U.S. Department of the Interior

U.S. Geological Survey 

Factors Related to Well Yield in 

the Fractured-Bedrock Aquifer of 

New Hampshire 

By Richard Bridge Moore, Gregory E. Schwarz, Stewart F. Clark, Jr., 

Gregory J. Walsh, and James R. Degnan 

U.S. Geological Survey Professional Paper 1660 

In cooperation with the 

New Hampshire Department of Environmental Services 
2002 

U.S. Department of the Interior

Gale A. Norton, Secretary 

U.S. Geological Survey

Charles G. Groat, Director 

The use of firm, trade, and brand names in this report is for identification purposes 

only and does not constitute endorsement by the U.S. Government. 

For sale by 

U.S. Geological Survey 

Information Services 

Box 25286, Federal Center 

Denver, CO 80225 

Library of Congress Cataloging-in-Publication Data 

Factors related to well yield in the fractured-bedrock aquifer of New Hampshire / by 

Richard Bridge Moore ... [et.al.]. 

p. cm. -- (U.S. Geological Survey professional paper ; 1660)


Includes bibliographical references.


ISBN 0-607-98453-8 (alk. paper)


1. Groundwater--New Hampshire. 2. Aquifers--New Hampshire. 3. Wells--New


Hampshire. I. Moore, Richard Bridge. II. Series.


GB1025.N45 F23 2002 

553.7'9'09742--dc21 2002020252 
CONTENTS 

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


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


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


Previous Investigations ................................................................................................................................................ 3


Acknowledgments........................................................................................................................................................ 4


Hydrogeologic Setting ........................................................................................................................................................... 4


Methods and Approach .......................................................................................................................................................... 5


Descriptions and Sources of Data................................................................................................................................ 6


Well Information................................................................................................................................................ 6


Bedrock Lithology ............................................................................................................................................. 7


Lineaments......................................................................................................................................................... 7


Surficial Material ............................................................................................................................................... 10


Topographic Settings and Characteristics.......................................................................................................... 10


Additional Variables Investigated at the Quadrangle Scale ............................................................................... 11


Creation of a Verification Data Set .............................................................................................................................. 16


Development of the Statewide Regression Model ....................................................................................................... 16


Factors Related to Well Yield ................................................................................................................................................ 19


Regression Model Equation......................................................................................................................................... 19


Regression Model Testing............................................................................................................................................ 22


Yield-Probability Forecasting ...................................................................................................................................... 23


Maps of Yield Probability............................................................................................................................................ 23


Comparison of Regional Variables with Statewide Variables...................................................................................... 27


Summary and Conclusions .................................................................................................................................................... 28


Selected References ............................................................................................................................................................... 31


Appendix A. Regression Model Construction ...................................................................................................................... 36


Appendix B. List of Variables Tested for the Regression Model ......................................................................................... 38


Reference Cited............................................................................................................................................................ 44


Appendix C. Regression Model Tests................................................................................................................................... 46


Test for an Endogenous Variable ................................................................................................................................. 46


Specification Test......................................................................................................................................................... 46


Residual Statistical Distribution .................................................................................................................................. 47


Residual Spatial Distribution and Spatial Analysis ..................................................................................................... 48


Comparison of Verification and Primary Data Sets..................................................................................................... 48


Finite Bias Test ............................................................................................................................................................ 50


Overall Statement ........................................................................................................................................................ 50


References Cited .......................................................................................................................................................... 51


FIGURES 

1.	 Map showing location of the Pinardville and Windham 1:24,000-scale quadrangles in southern 


New Hampshire where additional geohydrologic, fracture, and lineament data were collected ............................ 6


2. Graph showing percent of statewide bedrock well yields in New Hampshire............................................................ 7


3-6. Maps showing:


3. Major lithologic units in New Hampshire.......................................................................................................... 8


4. Simplified bedrock geologic map of the Pinardville quadrangle, New Hampshire ........................................... 13


5. Simplified bedrock geologic map of the Windham quadrangle, New Hampshire............................................. 14


6. Fracture-correlated lineaments in the Pinardville and Windham quadrangles, New Hampshire ...................... 15


Contents III
7.	 Graph showing use of the instrumental variable “YEAR,” and a two-stage regression model to separate 

demand-driven relations from physical relations between well depth (lnDEPTH) and yield (lnYIELD) 

used in this study of New Hampshire...................................................................................................................... 17 

8-11. Maps showing: 

8. Median household income for New Hampshire using the 1990 U.S. Bureau of Census data........................... 18


9.	 Model results of the probability of obtaining 40 gallons of water per minute from a 400-foot-deep well 

in New Hampshire ......................................................................................................................................... 24 

10-11.	 Model results of the probability of obtaining 40 gallons of water per minute or more from a 400-foot-deep 

well for the: 

10. Pinardville quadrangle, New Hampshire ........................................................................................................... 25


11. Windham quadrangle, New Hampshire ............................................................................................................. 26 

C1. Graph showing distribution of model residuals for New Hampshire shown as a (A) bell curve and 

(B) cumulative curve ............................................................................................................................................... 47


C2.	 Correlogram showing the spatial correlation of model residuals that have a low-resolution binning 

(100 cells across) for the Pinardville quadrangle, New Hampshire ........................................................................ 49 

PLATES 

1-2.	 Maps showing well-yield probabilities in fractured bedrock of the: 

1. Pinardville quadrangle, New Hampshire 

2. Windham quadrangle, New Hampshire 

TABLES 

1. Summary of well yield and lineament investigations considered for this study ..................................................... 3


2. Regression equation used by this study................................................................................................................... 20


3-4.	 Percentage of area identified as having a probability of obtaining 40 gallons per minute or more from a 


400-foot-well depth with detailed geologic mapping variables included and excluded for the:


3. Pinardville quadrangle, New Hampshire ........................................................................................................ 28


4. Windham quadrangle, New Hampshire .......................................................................................................... 28 

C1. Significant predictors in the specification test of the model used in this study ...................................................... 47 

C2. Numerical correlations of the model residuals for the Pinardville quadrangle, New Hampshire, for distance 

separations less than 1 kilometer ........................................................................................................................ 49


 CONVERSION FACTORS, ABBREVIATIONS, AND VERTICAL DATUM 

Multiply By To obtain 

foot (ft) 0.3048 meter (m) 

mile (mi) 1.609 kilometer 

square mile (mi2) 2.590 square kilometer 

gallons per minute (gal/min) 0.06309 liter per second 

gallons per minute per foot (gal/min/ft) 0.2070 liter per second per meter 

Vertical Datum: In this report “sea level” refers to the National Geodetic Vertical Datum of 1929 

(NGVD of 1929)—a geodetic datum derived from a general adjustment of the first-order level nets of 

both the United States and Canada, formerly called Sea Level Datum of 1929. 

