U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2012–5158

Prepared in cooperation with the West Dakota Water Development District, South Dakota 

Department of Environment and Natural Resources, and Lawrence County

Microgravity Methods for Characterization of GroundwaterStorage Changes and Aquifer Properties in the 

Karstic Madison Aquifer in the Black Hills of 

South Dakota, 2009–12
Front cover. Microgravity measurement at a gravity station in Spring Canyon with an outcrop of the Madison Limestone in the 

background. An inset of the of the measurement screen of the CG–5 relative gravimeter is shown in the lower left corner. 

This photograph was taken in the summer of 2012 by Jonathan McKaskey.

Back cover. Upper left: Measurement with the absolute gravimeter at a base station in the Doty focus area, S. Dak. 

Photograph by Jeff Kennedy. 

Upper center: Leveling the lower unit of the absolute gravimeter at a base station in the Doty focus area, S. Dak. 

Photograph by Karl Koth. 

Upper right: Measurement with the relative gravimeter at an absolute-gravity station on an outcrop of Precambrian 

rocks. Photograph by Jonathan McKaskey. 

Lower left: Relative-gravity measurement in the Doty focus area, S. Dak., with a future scientist in the background. 

Photograph by Jonathan McKaskey. 

Lower right: Absolute-gravity measurement at a station located on top of Terry Peak in the northern Black Hills.
Microgravity Methods for Characterization 

of Groundwater-Storage Changes and 

Aquifer Properties in the Karstic Madison 

Aquifer in the Black Hills of South Dakota, 

2009–12

By Karl R. Koth and Andrew J. Long

Prepared in cooperation with the West Dakota Water Development District, 

South Dakota Department of Environment and Natural Resources, and 

Lawrence County

Scientific Investigations Report 2012–5158

U.S. Department of the Interior

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

KEN SALAZAR, Secretary

U.S. Geological Survey

Marcia K. McNutt, Director

U.S. Geological Survey, Reston, Virginia: 2012

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Suggested citation:

Koth, K.R., and Long, A.J., 2012, Microgravity methods for characterization of groundwater-storage changes and 

aquifer properties in the karstic Madison aquifer in the Black Hills of South Dakota, 2009–12: U.S. Geological Survey 

Scientific Investigations Report 2012–5158, 22 p.

ISBN 978-1-4113-3453-3
iii

Acknowledgments

This study was conducted by the U.S. Geological Survey (USGS) South Dakota Water Science 

Center in Rapid City, South Dakota, in cooperation with the West Dakota Water Development 

District, South Dakota Department of Environment and Natural Resources, and Lawrence 

County. Special thanks are extended to Robert Carruth, Donald Pool, and Jeff Kennedy from 

the USGS Arizona Water Science Center for providing valuable advice in site installation and 

gravity-survey procedures and also for making the A–10 absolute gravimeter measurements. We 

are indebted to Mike Webring, Chad Ailes, and Ben Drenth of the USGS Crustal Geophysics and 

Geochemistry Science Center in Denver, Colorado, for maintaining the CG–5 relative gravimeter 

and for their help in developing a better understanding of gravimeters and gravity measurement 

techniques. 
iv

Contents

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

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

Introduction.....................................................................................................................................................2

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

Hydrogeology of the Study Area ........................................................................................................2

Methods of Study...........................................................................................................................................4

Absolute and Relative Gravity Measurements ................................................................................4

Methods for Determining Groundwater-Storage Change and Effective Porosity .....................5

Microgravity Survey Network Design...............................................................................................6

Gravity Station Design ................................................................................................................6

Focus Areas..................................................................................................................................6

Microgravity Measurement Methods ...............................................................................................8

Field Procedures..........................................................................................................................8

Sources of Error...........................................................................................................................9

Characterization of Groundwater-Storage Changes and Aquifer Properties ...................................10

Spring Canyon Focus Area................................................................................................................10

Doty Focus Area..................................................................................................................................10

Limestone Plateau Focus Area.........................................................................................................15

Limitations of Microgravity Methods .......................................................................................................16

Summary........................................................................................................................................................17

References Cited..........................................................................................................................................18

Appendix........................................................................................................................................................20

Figures

1. Map showing study area, geologic units, and data collection sites ...................................3

2. Map showing Doty focus area showing gravity stations and observation wells..............7

3. Graph showing groundwater-storage change determined from gravity data at

station TEU plotted with the hydrograph for observation well RG–Mdsn (PE–86A)

and streamflow for streamgage 06407500 from January 2010 to October 2011...............11

4. Graph showing groundwater-storage change determined from gravity data for

group 2 stations in the Doty focus area plotted with the hydrograph for observation 

well Doty–Mdsn (PE–95C).........................................................................................................12
v

5. Conceptual plot of change in water-table altitude and groundwater-storage 

change with divisions where the balance of inflow to outflow in the unsaturated 

zone changes ..............................................................................................................................13

6. Diagram showing effective porosity estimates for Madison aquifer beneath station

MN04 for vertical sections a–d determined for three different time periods...................13

7. Graph showing groundwater-storage change determined from gravity data for

group 1 stations in the Doty focus area plotted with the hydrograph for observation 

well Doty–Mdsn (PE–95C).........................................................................................................14

8. Water-level hydrograph for lake in Brooks Cave from 1988 to 1998 ..................................15

9. Graph showing groundwater-storage change determined from gravity data for 

station O27 plotted with the hydrograph for observation well Chyn–Xing–Mdsn

(LA–95B).......................................................................................................................................16

1–1. Graph showing relative-gravity measurements made at the U.S. Geological Survey

South Dakota Water Science Center in Rapid City, South Dakota, on September 19, 

2011, plotted with a fitted sixth-order polynomial .................................................................21

1–2. Graphs showing relative-gravity measurements for a survey conducted on 

June 9, 2011, at stations MN04, MN05, MN06, and MD01 plotted with a fitted 

fifth-order polynomial.................................................................................................................22

Conversion Factors

Inch/Pound to SI

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

inch (in.) 25.4 millimeter (mm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Flow rate

cubic foot per second (ft3

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

Acceleration

Galileo (Gal) 1 centimeter per second squared (cm/s2

)

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).

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

Symbols, Units of Measurement, Abbreviations, and 

Acronyms

b thickness of Bouguer slab

g accelerative force of gravity

G universal gravitational constant

∆g change in gravitational potential

∆h change in water-table altitude

M mass of the Earth

n number of averaged relative-gravity measurements

Op order of polynomial

r distance to the Earth’s center of mass

∆S change in groundwater storage

φ effective porosity

ρw density of water

∆ρrw change in density of the rock and water slab

> greater than

± plus or minus

μGal microgal

SDDENR South Dakota Department of Environment and Natural Resources
Microgravity Methods for Characterization of 

Groundwater-Storage Changes and Aquifer Properties in 

the Karstic Madison Aquifer in the Black Hills of South 

Dakota, 2009–12

By Karl R. Koth and Andrew J. Long

Abstract

A study of groundwater storage in the karstic Madison 

aquifer in the Black Hills of South Dakota using microgravity methods was conducted by the U.S. Geological Survey in 

cooperation with West Dakota Water Development District, 

South Dakota Department of Environment and Natural 

Resources, and Lawrence County. Microgravity measurements 

from 2009 to 2012 were used to investigate groundwaterstorage changes and effective porosity in unconfined areas of 

the Madison aquifer. Time-lapse microgravity surveys that 

use portable high-sensitivity absolute and relative gravimeters 

indicated temporal-gravity changes as a result of changing 

groundwater mass. These extremely precise measurements 

of gravity required characterization and removal of internal 

instrumental and external environmental effects on gravity 

from the raw data. The corrected data allowed groundwaterstorage volume to be quantified with an accuracy of about plus 

or minus 0.5 foot of water per unit area of aquifer. Quantification of groundwater-storage change, coupled with water-level 

data from observation wells located near the focus areas, also 

was used to calculate the effective porosity at specific altitudes 

directly beneath gravity stations. 

Gravity stations were established on bedrock outcrops 

in three separate focus areas for this study. The first area, the 

Spring Canyon focus area, is located to the south of Rapid 

City with one gravity station on the rim of Spring Canyon near 

the area where Spring Creek sinks into the Madison aquifer. 

The second area, the Doty focus area, is located on outcrops 

of the Madison Limestone and Minnelusa Formation to the 

northwest of Rapid City, and consists of nine gravity stations. 

The third area, the Limestone Plateau focus area, consists of a 

single gravity station in the northwestern Black Hills located 

on an outcrop of the Madison Limestone. An absolute-gravity 

station, used to tie relative-gravity survey data together, was 

established on a relatively impermeable bedrock outcrop 

to minimize groundwater-storage change at the reference 

location. 

Data from the three focus areas allow for interpretation 

of groundwater-storage characteristics using microgravity 

measurements. Gravity measurements, together with waterlevel data from an observation well located 2 miles from the 

Spring Canyon focus area and measured streamflow in Spring 

Creek, provided evidence that rapid groundwater-storage 

change, responding to changes in sinking streamflow over the 

recharge area of the aquifer, occurred in the Madison aquifer 

directly beneath the gravity station at Spring Canyon. This 

phenomenon likely was a result of groundwater movement 

through caverns, conduits, and fractures, which are common in 

karst aquifers. Spatially and temporally separated microgravity 

data for the Doty focus area indicated horizontal and vertical 

heterogeneity of effective porosity for the Madison aquifer. 

