not correlate to DB, then the rate of change of Wi (DWi /DT)

will not correlate to the rate of change of B (DB/DT). As a 

result, focusing time-interval selection about these inflection 

points will maximize the value of DB while not maximizing the 

value of DWi . The result will be that time intervals will have 

minimum values of DWi /DB (error) associated with them.

Time-interval boundaries are selected so that they are 

close enough to the inflection point to obtain the benefit of 

rapidly-changing barometric pressure but far enough apart to 

ensure that the barometric-pressure change associated with the 

time interval is significantly greater than measurement error. 

Time-interval-boundary selection becomes important when the 

shapes of barometric-pressure fluctuations, in time series, are 

not simple. Features to avoid include times when barometric 

pressure is slowly changing during a relatively long period of 

time. The boundaries for optimally selected time intervals will 

be situated between the inflection points and the extremes of 

the barometric-pressure fluctuation (for example, one boundary 

is between the minimum and an inflection point and the other 

boundary is between the inflection point and the maximum).

1 Graphical Method for Estimation of Barometric Efficiency from Continuous Data

1 3 5 7 9 11 13 15 17 19 21

AUGUST 2003

Barometric

pressure

Rate of 

barometricpressure

change

–32.5

–32.6

–32.7

–32.8

–32.9

–33.0

–33.1

–33.2

BAROMETRIC PRESSURE, IN FEET OF WATER

BAROMETRIC-PRESSURE CHANGE, 

IN FEET OF WATER PER 15-MINUTE INTERVAL

–0.003

–0.002

–0.001

0

0.001

0.002

0.003

Selected

time

intervals

Inflection

points

Figure 5. Example of time-interval selection about two inflection points during maximum rates 

of barometric-pressure change, well b4mwi, Air Force Plant 6, Marietta, Georgia, August, 2003. 

Barometric pressure is in the form of daily-moving averages.
A Graphical Method for Estimating 

Barometric Efficiency from 

Continuous Water-Level Data

This section describes a graphical method that uses continuous data to quickly determine if water levels are affected 

by barometric-pressure change and to estimate barometric 

efficiency in a manner that is resistant to large barometricpressure-independent water-level change. This section 

(1) discusses the theory behind the graphical method, 

(2) presents examples from actual continuous data of 

graphical features used to estimate the barometric efficiency, 

(3) describes two techniques for determining barometric efficiency from continuous data using the graphical method, and 

(4) provides an example application in the Piedmont at 

Air Force Plant 6, Marietta, Georgia.

Theory Behind the Graphical Method

A plot of nearly continuous water level (on the y-axis) as 

a function of nearly continuous barometric pressure (on the xaxis) will plot as a line curved into a series of connected elliptical loops (figs. 6–10). A complete fluctuation in baro­metric 

pressure, in time series (from minimum to maximum back to 

minimum, or from maximum to minimum back to maximum, 

fig. 6A), plots as a single loop (parabolic if open or elliptical 

if closed) on a graph of water level as a function of barometric pressure (closed loop in fig. 6B; open and closed loops in 

fig. 7B). Hereinafter loop shapes will be called elliptical.

The length of an elliptical loop seen on the graph, along 

the horizontal x-axis, can be mathematically represented as a 

compilation of two barometric-pressure changes ( DB). The 

double-bar over DB is used here to indicate that each value of 

change comes from four barometric-pressure values as shown 

in equation 30: 

DB B B B B j = + ( ) j j j j - + ( ) é

ë

ê ù

û

ú

1

2 1 4 2 3 , (30)

where 

 j is for the j

 th barometric-pressure

 fluctuation or loop 

and

 Bj1, Bj2, Bj3, and Bj4 are values of barometric pressure at

 times t

1

, t

2

, t

3

, and t

4

, respectively. 

Parts of a loop that are most apparent on a graph originate 

from barometric-pressure data that best estimates barometric 

efficiency. The four times in equation 30 (t

1

, t

2

, t

3

, and t

4

) are 

times that best define the slope of the loop as it appears on 

a graph. These four times happen to occur at the boundaries of times during which the barometric-pressure change is 

most rapid. As discussed in a previous section, time t

1

 occurs 

between the first maximum or minimum and the first inflection 

point, time t

2

 occurs between the first inflection point and the 

intermediate minimum or maximum, time t

3

 occurs between 

the intermediate minimum or maximum and the second inflection point, and time t

4

 occurs between the second inflection 

point and the second maximum or minimum (fig. 6).

The length of an elliptical loop seen on the graph, along 

the vertical y-axis, can be mathematically represented as a 

compilation of two water-level changes ( ) and can be 

calculated as follows:

DW W W W W j = + ( ) j j - + ( ) j j

é

ë

ê ù

û

ú

1

2 2 3 1 4 , (31)

where 

W1, W2, W3, and W4 are water levels during the

 same times as B1, B2, B3, 

 and B4, respectively. 

The order of times t

1

 and t

4

, and t

2

 and t

3

 in equation 30 is 

inverted with respect to equation 31 in order to follow the convention that an increase in barometric pressure is a negative 

change during a time interval. 

The plots of continuous data are for water level as a function of barometric pressure. As a result, the negative of the 

slopes of the elliptical loops on the graph will be the water-level 

change divided by the barometric-pressure change ( D D W B ) 

that is associated with barometric-pressure fluctuations.

Using figure 6 as an example, a barometric-pressure 

fluctuation occurs such that there is a first barometric-pressure 

minimum, time t

1

, an inflection point, time t

2

, an intermediate barometric-pressure maximum, time t

3

, another inflection 

point, time t

4

, and a second barometric-pressure minimum. 

The negative of the slope of the long axis of an elliptical 

loop will be D D W B or from equation 10 will be the 

sum of two ratios: D D W B b (the barometric efficiency) and 

D D W B i

 (the error). Values of D D W B usually vary for each 

loop (barometric-pressure fluctuation).

The negative of the slope of the preferred orientation of 

many elliptical loops, as viewed by an observer, is an estimate 

of barometric efficiency. The slope of the preferred orientation 

of many elliptical loops is approximately the median of the 

slopes of the major axes of the elliptical loops. The graphical 

method uses the same principle used in the median-of-ratios 

method; however, instead of using direct ratios from time 

intervals in the median-of-ratios method, negatives of the 

slopes of loops are used in the graphical method. The error 

in the estimate of barometric efficiency using the graphical 

method ( εgm ) will be:

egm = median W B ( ) D D ij j

, (32)

plus observational error.

Observational error is defined as the difference between 

the preferred orientation of the slopes of the loops as determined by the observer, and the actual median of the slopes of 

the loops. Because there is observational error, it is advisable 

that more than one person select the slope or that the same person selects the slope for the same data several times in order 

to minimize the effect of subjectivity. Also a long study period 

should be used (at least 60 days) to ensure that the loops that 

are affected by large amounts of DWi

 do not significantly contribute error to the barometric-efficiency estimate.