IV Contents 
Factors Related to Well Yield in the 

Fractured-Bedrock Aquifer of New Hampshire 

By Richard Bridge Moore, Gregory E. Schwarz, Stewart F. Clark, Jr., 

Gregory J. Walsh, and James R. Degnan 

Abstract 

The New Hampshire Bedrock Aquifer 

Assessment was designed to provide information 

that can be used by communities, industry, professional consultants, and other interests to evaluate 

the ground-water development potential of the 

fractured-bedrock aquifer in the State. The assessment was done at statewide, regional, and well 

field scales to identify relations that potentially 

could increase the success in locating high-yield 

water supplies in the fractured-bedrock aquifer. 

statewide, data were collected for well construction and yield information, bedrock lithology, 

surficial geology, lineaments, topography, and 

various derivatives of these basic data sets. 

Regionally, geologic, fracture, and lineament data 

were collected for the Pinardville and Windham 

quadrangles in New Hampshire. The regional 

scale of the study examined the degree to which 

predictive well-yield relations, developed as part 

of the statewide reconnaissance investigation, 

could be improved by use of quadrangle-scale 

geologic mapping. 

Beginning in 1984, water-well contractors 

in the State were required to report detailed 

information on newly constructed wells to the 

New Hampshire Department of Environmental 

Services (NHDES). The reports contain basic 

data on well construction, including six characteristics used in this study—well yield, well depth, 

well use, method of construction, date drilled, and 

depth to bedrock (or length of casing). The 

NHDES has determined accurate georeferenced 

locations for more than 20,000 wells reported 

since 1984. The availability of this large data set 

provided an opportunity for a statistical analysis 

of bedrock-well yields. Well yields in the 

database ranged from zero to greater than 

500 gallons per minute (gal/min). 

Multivariate regression was used as the 

primary statistical method of analysis because it is 

the most efficient tool for predicting a single 

variable with many potentially independent 

variables. The dependent variable that was 

explored in this study was the natural logarithm 

(ln) of the reported well yield. One complication 

with using well yield as a dependent variable is 

that yield also is a function of demand. An 

innovative statistical technique that involves the 

use of instrumental variables was implemented to 

compensate for the effect of demand on well 

yield. 

Results of the multivariate-regression 

model show that a variety of factors are either 

positively or negatively related to well yields. 

Using instrumental variables, well depth is 

positively related to total well yield. Other factors 

that were found to be positively related to well 

yield include (1) distance to the nearest 

waterbody; (2) size of the drainage area 

upgradient of a well; (3) well location in swales or 

valley bottoms in the Massabesic Gneiss 

Complex and Breakfast Hill Granite; (4) well 

proximity to lineaments, identified using highaltitude (1:80,000-scale) aerial photography, 

which are correlated with the primary fracture 

direction (regional analysis); (5) use of a cable 

tool rig for well drilling; and (6) wells drilled for 

commercial or public supply. Factors negatively 

Abstract 1
related to well yields include sites underlain by 

foliated plutons, sites on steep slopes sites at high 

elevations, and sites on hilltops. Additionally, 

seven detailed geologic map units, identified 

during the detailed geologic mapping of the 

Pinardville and Windham quadrangles, were 

found to be positively or negatively related to 

well yields. Twenty-four geologic map units, 

depicted on the Bedrock Geologic Map of New 

Hampshire, also were found to be positively or 

negatively related to well yields. 

Maps or geographic information system 

(GIS) data sets identifying areas of various yield 

probabilities clearly display model results. 

Probability criteria developed in this investigation 

can be used to select areas where other 

techniques, such as geophysical techniques, can 

be applied to more closely identify potential 

drilling sites for high-yielding (greater than 

40 gal/min) bedrock wells. 

To measure the added value of field-based 

methods for well-yield-probability forecasting, 

the model was run with and without the variables 

developed from the detailed geologic mapping for 

the Pinardville and Windham quadrangles. Four 

probability maps were produced for the two 

quadrangles, one with and one without the added 

variables. These maps clearly demonstrate the 

advantage of detailed geologic mapping when 

prospecting for new ground-water supplies in the 

fractured-bedrock aquifer of New Hampshire. 

INTRODUCTION 

Considerable population growth has resulted in 

New Hampshire in the last 40-50 years. This growth 

has led to a need for additional water resources in 

many communities. Many of these communities have 

limited sand and gravel aquifers, which generally are 

the most favorable aquifers for constructing high-yield 

wells in New Hampshire. Additional water resources 

can be found in the fractured-bedrock aquifer, which 

generally produce low yields to wells (a few gallons 

per minute). However, through intensive site analyses, 

high-yielding zones can be located in the aquifer. 

Throughout New Hampshire, the bedrock is crystalline. Fractured-bedrock aquifers in crystalline bedrock 

are among the least understood and quantified groundwater resources in the Nation. A major problem in 

evaluating ground-water availability in fractured 

bedrock is its extreme variability in water-bearing 

properties. 

The New Hampshire Bedrock Aquifer 

Assessment was done by the U.S. Geological Survey 

(USGS), in cooperation with the New Hampshire 

Department of Environmental Services (NHDES), 

Water Division, to provide information that can be 

used by communities, industry, professional consultants, and other interests to evaluate the ground-water 

development potential of the fractured-bedrock 

aquifer. This study was done at three scales— 

statewide, regional, and local—to quantify relations 

among well yields and bedrock data, physiographic 

setting, remotely sensed lineaments, and well characteristics in New Hampshire. The statewide scale was 

designed as a reconnaissance-level statistical analysis 

of the relations between bedrock well yield and 

bedrock type, lineament characteristics, topography, 

and other well-site and construction characteristics. 