One such example of this was indicated by water-level measurements at an observation well and gravity measurements at 

four gravity stations in the southeastern part of the Doty area, 

which were used to estimate effective porosity values ranging 

from greater than 0 to 0.18. A decrease in groundwater storage 

determined by microgravity measurements during the spring 

recharge period for five upgradient stations in the Doty focus 

area indicated the possibility of rapid release and downgradient cascading of perched groundwater. Evidence for similar 

phenomena was documented for Wind Cave and Brooks Cave 

in the Black Hills. Absolute-gravity measurements at the 

Limestone Plateau focus area confirmed the relation between 

water levels in an observation well and changes in groundwater storage. Comparison of these gravity measurements with 

water levels in a nearby observation well resulted in an effective porosity estimate of 0.02 for the Madison aquifer beneath 

the gravity station.
2 Microgravity Methods for Characterization of the Karstic Madison Aquifer in the Black Hills of South Dakota

Introduction

Microgravity methods were applied to the Madison 

aquifer in the Black Hills of western South Dakota as an 

exploratory pilot study to evaluate the potential utility of these 

methods for advancing the knowledge of groundwater-storage 

change, groundwater flow, and effective porosity in this karstic 

aquifer. Changes in the Earth’s gravitational field can be used 

to infer changes in groundwater storage caused by a change 

in mass from the addition or subtraction of groundwater. One 

factor that affects the gravitational force on the Earth’s surface 

is the density of the underlying mass. The most common cause 

of density change over time is a change in groundwater storage or groundwater mass change beneath a particular location, 

which can be quantified by measuring temporal changes in the 

gravitational force (Damiata and Lee, 2006; Christiansen and 

others, 2011). The gravitational force may be measured with 

portable gravimeters. 

Microgravity methods have been used in several applications to monitor changes in groundwater storage (Pool and 

Schmidt, 1997; Blainey and others, 2007; Pool and Anderson, 

2008; Jacob and others, 2010). These investigations of groundwater-storage change and specific yield using temporal-gravity 

methods were helpful in planning the study methods for the 

Madison aquifer and gave insight to data analysis techniques 

and considerations. Blainey and others (2007) estimated specific yield, storage change, and hydraulic conductivity using 

simulated data with a 1-microgal (μGal) noise level. Other 

studies of unconfined alluvial aquifers, such as those in Pool 

and Schmidt (1997) and Pool and Anderson (2008), determined temporal gravity changes using gravimeters. Pool and 

Schmidt (1997) used only relative gravimeters, whereas Pool 

and Anderson (2008) used absolute and relative gravimeters. 

The use of the highly portable CG–5 relative gravimeter 

allowed for gravity surveys that included a larger number of 

gravity stations than surveys using only an absolute gravimeter, which requires more time to deploy and make a measurement than a relative gravimeter. 

Aside from the study conducted by Jacob and others 

(2010) for the karstic Durzon aquifer in France, few microgravity studies have been documented for karstic or fractured 

aquifers such as those in the Black Hills. Because of the challenges related to characterizing these highly heterogeneous 

aquifers, microgravity methods that have not been applied 

may provide useful information that was not available using 

only data obtained from observation wells. Installing deep 

observation wells in competent bedrock is costly, whereas 

microgravity methods can be used to monitor changes in 

groundwater storage in areas where observation wells cannot be installed because of cost or access. Observation wells 

and microgravity monitoring provide different but related 

information about groundwater. Observation wells monitor 

changes in hydraulic head, which can be done continuously 

with recorders (Cunningham and Schalk, 2011). Pumping 

tests using these observation wells can be used to infer aquifer 

properties such as transmissivity, specific yield, and effective 

porosity (Heath, 2004). Microgravity measurements can be 

used to monitor changes in groundwater-storage volume, or 

mass. Coupling groundwater-storage change determined from 

microgravity measurements with changes in hydraulic head 

at observation wells permits estimates of effective porosity. 

Quantification of groundwater storage based on groundwater 

levels alone is difficult because effective porosity is largely 

unknown. Therefore, microgravity methods provide information that cannot be obtained by other methods. For example, 

these microgravity methods were used to estimate the effective 

porosity at different altitudes in the Madison aquifer beneath a 

gravity station. 

 The U.S. Geological Survey in cooperation with the 

West Dakota Water Development District, South Dakota 

Department of Environment and Natural Resources, and Lawrence County conducted a pilot study from 2009 to 2012 to 

evaluate the potential utility of microgravity methods for the 

Madison aquifer in the Black Hills of western South Dakota. 

Objectives of this study were to (1) measure and assess the 

characteristics of groundwater-storage changes in unconfined 

areas of the Madison aquifer; (2) determine effective porosity for areas where observation wells exist; and (3) assess the 

horizontal and vertical heterogeneity of effective porosity at 

selected locations. 

Purpose and Scope

The purpose of this report is to document the results of 

the application of microgravity methods and provide analyses 

of storage changes and aquifer properties based on microgravity measurements made from 2009 to 2012. Although the 

primary focus of this report is the Madison aquifer, analysis 

of microgravity data for the overlying Minnelusa aquifer also 

is included. Estimates of effective porosity and the heterogeneity thereof provided in this report are useful for watermanagement considerations and additional investigations, such 

as groundwater modeling. Methods to monitor groundwaterstorage changes directly and without observation wells are 

described as well as phenomena unique to karst aquifers that 

were detected by microgravity methods. This report provides 

useful information about microgravity measurement methods 

and inherent challenges in analyzing high-precision measurements of Earth’s gravitational field. Field procedures and 

methods of analysis for microgravity data are described, which 

may provide a guide for future microgravity studies.

Hydrogeology of the Study Area

The study area is located in the Black Hills of western 

South Dakota. The Black Hills are a doubly-plunging fold 

structure with Paleozoic-age sedimentary formations dipping 

radially outward on the flanks and overlying Precambrian-age 

rocks (fg. 1). Detailed geologic and hydrogeologic descriptions of the region can be found in Gries and Martin (1981), 

Strobel and others (1999), and Redden and DeWitt (2008). 
Introduction 3

Xh

Xu

Xu

Xu

O_wd

MDme

MDme

P*m

P*m

Base from U.S. Geological Survey digital data, 1997, 1:100,000

Universal Transverse Mercator projection, Zone 13

Geology modified from Strobel and others, 1999

104°00’ 103°40’ 103°20’

44°20’

44°00’

Limestone 

Plateau

focus 

area

Spring Canyon

focus area

Doty focus area

O27

TEU

VLR2

HP2

WR1

RG-Mdsn

Doty-Mdsn

Chyn-Xing-Mdsn

06407500

SOUTH DAKOTA

Black Hills

Study area

Rapid City

EXPLANATION

HP2 Absolute-gravity station and identifier

Location of Brooks Cave

U.S Geological Survey streamgage and identifier

Madison aquifer unconfined 

(water table) area (Long 

and Putnam, 2002)

Minnelusa aquifer unconfined 

(water table) area (Long and 

Putnam, 2002)

Focus area

Relative-gravity station and identifier

06407500

RG-Mdsn

TEU

0 1 2 3 4 5 KILOMETERS

0 1 2 3 4 5 MILES

Rapid

Spring

Creek

Creek

Spearfish Creek

Pactola

Reservoir

Sheridan Lake

Deerfield

Lake

Castle

Creek

Elk Creek

Boxelder

Creek

Geologic units

MDme

O_wd

Xu

Xh

Minnelusa Formation (Lower Permian 

and Pennsylvanian)

Madison (Pahasapa) Limestone (Lower 

Mississippian) and Englewood 

Limestone (Lower Mississippian 

and Upper Devonian)

Whitewood Limestone (Upper Ordovician), 

Winnepeg Formation (Middle 

Ordovician), and Deadwood Formation 

(Lower Ordovician and Cambrian)

Harney Peak Granite (Early Proterozoic)

Precambrian rocks, undifferentiated (Early 

Proterozoic and Late Archean)

P*m

South Dakota Department of Environment and 

Natural Resources observation well 

completed in the Madison aquifer and 

identifier

Rapid

City

Figure 1. Study area, geologic units, and data collection sites.
4 Microgravity Methods for Characterization of the Karstic Madison Aquifer in the Black Hills of South Dakota

The primary focus of this study is the Madison aquifer, but the 

overlying Minnelusa aquifer also is included in the analysis.

The Minnelusa aquifer is contained within the 

Pennsylvanian- and Permian-aged Minnelusa Formation. The 

Minnelusa Formation ranges from 375 to 1,175 feet (ft) thick 

in the Black Hills (Carter and others, 2003). Two different lithologic regimes divide the Minnelusa Formation into upper and 

lower units: the upper unit contains thick sandstone with thin 

limestone, dolomite and mudstone, and the lower unit has less 

sandstone and more shale, limestone, and dolomite (Bowles 

and Braddock, 1963). Siltstone, gypsum, and anhydrite also 

may be present in the lower unit.

Underlying the Minnelusa aquifer is the Madison aquifer, 

which is contained within the Mississippian-age Madison 

Limestone, locally named the Pahasapa Limestone, which 

is a massive, slabby limestone that contains both dolostone 

and limestone and has a large secondary porosity component 

consisting of fractures and solution openings. This porosity 

component commonly is contained in the upper 100 to 200 ft 

of the Madison Limestone, which is a well-developed karst 

aquifer with large networks of groundwater conduits and pore 

spaces that result from dissolution of carbonate rocks (Greene, 

1993). The Madison Limestone ranges from less than 200 to 

1,000 ft thick in the Black Hills region (Carter and others, 

2003). 

Underlying the Madison Limestone are the Devonianand Mississippian-age Englewood Limestone and the 

Cambrian- and Ordovician-age Deadwood Formation. The 

Englewood Limestone is relatively thin, ranging from 0 to 

60 ft thick, and may be considered as part of the Madison 

Limestone (Gries and Martin, 1981). In the northern part of 

the study area, the Ordovician-age Whitewood Limestone 

and Winnipeg Formation are present between the Englewood 

Limestone and Deadwood Formation. Hydrogeologically, 

these formations may be considered part of the underlying Deadwood Formation. The Deadwood Formation is 0 to 

500 ft thick, and is the oldest and least porous of the sedimentary rocks described in this report (Carter and others, 2003). 

Underlying the Whitewood Limestone, Winnipeg Formation, 

and Deadwood Formation are Precambrian-age metamorphic 

and igneous rocks.