A Graphical Method for Estimating Barometric Efficiency from Continuous Water-Level Data 1

DW
1 Graphical Method for Estimation of Barometric Efficiency from Continuous Data

If there is a constant barometric-pressure-independent 

water-level change with time at a rate of τ, DWi will originate 

from an asymmetry in the barometric-pressure fluctuation 

with respect to time. Replacing DW with DWi in equation 31:

DW W W W W ij = + ( ) ij ij ij ij - + ( ) é

ë

ê ù

û

ú

1

2 2 3 1 4 , (33)

where 

 Wi1, Wi2, Wi3, and Wi4 are water levels that 

are not caused by barometric-pressure 

change during the same times as B1, B2, 

B3, and B4, respectively. 

If (t

2 – t1

) and (t

4

 – t

3

) are the same for a specified barometricpressure fluctuation, that is, the time intervals about the 

inflection points or “limbs” of the loop are the same with 

respect to time, and the rate of barometric-pressureindependent water-level change is constant, DWi

 will be 

zero. If the limbs of the loop are not equal and the rate of 

barometric-pressure-independent water-level change is constant, the value of DWi

 will be half of the difference in the 

time between the two limbs multiplied by the rate of 

barometric-pressure-independent water-level change 

with time (τ). Therefore, in the special case of a constant 

barometric-pressure-independent water-level change 

(const-DWi

 in other methods), the error in the estimate of 

barometric efficiency using the graphical method will be:

 e

t

gm

j j j j

j

median

t t t t

B

=

( ) + - + ( ) é

ë

ê ù

û

ú

æ

è

ç

2 2 3 1 4

D

ç

ç

ç

ç

ç

ç

ç

ç

ö

ø

÷

÷

÷

÷

÷

÷

÷

÷

÷

÷÷

, (34)

plus observational error. The term

j is for the j

 th barometric-pressure fluctuation; 

t

1

, t

2

, t

3

, and t

4 are times as used in equations 30, 31, and 33; 

	 τ is the rate of barometric-pressure-independent 

water-level change with time; 

HYPOTHETICAL BAROMETRIC PRESSURE, 

IN FEET OF WATER

HYPOTHETICAL WATER LEVEL, IN FEET

A.

B.

30.7

30.9

31.1

31.3

31.5

31.7

31.9

32.1

32.3

32.5

30.9

31.0

31.1

31.2

31.3

31.4

31.5

31.6

31.7

32.0 32.2 32.4 32.6 32.8 33.0 33.2 33.4 33.6

HYPOTHETICAL BAROMETRIC PRESSURE, IN FEET OF WATER

32.0

32.2

32.4

32.6

32.8

33.0

33.2

33.4

33.6

TIME

TIME

t

4

t

4

t

2

t

3

t

1

t

1

Second

minimum

Second

minimum

First

minimum

First

minimum

First

inflection

point

First

inflection

point

Intermediate

maximum

Intermediate

maximum

Second

inflection

point

Second

inflection

point

t

2 t

3

Figure 6. Example of (A) time series of hypothetical water level and barometric pressure, and 

(B) hypothetical water level as a function of barometric pressure. Barometric efficiency equals 0.50.
and 

DBj is the barometric-pressure change as defined 

in equation 30. 

Examples of Elliptical Loops in Graphs

Determining the slope of the preferred orientation of 

loops requires data frequency to be high enough that shapes 

of the water-level and barometric-pressure fluctuations are 

defined by the data. Figures 7A–10A are time-series plots of 

water level and barometric pressure for select wells at Air 

Force Plant 6. Figures 7B–10B are plots of nearly continuous 

water level as a function of nearly continuous barometric pressure. Long-term trends have been removed. Data presented in 

this report are in the form of centered daily-moving averages. 

Figure 7 shows fluctuations of barometric pressure and 

water level of a well (ob202b) with a barometric efficiency 

as estimated from the graphical-loop method of 0.703. In the 

time-series graph (fig. 7A), there is a correlation between

barometric-pressure changes and water-level changes. Peaks 

and troughs of water level tend to coincide with troughs and 

peaks of barometric pressure, respectively. A plot of nearly 

continuous water level as a function of barometric pressure 

(fig. 7B) shows several loops, each representing a barometricpressure fluctuation. The slopes of the loops vary slightly, with a 

slope of the preferred orientation of –0.703.

Figure 8 shows the fluctuations of barometric pressure 

and water level of a well (mwg05) with a barometric efficiency 

as estimated by the graphical-loop method of 0.379. Although 

the barometric efficiency is lower than that of well ob202b, the 

time-series graph (fig. 8A) still shows a correlation between 

barometric pressure and water level. Peaks and troughs of 

water levels from well mwg05 are smaller than those of well 

ob202b; however, they still tend to coincide with barometricA Graphical Method for Estimating Barometric Efficiency from Continuous Water-Level Data 1

A.

B.

Aug 2 Aug 17 Sept 1 Sept 16 Oct 1

2003

–0.1

–0.1

–0.2 –0.3

–0.2

0

0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

RELATIVE BAROMETRIC PRESSURE ABOVE AND

BELOW (–) INITIAL MEASUREMENT, IN FEET OF WATER

RELATIVE BAROMETRIC PRESSURE ABOVE AND BELOW (–) INITIAL MEASUREMENT, IN FEET OF WATER

–0.10

–0.05

–0.20

–0.15

0.10

0.20

0.15

0.05

0

–0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4

RELATIVE WATER LEVEL ABOVE AND BELOW (–) INITIAL MEASUREMENT, IN FEET

Slope of preferred 

orientation of 

loops = –0.703

Figure 7. (A) Time series of water level and barometric pressure, and (B) water level as a function of 

barometric pressure for well ob202b, August 2–October 6, 2003. Barometric-efficiency estimate from the 

graphical method is 0.703; initial water-level and barometric-pressure measurements are normalized to 

zero. Data are in the form of daily-moving averages. See figure 1 for well location.
16 Graphical Method for Estimation of Barometric Efficiency from Continuous Data

pressure troughs and peaks, respectively. The plot of nearly 

continuous water level as a function of barometric pressure 

(fig. 8B) shows most of the loops nested within one large loop. 

Most of the loops have a slope close to the slope of the preferred 

orientation of –0.379.

Figure 9 shows fluctuations of barometric pressure and 

water level of a well (ob208c) with a barometric efficiency as 

estimated by both the Clark (1967) method and the graphicalloop method near 0.16. In the time-series graph (fig. 9A), the 

correlation between water level and barometric pressure is not 

apparent. The plot of nearly continuous water level as a function of barometric pressure (fig. 9B) shows a number of loops 

with a slope of preferred orientation of –0.162. A rain event 

near September 22, 2003, is apparent in figure 9A as a spike, 

and in figure 9B as a loop with a different slope than those of 

other loops. Thus, the slope of the loop from the rain event is 

ignored in the analysis.

Figure 10 shows fluctuations of barometric pressure and 

water level of a well (mwg11) under strong surface influence 

and with a barometric efficiency of near zero. The barometric 

efficiency estimated by the graphical method was 0.027. In 

the time-series graph (fig. 10A), well mwg11 responds to rain 

events. No correlation is apparent between the water level and 

barometric pressure. In the plot of nearly continuous water 

level as a function of barometric pressure (fig. 10B), most of 

the loops have a preferred orientation that is nearly horizontal. 

The slope of the preferred orientation of the loops was measured as –0.027. Several loops have a positive slope ( D D W B

less than zero). The positive sloping loops are the result of 

rainfall events (spike in water level followed by a lengthy 

recession curve) coinciding with some barometric-pressure 

changes (fig. 10A).