The regional-scale analysis was similar, except that 

field-based geologic-data collection and additional 

lineament data were added to assess what effect 

quadrangle-scale (1:24,000) data had on well-yield 

relations. Local assessments (usually at a well-field 

scale) were done by use of geophysical tools to 

identify the location and orientation of discrete zones 

of fracture in the bedrock that are referred to as 

“fracture zones” in this report. 

Purpose and Scope 

This report presents results of statistical 

analyses conducted for the statewide and regional 

scale investigations of the bedrock-aquifer assessment. 

Results of the well-field assessments are presented in a 

companion report by Degnan and others (2001). 

The NHDES has a large data set of bedrock-well 

locations, yields, and other characteristics that are 

adequate for an exploratory, statistical analysis of 

hydrogeologic and construction factors associated 

with well yields. There are many possible factors (site 

and well characteristics) that relate to high-yield well 

sites. For the statewide analysis, available well data 

and remotely sensed data were compiled. For the 

regional analysis, fracture data and general geologic 

2 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
and hydrogeologic information were collected and 

compiled for two 1:24,000-scale New Hampshire 

quadrangles. The purpose of the more intensive 

regional-scale compilation was to determine the 

degree to which predictive well-yield relations, 

developed as part of the statewide reconnaissance 

investigation, could be improved with quadranglescale geologic mapping. Data were collected through 

bedrock mapping and supplemental hydrogeologic 

observation. The Pinardville and Windham quadrangles were chosen for regional analysis because of the 

large number of wells located in each quadrangle and 

the variety of geologic settings in the quadrangles. 

This study was not intended to explain the 

occurrence and nature of water-bearing bedrock 

fractures in New Hampshire, nor the nature of water 

movement in the fracture system. Understanding the 

occurrence and distribution of fractures in crystalline 

bedrock and associated water movement are being 

intensively evaluated at research sites (Shapiro and 

Hsieh, 1991) and cannot be determined statewide or 

regionally with the present state of the science. The 

statistical identification of hydrologic and other 

factors that are related to bedrock well yields 

statewide and regionally, will, however, provide 

information to further the understanding of these 

processes. 

Previous Investigations 

Large-scale analyses of bedrock-aquifer yields 

are few, and especially are limited in crystalline 

bedrock settings in the northeastern United States. 

Investigators have successfully related geohydrologic 

factors, particularly lineaments, to well yields in the 

carbonate rocks of Pennsylvania (for example, 

Knopman, 1990; Siddiqui and Parizak, 1971). One of 

the largest studies (Daniel, 1989; Daniel and others, 

1983) assessed hydrogeologic characteristics and 

yields of 6,200 bedrock wells in the Piedmont and 

Blue Ridge provinces of North Carolina. These 

investigations found that wells in draws or valleys 

have average yields three times those of wells on hills 

and ridges. Wells in the most productive hydrogeologic units had average yields twice those of wells in 

the least-productive units. One of the investigations, 

directly applied to fractured-bedrock aquifers in the 

Northeast, was performed by Mabee and others (1994) 

on a 17-mi2

 island on the Maine coast. This study 

examined 35 bedrock wells and found that aquifer 

transmissivity, normalized by well depth, was 

positively related to well proximity to fracturecorrelated lineaments, which are lineaments trending 

in the same direction as fractures observed in the 

bedrock. A number of these studies (table 1) measured 

the degree to which wells near lineaments represented 

a population of high yields. 

Table 1. Summary of well yield and lineament investigations considered for this study 

[>, greater than; NHDES, New Hampshire Department of Environmental Services; NA, not available] 

Number of Confidence 

Study/imagery used Region/units tested Measure of well 

productivity 

Total 

number 

of wells 

Number 

of wells 

near lineaments 

wells 

beyond 

buffer 

zone 

around 

the samples represent, two 

separate 

populations 

lineaments (in percent) 

Siddiqui (1969); Siddiqui and Pennsylvania carbonate rocks Normalized 45 18 27 >99 

Parizek (1971 and 1974) transmissivity 80 53 

Low-altitude aerial photography 

Mabee and others (1994) Maine metasedimentary Normalized 35 20 15 30 

High-altitude areal photogra­ transmissivity 

phy and side-looking radar 

Mabee and others (1994) Maine “Fracture-correlated” linea- Normalized 35 7 28 89 

High-altitude areal photogra­ ments (involving extensive addi­ transmissivity 

phy and side-looking radar tional fracture-fabric field work) 

Preliminary report of present Plates 1-6, covering southern New Reported yield 11,212 NA NA 99.9 

investigation (Moore and others, Hampshire, “migmatites” and unfoli1998) Low- and high­ ated plutons (involving no field work 

altitude areal photography beyond NHDES locating wells) 

INTRODUCTION 3
Most of the investigations mentioned in table 1 

used lineament analysis to identify potential bedrockfracture zones or bedrock-solution-channel enhancements in carbonate bedrock. Numerous investigations 

refer to lineament analysis as “fracture-trace analysis”; 

however, field observations of fractures is desirable 

for fracture-trace analysis. A lineament is defined as a 

linear pattern, seen on aerial photographs and other 

remotely sensed imagery, that meet established criteria 

for features that may be the result of underlying zones 

of fractured bedrock (Clark, Moore, and others, 1996). 

Despite the use of rigid criteria, any given lineament 

may not be underlain by a fracture zone. Anthropogenic remnants such as an abandoned woods road, rail 

lines, or right-of-ways, could be identified as a 

lineament but may not be related to fractures. 

Lineament data also are limited because of the 

accuracy of lineament locations, which varies with the 

scale of the imagery. A lineament analysis, is 

therefore, a preliminary analysis whereby lineaments 

are identified on remotely sensed imagery, and can 

only be confirmed as fracture related through 

subsequent field observation. Blanchet (1957) 

described patterns observed on aerial photographs, 

which were related to fracturing in the Earth’s crust. 

One of the first thorough treatments of lineament 

analysis was described by Lattman (1958) who 

provided much of the framework for techniques often 

applied currently (2001). In New England, many 

consulting firms commonly use some form of 

lineament analysis, in combination with other investigative techniques, to locate high-yield well sites in 

fractured-bedrock aquifers. The specific techniques 

applied, however, generally are considered to be 

proprietary and descriptions are often incomplete or 

not available. 