Recharge to the Madison and Minnelusa aquifers occurs 

as infiltration of direct precipitation and sinking streams 

(streamflow recharge) on exposed areas of these aquifers. 

Recharge primarily occurs during the wettest parts of the year 

from March through July. Groundwater flow generally is radially outward from recharge areas and in a down-dip direction. 

The water table of a karst aquifer, such as the Madison aquifer, 

may not be a smooth surface typical of other aquifers but can 

be better characterized as multiple water surfaces at different altitudes in numerous caves or other solution openings. 

As sinking streams or precipitation infiltrates the aquifer, 

these openings can overfill and water can flow into adjacent 

openings at lower altitudes, resulting in a procession of water 

cascading from one opening to another in a downgradient 

direction (Long and others, 2012). Perched water also exists in 

caves in karst aquifers.

Methods of Study

This section includes a brief introduction to the measurement of gravity, or gravimetry. The primary objective of the 

first part of this section is to describe the utility of absolute 

and relative gravimeters in groundwater-storage monitoring 

and to provide a short derivation of the theory of gravity measurement. The design of the gravity station network and the 

methods for microgravity measurements and analyses specific 

to this study are described in the second part of this section. 

Absolute and Relative Gravity Measurements

The Earth’s gravitational field varies across the surface 

and is affected by altitude and the density of the underlying rock and groundwater. Two types of gravimeters exist: 

absolute and relative (Pool and Anderson, 2008). A measurement made with an absolute gravimeter indicates the absolute 

accelerative force of gravity at the measurement location, 

which is referred to hereafter as g. The unit of measure for g

is the Gal, equal to 1 centimeter per second squared (cm/s2). 

High-precision gravity methods required for groundwater 

applications are referred to as microgravity methods, and thus 

g is reported in microgals (μGal, or 10-6 Gal) in this report. A 

relative gravimeter cannot be calibrated to measure g directly 

because of uncertain long-term instrument drift. Relativegravity measurements made at two stations, however, indicate 

the difference in g between these stations. Therefore, one use 

of relative-gravity surveys is to measure spatial differences in 

the Earth’s gravitational field. Measurements made during a 

relative-gravity survey can be converted to g values if one of 

the relative-gravity measurements is made at a station where 

g has been measured by an absolute gravimeter. A relative 

gravimeter is more portable than an absolute gravimeter and 

therefore was the primary gravimeter used in this study. All 

relative-gravity measurements were converted to g values, and 

by doing so the temporal change in the gravitational force at 

each station between surveys was quantified and compared 

over the entire period of the study. 

Measurements of g were made with a Micro-g Lacoste, 

Inc., A–10 absolute gravimeter. This meter measures the 

acceleration of a free-falling corner cube test mass contained inside a drag-free vacuum chamber. A laser interferometer measures the position of the falling mass, and 

a rubidium oscillator provides a time reference. At an 

optimal measurement site, the A–10 absolute gravimeter 

measures g with an accuracy of plus or minus (±) 10 μGal 

(http://www.microglacoste.com/a10.php). This uncertainty 

may increase depending on site conditions, that is, the uncertainty may be greater than ± 10 μGal.
Methods of Study 5

Relative-gravity measurements were made at stations 

over unconfined parts of the Madison and Minnelusa aquifers using the Scintrex CG–5 Autograv. The CG–5 relative 

gravimeter measures relative differences in gravitational 

attraction using a fused-quartz elastic system. This system 

operates by balancing the elongation of a fused-quartz proof 

mass with an electrostatic restoring force, where the restoring 

force is equal to the relative gravity value. The displacement 

of the proof mass is measured using a capacitive displacement transducer. The displacement transducer signal drives a 

feedback circuit that controls the magnitude of the electrostatic 

restoring force. The Scintrex CG–5 gravimeter has a measurement resolution of 1 μGal with an accuracy of ±5 μGal (http://

scintrexltd.com/internal.php?storeCategoryID=1&subcatID=

9&s_page=Gravity#CG-5).

Methods for Determining Groundwater-Storage 

Change and Effective Porosity

Microgravity methods were used to determine temporal 

changes in groundwater mass. It was assumed that the only 

mass changes, and therefore, changes in g, occurring beneath 

measurement stations resulted from changes in groundwater 

storage. Because little groundwater-storage change occurs in 

confined aquifers, all estimated groundwater-storage changes 

were assumed to occur only in unconfined aquifers, generally the uppermost aquifer beneath a measurement station. A 

brief explanation of the theory of microgravity analysis and a 

derivation of applicable equations follow (Pool and Eychaner, 

1995).

The generalized form of Newton’s Law of Gravitation 

states that the acceleration resulting from the Earth’s mass 

is proportional to this mass and inversely proportional to the 

square of the distance to the center of mass:

g = GM/r2, (1)

where 

g is the accelerative force of the Earth’s 

gravitational field, 

G is the universal gravitational constant, 

M is the mass of the Earth, and 

r is the distance to the Earth’s center of mass.

A one-dimensional approximation of Newton’s law may 

be made by assuming the lateral extent of the fluctuating 

groundwater mass is much larger than the vertical distance 

to the groundwater mass. In this application, the changing 

groundwater mass occurs in saturated and unsaturated zones 

and is assumed to be in the form of a slab of infinite extent. 

Because this slab contains rock and water, its density changes 

as groundwater-storage changes. This approximation is 

referred to as the Bouguer slab equation (Telford and others, 

1990):

Δg = 2πGΔρrwb, (2)

where 

Δg is the change in the gravitational potential 

resulting from a change in groundwater 

storage mass, 

Δρrw is the change in density of the rock and water 

slab, and

b is the thickness of the slab.

A beneficial consequence of using the Bouguer slab equation 

is that Δg is independent of the distance to the groundwater 

table because equation 2 contains no distance term. 

In this derivation, the thickness of the Bouguer slab (b) 

is assumed to be the entire thickness of the volume in which 

groundwater-storage change occurs. For the special case in 

which the groundwater-storage change in the unsaturated zone 

is zero, b is equal to the change in water-table altitude Δh, and 

the change in density then is calculated as the effective porosity (φ) times the density of water ρw:

Δρrw = φρw (3)

Substituting equation 3 into equation 2 for the special case 

where Δh equals b and combining the constant parameters 

yields 

Δg = 0.4192φΔh, (4)

where 

Δg is the change in the gravitational potential in 

microgals,

	 φ is the effective porosity,

Δh is the change in water-table altitude in 

centimeters, and

Δg = 12.77φΔh, (5)

where 

Δg is the change in the gravitational potential in 

microgals,

	 φ is the effective porosity,

Δh is the change in water-table altitude in feet.

Also for the special case of Δh = b, the effective porosity can 

be expressed as

	 φ = ΔS/Δh, (6)

where 

φ is the effective porosity,

ΔS is the groundwater-storage change expressed 

as the change in the vertical height of 

water in feet with no rock present, and

Δh is the change in water-table altitude in feet.
6 Microgravity Methods for Characterization of the Karstic Madison Aquifer in the Black Hills of South Dakota

For example, for a 10-ft increase in the water table in an 

aquifer with 10 percent effective porosity, ΔS would be equal 

to 1 ft, assuming that no mass change in the unsaturated zone 

occurred. Combining equations 5 and 6 yields

ΔS = Δg/12.77 (7)

Rearranging equation 5 yields the final equation for the 

effective porosity of an unconfined aquifer at the water table 

as a function of the change in gravitational potential (Δg), 

in microgals, and water-level change (Δh), in feet, for cases 

when Δh = b:

	 φ = Δg/12.77Δh (8)

Microgravity Survey Network Design

This section describes the general requirements for 

gravity station design. Detailed descriptions of the survey 

networks (focus areas) used in this study are also provided in 

the section.

Gravity Station Design

An absolute-gravity station on the Harney Peak Granite 

(station HP2; fig. 1) was established in 2009 and measured 

with the absolute gravimeter annually in the summers of 

2009–11 and the spring of 2012. This location was selected 

because the granite has very low porosity. The measurements 

of g at absolute-gravity station HP2 showed that values of g at 

the station varied by a maximum of 21 μGal (2009–12), which 

is within ±10.6 μGal of uncertainty in A–10 absolute gravimeter field measurements of g at station HP2. These measurements are consistent with the assumption that there was little 

or no groundwater-storage change at this location. However, 

this station was not used for relative gravity surveys because 

the CG–5 relative gravimeter had odd drift that resulted in 

large uncertainties when measuring at station HP2. The drift in 

these surveys might have resulted from the very winding road 

that leads to station HP2 that caused abnormally high transportation drift in the relative gravimeter. A new absolute-gravity 

station, WR1, was established in 2012 along a straighter road 

crossing the Harney Peak Granite for use in future microgravity studies in the Black Hills area. 

As a result of the large uncertainty in relative gravity 

surveys at station HP2, another absolute-gravity station, 

VLR2, was established on an outcrop of low-permeability 

Precambrian rocks with an access road (fig. 1). This location 

is at the top of a ridge of resistant rocks where precipitation 

forms in pools with little infiltration. Because of the apparent 

low permeability at this location, the subsurface was assumed 

to have little change in groundwater storage, and therefore, 

little change in the gravitational field. An important criterion 

restricting the selection of an absolute-gravity station location 

is that the station needs to be accessible by vehicle for measurements with the A–10 absolute gravimeter. Placement and 

orientation of the A–10 absolute gravimeter were recorded 

using paint markers, survey pins, and photographs in the field.