A.

B.

Aug 2 Aug 17 Sept 1 Sept 16 Oct 1

2003

–0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4

–0.1

–0.1 –0.3

–0.2

0

0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

–0.10

–0.05

–0.20

–0.15

0.10

0.20

0.15

0.05

0

0.4

0.5

0.6

RELATIVE WATER LEVEL ABOVE AND BELOW (–) INITIAL MEASUREMENT, IN FEET

RELATIVE BAROMETRIC PRESSURE ABOVE AND

BELOW (–) INITIAL MEASUREMENT, IN FEET OF WATER

RELATIVE BAROMETRIC PRESSURE ABOVE AND BELOW (–) INITIAL MEASUREMENT, IN FEET OF WATER

Slope of preferred 

orientation of 

loops = –0.379

Figure 8. (A) Time series of water level and barometric pressure, and (B) water level as a function of 

barometric pressure for well mwg05, August 2–October 6, 2003. Barometric-efficiency estimate from the 

graphical method is 0.379; initial water-level and barometric-pressure measurements are normalized to 

zero. Data are in the form of daily-moving averages. See figure 1 for well location.
Determining the Slope of the 

Preferred Orientation of Loops

Two techniques to determine the slope of the preferred 

orientation of all loops within a given graph are presented 

herein. The first technique is to orient a straight line parallel to 

the preferred orientation of the loops (straight lines in figs. 7B, 

8B, 9B, and 10B). The negative of the slope of the straight line 

is the estimate of barometric efficiency. The second technique 

is to adjust a correction factor to modify the slope of the 

preferred orientation of the loops of corrected water level as 

a function of barometric pressure. The correction factor that 

modifies the slope to zero is the estimate of barometric efficiency. The second technique is discussed in detail below.

The second technique uses a spreadsheet and an accompanying graph. Figure 11A shows the spreadsheet including a 

graph, on the right side of the spreadsheet, which is a plot of 

nearly continuous, centered daily-moving-average water level 

as a function of nearly continuous, centered daily-movingaverage barometric pressure. Loops with a preferred orientation are apparent on the graph. The slope of the preferred 

orientation of the loops is less than zero, indicating that the 

well is noticeably affected by barometric-pressure change.

The columns within the spreadsheet (fig. 11A) are date 

and time (columns A and B, respectively); barometric pressure, in feet of water, and centered daily-moving average of 

barometric pressure (columns D and E, respectively); water 

level in well ob202b (column G); water level after being 

A Graphical Method for Estimating Barometric Efficiency from Continuous Water-Level Data 1

0

A.

Aug 2 Aug 17 Sept 1 Sept 16 Oct 1

2003

–0.1

–0.1 –0.3

–0.2

0

0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.6

RELATIVE BAROMETRIC PRESSURE ABOVE AND

BELOW (–) INITIAL MEASUREMENT, IN FEET OF WATER

B.

–0.3 –0.2 –0.1 0 0.2 0.3 0.4

–0.10

–0.05

–0.20

–0.15

0.10

0.20

0.15

0.05

RELATIVE WATER LEVEL ABOVE AND BELOW (–) INITIAL MEASUREMENT, IN FEET

0.1

RELATIVE BAROMETRIC PRESSURE ABOVE AND BELOW (–) INITIAL MEASUREMENT, IN FEET OF WATER

Rain event

Rain event

Slope of preferred 

orientation of 

loops = –0.162

Figure 9. (A) Time series of water level and barometric pressure, and (B) water level as a function of 

barometric pressure for well ob208c, August 2–October 6, 2003. Barometric-efficiency estimate from the 

graphical method is 0.162; initial water-level and barometric-pressure measurements are normalized to 

zero. Data are in the form of daily-moving averages. See figure 1 for well location.
1 Graphical Method for Estimation of Barometric Efficiency from Continuous Data

corrected for a long-term trend (column I); water level from 

column I that is “corrected” for barometric-pressure change 

using a proposed value of barometric efficiency (column J); 

and, finally, centered daily-moving average of water level from 

column J (column K). Columns C, F, and H are blank. Data 

from columns E and K are not visible on fig. 11A.

Each cell in column J has the following formula derived 

from equation 3:

W W m B B ( ) ( ) j j uncorr = - - ( ) 0 ( )j , (35)

where 

j is for the j

 th row (related to a 

15-minute interval); 

W(j) is the water level, in column J; 

W(j)uncorr is the water level that has been corrected 

for a long-term trend but has not been 

corrected for barometric pressure, in 

column I; 

m is the proposed barometric efficiency, 

input into cell J2; 

B(j) is the barometric pressure, in column D; 

and 

B0 is the reference barometric pressure 

(32.836 feet of water).

With the correction factor m (cell J2) set to zero, the graph is 

inspected and is determined to have a preferred orientation of 

loops with a negative slope.

B.

–0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4

–0.1

–0.3

–0.2

–0.4

–0.5

0

0.2

0.1

A.

Aug 2 Aug 17

Rain event

Sept 1 Sept 16 Oct 1

2003

–0.1

–0.1

–0.3

–0.2

–0.2

–0.3

–0.4

–0.5

0

0.1

0

0.1

0.2

0.2

0.3

0.4

RELATIVE BAROMETRIC PRESSURE ABOVE AND

BELOW (–) INITIAL MEASUREMENT, IN FEET OF WATER

RELATIVE BAROMETRIC PRESSURE ABOVE AND BELOW (–) INITIAL MEASUREMENT, IN FEET OF WATER

RELATIVE WATER LEVEL ABOVE AND BELOW (–) INITIAL MEASUREMENT, IN FEET

Slope of preferred 

orientation of 

loops = –0.027

Figure 10. (A) Time series of water level and barometric pressure, and (B) water level as a function of 

barometric pressure for well mwg11, August 2–October 6, 2003. Barometric-efficiency estimate from the 

graphical method is 0.027; initial water-level and barometric-pressure measurements are normalized to 

zero. Data are in the form of daily-moving averages. See figure 1 for well location.
A Graphical Method for Estimating Barometric Efficiency from Continuous Water-Level Data 1

–0.04

Variance of

water-level

data corrected

for long term trend

Variance of

uncorrected

water-level data 

0.0

0.603 0.007

Water level in

well ob202b, 

in feet above

above NAVD 88

Proposed

barometric efficiency

Proposed

barometric

efficiency

0.65

Proposed

barometric

efficiency

0.85

Proposed

barometric

efficiency

0.25

Proposed

barometric

efficiency

0.50

Factor used to remove barometric 

pressure effects (Cell J2)

Factor used to remove long-term 

trend (Cell I2)

A.

B.

D.

C.

E.

Water level with no seasonal trend

Water level with seasonal trend

Figure 11. Screen captures of software display showing (A) spreadsheet and selected proposed barometric efficiency 

(BE) values. The spreadsheet is used to estimate the BE by using a proposed BE (a correction factor) to zero the slope 

of the preferred orientation of the loops on the graph. BE = 0.0 (shown in green). Selected proposed BE values (shown in 

green) are (B) 0.25, (C) 0.50, (D) 0.65, and (E) 0.85.
Increasing the correction factor m from zero will increase 

the slope of the preferred orientation of the elliptical loops 

on the graph. Figure 11B shows that a proposed barometric 

efficiency of 0.25, assigned to cell J2, increases the slope of 

the preferred orientation of the loops (fig. 11B) compared to a 

proposed barometric efficiency of zero (fig. 11A). Figure 11C

shows that a proposed barometric efficiency of 0.50 makes 

the slope of the preferred orientation closer to horizontal. 