Clark, Moore, and others (1996) describe the 

lineament-analysis methods used in the New 

Hampshire Bedrock Assessment. Some of these 

methods previously were used by Blanchet (1957), 

Lattman (1958), Brown (1961), Daniel (1989), and 

Mabee and others, (1994), but no studies integrated 

results of statewide, regional, and local-scale investigations based on the variety of data compiled for the 

New Hampshire bedrock-aquifer statistical analysis. 

Acknowledgments 

The authors thank the individuals who contributed their time and effort offering advise and 

assistance. Thanks go especially to the following 

individuals: from New Hampshire Department of 

Environmental Services, Frederick Chormann, 

John Cotton, Philip O’Brien, Eugene Boudette, and 

David Wunsch; from the University of New 

Hampshire, Mathew Davis, Wallace Bothner, and 

Lawrence Dingman; from Dartmouth College, 

John Lyons; from the Maine Geological Survey, 

Marc Loiselle; from the University of Massachusetts, 

Stephen Mabee; from the private sector, Gary Smith 

and Bradford Caswell; and from the U.S. Geological 

Survey, Thomas Armstrong, Lawrence Drew, 

James Borchers, Paul Hsieh, Ben Morgan, 

Jack Schuenemeyer, Allen Shapiro, Charles Daniel 

and Jeffrey Starn. 

HYDROGEOLOGIC SETTING 

New Hampshire’s landscape is underlain by 

bedrock, which makes up the crust of the Earth, and by 

surficial deposits, most of which were left behind by 

the continental ice sheet during glaciation. The trend 

of major bedrock units that form the regional 

structural pattern in New Hampshire generally is 

north-northeast. The bedrock that underlies New 

Hampshire is composed of metamorphic and plutonic 

rocks (Lyons and others, 1997). The geologic history 

of New Hampshire’s bedrock materials, their structure, 

and the tectonic forces and processes that formed them 

is part of the history of the Appalachian Mountains. 

Details of this history can be found in Hatcher and 

others (1989) and Boudette (1990). 

Plutonic rocks, locally common in New 

Hampshire, are composed of interlocking minerals and 

have little or no primary pore space. Likewise, 

primary porosity of the parent materials of metamorphic rock in New Hampshire has been eliminated by 

metamorphism. Sedimentary and igneous rocks are the 

parent materials for many of the metamorphic rocks in 

New Hampshire. These rocks have been subjected to a 

wide variety of conditions of temperature and 

pressure, which formed differing assemblages of 

minerals and textures and eliminated voids by 

compaction and recrystallization. 

4 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
Tectonic processes, which accompanied 

metamorphism and mountain building, folded and 

faulted New Hampshire’s rocks. Tectonic processes 

created many fractures, faults, and voids and, at the 

same time, locally sealed some of these structures 

(Swanson, 1988). 

Despite a complex history of sedimentation and 

multiple events of deformation, metamorphism, and 

intrusion, New Hampshire’s bedrock can be viewed 

from a broad hydrologic perspective. Water moves 

through, and is stored in, open fractures. The size, 

number, distribution, and degree of interconnection of 

fractures are highly variable; in general, however, 

fractures are few and, when present, generally are 

presumed to decrease in size and number with depth. 

Thus, the overall storage capacity of bedrock is small 

and tends to decrease with depth (U.S. Geological 

Survey, 1984, p. 304). Wells that penetrate bedrock 

commonly yield dependable supplies of water suitable 

for single-family domestic needs, and, for this 

purpose, bedrock is a principal aquifer. Zones where 

bedrock is extensively fractured may yield large 

quantities of water. Many small water systems that 

serve residential developments use bedrock wells, and 

in the past two decades, the application of exploration 

technology has enabled more municipal water-supply 

systems to use the bedrock aquifer. 

Because ground-water flow processes in 

fractured crystalline bedrock are complex, developing 

a more complete understanding of these processes, and 

methods to characterize them, is a priority of USGS 

research. The USGS has been conducting investigations at its national fractured-bedrock research site at 

the Hubbard Brook Experimental Forest in Thorton, 

N.H., since 1990 (Shapiro and Hsieh, 1991; Hsieh and 

others, 1993). Research conducted at this site has 

included developing and utilizing state-of-the-art 

geologic, geochemical, geophysical, and hydrologic 

methods to gain a better understanding of groundwater-flow processes in fractured crystalline bedrock. 

METHODS AND APPROACH 

The approach of this study was to relate bedrock 

well yields in New Hampshire to factors known to 

affect well yields in other parts of the country. These 

factors include bedrock type, lineament characteristics, physiography, and other well-site and construction characteristics on a statewide scale. This analysis 

was done by use of a multivariate-regression model 

with instrumental variables. Emphasis was placed on 

the use of available data and remotely sensed information, with some additional field-data collection 

involving two geologically diverse 7.5-minute 

quadrangles. Lineament investigations, as previously 

discussed, were useful indicators of high-yield wells in 

crystalline bedrock. Other geologic and site characteristics that were significantly related to well yield also 

were used in a predictive mode to identify areas with 

various probabilities of obtaining a selected yield. The 

approach incorporates the use of statewide and 

regional information to evaluate the added benefit of 

additional data collection. The following sections 

describe the sources and treatment of data and 

development of the statistical approach. 

In investigating factors that relate to high-yield 

well sites, many possible factors (referred to as 

variables in this report) were examined. Statewide data 

sets were compiled from existing sources or were 

created for this study. The primary information that 

was analyzed included well information, bedrock 

lithology, surficial geology, lineaments, topography, 

and various derivatives of information in those data 

sets. In addition, data were collected at a more detailed 

scale for the areas contained in the Pinardville and 

Windham quadrangles, New Hampshire (fig. 1). 

These data are discussed in detail in the following 

paragraphs. 

The relation of multiple variables to bedrockwell yields initially was examined by simple regression techniques to compare preliminary analyses with 

other investigations and to help streamline more 

detailed regression-model analyses involving the use 

of instrumental variables. The relation of selected 

individual variables to yields also was examined by 

simple, bivariate, statistical analyses. 