Relative-gravity station locations were selected on the 

basis of site stability, underlying geologic unit, and accessibility. Resistance to erosion and ground movement made bedrock 

outcrops a favorable platform on which to establish gravity 

stations. At station RCPC–4, where no bedrock was exposed, 

a concrete platform was constructed on top of an existing 

land-survey marker consisting of a 7-ft deep concrete column, 

which extends below the frost line to ensure that frost heave 

does not perturb the station altitude (fig. 2). At most stations, 

three indentations each about 0.5 inch (in.) in diameter and 

0.3 in. deep were drilled into the bedrock station platform and 

used to position the leveling tripod on which the CG–5 relative 

gravimeter was placed during measurements. These positioning indentations maintained consistent orientation of the 

gravimeter and aided in leveling the instrument during measurements. Paint markers also were used to help identify and 

locate stations during field surveys and to indicate the direction that the gravimeter was oriented during measurements.

Focus Areas

Gravity stations were established in three separate focus 

areas for this study. The water table of the Madison aquifer 

was assumed to exist beneath all gravity stations, except 

absolute-gravity stations VLR2, HP2, and WR1 (fig. 1). All 

measureable groundwater-storage change was assumed to 

occur in unconfined areas of the Madison aquifer, with the 

addition of the unsaturated zone of the Minnelusa aquifer in 

some areas. Underlying confined aquifers were assumed to 

have negligible groundwater-storage change.

The survey network for the Spring Canyon focus area 

consists of one relative-gravity station on the rim of Spring 

Canyon (station TEU) and the absolute-gravity station 

VLR2 located 5 miles (mi) to the west (fig. 1). The South 

Dakota Department of Environment and Natural Resources 

(SDDENR) observation well RG–Mdsn, also called PE86A, 

is open to the Madison aquifer from 1,194 to 1,220 feet 

below the land surface and is located 2 mi to the east of station TEU (U.S. Geological Survey, 2012). Recharge to the 

Madison and Minnelusa aquifers in this area primarily occurs 

as streamflow recharge from Spring Creek, which sinks into 

these aquifers at several locations where the stream crosses 

outcrop areas, with a maximum estimated rate of 21 cubic 

feet per second (ft3

/s) into the Madison aquifer (Hortness and 

Driscoll, 1998; Long and Putnam, 2002). A streamgage for 

Spring Creek (streamgage 06407500; U.S. Geological Survey, 

2012) is located upstream from where the stream sinks (fig. 1). 

Groundwater flow generally is to the east and northeast 

(Greene, 1993).

In the Doty focus area, located on the eastern side of the 

Black Hills (fig. 1), the sedimentary formations that make up 

the hydrogeologic units generally dip to the east (Redden and 
Methods of Study 7

Figure 2. Doty focus area showing gravity stations and observation wells.

103°26’ 103°24’ 103°22’ 103°20’

44°10’

44°08’

Base from U.S. Geological Survey Geology modified from Strobel and others, 1999

digital data, 1997, 1:100,000

Universal Transverse Mercator projection

Zone 13

Xu

MDme

0 0.5 1 MILE

0 1 KILOMETER 0.5

EXPLANATION

Geologic units

MDme

O_wd

Xu

Minnelusa Formation (Lower Permian and Pennsylvanian)

Madison (Pahasapa) Limestone (Lower Mississippian) 

and Englewood Limestone (Lower Mississippian 

and Upper Devonian)

Whitewood Limestone (Upper Ordovician), Winnepeg 

Formation (Middle Ordovician), and Deadwood Formation 

(Lower Ordovician and Cambrian)

Precambrian rocks, undifferentiated (Early Proterozoic and 

Late Archean)

P*m

O_wd

P*m

P*m

MN06

MN05

MN04

MN03

MN01

MD03 MD02 MD01

RCPC-4

Doty-Mdsn

Group 1

Group 2

South Dakota Department of 

Environment and Natural 

Resources observation well 

completed in the Madison 

aquifer and identifier 

Gravity station and identifier

Doty-Mdsn

MNO4

Group boundary and name Madison aquifer unconfined (water table) 

area (Long and Putnam, 2002)

Minnelusa aquifer unconfined (water table) 

area (Long and Putnam, 2002)

Structural features

Anticline—Showing trace of axial 

plane. Dashed where approximately 

located

Syncline—Showing trace of axial 

plane. Dashed where approximately 

located

Blackhawk Creek

Bogus Jim Creek

Boxelder Creek

Group 2

DeWitt, 2008). Bisecting the Doty focus area from the northwest to the southeast is a fold couple consisting of a syncline 

and anticline (fig. 2). Evidence for a fold is apparent from the 

geologic contacts between the Madison Limestone and Minnelusa Formation. The presence of these structures is likely to 

affect the groundwater-storage characteristics of the area. 

Drilling records for SDDENR observation well Doty–

Mdsn, also called PE95C (fig. 2), indicate that the Minnelusa 

Formation is 335 ft thick near the southeastern gravity stations 

in the Doty focus area. Because of the easterly dip of this 

formation in the area, the Minnelusa Formation diminishes 

in thickness to the west until it contacts the Madison outcrop, 

where the thickness of the Minnelusa Formation is zero. The 

Madison Limestone is about 455 ft thick near the southeastern 

gravity stations in the Doty focus area and thins to the west in 

the outcrop area (Redden and DeWitt, 2008).

The survey network for the Doty focus area consists of 

nine relative-gravity stations located within outcrop areas of 
8 Microgravity Methods for Characterization of the Karstic Madison Aquifer in the Black Hills of South Dakota

the Madison Limestone and Minnelusa Formation (fig. 2); 

the absolute-gravity station VLR2 is located 9 mi south of the 

Doty focus area (fig. 1). Relative-gravity stations were placed 

at approximately 0.5-mi intervals overlying the water-table 

area of the Madison aquifer. The network of survey stations 

stretches over about 3.3 mi trending from the northwest to 

the southeast. Observation well Doty–Mdsn is open to the 

Madison aquifer from 375 to 415 feet below the land surface 

(U.S. Geological Survey, 2012). Recharge in this area primarily occurs as infiltration of direct precipitation. Streamflow 

recharge also occurs about 1 mi south of the gravity stations 

where Boxelder Creek sinks into the Madison and Minnelusa 

aquifers over a stretch of several miles as it crosses outcrops 

of the Madison and Minnelusa Formations (fig. 2).

Areas where the Madison and Minnelusa aquifers are 

unconfined and have water tables at the base of an unsaturated 

zone were estimated by Long and Putnam (2002; figs. 1 and 

2). These authors also indicated that the boundaries of these 

zones change as water-table altitudes (or hydraulic heads as 

used for well hydrographs in this report) change because of 

the dip angles of the aquifers, and therefore, these boundaries are only approximations. A preferential southeasterly 

groundwater-flow path exists in the Madison aquifer approximately along the axis of the syncline shown in figure 2 (Long 

and Putnam, 2002). Group 1 gravity stations are located near 

the upgradient end of this flow path on or near the outcrop of 

the Madison Limestone (fig. 2). Group 2 gravity stations are 

located to the southeast of group 1 within or near this flow 

path and on the outcrop of the Minnelusa Formation (fig. 2). 

The Limestone Plateau focus area in the northwestern 

part of the study area consists of one absolute-gravity station (O27) on an outcrop of the Madison Limestone (fig. 1). 

SDDENR observation well Chyn–Xing–Mdsn, also called 

LA–95B, is located 0.3 mi northeast of gravity station O27 

and is open to the Madison aquifer from 350 to 411 feet below 

the land surface (U.S. Geological Survey, 2012). Recharge in 

this area primarily occurs as infiltration of direct precipitation, 

and groundwater flow in the Madison aquifer generally is to 

the north (Greene and others, 1998). 

Microgravity Measurement Methods

This section describes the methods used during microgravity field surveys and the errors induced by internal and 

external effects on the gravimeters. Solutions for removing 

errors and increasing measurement repeatability also are 

described.

Field Procedures

Field surveys conducted in the Doty and Spring Canyon 

focus areas consisted of one relative-gravity base station (the 

base station may be a relative or absolute-gravity station) and 

from one to five relative-gravity stations. For the Limestone 

Plateau focus area, measurements of g were made at absolutegravity station O27 with the A–10 gravimeter only. During 

a field survey with the CG–5 gravimeter, the first station 

measured was defined as the base station. The remaining 

relative-gravity stations were then measured to create a closed 

loop. The relative-survey loop, starting with the base station, 

was repeated in the same order from one to three times (a 

total of as many as four loops) during a field survey to ensure 

measurement repeatability and to help assess measurement 

accuracy. Each time a station was visited and measured is 

hereafter referred to as a station occupation. In most cases, the 

survey base station was occupied at least three times during 

the survey period to ensure measurement repeatability. Each 

gravity station occupation consisted of three 117-second measurements during which data were continuously collected at a 

6-hertz sample rate. These three measurements were averaged 

to obtain the tide-corrected gravity value for a given station 

occupation, which was later processed to remove instrument 

drift. 

The CG–5 gravimeter was transported in the passenger 

seat of a vehicle, open to the ambient air temperature, and 

buckled securely in place with a seatbelt. Special care was 

taken to avoid jarring the instrument while in transit or when 

moving the instrument from the vehicle to the survey station. 

Also, the instrument was lifted, turned, and set down slowly 

to minimize acceleration effects on the instrument. Whenever 

possible, a shelter was deployed over the gravity station during the entire survey period to shelter the gravimeter and rock 

outcrop from wind and sun exposure. For surveys including 

several gravity stations, the shelter was moved and deployed 

during measurements at all of the gravity stations. The CG–5 

gravimeter also was allowed to stabilize for about 5 minutes 

at the survey station before gravity measurements were made. 

This procedure ensured the consistency of relative-gravity 

measurements and minimized measurement errors. 

Absolute-gravity measurements were collected using the 

A–10 absolute gravimeter. A sport-utility vehicle supplied continuous power to the gravimeter and housed the data collection and processing instrument suite for the A–10 gravimeter. 