At the point that the correction factor m creates a preferred 

orientation on the graph that is horizontal (fig. 11D, proposed 

barometric efficiency of 0.65, slope is near zero), the proposed 

barometric efficiency will be the estimate of barometric efficiency of the well for that period of record. Further increasing 

the correction factor m would create a preferred orientation 

with a positive slope (fig. 11E, proposed barometric efficiency 

of 0.85), indicating that the proposed barometric efficiency is 

greater than the actual barometric efficiency.

Application to a Site in the Piedmont Province, 

Air Force Plant 6, Marietta, Georgia

The Air Force Plant 6 study site is located on about 

150 acres in northern Georgia in the Piedmont Province 

(fig. 1), which is underlain by fractured-crystalline metamorphic rock with igneous intrusions. The crystalline rock yields 

limited amounts of water, except where fractures connect to 

surface water. The rock is weathered toward the surface and 

is covered by a regolith of saprolitic material referred to as 

the overburden. A transition zone of intermediate weathering 

lies between the crystalline rock and the overburden. Some 

monitor wells are screened in the overburden and transition 

zone. Most monitor wells are open to the fractured-crystalline rock. Deeper wells open to the fractured-crystalline rock 

are under more confined conditions than wells screened in the 

overburden. Wells open to the fractured-crystalline rock 

have higher barometric efficiencies than wells screened in 

the overburden.

The aquifer test that was conducted during August and 

September 2003 occurred as three steps of constant discharge. 

The total pump duration lasted 889 hours (37 days). The 

aquifer test was performed in fractured-crystalline rock. During the test, pressure transducers monitored water levels in 

45 wells and a retention pond. Two barometers also monitored 

barometric pressure. The frequency of record was once every 

15 minutes. Monitoring continued months after the end of 

the aquifer test. Beginning about 2 days after the start of the 

test, barometric pressure rose and fell during a period of about 

5 days by a maximum pressure of 10 millimeters of mercury 

(mm of Hg) (equivalent to about 0.43 foot of water). Several 

wells responded with a water-level decline followed by a 

water-level increase. The barometric effects on water level 

had to be removed to better interpret the aquifer-test data. 

The graphical method was used to assess which wells had 

water levels affected by barometric-pressure changes during 

the constant-discharge aquifer test and to estimate barometric 

efficiency. The centered daily-moving average of water level 

was plotted as a function of centered daily-moving-average 

barometric pressure. Long-term, barometric-pressureindependent water-level change was removed from the 

data by using equations 28 and 29. The plots of water level 

as a function of barometric pressure from wells contained 

elliptical loops with an observable slope of preferred orientation (see figs. 7–10 for examples). Slopes of the preferred 

orientation were determined using the two previously 

mentioned techniques. The negative of the slopes yielded 

estimates of barometric efficiency.

Barometric-efficiency estimates of the graphical method 

are listed in table 2 with those of other methods. Barometricefficiency estimates from the graphical method, using both 

techniques, ranged from –0.138 to 0.703. The graphical 

method provided a rapid assessment of whether a well was 

affected by barometric-pressure change and also provided a 

rapid estimate of barometric efficiency. Using a correction 

factor to make the slope of the preferred orientation of the 

elliptical loops equal zero (technique 2, previously discussed) 

provided more consistent and reproducible results than the 

direct measurements of the preferred orientation of the 

elliptical loops (technique 1). 

For a given data set, a wide range of proposed barometric 

efficiencies should be used with technique 2 in order to identify the slope of the preferred orientation of the loops. In some 

cases, a broad range (for example, from –3 to 3) will help. 

Unlike the other methods that are reported to three decimal 

places, technique 2 of the graphical method is reported to 

only two decimal places.

Comparison of Methods

This section compares the graphical method with four 

other methods. The performance of the methods were assessed 

in two ways: (1) the number of estimates that were outside the 

domain of the barometric efficiency (less than zero or greater 

than one), and (2) the results from six control wells where 

barometric efficiency is determined to be zero based on a 

strong surface influence on water levels. The two barometricefficiency methods that used the median values of water-level 

change divided by barometric-pressure change appeared to be 

most resistant to barometric-pressure-independent water-level 

change. Details are discussed below.

Barometric efficiencies were estimated for 45 wells at 

Air Force Plant 6 using a total of eight versions of five methods (table 2). Two data sets were used for the Clark method: 

(1) 15-minute data and (2) inflection-centered time intervals. 

The ordinary-least-squares method was used on both water 

level and barometric pressure, and water-level change and 

barometric-pressure change. Study periods were from August 

through October 2003 for wells that did not respond to the 

constant-discharge aquifer test. Study periods were from October to December 2003 for wells that did appear to respond to 

the constant-discharge aquifer test. Up to 20 time intervals 

0 Graphical Method for Estimation of Barometric Efficiency from Continuous Data
Comparison of Methods 1

Table 2. Barometric efficiencies of wells using the average-of-ratios, median-of-ratios, Clark, slope, and graphical methods, 

Air Force Plant 6, Marietta, Georgia, 2003.

[n, number of time intervals for methods that used the inflection-centered selection technique; inflection-centered selection technique, time intervals were 

selected during times when barometric-pressure change was most rapid; OLS, ordinary least squares water level plotted on the y-axis and barometric pressure 

plotted on the x-axis; OLS D, ordinary least squares with water-level change (DW) plotted on the y-axis and barometric-pressure change (DB) plotted on the 

x-axis; Technique 1, directly fitting a line to the preferred orientation of the major axes of the elliptical loops; Technique 2, using a correction factor to make the 

slope of the preferred orientation of the major axes of the elliptical loops be zero. Shading is for wells that are determined to have a barometric efficiency that is 

near zero based on surface influences on water levels. Theoretically, barometric efficiency should be between zero and one]