Regression was selected as the primary method 

of analysis. As such, the established model is a 

stochastic (statistical) model rather than a processoriented geologic model. A process-oriented model 

(modeling flow in known fractures) was not possible 

because the necessary detailed geologic information, 

such as a statewide data set on fractures, is not 

available. Variables examined included categorical 

variables (such as bedrock type), as well as continuous 

numerical variables such as the slope of the land 

surface. A nested model was used, where detailed 

geologic data from the quadrangles were applied 

METHODS AND APPROACH 5
Figure 1. Location of the Pinardville and Windham 1:24,000-

scale quadrangles in southern New Hampshire where 

additional geohydrologic, fracture, and lineament data were 

collected. 

where available or nested in the statewide model. The 

model was used to evaluate the simultaneous effect of 

many variables and the interaction of those variables, 

and provided confidence intervals. These intervals can 

be used to predict the probabilities of obtaining a 

given yield given different site and well characteristics. 

Descriptions and Sources of Data 

This study required the development of 

geographic information system (GIS) compatible data 

sets by use of ARC/INFO. All data sets used were the 

most detailed available statewide. Many variables 

were considered, however, only 43 were found to be 

significant and are included in the final regression 

model. 

Well Information 

Detailed information on water wells constructed 

since 1984 is collected and maintained in a computer 

database by NHDES. Many of these wells have been 

field-inventoried to obtain accurate locations. Wells 

are categorized as to whether they are used for 

domestic, public supply, commercial, or other uses. 

Nearly 98 percent of the all wells in the database were 

drilled for domestic use. Basic well data are reported 

by drillers and include six characteristics used in this 

study—well yield, well depth, well use, method of 

construction, date of construction, and depth to 

bedrock (or length of casing). Well diameter was not 

included in this database. However, nearly all of the 

domestic wells are 6 inches in diameter. A georeferenced digital database is maintained by the State 

where location coordinates were determined either by 

plotting and digitizing from a map at 1:24,000 or 

1:25,000 scale or by differentially corrected Global 

Positioning System (GPS) measurement. Well 

locations are accurate to ±100 ft. 

The well data set collected by NHDES and used 

in this study contains records of 20,308 wells drilled 

from 1984 to 1998, and subsequently field located. A 

randomly selected subset (20 percent or 4,050 wells) 

of the available well data were reserved for verification tests of the model. The availability of this large 

data set provided an opportunity for statistical analysis 

of variables considered to be related to well yields. 

Specific components of the database used in this study 

are described in the following paragraph with respect 

to their use for statistical analysis. 

Well yields in the database ranged from zero to 

greater than 500 gal/min. Yields are reported by well 

drillers and are determined by a variety of methods at 

the time the well is drilled. Yields generally are 

determined as the rate of water that can be airlifted on 

a continuous, short-term (generally tens of minutes) 

basis. Low yields may be accurately measured by 

timing the pumpage of water into a known volume. 

Moderate- to high-reported yields often are quantified 

by discharging water through a rated channel and are 

likely to be rough approximations of the true yield. 

Such methods introduce an unquantifiable variance in 

measured yields but are acceptable for statewide or 

regional evaluations involving large numbers of wells. 

Yields equal to or greater than 40 gal/min were considered high in this investigation. High-yielding wells 

constitute nearly 10 percent of the well population 

(fig. 2). The yields of public-supply wells typically are 

quantified by an aquifer test of hours to days in 

duration and represent a more rigorous estimate of 

yield than that reported for domestic wells by the 

drillers. The dependent variable that was statistically 

explored in this study was the natural log of the 

reported well yield. Other possible dependent 

variables considered were the yield and yield per foot 

of open hole; however, the natural log of well yield 

was selected because the population of well yields is 

log normally distributed. 

6 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
A problem in using well yield as the dependent 

variable in a multivariate-regression analysis is that 

yield is a function of a well owner’s water needs. 

Generally, a well only is drilled as deep as is needed to 

meet a specific targeted yield. Thus, the reported yield 

is not the maximum potential yield for that site. As 

well yield is, in part, a function of demand, an 

advanced statistical technique involving instrumental 

variables must be used for a more accurate estimate of 

yield. 

The year each well was drilled can be included 

as a surrogate for demand in the model because 

demand for water (and the depths to which well 

owners have been willing to pay to drill to obtain more 

water) has increased in New Hampshire. The effect of 

time was observed using data from the Pinardville 

quadrangle (Lawrence Drew, U.S. Geological Survey, 

oral commun., 2000). Drillers confirmed that well 

depth increased to meet this demand (Terry Swain, 

Capital Well Co., oral commun., 2000). 

Depth, reported by the drillers, likely is accurate 

to the nearest foot. Depth also is a function of demand 

and is inversely related to well yield. Wells are drilled 

deeper at low-yield sites and drilling is stopped when a 

desired yield is reached. This action makes depth an 

“endogenous variable,” and, for this reason, depth is 

inversely related to well yield. 

Data on the method of well construction 

essentially are limited to two general categories; cable 

tool and rotary. Bedrock wells drilled with a cable-tool 

drill rig can be identified in the NHDES database, but 

all other methods, such as the various forms of rotary 

drilling, were not differentiated. As cable-tool 

construction is different from the other techniques, it is 

reasonable to test the effects of cable tool in comparison to other methods of construction in the regression 

model. With cable-tool construction, the drill bit is 

pounded into the bedrock perhaps opening up or 

enhancing fractures. An indicator variable was used to 

identify whether or not each well was constructed by 

use of a cable-tool drill rig. 

Bedrock Lithology 

Bedrock lithology was assigned using a digital 

data set of the Bedrock Geologic Map of New 

Hampshire (Lyons and others, 1997). This data set 

contains 174 mapped bedrock units and was compiled 

at a scale of 1:250,000. 

Figure 2. Percent of statewide bedrock well yields in 

New Hampshire. 

In addition to the 174 mapped bedrock units, 

7 generalized categories, covering all the rocks of 

New Hampshire, were created with input from 

Dr. Wallace Bothner (University of New Hampshire, 

written commun., 1998). Variables identifying these 

groupings were created to broadly categorize similar 

rock types. Some mapped geologic units depicted on 

the State bedrock map have limited areal extent, yet 

have properties such as origin, texture, and degree of 

metamorphism common to other mapped units. 