As described in the “Gravity Station Design” section of this 

report, the location of absolute-gravity station VLR2 was 

selected to be where groundwater-storage change was likely 

to be small in an area of low-porosity rock. Measurements 

were made at station VLR2 in July 2010 and June 2011 and 

indicated an increase of 17 μGal during this period, corresponding to an apparent increase in groundwater storage (ΔS) 

of 1.33 ft. Considering the potential measurement error of the 

A–10 gravimeter (±10 μGal), this groundwater-storage change 

could be smaller or larger than this value. To account for this 

unexpected apparent change in storage, a linear interpolation 

between the two measurements at station VLR2 was used to 

estimate the g value at this station for other measurement periods, assuming a linear change in groundwater level. 

These interpolated g values at station VLR2 were used to 

determine g values at relative-gravity stations. This was done 
Methods of Study 9

by making relative-gravity measurements at station VLR2 

and a relative-gravity base station and subtracting the value 

of these two measurements to obtain the difference in gravity 

between the two stations. Measuring this difference with the 

CG–5 gravimeter is referred to as a connecting survey because 

this allows all measurements made during a relative-gravity 

survey to be converted to g values. A connecting survey 

between the base station in a relative survey loop and the 

absolute-gravity station VLR2 was performed within 1 week 

of a relative-gravity survey. The field procedures during a 

connecting survey were conducted in the same manner as any 

other relative-gravity survey, except only two stations were 

measured. The difference in the gravitational field between 

the two stations in the connecting survey as measured by 

the CG–5 gravimeter also is the difference in g between the 

two stations. Therefore, this difference was subtracted from 

the interpolated g value at station VLR2 for the date of the 

connecting survey to obtain the g value at the relative-survey 

base station. For relative surveys occurring before or after the 

two absolute-gravity measurements at station VLR2 during 

July 2010 and June 2011, g values for station VLR2 were 

extrapolated using the slope of the interpolated line between 

the two measurements at station VLR2. 

Sources of Error

Gravitational attraction of the Sun and Moon cause 

harmonic deformation of the Earth’s crust, which is referred to 

as the Earth tide. These transient tidal effects, with amplitudes 

of 100 μGal or more, appear in gravity data. The resulting 

ground motion (lithospheric flexure) is assumed to be continuous over the entire expanse of a survey area, which allows 

all measurement stations to be corrected by a single temporal 

function that represents this tidal motion for each field survey. 

The CG–5 gravimeter uses internal Earth-tide modeling software to calculate the theoretical Earth tide and remove it from 

the recorded measurement. Modeled Earth-tide corrections 

calculated by the CG–5 gravimeter are accurate to less than 

±1 μGal and are a minimal source of error.

Other sources of measurement error include instrument 

drift and changes in barometric pressure, which may cause 

perturbations in gravity measurements that must be removed 

from the final data if the objective is to isolate the effects of 

groundwater-storage change (∆S). The dominant component of 

internal instrument drift for the CG–5 gravimeter is approximately linear in nature and may be monitored and quantified 

during periods between relative-gravity surveys while the 

gravimeter is stationary. Changes in barometric pressure cause 

atmospheric pressure loading of the CG–5 gravimeter, which 

has a small effect on measurements. Variations in barometric 

pressure cause changes of ±0.3 μGal per hectoPascal (Jacob 

and others, 2010). 

In addition to linear instrument drift in CG–5 gravimeter 

measurements, nonlinear instrument drift also exists, which 

may be induced by temperature variations in the gravimeter 

housing and tripod, voltage fluctuations, spring hysteresis, 

atmospheric pressure changes, and instrument transport 

(Torge, 1989). Sheltering the gravimeter and gravity stations 

prevented sun exposure from heating the outcrop, gravimeter, 

and tripod and causing tilts attributable to thermal expansion 

in the gravimeter housing and tripod. Wind induced error also 

was minimized by sheltering the gravimeter during measurements. Wind also may cause ground vibrations by causing 

trees to sway, which results in movement of the roots and 

surrounding ground. Therefore, gravity stations with trees in 

proximity were susceptible to interference from vibrations 

transmitted into the ground that were observed as an increase 

in standard deviation in the raw (unaveraged) data. The tripod 

was leveled with the front and right rear legs only so that 

the height of the left rear leg remained at a constant height 

throughout the survey. This eliminated error from instrument 

altitude changes, which are equal to about 7.5 μGal per inch of 

altitude change. Instrument height was recorded and compared 

at each base-station occupation to ensure consistent instrument height. Allowing the CG–5 gravimeter a short period to 

stabilize at the survey station after transportation considerably 

increased the repeatability of relative-gravity measurements. 

In the time between gravity surveys, the CG–5 gravimeter 

was stored in a quiet location, remained on power, and leveled 

on the tripod to minimize spring hysteresis effects that occur 

when the sensor is out of level for extended periods.

Modeling techniques to correct for nonlinear drift as 

described by Torge (1989) include the fitting of polynomial 

or spline curves to measurement data for a gravity survey. A 

method based on these techniques was applied to measurements made with the CG–5 gravimeter for this study and is 

described in the appendix. The uncertainty determined by 

repeat measurements made with the CG–5 gravimeter ranged 

from less than 1 μGal to 9 μGal, corresponding to an uncertainty in ΔS of 0.25 to 8.5 in. The total uncertainty between 

two measurements was determined by summing, in quadrature, the individual uncertainties in each measurement. On 

average, the corrected data allowed groundwater-storage volume to be quantified with an accuracy of about plus or minus 

0.5 ft of water per unit area of aquifer. Linear and nonlinear 

drift corrections were not necessary for the A–10 absolute 

gravimeter because of the absence of measurement drift in 

free-fall gravimeters and internal correction for Earth tides, 

ocean loading (lithospheric flexure caused by ocean tides), 

polar motion (precession of the polar axis), and changes in 

barometric pressure. 
10 Microgravity Methods for Characterization of the Karstic Madison Aquifer in the Black Hills of South Dakota

Characterization of GroundwaterStorage Changes and Aquifer 

Properties

This section presents the results of the microgravity measurements made during 2009–12 for this study. Quantitative 

and qualitative hydrologic analyses based on these measurements for the Spring Canyon, Doty, and Limestone Plateau 

focus areas are included in the section.

Spring Canyon Focus Area

Comparison of gravity data in the Spring Canyon focus 

area (fig. 1) collected in 2010 and 2011 at station TEU with 

water-level data for well RG–Mdsn and streamflow measurements for streamgage 06407500 indicate that groundwaterstorage change (ΔS) reflects temporal patterns in streamflow 

recharge to the Madison aquifer (fig. 3). Recharge from Spring 

Creek accounts for greater than 90 percent of the total recharge 

in this area (Long and Putnam, 2002). Groundwater storage 

at the Spring Canyon site between April 7, 2010, and September 25, 2011, varied by a maximum of 3.4 ft of water per unit 

area of aquifer. The groundwater-storage change determined 

from gravity measurements is similar to the cycle of seasonal 

changes in streamflow recharge but different from changes in 

hydraulic head at the well RG–Mdsn, which is downgradient 

from the recharge area (fig. 3). Change in hydraulic head at 

well RG–Mdsn responds to upgradient groundwater-storage 

change in the Madison aquifer beneath station TEU in the 

streamflow recharge area. Figure 3 indicates that seasonal 

groundwater-storage changes at the streamflow recharge 

area have a damped response in the hydraulic head at well 

RG–Mdsn. Groundwater-storage change determined from 

gravity measurements in the recharge area correspond to 

increasing flows in Spring Creek in March through May. The 

estimated maximum streamflow recharge rate of 21 ft3

/s for 

the Madison aquifer (Hortness and Driscoll, 1998) is shown 

in figure 3. Streamflow in excess of this rate is assumed to 

flow downstream and not sink into the Madison aquifer. 

Rapid increases in groundwater storage occurred just after the 

streamflow exceeded the estimated maximum recharge rate 

and decreased soon after streamflow decreased below this rate. 

The quick response of groundwater-storage change to increasing streamflow recharge in the Spring Canyon focus area is 

consistent with the presence of karst features, such as caverns 

and conduits, in this area because these features allow rapid 

and focused recharge. Conduits then allow the rapid transport 

of this streamflow recharge away from the recharge area (Putnam and Long, 2007), which allows the groundwater-storage 

to decline rapidly after streamflow recharge declines. Unsaturated zone storage is assumed to be small in this area because 

these large conduits allow rapid transport of water from the 

sinking stream downward to the water table. 

This rapid response of groundwater-storage change to 

streamflow recharge in the recharge area is not reflected by 

rapid hydraulic-head response at well RG–Mdsn (fig. 3). 

Hydraulic head measured in this well indicates a damped 

response compared with groundwater-storage change in the 

recharge area. This phenomenon is similar to quickly injecting 

water into a well (a slug test) and measuring the water level 

at the injection well and a proximal observation well. In this 

example, the observation well, which is analogous to well 

RG–Mdsn, will have a damped response in comparison to the 

injection well, which is analogous to the groundwater level 

at the stream sink. This difference in the hydraulic response 

between the recharge area and a downgradient area could not 

have been determined without gravity measurements at the 

recharge area because an observation well does not exist at 

that location. 

Doty Focus Area

All gravity stations in the Doty focus area are located 

where the Madison aquifer water table is assumed to be 

unconfined (fig. 2). Beneath the group 1 gravity stations, the 

unsaturated zone of the Madison aquifer exists above the 

water table. At stations MN05 and MN06, which are located 

on Minnelusa Formation outcrop but near the Madison Limestone outcrop (fig. 2), a small thickness of the unsaturated 

zone of the Minnelusa Formation also overlies the Madison 

aquifer but is assumed to contribute little to the total groundwater-storage change for these stations. The unsaturated zone 

beneath the group 2 gravity stations, all of which are located 

on the Minnelusa Formation outcrop (fig. 2), consists of the 

unsaturated parts of the Madison and Minnelusa aquifers. 

The calculation of effective porosity (φ) in equation 8 

applies when groundwater-storage change in the unsaturated 

zone is small and can be neglected, which occurs during 

periods when infiltration to the unsaturated zone is approximately equal to outflow. This equilibrium condition for the 

unsaturated zone is most likely to occur during the wettest part 

of the year because of frequent input of water, which percolates downward into the saturated part of the aquifer. This 

period begins after the start of the spring rains and ends before 

recharge decreases in mid to late summer. 