Well name n

Average 

of ratios

Median

of ratios Clark1 Slope2

Graphical Standard 

deviation 

of all 

methods

OLS D OLS

Inflection-centered 

selection technique

All 15-minute 

increments

Inflectioncentered 

selection 

technique

Inflection-centered 

selection technique

Technique 1Technique 2

b4mwh 20 0.236 0.270 0.263 0.230 0.222 0.219 0.353 0.29 0.045

b4mwi 20 0.265 0.277 0.287 0.259 0.256 0.254 0.436 0.29 0.060

b4mwq 19 0.532 0.551 0.501 0.494 0.446 0.435 0.555 0.52 0.045

mw-00-102a 20 0.056 0.065 –0.005 0.032 0.006 0.163 0.000 0.00 0.057

mw-00-102b 20 –0.046 0.004 –0.008 –0.066 –0.087 0.097 0.000 0.00 0.056

mwg05 20 0.357 0.350 0.368 0.348 0.335 0.264 0.379 0.36 0.035

mwg10 20 0.364 0.350 0.370 0.354 0.341 0.274 0.387 0.37 0.034

mwg11 20 –0.156 –0.040 –0.093 –0.203 –0.230 –0.030 0.027 0.06 0.106

mwg21 20 0.123 0.128 0.141 0.138 0.153 0.182 0.169 0.14 0.020

mwg5-5r 20 0.541 0.571 0.538 0.524 0.492 0.369 0.584 0.50 0.067

mwos01 20 0.112 0.164 0.143 0.104 0.083 0.036 0.134 0.12 0.039

mwos02 12 0.192 0.128 0.132 0.174 0.160 0.226 0.162 0.13 0.034

mwos03 20 –0.025 –0.055 0.010 –0.018 –0.020 –0.103 0.076 0.07 0.060

mwos04 20 –1.270 –0.352 –0.769 –0.930 –0.677 –0.877 –0.045 0.00 0.447

mwos09 20 0.108 0.012 0.094 0.050 0.004 0.185 0.049 0.04 0.059

mwos10 20 0.068 0.028 0.059 0.032 0.003 0.168 0.021 0.01 0.053

mwos11 20 0.080 0.021 0.087 0.048 0.025 0.071 0.060 0.07 0.025

mwos12 20 0.089 0.035 0.094 0.050 0.020 0.132 0.070 0.05 0.036

mw-w2 20 0.541 0.502 0.503 0.493 0.458 0.392 0.522 0.48 0.046

ob202a 20 0.122 0.014 0.046 0.038 0.013 –0.080 0.000 0.00 0.056

ob202b 20 0.698 0.727 0.677 0.662 0.627 0.453 0.703 0.65 0.086

ob203a 12 0.408 0.391 0.362 0.386 0.370 0.346 0.458 0.39 0.034

ob203b 12 0.401 0.381 0.341 0.377 0.363 0.332 0.425 0.34 0.033

ob203c 20 –0.894 –0.566 –0.682 –0.992 –1.003 1.565 –0.138 –0.13 0.850

ob204b 20 0.051 0.215 0.146 0.046 0.046 0.038 0.341 0.28 0.119

ob204c 20 0.019 0.211 0.111 0.019 0.019 –0.049 0.257 0.26 0.122

ob205a 20 –0.021 0.213 0.078 –0.040 –0.050 –0.282 0.212 0.25 0.180

ob205b 20 0.043 0.043 0.055 0.050 0.058 0.059 0.063 0.05 0.007

ob208a 20 0.327 0.317 0.335 0.299 0.269 0.205 0.393 0.28 0.055

ob208b 18 0.244 0.202 0.247 0.206 0.172 0.170 0.288 0.25 0.042

ob208c 20 0.179 0.133 0.165 0.140 0.104 0.124 0.162 0.17 0.026

ob209a 20 0.030 0.177 0.112 0.024 0.020 –0.002 0.284 0.19 0.104

pmw7d 20 0.279 0.269 0.272 0.270 0.265 0.239 0.272 0.27 0.012

rw201 20 0.147 0.151 0.096 0.084 0.041 –0.047 0.101 0.09 0.063

rw202 20 0.705 0.749 0.686 0.668 0.634 0.452 0.687 0.67 0.089

rw203 12 0.412 0.378 0.323 0.379 0.360 0.309 0.443 0.42 0.047

rw204 20 –0.136 –0.024 –0.055 –0.113 –0.074 –0.008 0.007 0.00 0.054

rw205 20 –0.834 0.071 –0.127 –0.429 –0.218 –0.064 0.134 0.16 0.334

rw207 20 –1.012 –0.721 –0.879 –1.103 –1.083 –0.599 0.013 0.00 0.454

sct1 19 –0.053 0.040 –0.040 –0.100 –0.150 0.259 0.013 0.00 0.123

sct3 20 0.120 0.136 0.187 0.112 0.070 0.442 0.076 0.09 0.122

sct4 20 0.069 0.068 0.097 0.051 0.024 0.162 0.070 0.04 0.042

sct5 20 –0.116 –0.046 0.004 –0.193 –0.212 0.119 0.020 0.04 0.117

sct6 20 0.210 0.197 0.194 0.166 0.141 0.121 0.256 0.14 0.045

sct7 20 0.033 0.037 0.046 0.019 0.006 0.018 0.066 0.05 0.020 1

Clark, 1967

2

Ferris and others, 1962
Graphical Method for Estimation of Barometric Efficiency from Continuous Data

were selected about inflection points of barometric-pressure 

fluctuations for use in the average-of-ratios, median-of-ratios, 

Clark, and slope methods. The sample size used for ordinary 

least squares on water level and barometric pressure was twice 

that used for other methods using values from both the beginning and end of each time interval.

A total of 360 estimates of barometric efficiency are listed 

in table 2. Values ranged from –1.270 to 1.565. Results indicate 

that method consistency varied by well. The standard deviations of method estimates by well ranged from 0.007 to 0.850.

Barometric-Efficiency Estimates Compared to 

the Domain of the Barometric Efficiency

Method performance was partly assessed by the relative 

number of barometric-efficiency estimates that were less than 

zero or greater than one, outside the domain of the barometric 

efficiency. While estimates that are between zero and one are 

not necessarily correct, estimates of less than zero or greater 

than one are known to be incorrect. Table 3 lists the number of estimates that were less than zero or greater than one, 

which was about 18 percent (64) of all estimates. The methods 

with the least number of estimates outside the domain of the 

barometric efficiency were the two techniques of the graphical 

method followed by the median-of-ratios method. The Clark 

method using 15-minute data had fewer estimates outside 

of the domain than the Clark method using the data from 

20 inflection-centered time intervals.

Barometric-Efficiency Estimates of Control Wells

Six of the 45 wells were determined to have barometric 

efficiencies close to zero based on strong surface influences on 

water levels. Table 4 lists the depths to the tops and bottoms of 

well openings and justification for determining whether the well 

has a barometric efficiency near zero (surface influence). There are 

two lines of evidence that can potentially indicate that a well is not 

affected by barometric-pressure changes: (1) top of screen is above 

the water table and (2) well opening is connected to a surfaceTable 3. Number of estimates, by method, that were outside the domain of the barometric efficiency for 45 wells at Air Force Plant 6, 

Marietta, Georgia, 2003.

[>, greater than; <, less than; –, negative; 15-minute, all 15-minute increments were used; OLS, ordinary least squares best-fit technique was used on the 

slope method2. Inflection centered, time intervals were selected about inflection points of barometric-pressure fluctuations; DW, water-level change on the 

y-axis; DB, barometric-pressure change on the x-axis; W, water level on the y-axis; B, barometric pressure on the x-axis; technique 1, direct measure of the 

slope of the preferred orientation of elliptical loops; technique 2, use of a proposed barometric efficiency to make the slope of the preferred orientation of

elliptical loops be zero]

Method name Type of data Number of barometric-efficiency estimates

>1 <0 <–0.05 <–0.10 <–0.50

Average of ratios Inflection centered 0 11 8 7 4 

Median of ratios Inflection centered 0 7 4 3 2

Clark1 15-minute 0 9 6 4 3 

Clark1 Inflection centered 0 11 9 7 3

Slope2

, OLS on DW and DB Inflection centered 0 11 9 7 3

Slope, OLS on DW and DB Inflection centered 1 11 6 4 2

Graphical method—Technique 1 15-minute 0 2 1 1 0

Graphical method—Technique 2 15-minute 0 1 1 1 0

Total 1 63 44 34 17 

1

Clark, 1967

2

Ferris and others, 1962
water feature. Control need not be limited to wells. The water levels in the retention pond or any other 15-minute data set not correlated to barometric pressure could have been used as a control.