Grouping these lithologies together into common 

categories allows for statistical analysis of areas that 

would otherwise have had too few well-data points to 

be included individually as independent variables. 

These seven major lithologic units with similar 

lithologic characteristics are (1) nonfoliated to weakly 

foliated igneous rocks, (2) well-foliated igneous rock 

(well-foliated rocks of the New Hampshire Plutonic 

Suite and Oliverian Plutonic Suite), (3) fine-grained, 

tightly bonded metasedimentary rocks, (4) fine to 

medium-grained granular-metasedimentary rocks, 

(5) gray-black graphitic micaceous metapelite locally 

interbedded with calcareous or siliceous rock, 

(6) volcanic rock and associated metapelite, and 

(7) rocks of the Massabesic Gneiss Complex and 

Breakfast Hill Granite (fig. 3). 

Lineaments 

A statewide data set of lineaments was created 

for this project following the methods presented in 

Clark, Moore, and others (1996). The imagery used 

came from four remotely sensed platforms. These 

platforms included Landsat imagery (1:1,000,000 

enlarged to 1:250,000), side-looking airborne radar 

(SLAR) (1:250,000), high-altitude aerial photography (approximately 1:80,000), and low-altitude 

METHODS AND APPROACH 7
Figure 3. Major lithologic units in New Hampshire adapted from Lyons and others (1997). 

8 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
photography (approximately 1:40,000, enlarged to 

1:20,000). In addition to the imagery mentioned 

above, the regional-scale analysis included colorinfrared imagery (approximately 1:40,000) and 

topography from standard (7.5 minute) topographic 

maps (1:24,000). Data at different scales were 

analyzed by different analysts in order to maximize the 

reproducibility of lineament identifications (Clark, 

Moore, and others, 1996). This analysis was done by 

selecting just those lineaments identified by two 

different observers. The imagery was examined and 

lineaments were recorded on mylar overlays by the 

observers working independently. Results were 

compared and placed in the following three categories: 

(1) blind comparisons—matching lineaments drawn 

by independent observers; (2) confirmed 

lineaments—lineaments drawn by one observer and 

confirmed by another observer; and (3) rejected 

lineaments—lineaments drawn by one observer and 

not confirmed by another observer. All blind or 

confirmed lineaments were digitized into a GIS 

database for use in the statistical analysis. These 

features were either plotted on scale-stable base maps 

that were later mounted on a digitizing tablet and 

digitized, or were plotted directly on imagery overlays 

that were scanned later and rectified to allow onscreen digitizing. 

Lineament data are limited because (1) the 

accuracy of lineament locations varies with the source 

and scale of the imagery, (2) lineaments can be 

caused by cultural rather than geologic features, 

(3) lineaments related to bedrock features are most 

easily observed in areas where the surficial material is 

thin (areas where overburden is thin are known to have 

low-yielding bedrock), and (4) lineaments theoretically are restricted to steeply dipping fractures or 

fracture zones that produce straight-line intersections 

with the earth surface. As lineaments are more readily 

found in the hilly shallow-to-bedrock areas, 

establishing a relation to well yield is difficult because 

of inaccuracies in the georeferenced locations of the 

wells and the lineaments, and because these shallowto-bedrock areas also are associated with relatively 

low yields. 

For each well site, distance to lineaments and 

lineament characteristics were compiled for use in the 

statistical analyses. Distance from wells to lineaments 

was coded by identifying whether or not the wells 

were in a “buffer zone” surrounding each lineament. A 

buffer zone of 100 ft was selected by trial-and-error, 

considering the accuracy of the well and lineament 

locations. The accuracy of lineament locations is 

approximately ±80 for aerial photography and ±150 ft 

for the Landsat imagery. Well locations are estimated 

to have an accuracy of ±100 ft. Selecting a buffer zone 

less than 100 ft would, thus, have little or no meaning. 

The buffer zone selected (100 ft) is consistent with the 

findings of Mabee (1992, p. 106), who selected 

(by trial-and-error) an optimal buffer zone, for statistical analysis, of 98 ft (30 m). 

Lineaments identified from SLAR were 

excluded from the analysis because of difficulties in 

positioning the lineaments even to an accuracy of 

±100 ft. The SLAR data, compiled on 1:250,000 scale 

maps, were located to approximately ±200 ft. 

Additionally, because SLAR relies on shadowing 

effects, there may be a tendency for the apparent 

location of the lineament (the shadow) to be off-center 

from the bottom of the valleys or swales that cause the 

feature. Linear cultural features, such as road cuts, rail 

lines, right of ways, and tree lines at the edges of open 

spaces, can be prominent on SLAR imagery (more so 

than on other imagery) and could be mistaken for 

geologic lineaments. The SLAR imagery was 

carefully examined to avoid misidentification; 

however, the difficulties in positioning the SLAR 

lineaments precluded its use. 

Additional variables were created to evaluate 

interactions between lithology and lineaments. Well 

yields in relation to lineaments or fractures can differ 

between lithologies because the character of fractures 

and the extent of the affected area near fracture traces 

may differ between lithologic units. For example, 

weathered fracture zones in plutons may contain 

permeable “grus,” a granular fragmental product of 

weathering of granitic rocks (Bates and Jackson, 

1980), but in metasediments may contain clay-rich 

low-permeability saprolite. The character of fractures 

and the extent of affected areas near fracture traces 

also can differ between bedrock types. In plutonic 

rocks, for example, there tend to be more sheeting 

fractures, which could be hydrologically connected to 

vertical fracture zones. 

The effect of the orientation of the lineaments 

was examined late in the process of building the 

regression model to minimize the number of possible 

variables. The strike of the nearest lineament within 

100 ft of each well was computed and binned into 

eighteen 10-degree-angle categories. Eighteen angle 

categories multiplied by numerous lithologies would 

result in an inordinate number of variables with the 

likely possibility of spurious results. 

METHODS AND APPROACH 9
A lineament-density database for New 

Hampshire was created by identifying the density of 

lineaments within a 1,000-foot radius around each 

well for each lineament platform (Clark, Moore, and 

others, 1996, p. 10, plates 1-11. Plate areas 1-10 

include all of New Hampshire south of 44 degrees 

latitude and plate 11 is just to the north and includes 

most of the eastern part of the White Mountains). 