Groundwater flow in the unsaturated zone is analogous 

to a leaky, sand-filled bucket. As water is slowly added to the 

top, the sand becomes partially saturated as water percolates 

downward and leaks out the bottom. If inflow and outflow balance, storage in the partially saturated sand is at equilibrium. It 

is uncertain how closely this equilibrium state is approximated 

in the unsaturated zone of the aquifer during wet seasons 

(March–July), but a comparison of gravity data to hydraulic 

heads at the observation well provides insight. Periods of 

relatively steady rise in the hydraulic head at well Doty–Mdsn 

were observed during wet seasons; increases in groundwater-storage as determined by gravity measurements were 
Characterization of Groundwater-Storage Changes and Aquifer Properties 11

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0

50

100

150

200

250

300

350

400

Groundwater-storage change (

Δ

S) determined 

from gravity, in feet

Streamflow, in cubic feet per second

3,500

3,505

3,510

3,515

3,520

3,525

3,530

3,535

3,540

3,545

Hydraulic head, in feet above

NAVD 88

Hydraulic head for observation well RG-Mdsn, 

continuous measurements (data from South 

Dakota Department of Environment and 

Natural Resources, written commun., 2011)

Hydraulic head for observation well RG-Mdsn, 

manual measurements (data from South 

Dakota Department of Environment and 

Natural Resources, written commun., 2011)

EXPLANATION

Date

January 2010

February 2010

March 2010

April 2010

May 2010

June 2010

July 2010

August 2010

September 2010

October 2010

November 2010December 2010

January 2011

February 2011

March 2011

April 2011

May 2011

June 2011

August 2011

September 2011

October 2011

July 2011

Spring Creek flow at streamgage 06407500 

(U.S. Geological Survey, 2012)

Estimated maximum streamflow recharge rate

Groundwater-storage change for station TEU

Figure 3. Groundwater-storage change (ΔS) determined from gravity data at station TEU plotted with the hydrograph for observation 

well RG–Mdsn (PE–86A) and streamflow for streamgage 06407500 from January 2010 to October 2011.

proportional to these hydraulic head increases for group 2 

gravity stations (figs. 2 and 4). 

When recharge subsides, water in the unsaturated 

zone continues to percolate downward, and a decline in 

groundwater-storage results. During the dry seasons (August–

February), total groundwater-storage (saturated and unsaturated zones) declines by a larger amount per foot of waterlevel (hydraulic-head) change than the groundwater-storage 

increases during wet seasons because unsaturated-zone storage 

is declining during the dry seasons. Therefore, an equilibrium 

does not exist because inflow to the unsaturated zone is less 

than outflow unless storage in the unsaturated zone is zero, 

which is unlikely. This is illustrated conceptually in figure 5, 

where ∆S declines more quickly during periods of declining 

water levels than ∆S increases during periods of increasing water levels because of the decrease in unsaturated-zone 

storage. A near-equilibrium state is assumed to occur during 

periods of increasing water levels when the slope of the waterlevel change is nearly constant (inflow equals outflow; fig. 5). 

During periods of declining water levels or when the waterlevel change is not constant, unsaturated-zone storage is not at 

a steady state (inflow less than or greater than outflow; fig. 5). 

This assessment is consistent with the groundwaterstorage change at station MN04 (fig. 4) for October 18, 2010, 

that was low in relation to the water level when compared 

with the groundwater-storage change at station MN04 for 
12 Microgravity Methods for Characterization of the Karstic Madison Aquifer in the Black Hills of South Dakota

Figure 4. Groundwater-storage change (∆S) determined from gravity data for group 2 stations (MN01, MN03, MN04, and RCPC–4) 

in the Doty focus area plotted with the hydrograph for observation well Doty–Mdsn (PE–95C). 

Hydraulic head for observation well Doty-Mdsn, 

continuous measurements (data from South 

Dakota Department of Environment and 

Natural Resources, written commun., 2011)

Hydraulic head for observation well Doty-Mdsn, 

manual measurements (data from South 

Dakota Department of Environment and 

Natural Resources, written commun., 2011)

April 2010

May 2010

June 2010

July 2010

August 2010

September 2010

October 2010

November 2010

December 2010

January 2011

February 2011

March 2011

April 2011

May 2011

June 2011

July 2011

August 2011

September 2011

October 2011

-2.0

-2.5

-1.5

-0.5

-1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3,846

3,847

3,848

3,849

3,850

3,851

3,852

3,853

3,854

3,855

3,856

3,857

3,858

3,859

3,860

3,861

3,862

3,863

3,864

EXPLANATION

Groundwater-storage change (

Δ

S) determined from gravity, in feet

Period 1

Period 2 Period 3

0.12

0.07

0.05

0.02

0.43

0.16

July 15

October 18

RCPC-4

MN03

MN01

MN04

Well level

Hydraulic head, in feet above NAVD 88

Date

Groundwater-storage change at station MN01

Groundwater-storage change at station MN03

Groundwater-storage change at station MN04

Groundwater-storage change at station RCPC-4

Estimates of effective porosity for measurement 

periods 1, 2, and 3

0.02

July 15, 2010, in relation to the water level at that time. Therefore, the groundwater-storage change for October 18, 2010, 

probably resulted from a decrease in unsaturated-zone storage 

prior to this measurement.

Therefore, periods of steady increase in groundwater 

levels were assumed to provide the best estimates of effective 

porosity. Specifically, periods 1, 2, and 3 (fig. 4) are assumed 

to provide the best estimates of effective porosity as calculated 

by equation 8 because of the steady increase in water level 

(hydraulic head) for well Doty–Mdsn during these periods. 

Period 2 may provide the best estimate because it occurs 

well after the onset of spring recharge and before recharge 

begins to subside, which is presumably the most likely time 

for steady-state conditions in the unsaturated zone. The 

estimated effective porosity values for the Madison aquifer 

beneath station MN04 are 0.16, 0.02, and 0.43 for periods 1, 

2, and 3, respectively. Errors in effective porosity estimations 

were evaluated using the maximum uncertainty in g values 

determined from field surveys with a minimum error no less 

than the published ±5 μGal accuracy of the CG–5 gravimeter 

and an uncertainty of ±0.5 ft for water-level observations for 

corresponding relative-gravity surveys. These errors represent 

the maximum possible uncertainty in measurements and were 

propagated using the techniques described by Taylor (1982). 

On the basis of this error analysis, possible ranges for these 

effective porosity estimates are 0.10 to 0.24, greater than (>) 0 

to 0.16, and 0.27 to 0.59 for periods 1, 2, and 3, respectively. 

Differences in these estimates indicate vertical heterogeneity 
Characterization of Groundwater-Storage Changes and Aquifer Properties 13

because the water-table changes for these three periods 

occurred at different altitudes. Period 1 also was assumed to 

provide useful estimates of effective porosity for gravity stations MN01, MN03, and RCPC–4 (fig. 4) with values from 

> 0 to 0.11, > 0 to 0.14, and 0.06 to 0.18, respectively. These 

values are complementary to the estimate of average effective 

porosity from Greene (1993) of 0.35 for the Madison aquifer. 

The estimate in Greene (1993) was made using resistivity logs 

from Madison aquifer wells over a 600-ft thick section of the 

Madison aquifer. 

Average effective porosity values estimated for the 

Madison aquifer beneath station MN04 for the three periods 

correspond to three overlapping vertical sections (fig. 6). 

The altitude of these vertical sections at station MN04 were 

determined using potentiometric surfaces estimated by Strobel 

and others (2000) to determine water levels at station MN04 

that correspond to water levels at well Doty–Mdsn for the 

three periods. Estimates represent average effective porosity values for each vertical section, but heterogeneity could 

exist within sections. If estimates from these three periods are 

combined, then additional information related to the vertical 

heterogeneity of effective porosity is obtained. Combining the 

three periods results in new information for four subsections 

derived from the original three overlapping sections (fig. 6; 

sections a–d). If the estimate of 0.02 for period 2 (section b) is 

combined with the estimate of 0.16 for period 1 (average for 

sections a–c), then the average effective porosity of sections a 

and c is 0.27 based on an average calculation weighted by the 

vertical length of each subsection. Additional measurements 

corresponding to other overlapping vertical sections would 

provide a basis on which to determine the effective porosity Increasing 

Δh and 

Δ

S

Inflow = outflow

Inflow < outflow

Inflow > outflow

Inflow = outflow

ΔS

Δh

Wet season Dry season Wet season

Δ

S = Saturated and unsaturated 

zone storage

Unsaturated zone

water balance

EXPLANATION

> Greater than

< Less than

= Equal to December January February

March

April

May

June

July

August

September

October

November

December

January

February

March

April

May

June

July

August

September

October

Figure 5. Conceptual plot of 

change in water-table altitude 

(Δh) and groundwater-storage 

change (ΔS) with divisions 

where the balance of inflow to 

outflow in the unsaturated zone 

changes.

Figure 6. Effective porosity estimates for Madison aquifer 

beneath station MN04 for vertical sections a–d determined 

for three different time periods. Altitude, in feet above NAVD 88

4,092

4,091

4,090

4,089

4,088

4,087

4,086

4,085

4,084

4,083

4,082

1 2 3

d

0.43

c

b 0.16

a

0.02

Vertical

section

Period corresponding to figure 4
14 Microgravity Methods for Characterization of the Karstic Madison Aquifer in the Black Hills of South Dakota

for additional vertical subsections at this location, possibly 

isolating zones of very high porosity that might be interpreted 

as solution openings in the Madison aquifer.