Water levels in the six control wells do not appear to 

respond to barometric-pressure change. Instead the wells 

respond to surface influences such as water-level fluctuations 

in a retention pond, evapotranspiration, or rain events. The top 

of the well screen for mwos04 usually is above the water table; 

thus, this well is not influenced by barometric effects. While a 

barometric-pressure change may take time to migrate through 

the unsaturated zone, the pressure may be able to translate to 

the well quickly when the well is directly connected to surface 

water. Wells rw207 and ob203c were connected to a nearby 

retention pond. Water levels in these two wells closely followed 

those in the retention pond (fig. 12). Surface-water fluctuations 

in the pond were caused by storm runoff and a pump station. 

During the growing season, water levels in wells mwg11 and 

mwos10 declined during daylight hours and rebounded during the night indicating evapotranspiration effects. Because 

evapotranspiration occurs close to the surface, it is assumed that 

these two wells are directly connected to soil moisture and were 

not affected by barometric-pressure fluctuations. Wells mwos10 

and sct5 appeared to be responding to nearby Rottenwood 

Creek; however, no stage data were collected at the site. 

Comparison of Methods

July 28 Aug 7 Aug 17 Aug 27 Sept 6 Sept 16 Sept 26 Oct 6 Oct 16

2003

1,012

1,014

1,016

1,018

1,020

1,022

1,024

WATER LEVEL, IN FEET ABOVE NAVD 88

0

2

4

6

8

10

12

WATER LEVEL IN POND, IN FEET ABOVE

PRESSURE TRANSDUCER

Well ob203c

Well rw207

Pond

Table 4. Control wells with barometric efficiencies determined to be zero based on surface influences, Air Force Plant 6, 

Marietta, Georgia, 2003.

[TZ, transition zone, partially weathered bedrock; overburden, fully weathered bedrock and fill]

Well 

name

Top of opening 

(feet below 

land surface)

Bottom of opening 

(feet below 

land surface)

Hydrogeologic 

unit Surface influence

mwos04 13 23 Overburden Top of well screen is above the water table.

rw207 85 200 TZ Directly connected to a retention pond.

ob203c 24.2 29.2 Overburden Very shallow, connected to a retention pond.

mwg11 65 95 TZ Evapotranspiration dominates water levels during

the growing season.

mwos10 66.5 97 Bedrock Evapotranspiration dominates water levels during 

the growing season; might be connected to stream.

sct5 57 77 Overburden Unconfined response to rain events; might be 

connected to stream.

Figure 12. Water levels in wells rw207 and ob203c, and in a nearby pond, Air Force Plant 6, Marietta, Georgia, 2003. 

See figure 1 for well and pond locations.
Graphical Method for Estimation of Barometric Efficiency from Continuous Data

Estimates of barometric efficiency for the six control wells 

are summarized in table 5. Estimates for all methods ranged from 

–1.270 to 0.068. Excluded from table 5 are results of the Clark 

method using the inflection-centered time intervals, technique 1 

of the graphical method, and the ordinary least squares of water 

level as a function of barometric pressure. Table 5 also lists variances of DW and DB for the 19–20 time intervals that were used 

in the average-of-ratios, median-of-ratios, and slope methods. 

The two methods that used the median values of DW/DB

or D D W B from time intervals or barometric-pressure 

fluctuations, respectively, appeared to be most resistant to 

barometric-pressure-independent water-level change. The 

graphical method was particularly resistant to large amounts of 

barometric-pressure-independent water-level change. Because 

barometric efficiencies for the six control wells are assumed to 

be zero, the barometric efficiency estimate of a given method 

is equivalent to the error in the barometric efficiency estimate 

for that method. The closer an estimate is to zero, the smaller 

is the error. The error of each method was determined from the 

average and standard deviation of the barometric efficiencies 

of the six control wells. The method with the smallest average 

and standard deviation was the graphical method (–0.003 and 

0.067, respectively) followed by the median-of-ratios method 

(–0.283 and 0.313, respectively).

If barometric efficiencies for all of the wells are zero, 

the variance of DW is equivalent to the variance of DWi

. The 

ratio of the variance of DWi divided by the variance of DB

appears to be a predictor of the amount of error that might 

occur in the estimates (see example in fig. 13). The wells 

with the highest ratio of the variance of DWi divided by the 

variance of DB (wells rw207, ob203c, and mwos04) had the 

largest average error (–0.739, –0.655, and –0.614, respectively). The plots of wells rw207, ob203c, and mwos04 were 

particularly difficult to objectively determine barometric 

efficiency using the graphical method. 

Table 5. Estimates of barometric efficiency from different methods for wells determined to have barometric efficiencies of near zero 

based on surface influences, Air Force Plant 6, Marietta, Georgia, 2003.

[n, sample size or number of time intervals used for the slope2, average-of-ratios, and median-of-ratios methods; DB, the barometric-pressure change during a 

time interval; DW, water-level change in a well during a time interval. With the barometric efficiencies of the wells determined to be zero, DW is equivalent to 

DWi, the barometric-pressure-independent water-level change; VW/VB, the ratio of the variance of DW to the variance of DB for n number of time intervals; 

OLS, ordinary least squares water level plotted on the y-axis and barometric pressure plotted on the x-axis; average, the average estimate of barometric efficiency of a method for the six control wells; standard deviation of the estimate of barometric efficiency of a method for six control wells; 15-minute increments 

were used for the Clark1 method; –, negative]

Well 

name n

Variance 

of DB

(VB, 

in feet)

Variance 

of DW

(VW, in 

feet)

VW/VB

Barometric-efficiency estimates

Average 

of ratios

Median of 

ratios Clark1 Slope 2

(OLS) Graphical

Average 

barometricefficiency 

estimate, 

by well

mwg11 20 0.030 0.011 0.383 –0.156 –0.040 –0.093 –0.230 0.06 –0.092

mwos04 20 0.030 0.098 3.276 –1.270 –0.352 –0.769 –0.677 0.00 –0.614

mwos10 20 0.030 0.003 0.092 0.068 0.028 0.059 0.003 0.01 0.034

ob203c 19 0.031 0.113 3.636 –0.894 –0.566 –0.682 –1.003 –0.13 –0.655

rw207 20 0.030 0.137 4.459 –1.012 –0.721 –0.879 –1.083 0.00 –0.739

sct5 20 0.030 0.049 1.631 –0.116 –0.046 0.004 –0.212 0.04 –0.066

Average –0.563 –0.283 –0.393 –0.534 –0.003

Standard 

deviation 0.561 0.313 0.427 0.453 0.067

1

Clark, 1967

2

Ferris and others, 1962,
For non-control wells, the variance in values of the ratio 

of DW to DB can be used to determine about how much error 

might occur in the methods. The larger the variance of ratios, 

the more probable the error will be significant. The variability 

in the slopes of the loops can be used to determine how much 

error might occur in the graphical method. The r-squared for 

the ordinary least squares can be used to determine the probable error in estimations from the slope method.