Lineament density was equal to the total length of all 

lineaments within 1,000 ft of each well divided by the 

area of this circle. Contrary to expectations, lineament 

density was found to be negatively correlated to well 

yield in these preliminary analyses. Thus, areas with 

high lineament densities are most likely in areas of 

thin overburden, which are inversely related to well 

yield. (Fracture zones beneath thick overburden are 

less apt to be observed). Because of this negative 

relation, and the availability of better predictors of 

areas that have shallow depths to bedrock, lineament 

density was not used in further well-yield analyses. 

Similarly, the distances from the wells to where 

lineaments intersect one another were examined for 

five test quadrangles. Unlike positive relations found 

elsewhere in carbonate bedrock, no significant 

relations were found in the New Hampshire setting; 

therefore, intersections were not used in further 

analyses. 

Surficial Material 

Surficial material such as stratified-drift 

aquifers possibly affect bedrock well yield through 

(1) increased permeability or the ability to supply 

water to the bedrock aquifer, (2) saturated overburden 

at the bedrock interface, and (3) greater occurrence of 

stratified-drift aquifers in valleys. New Hampshire 

lacks a statewide GIS data set of overburden types; 

however, a statewide stratified-drift aquifer data set is 

available and can be applied to the regression model 

by use of a dichotomous indicator (0 or 1 binary) 

variable. Overburden type was obtained from 

maps of the extent of stratified-drift aquifers in 

New Hampshire, which were mapped as part of a 

statewide series of investigations (Medalie and Moore, 

1995; Moore and others, 1999). Areas outside the 

mapped stratified-drift aquifers can be classified as till 

and (or) bedrock. An indicator data set (1 or 0) was 

used to identify whether a well penetrates through 

stratified drift or not. No further differentiation of 

surficial materials is available on a statewide basis. 

The thickness of non-stratified-drift deposits is 

not mapped statewide but can be determined or 

estimated at individual well sites. Most well drillers’ 

reports contain the depth to bedrock determined 

during drilling. Where these data are incomplete, the 

length of casing minus 12 ft can be used as a surrogate 

for depth of overburden. Twelve feet is the median 

depth that casings are set into bedrock based on 

18,330 wells statewide. Drillers prefer to end the 

casing well in competent bedrock to avoid the upper 

few feet, which may be weathered or fractured. 

Because there is no statewide GIS data set of 

overburden thickness, this variable could not be used 

in a predictive mode. 

Topographic Settings and Characteristics 

Data were compiled to provide various sets of 

topographic and physiographic information that may 

be associated with bedrock well-yield characteristics. 

Topographic settings can be derived from analysis of 

USGS Digital Elevation Models (DEMs). DEMs 

consist of a sampled array of elevations for a number 

of ground positions at regularly spaced intervals. The 

DEM data for 7.5-minute units correspond to the 

USGS 1:24,000 and 1:25,000-scale topographic 

quadrangle map series. Each 7.5-minute DEM is based 

on 30 x 30-m data spacing with the Universal 

Transverse Mercator (UTM) projection. The 1-degree 

DEMs, with 3 x 3-arc-second data spacing (or roughly 

100-m spacing) provides coverage in 1 x 1-degree 

blocks. The 1-degree DEMs also are referred to as 

1:250,000-scale DEM data. Land-surface elevations of 

wells determined from the 7.5-minute DEMs were 

included in the model. Lower well yields are expected 

in high elevations as a result of greater depths to water, 

less saturated thicknesses, and(or) the predominance 

of competent rock with few fractures. 

Topographic setting also can be analyzed 

regionally to identify, for example, the effects of major 

river valleys. This analysis is done by use of 1:250,000 

scale DEMs. Effects of more localized features, such 

as swales or cuts in ridgelines, can be assessed using 

1:24,000-scale DEMs. Slope, or the rate of change in 

elevation surrounding a well, was tested at both scales. 

Slope at the 1:24,000-scale can be used to compare the 

change in elevations between neighboring DEM 

elevation points about 100 ft away from one another. 

Slope at the 1:250,000-scale DEM is a more regional 

measurement. Slope expressed as a percent is 

10 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
calculated in ARC/INFO relative to the direction of 

maximum change in elevation. Assuming similar 

results to those for North Carolina (Daniel, 1989), 

slopes or hillsides are expected to have average well 

yields, and valley bottoms or depressions are expected 

to have above-average well yields. Wells on hills and 

ridges are expected to have below-average yields. 

Curvature, calculated as the second derivative of 

the land-surface elevation, provides another quantifiable measure of topographic setting. The curvature 

function used in this study fits a fourth order polynomial surface to each 3 x 3 block of DEM data points 

(referred to as GRID cells in ARC/INFO) surrounding 

and including the central DEM point for the calculated 

curvature. Negative values indicate that the land 

surface is concave upward (for example, a swale or 

valley bottom) and positive values indicate the land 

surface is concave downward (for example, a hilltop). 

Again, use of the 1:250,000-scale DEM results in a 

coarse resolution with a much more generalized 

regional depiction of the land surface, and cannot be 

used to detect small features such as depressions in 

open areas or small gaps in ridges. Curvature at the 

1:24,000-scale, however, will not show large physiographic features at the well site. Sites where the 

topography was concave upward (especially with 

gentle slopes) represented locations that were eroded 

more than the surrounding rock and may be underlain 

by fractures. Conversely, hilltops, which are concave 

downward, are more resistant to erosion and may be 

related to a decrease in fracture density and well yield. 

Categorical variables were created for combinations of slope and curvature. These variables are 

“dichotomous,” in that they indicate with a 0 or a 1 

whether a given condition is met. For example, the 

quartile of the well population with steepest site slopes 

was identified as being on hill slopes (dichotomous 

variable is set equal to 1). The cross between the most 

negative quartile of the population, for curvature, with 

the quartile of the population with the most gentle 

slopes, was used to identify swales or valley bottoms. 

Drainage area, upgradient from each well site, 

was determined by use of the 1:24,000-scale DEM and 

a computer process known as “flow accumulation.” 