For group 1 gravity stations (fig. 2), a large decrease in 

groundwater storage occurred during period 1, a time during which the hydraulic head in well Doty–Mdsn was rising 

(fig. 7). Although the groundwater-storage decline is opposite 

to what was expected for the spring recharge period, data for 

all five of these gravity stations were consistent, indicating 

that measurements probably were not in error. This storage 

decline might be explained by possible outflow of perched 

groundwater or the downgradient cascading of groundwater from cave to cave as described in the “Hydrogeology of 

the Study Area” section. As an example, an event similar to 

this occurred in the southern Black Hills in 1996 in Wind 

Cave, which is contained in the Madison Limestone. During 

this event, a cave lake, which previously was dry for many 

years, appeared in one of the passages within a timeframe of 

1 month and presumably was the result of the rapid emptying 

of perched water from a proximal cave opening at a higher 

altitude (Long and others, 2012). Because of its sudden 

appearance, the lake was named What the Hell Lake. Another 

example is illustrated by the water-level record for a lake in 

Brooks Cave located west of Rapid City and contained in the 

Madison Limestone; the water level rose during spring and 

remained relatively constant at an altitude of about 3,464 ft 

until declining during each fall season from 1988 until 1992 

(Driscoll and others, 2000; fig. 8). This altitude is interpreted 

as the altitude at which water from this cave lake spilled out 

into the surrounding aquifer because of an outlet that existed 

Period 1

MD03

MN06

MD01

MN05

MD02

Groundwater-storage change (

Δ

S) determined from gravity, in feet

Hydraulic head, in feet above NAVD 88

Date

April 2010

May 2010

June 2010

July 2010

August 2010

September 2010

October 2010

November 2010

December 2010

January 2011

February 2011

March 2011

April 2011

May 2011

June 2011

July 2011

August 2011

September 2011

October 2011 -5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

3,847

3,846

3,848

3,849

3,850

3,851

3,852

3,853

3,855

3,854

3,856

3,857

3,858

3,859

3,860

3,861

3,862

3,864

3,863

Well level

Hydraulic head for observation well Doty-Mdsn,

continuous measurements (data from South 

Dakota Department of Environment and 

Natural Resources, written commun., 2011)

Hydraulic head for observation well Doty-Mdsn,

manual measurements (data from South 

Dakota Department of Environment and 

Natural Resources, written commun., 2011)

EXPLANATION

Groundwater-storage change at station MD01

Groundwater-storage change at station MD02

Groundwater-storage change at station MD03

Groundwater-storage change at station MN05

Groundwater-storage change at station MN06

Figure 7. Groundwater-storage change (∆S) determined from gravity data for group 1 stations (MD01, MD02, MD03, MN05, and 

MN06) in the Doty focus area plotted with the hydrograph for observation well Doty–Mdsn (PE–95C).
Characterization of Groundwater-Storage Changes and Aquifer Properties 15

at this altitude. Also, water probably seeped out continuously, 

possibly through clay sediments that filled other outlets, which 

resulted in water-level declines during the fall seasons when 

recharge declined. After 1992, this annual cycle changed and 

is characterized by multiple rapid declines annually, possibly resulting from periodic outflows through outlets that 

were intermittently filled with clay sediments or other debris. 

Increases in precipitation rates during the early 1990s (Driscoll 

and others, 2000) might have resulted in changes to the 

hydraulics controlling outflow from this lake.

Rapid emptying of groundwater from a cave or network 

of connected caves could result in a total groundwater-storage 

change that is dominated by the storage decline in these caves. 

For example, if the water level in a cave declined by 10 ft, 

then ∆S at this location would be -10 ft because the effective 

porosity is 1.0 inside the cave, whereas if the water table 

increased by 10 ft in an adjacent area where the effective 

porosity is 0.1, then ∆S would be only +1 ft. An event such as 

this could explain the large decline in total groundwater storage during period 1 for group 1 gravity stations (figs. 2 and 7). 

Limestone Plateau Focus Area

Measurements of g at absolute-gravity station O27, 

which is at the highest altitude of any of the gravity stations 

in this study, were made annually in the summers of 2009–11 

and the spring of 2012 using the A–10 gravimeter. The change 

in groundwater-storage determined from gravity during these 

periods was similar to the change in the water level in well 

Chyn–Xing–Mdsn (fig. 9). Little groundwater-storage or 

water-level change occurred between September 2009 and 

July 2010 (fig. 9). Following the same procedure as in the 

Doty focus area for periods of increasing groundwater levels 

(hydraulic head), the effective porosity was estimated by using 

equation 8 for the period between July 2010 and June 2011, 

which resulted in an estimated effective porosity ranging from 

0.01 to 0.03 in the Madison aquifer beneath gravity station 

O27 for the vertical section in which the water level increased 

during that period. This estimate assumes that groundwaterstorage change in the unsaturated zone was small. Karst aquifers commonly are highly heterogeneous, and much higher 

values of effective porosity are expected to occur elsewhere in 

the Limestone Plateau area. 

Figure 8. Water-level hydrograph for lake in Brooks Cave from 1988 to 1998 (modified from Driscoll and others, 2000).

10

12

14

16

18

20

22

24

1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997

Water level, in feet below reference point

Year
16 Microgravity Methods for Characterization of the Karstic Madison Aquifer in the Black Hills of South Dakota

Limitations of Microgravity Methods

Microgravity methods have advantages over traditional 

groundwater-monitoring techniques because measurements are 

made without the need for invasive and time consuming measures to drill a well into the formation in question. Microgravity also allows these storage-change measurements to be made 

at nearly any location over the aquifer. However, the ability to 

measure groundwater-storage change indirectly with microgravity does have limitations. As described in the “Methods 

for Determining Groundwater-Storage Change and Effective 

Porosity” section of this report, the microgravity methods 

used in this investigation do not supply information about 

the altitude beneath the gravity station where groundwaterstorage change is occurring; microgravity data only provide 

estimates of total groundwater-storage change at a particular 

location. Additionally, microgravity data alone do not allow 

for estimation of aquifer properties. In order to make estimates 

of effective porosity, microgravity data must be paired with 

other hydrologic data such as hydraulic-head measurements in 

observation wells. As described in the “Spring Canyon Focus 

Area” section of this report, the corresponding hydraulic-head 

measurements may not reflect the nature of the groundwater 

flow beneath a gravity station and upgradient from the 

observation well. This is especially true in areas where karst 

topography exists. 

Another factor that limits the certainty of gravity 

measurements is the stability of absolute-gravity reference 

stations, which are assumed to be in areas of little to no 

groundwater-storage change. In this study, g was measured at 

absolute-gravity station VLR2 three times during the study to 

track the stability of the reference station. These measurements 

showed a linear increase in g at station VLR2. This change 

is likely a result of station VLR2 being located on a ridge 

top, which makes the measurement sensitive to the nearby 

region downhill from the station altitude. Therefore, changes 

in surface-water volume in the adjacent valleys may lead to 

changes in g at station VLR2. This change was addressed by 

linearly interpolating between the two measurements at station VLR2 for each relative-gravity connecting survey. The 

third measurement was made in the spring of 2012 after the 

data analysis was complete and therefore was not included in Groundwater-storage change (

Δ

S) determined from gravity, in feet

July 2009

August 2009

September 2009

October 2009

November 2009

December 2009

January 2010

February 2010

March 2010

April 2010

May 2010

June 2010

July 2010

August 2010

September 2010

October 2010

November 2010

December 2010

January 2011

February 2011

March 2011

April 2011

May 2011

June 2011

July 2011

August 2011

September 2011

October 2011

November 2011

December 2011

January 2012

February 2012

March 2012

April 2012

May 2012

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

5,800

5,805

5,810

5,815

5,820

5,825

5,830

5,835

5,840

EXPLANATION

Hydraulic head at Chyn-Xing-Mdsn well, 

manual measurements (data from South 

Dakota Department of Environment and 

Natural Resources, written commun., 2011)

Groundwater-storage change at station O27

Date

Hydraulic head, in feet above NAVD 88

Figure 9. Groundwater-storage change (∆S) determined from gravity data for station O27 plotted with the hydrograph for 

observation well Chyn–Xing–Mdsn (LA–95B).
Summary 17

the final data; however, the measurement confirmed the linear 

increase in gravity at station VLR2. A new absolute-gravity 

station on the Harney Peak Granite (WR1; fig. 1) has been 

established for future gravity studies in the Black Hills region 

to address the apparent instability at station VLR2. The new 

station (WR1) is located in an area that is geologically similar 

to the area for absolute-gravity station HP2, and therefore, 

little change in g from groundwater-storage fluctuations is 

expected at station WR1. The A–10 absolute gravimeter will 

be used to verify that groundwater-storage change at this station is zero or negligible.

Summary

Microgravity methods were used to estimate 

groundwater-storage change in the karstic Madison aquifer 

in the Black Hills of South Dakota for a study conducted by 

the U.S. Geological Survey in cooperation with West Dakota 

Water Development District, South Dakota Department of 

Environment and Natural Resources, and Lawrence County. 

The microgravity data obtained from 2009 to 2012 were used 

to investigate groundwater-storage changes and effective 

porosity in unconfined areas of the Madison aquifer. Groundwater-storage change determined from microgravity measurements is expressed as the change in the vertical height of water 

with no rock present. Time-lapse microgravity surveys that 

used portable high-sensitivity absolute and relative gravimeters indicated temporal-gravity changes as a result of changing 

groundwater mass. These extremely precise measurements 

of gravity required characterization and removal of internal 

instrumental and external environmental effects on gravity 

from the raw data. The corrected data allowed groundwaterstorage volume to be quantified with an accuracy of about plus 

or minus 0.5 foot of water per unit area of aquifer. Quantification of groundwater-storage change, coupled with water-level 

data from observation wells located near three focus areas, 

also was used to calculate the effective porosity at specific 

altitudes directly beneath gravity stations. These estimates 

demonstrate the utility of absolute- and relative-gravity measurements in groundwater-storage monitoring applications.