In the case of a large amount of DWi , many measures 

of DW and DB will be required in order to obtain a reliable 

estimate of barometric efficiency. Methods that use multiple 

values of DW and DB to estimate barometric efficiency are 

superior to the method that estimates baro­metric efficiency 

from a single ratio (DW/DB). Still, table 5 demonstrates that 

large amounts of DWi within the study period can cause large 

errors for most methods. Surface influences on water levels 

in the control wells created a large amount of DWi.

Water levels in wells ob203c and rw207 correlated to 

surface-water fluctuations from the retention pond, as can 

be seen in figure 12 when peaks and troughs coincided with 

time. Water levels for wells ob203c and rw207 were then 

corrected for barometric-pressure-independent water-level 

change in the form of surface-water fluctuations (DWs) using 

a modified Taylor series:

W W a S c t corr t uncorr j t j

j

j

( ) = - - ( ) ( ) ( )

=

å1

6

, (36)

where 

W(t)corr is the water level corrected for surface-water 

fluctuations of the retention pond at time t; 

W(t)uncorr is the water level not corrected for surfacewater fluctuations, at time t; 

S(t) is the surface water in the retention pond, 

at time t; 

and 

 aj and cj for wells rw207 and ob203c are 

listed in table 6. 

Values of aj and cj were determined in a manner similar to

technique 2 of the graphical method (see example in fig. 14). 

Removing the effects of surface-water fluctuations removed 

more than 95 percent of the water-level variance for the two 

wells (table 7).

Table 7 compares the estimates of barometric efficiency 

for wells rw207 and ob203c before and after surface-water 

fluctuations were removed from the data. Removing the 

surface-water fluctuations greatly reduced the ratio of the 

variance of DWi

 to the variance of DB and greatly reduced the 

error of most methods. Error reduction usually was more than 

an order of magnitude. Only the graphical method had a slight 

increase in the error after the data were corrected for surfacewater fluctuations due to the resulting greater precision in the 

graphical method. The reader is reminded that another set of 

wells or the same wells but with different study periods will 

yield slightly different results.

Comparison of Methods

0 1 2 3 4 5

RATIO OF VARIANCE ( W) TO VARIANCE ( B)

0

0.2

0.4

0.6

0.8

1.0

1.2

DISCREPANCY IN ESTIMATE OF

BAROMETRIC EFFICIENCY, –

mwg11 sct5

mwos10

mwos04

ob203c

rw207

Table 6. Values used to correct wells rw207 and ob203c for 

barometric-pressure-independent water-level change in the 

form of surface-water fluctuations from a nearby retention 

pond, Air Force Plant 6, Marietta, Georgia, 2003. See figure 1 

for well locations.

[NA, not applicable; variables are located in equation 36]

rw207 ob203c

aj value cj value aj value cj value

a1 0.6 c1 2 a1 0.495 c1 2

a2 0.01 c2 2 a2 –0.04 c2 2

a3 –0.05 c3 2 a3 –0.05 c3 2

a4 0.0067 c4 1.65 a4 0.0067 c4 1.65

a5 0 c5 NA a5 0 c5 NA

a6 0.0003 c6 2.02 a6 0.0003 c6 2.02

Figure 13. The discrepancy in the estimate of the 

barometric efficiency as a function of the ratio of 

the variance of DW to the variance of DB. The 

discrepancy is the absolute value of the difference 

between the actual barometric efficiency (α, 

determined to be zero) and the estimate of the 

barometric efficiency (αˆ ) using the slope method 

(the ordinary least squares best-fit to the slope 

of DW as a function of DB). Nineteen to 20 time 

intervals (19 to 20 values of DW and corresponding 

DB) were used. See figure 1 for well locations.
6 Graphical Method for Estimation of Barometric Efficiency from Continuous Data

Table 7. Estimates of barometric efficiency from different methods for wells ob203c and rw207 before and after water levels were 

corrected for surface-water fluctuations in a nearby retention pond, Air Force Plant 6, Marietta, Georgia, 2003.

[n, sample size or number of time intervals used for the slope2, average-of-ratios, and median-of-ratios methods; DB, the barometric-pressure change during a 

time interval; DW, water-level change in a well during a time interval. With the barometric efficiencies of the wells determined to be zero, DW is equivalent to 

DWi, the barometric-pressure-independent water-level change; VW/VB, the ratio of the variance of DW to the variance of DB for n number of time intervals; 

OLS, ordinary least squares water level plotted on the y-axis and barometric pressure plotted on the x-axis; uncorr, uncorrected water-level data; corr, waterlevel data were corrected for surface-water fluctuations in a nearby retention pond; 15-minute increments were used for the Clark1

 method; –, negative]

Well name n

Variance 

of DB

(VB, 

in feet)

Variance 

of DW

(VW, in 

feet)

VW/VB

Barometric-efficiency estimates

Average of 

ratios

Median of 

ratios Clark1 Slope 2

(OLS) Graphical

Average 

barometricefficiency 

estimate, 

by well

ob203c

uncorr 19 0.031 0.113 3.636 –0.894 –0.566 –0.682 –1.003 –0.13 –0.655

ob203c

corr 19 0.031 0.005 0.151 0.084 –0.050 0.132 –0.064 0.06 0.032

rw207

uncorr 20 0.030 0.137 4.459 –1.012 –0.721 –0.879 –1.083 0.00 –0.739

rw207

corr 19 0.031 0.001 0.040 0.032 –0.020 –0.006 –0.049 –0.02 –0.013

1

Clark, 1967

2

Ferris and others, 1962

1,019.5

1,020.0

1,020.5

1,021.0

1,021.5

1,022.0

1,022.5

SURFACE-WATER LEVEL OF RETENTION POND, IN FEET ABOVE TRANSDUCER

0 1 2 3 4 5

WATER LEVEL, IN FEET ABOVE NAVD 88

Uncorrected for surface water

Corrected for surface water

Figure 14. Water level in well rw207 as a function of surface-water level in a nearby retention pond, 

Air Force Plant 6, Marietta, Georgia, August 5–October 6, 2003. See figure 1 for well location.
Summary

Summary

Barometric efficiency is the water-level change caused 

by a barometric-pressure change divided by that barometricpressure change. Water-level change that is caused by 

barometric-pressure change cannot be directly measured. 

The measurable water-level change in a well can be separated 

into two components: water-level change caused by 

barometric-pressure change and water-level change not 

caused by barometric-pressure change, hereinafter referred to 

as “barometric-pressure-independent water-level change.” 

The source of error in the estimate of barometric efficiency is the barometric-pressure-independent water-level 

change. The key to accurate estimation of barometric efficiency 

is to reduce the error-causing effect of barometric-pressureindependent water-level changes from the estimation. Some 

barometric-pressure-independent water-level change can be 

removed prior to estimating barometric efficiency because it 

can be identified and avoided or quantified and removed. After 

avoiding or removing as much barometric-pressure-independent 

water-level change as possible, some barometric-pressureindependent water-level change will remain in the data.