Gravity stations were established on bedrock outcrops 

in three separate focus areas for this study. The first area, the 

Spring Canyon focus area, is located south of Rapid City with 

one gravity station on the rim of Spring Canyon near the area 

where Spring Creek sinks into the Madison aquifer. A U.S. 

Geological Survey streamgage located upstream from this 

location allowed for measurements of streamflow over the 

recharge area. The second area, the Doty focus area, is located 

on outcrops of the Madison Limestone and Minnelusa Formation to the northwest of Rapid City, and consists of nine gravity stations. These stations were located on the land surface 

above the water table of the unconfined Madison aquifer, in an 

area that included part of the unconfined Minnelusa aquifer. It 

was assumed that most of the measured groundwater-storage 

change occurred at or near the water table and therefore primarily in the Madison aquifer. The third area, the Limestone 

Plateau focus area, consists of a single gravity station located 

in the northwestern Black Hills on an outcrop of the Madison 

Limestone. This study area is located at the highest altitude of 

the three study areas, and recharge likely occurs as a result of 

direct infiltration of precipitation. An absolute-gravity station, 

used to tie relative-gravity survey data together, was established on a relatively impermeable bedrock outcrop to minimize groundwater-storage change at the reference location. 

Data from the three focus areas allowed for interpretation of groundwater-storage characteristics using microgravity 

measurements. Results indicated that rapid groundwaterstorage change occurred in an unconfined area near a sinking 

stream in Spring Canyon as a result of aquifer recharge from 

the stream. This rapid groundwater-storage change likely was 

a result of groundwater movement through caverns, conduits, 

and fractures, which are common in karst aquifers. Estimates 

of effective porosity were not possible at this location because 

the response to recharge at the observation well was different 

than that of the Spring Canyon gravity station. Groundwaterstorage change at the Spring Canyon site between April 7, 

2010, and September 25, 2011, varied by a total of 3.4 feet of 

water per unit area of aquifer.

Spatially and temporally separated microgravity data for 

the Doty focus area indicated horizontal and vertical heterogeneity of effective porosity. Water-level measurements at an 

observation well and gravity measurements at four gravity 

stations in the southeastern part of the Doty area (group 2) 

were used to estimate effective porosity values ranging from 

greater than 0 to 0.18. Repeat measurements at one station in 

this group (MN04) during periods when unsaturated-zone storage was assumed to be zero allowed for estimations of effective porosity in three vertical sections at different altitudes. 

These measurements were combined to further isolate vertical 

differences of effective porosity, which ranged from greater 

than 0 to 0.59. A decrease in groundwater storage determined 

by microgravity measurements during the spring recharge 

period for five upgradient stations in the northwestern part of 

the Doty focus area (group 1) indicated the possibility of a 

rapid release and downgradient cascading of perched groundwater. Evidence for similar phenomena was documented for 

Wind and Brooks Caves (both of which are contained in the 

Madison Limestone) in the Black Hills. 

Absolute-gravity measurements in the Limestone Plateau 

focus area confirmed the relation between water levels in 

observation wells and groundwater-storage change. Comparison of groundwater-storage change determined from gravity 

measurements with water levels in a nearby observation well 

resulted in an effective porosity estimate ranging from greater 

than 0 to 0.03 for the Madison aquifer beneath the gravity 

station. 

The microgravity method offers advantages over traditional groundwater-monitoring techniques because it is 

noninvasive and allows for measurements at nearly any location over the unconfined aquifer. Microgravity measurements 
18 Microgravity Methods for Characterization of the Karstic Madison Aquifer in the Black Hills of South Dakota

are particularly informative when combined with additional 

hydrologic data, such as hydraulic-head measurements in 

observation wells, which in certain cases allows for estimation 

of effective porosity in the underlying aquifer. The stability 

of absolute-gravity stations is an important factor limiting the 

quality of microgravity data, and therefore, it is essential to 

choose a suitable site for an absolute-gravity reference station 

and to monitor the stability of the accelerative force of gravity

at that location.

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(Also available at http://pubs.usgs.gov/ha/ha745d/.) 

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20 Microgravity Methods for Characterization of the Karstic Madison Aquifer in the Black Hills of South Dakota

Appendix

Relative-gravity data collected continuously at the 

U.S. Geological Survey (USGS) South Dakota Water Science 

Center in Rapid City, South Dakota, on September 19, 2011, 

indicated that small, nonlinear internal drift (a combined effect 

of spring hysteresis and small discontinuities in the measurement function), with an amplitude ranging from approximately 

3 to 25 microgals (μGal), is still present in temporal relativegravity data after removal of the theoretically modeled Earth 

tides; that is, the nonlinear drift must be removed from the 

data. The magnitude of drift varies from day-to-day depending 

on handling and transport conditions and can only be determined empirically. An example of this effect is illustrated in 

figure 1–1. In this figure, continuous 117-second interval relative-gravity data are averaged into an aggregate of 18-minute 

intervals, which reveals residual perturbations with amplitudes 

of about 3 μGal (0.003 milligals). The averaged 18-minute 

interval data resembles the time interval between field survey 

measurements and establishes a baseline for the nonlinear 

stationary drift. A sixth-order polynomial was fit to and plotted 

with these data and may be used to remove the nonlinear drift 

from this dataset (Montgomery and Runger, 2007).

All of the field data collected during a survey were used 

collectively to estimate the nonlinear drift. After the averaged 

relative-gravity value for each station and each occupation 

was determined, a single polynomial curve was fit to the timeseries data for each station. This curve represents a combination of internal instrument drift and variable atmospheric 

loading during a survey. To ensure a polynomial fit that is 

constrained by observed data, the order of the polynomial 

must be smaller than the number of data points. For this study, 

the order of the polynomial was limited to 60 percent or less 

of the number of data points, with a maximum polynomial 

order of six. The order of the polynomial fit for a survey loop 

is determined from:

Op ≤ 0.6*n (A1)

and

1 ≤ Op ≤ 6 (A2)

where 

Op is the order of the polynomial, and

n is the number of averaged relative-gravity 

measurements.

For example, a survey loop conducted on June 9, 2011, 

consisted of a base station and three relative-gravity stations. 

The survey loop was repeated three times for a total of 12 data 

points for curve fitting. The averaged relative-gravity values 

at each gravity station (MN04, MN05, MN06, and MD01) 

are plotted in figure 1–2. Although the 12 measurements 

obtained during this survey allow for the use of a sixth-order 

polynomial, an acceptable fit was achieved using a fifth-order 

polynomial, which is shown on figure 1–2. The curve fit for 

each station was allowed to have a different y-intercept of 

the polynomial to account for different accelerative force of 

gravity (g) values between stations. Figure 1–2 (all stations) 

shows relative-gravity data for all of the gravity stations in the 

survey loop shifted to the base station y-intercept and plotted 

with the fitted fifth-order polynomial, which clearly shows the 

character of the nonlinear drift throughout the survey period. 

It is important to note that the amplitude of the nonlinear drift 

is larger in the field survey compared to the survey conducted 

at the USGS South Dakota Water Science Center because the 

gravimeter was handled during the field survey, thus contributing the added amplitude of nonlinear transport drift to the 

fitted function.

The fitted nonlinear drift function was removed from the 

relative-gravity data by interpolating the amount of drift based 

on the time of the individual relative-gravity measurements. 

The drift corrected measurement was obtained by subtracting 

the interpolated value from the field measurement. The result 

is a linear function of relative-gravity measurements. The 

deviation from the mean of the drift corrected relative-gravity 

data determines the uncertainty in the final gravity values for 

each survey.
Appendix 21

19:00 19:15 19:30 19:45 20:00 20:15 20:30 20:45 21:00 21:15 21:30 21:45 22:00 22:15 22:30 22:45

5,204.396

5,204.400

5,204.401

5,204.402

5,204.397

5,204.398

5,204.399

Relative-gravity values at survey station, in milligals

Time (hour:minute)

EXPLANATION

Relative-gravity measurements at the U.S. Geological 

Survey South Dakota Water Science Center

Sixth-order polynomial best-fit line

Figure 1–1. Relative-gravity measurements made at the U.S. Geological Survey South Dakota Water Science Center in Rapid City, 

South Dakota, on September 19, 2011, plotted with a fitted sixth-order polynomial.
22 Microgravity Methods for Characterization of the Karstic Madison Aquifer in the Black Hills of South Dakota

8:15 8:30 8:45 9:00 9:15 9:30 9:45 10:00 10:15 10:30 10:45 11:00 11:15 11:30 11:45

Relative-gravity values at survey station, in milligals

Time (hour:minute)

E. All stations

D. MD01

C. MN06

B. MN05

A. MN04

5,073.605

5,073.600

5,073.595

5,073.590

5,073.585

5,073.580

5,069.655

5,069.660

5,069.665

5,069.670

5,069.675

5,069.680

5,073.199

5,073.204

5,073.209

5,073.214

5,073.219

5,073.224

5,071.150

5,071.145

5,071.140

5,071.135

5,071.130

5,071.125

5,073.605

5,073.600

5,073.595

5,073.590

5,073.585

5,073.580

EXPLANATION

Fifth-order polynomial 

best-fit line

Relative-gravity 

measurement at:

Station MN04

Station MN05

Station MN06

Station MD01

Figure 1–2. Relative-gravity measurements for a survey conducted on June 9, 2011, at stations MN04, MN05, MN06, and MD01 

(A–D, respectively) plotted with a fitted fifth-order polynomial. Plot E shows all gravity measurements on June 9, 2011, shifted to the 

base station (MN04) intercept and plotted with the fitted fifth-order polynomial.
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For more information concerning this publication, contact: 

Director, USGS South Dakota Water Science Center 

1608 Mt. View Road, Rapid City, SD 57702 

(605) 394-3200

Or visit the South Dakota Water Science Center Web site at: 

http://sd.water.usgs.gov/
Koth and Long—Microgravity Methods for Characterization of the Karstic Madison Aquifer in the Black Hills of South Dakota—SIR 2012–5158

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