Methods that estimate barometric efficiency do so in 

the presence of remaining barometric-pressure-independent 

water-level change. The best methods for estimating barometric efficiency use many measurements of water-level change 

and corresponding barometric-pressure change from many 

time intervals and rely on barometric-pressure-independent 

water-level change to not be correlated with barometric-pressure change. With many measurements of water-level and 

barometric-pressure change, the constant relation between the 

water-level change that is caused by the barometric-pressure 

change and the barometric-pressure change becomes more 

apparent while the lack of correlation between the barometricpressure-independent water-level change and the barometricpressure change becomes less apparent. A correlation between 

barometric-pressure-independent water-level change and 

barometric-pressure change will tend to produce a systematic 

error. A constant barometric-pressure-independent water-level 

change with time will cause a random error in the estimate of 

barometric efficiency.

Time intervals used in methods should be selected such 

that the barometric-pressure-independent water-level change 

is as small as possible compared to the barometric-pressure 

change. Focusing time-interval selection about the inflection 

points of barometric-pressure fluctuations (when barometricpressure change is the most rapid) will maximize the value 

of the barometric-pressure change while not maximizing the 

barometric-pressure-independent water-level change. The result 

will be that time intervals will have minimum values of error.

A graphical method was developed that uses continuous 

water-level and barometric-pressure data to estimate barometric efficiency. A plot of nearly continuous water level (on the 

y-axis) as a function of nearly continuous barometric pressure 

(on the x-axis) will plot as a line curved into a series of connected elliptical loops. Each loop represents a barometricpressure change associated with a barometric-pressure fluctuation. The negative of the slope of the major axis of an elliptical 

loop will be the ratio of water-level change divided by barometric-pressure change, which is the sum of two ratios—the 

water-level change caused by the barometric-pressure change 

divided by the barometric-pressure change (the barometric efficiency) and the barometric-pressure-independent water-level 

change divided by the barometric-pressure change (the error). 

The negative of the slope of the preferred orientation of 

many elliptical loops is an estimate of the barometric efficiency. The slope of the preferred orientation of many elliptical loops is approximately the median slope of the major axes 

of the elliptical loops. If the barometric-pressure-independent 

water-level change does not correlate with barometric-pressure 

change, the probability that the error will be greater than 

zero will be the same as the probability that it will be less than 

zero. As a result, the negative of the median of the slope for 

many loops will be close to the barometric efficiency. Because 

there is observational error in using the graphical method, it 

is advisable that more than one person select the slope or that 

the same person fits the same data several times in order to 

minimize subjectivity. Also, a long study period should be 

used (at least 60 days) to ensure that loops that are affected by 

large amounts of barometric-pressure-independent water-level 

change do not significantly contribute error to the barometricefficiency estimate.

Water-level and barometric-pressure data were collected 

at Air Force Plant 6, Marietta, Georgia, as part of groundwater characterization efforts. The study site is located on 

about 150 acres in northern Georgia in the Piedmont, which 

is underlain by fractured-crystalline metamorphic rock with 

igneous intrusions.

An 889-hour (37-day) constant-discharge aquifer test 

was conducted in the fractured-crystalline rock during August 

and September 2003. Pressure transducers installed in more 

than 45 observation wells and a retention pond monitored 

water levels during the test. Two barometers also monitored 

barometric pressure. The frequency of record was once every 

15 minutes. Monitoring continued months after the end of the 

aquifer test. The barometric effects on water level had to be 

removed to better interpret the aquifer-test data.

The graphical method was used to assess which wells had 

water levels affected by barometric-pressure changes. Longterm, barometric-pressure-independent water-level change was 

removed from the data. The centered daily-moving average of 

water level was plotted as a function of centered daily-movingaverage barometric pressure. Barometric-efficiency estimates 

from the graphical method ranged from –0.138 to 0.703. The 

graphical method provided a rapid assessment of whether or 

not a well was affected by barometric-pressure change and also 

provided a rapid estimate of barometric efficiency. Barometric 

efficiency was best determined by using a correction factor to 

zero the slope of the preferred orientation of loops in corrected 

water-level data as a function of barometric pressure. The correction factor that zeroed the slope of the preferred orientation 

of the loops was the estimate of the barometric efficiency.
Graphical Method for Estimation of Barometric Efficiency from Continuous Data

The graphical method was compared to the average-ofratios, median-of-ratios, Clark, and slope methods. Estimates 

of barometric efficiency of the four other methods ranged 

from –1.270 to 1.565. The performance of the methods were 

assessed in two ways: (1) the number of estimates that were 

outside the domain of the barometric efficiency (less than zero 

or greater than one), and (2) the results from six control wells 

where the barometric efficiency is determined to be near zero 

based on a strong surface influence on water levels. 

The two methods that used the median values of waterlevel change divided by barometric-pressure change appeared 

to be most resistant to barometric-pressure-independent waterlevel change. The graphical method was particularly resistant to 

large amounts of barometric-pressure-independent water-level 

change, having an error that was less than the other four methods. The methods with the least number of estimates outside the 

domain of the barometric efficiency were the graphical method 

followed by the median-of-ratios method. Six of the 45 wells 

were determined to have a barometric efficiency near zero 

based on strong surface influences on water levels. Because 

the barometric efficiencies for the six control wells are determined to be zero, the barometric efficiency estimate of a given 

method is equivalent to the error in the barometric efficiency 

estimate for that method. The methods with the least average 

and standard deviation (error) for the six control wells were the 

graphical method (–0.003 and 0.067, respectively) followed by 

the median-of-ratios method (–0.283 and 0.313, respectively). 

For non-control wells, the variance in values of the ratio 

of DW to DB can be used to determine about how much error 

might occur in the methods. The larger the variance of ratios, 

the more probable the error will be significant. The variability 

in the slopes of the loops can be used to determine how much 

error might occur in the graphical method. The r-squared for 

the ordinary least squares can be used to determine the probable error in estimations from the slope method.

Methods that use multiple values of water-level change 

and barometric-pressure change to estimate barometric 

efficiency are superior to the method that estimates barometric efficiency from a single ratio of water-level change 

to barometric-pressure change. Still, large amounts of barometric-pressure-independent water-level change within the 

data can cause large errors for most methods. In the case of a 

large amount of barometric-pressure-independent water-level 

change, many measures of water-level change and barometricpressure change will be required to obtain a reliable estimate 

of barometric efficiency. 

Two of the six control wells had a large ratio of the variance 

of barometric-pressure-independent water-level change to the variance of barometric-pressure change. Estimates from the averageof-ratios, median-of-ratios, Clark, and slope methods for these 

wells were largely in error. Water levels in these wells were corrected for surface-water fluctuations, thereby reducing the amount 

of barometric-pressure-independent water-level change. Estimates 

for these two wells, after surface-water effects were removed, 

had errors that were usually more than an order of magnitude 

smaller than estimates from the uncorrected water-level data.

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Manuscript approved for publication, June 5, 2007

Prepared by USGS Georgia Water Science Center

Edited by Patricia L. Nobles

Graphics by Bonnie J. Turcott

Layout by Caryl J. Wipperfurth

For more information concerning the research in this report, contact

USGS Georgia Water Science Center, Atlanta

telephone: 770-903-9100