First, isolated low points (“sinks”) surrounded by high 

points are filled in. These points typically represent an 

artifact of the DEM. Next, the downslope direction 

between all DEM cells was determined, and the 

number of cells upgradient of every cell was 

determined (including that cell). This flow accumulation was done statewide to measure the drainage area 

of each well site. 

The distance of each well to the nearest 

waterbody was determined by use of the 1:24,000-

scale Digital Line Graphs (DLG) of hydrography. For 

this method, only shorelines and perennial streams 

were used to compute the distance. Wetlands and 

intermittent streams were excluded because they are 

depicted less accurately and consistently on the DLGs 

than are open perennial waterbodies. 

Additional Variables Investigated at the 

Quadrangle Scale 

Additional geologic, fracture, and lineament 

data were collected for the Pinardville and Windham 

quadrangles (fig. 1). These quadrangles were selected 

because they represent different geohydrologic 

settings, and because these quadrangles contain the 

largest number of georeferenced bedrock wells 

anywhere in the State of New Hampshire. Different 

degrees of metamorphism have affected the rocks 

present in these quadrangles with much of the 

Pinardville quadrangle containing the Massabesic 

Gneiss (migmatitic) Complex. In the Pinardville 

quadrangle, there are 1,682 bedrock wells with a 

complete set of well information in the database, and 

for the Windham quadrangle, there are 1,504 wells. 

These data provided a large enough data set to 

evaluate the effects of additional variables, such as 

fracture-correlated lineaments, lineaments identified 

from topographic maps and color infrared imagery, 

and additional variables identified by use of 

quadrangle-scale geologic mapping. 

Topographic lineaments, drawn on a 

1:48,000-scale topographic base reduced from the 

1:24,000-scale topographic maps, and lineaments 

drawn on 1:58,000-scale color infrared photographs 

were added to the lineament database for the Windham 

and Pinardville quadrangles. Topographic maps are a 

readily available platform commonly used for 

lineament identification. Color-infrared photography 

especially is sensitive to anomalies in vegetation 

growth, which may be related to underlying fracture 

zones. Vegetation type and intensity can vary with 

ground-water recharge and discharge zones. The same 

procedures described in Clark, Moore, and others 

(1996) were followed in this process. Color-infrared 

photographs with overlays were rectified and both sets 

of lineaments were digitized into GIS data sets for use 

in the statistical analysis. 

METHODS AND APPROACH 11
Geologic mapping at the 1:24,000 scale in the 

Pinardville quadrangle (Thomas R. Armstrong and 

William C. Burton, U.S. Geological Survey, written 

commun., 2000) and Windham quadrangle (Walsh and 

Clark, 1999) provided an accurate representation of 

lithologies and data on brittle fracture and ductile 

structure orientation for correlation with lineaments. 

These maps accurately depict geologic units, contacts, 

fault, axial trace, metamorphic isograd, dike and other 

features, which were tested relative to well yields. 

Simplified versions of the maps showing geologic 

units are shown in figures 4 and 5. 

The geologic units mapped at the quadrangle 

scale include numerous subdivisions of the geologic 

units shown on the Bedrock Geologic Map of New 

Hampshire (Lyons and others, 1997). The digital GIS 

coverage of the State map, used for development of 

the regression model, was adjusted where contacts 

mapped at the quadrangle scale separate units shown 

on the State map. 

Other geologic features in the quadrangles 

similarly were assessed as the lineaments in the 

regression model. These features include contacts, 

faults, dikes, and axial planes of folds large enough to 

be mapped at a scale of 1:24,000. Indicator variables 

were created to indicate whether a well was within 

100 ft of one of these features. Geologic contacts, 

faults, axial traces, metamorphic isograds, and dikes 

also were matched with lineaments in the Windham 

and Pinardville quadrangles. Lineament segments that 

fall on, or are parallel within 200 ft of these features, 

were identified and tested to determine if they were 

associated with high-yield wells. 

Information on low-angle jointing was not 

available statewide but was collected for the two 

quadrangles. The occurrence of low-angle joints near 

the anticlinal axis of the Massabesic Gneiss Complex 

(fig. 4) is a prime example of where these features can 

be related to well yield. An indicator variable was 

created to test whether or not wells within 2,000 ft of 

the mapped axis of this broad anticlinal feature had 

high yields. 

Whether some lineaments are more likely to be 

related to fractures than others has prompted the 

search for lineaments that are “fracture correlated.” 

Mabee and others (1994) recognized the need to filter, 

or reduce, lineament data sets before correlating these 

remotely sensed features with well yield. Their 

method for identifying fracture-correlated lineaments 

is based on identifying fracture domains. Every 

fracture at a few select outcrops (fracture stations) was 

sampled and the geographic extent of the fractures was 

identified by comparing fracture families at each 

fracture station. Fracture families, as defined by 

Mabee and Hardcastle (1997, p. 24), are “a set of 

fractures that have similar orientations.” A comparison 

of studies done at different scales by Mabee and others 

(1994) and Mabee and Hardcastle (1997) shows that 

the sizes of fracture domains appear to be related to 

the spacing of fracture stations. Fracture domains, 

therefore, appear to be scale-dependent and the 

domain boundaries can be arbitrary if they cannot be 

correlated to a specific geologic feature. 

In the New Hampshire Bedrock Aquifer Assessment, fracture families were derived using fracture 

data from the entire mapped area rather than just 

selected stations. Walsh and Clark (2000) compared 

mapped fracture orientations in the Windham 

quadrangle to fracture orientations at selected stations 

(large outcrops in the Windham quadrangle where they 

collected additional data). The results show a correlation between station fractures and mapped fractures 

for the quadrangle. However, about one-ninth of the 

quadrangle could not be correlated and the mapped 

fractures surrounding a given fracture station may 

contain fracture families not represented by the 

fracture-station data. For the development of the 

regression model, fracture-correlated lineaments are 

based on the geographically, more extensively mapped 

fracture data. 

Lineament data were analyzed to see if geologic 

mapping at the quadrangle scale could identify a set of 

lineaments with improved correlation to high-yield 

wells. The lineaments were categorized in the model 

in the following three ways: (1) unfiltered lineaments 

that include data from the entire statewide database; 

(2) filtered, domain-based fracture-correlated 

lineaments; and (3) filtered, discrete-analysis-based 

fracture-correlated lineaments (fig. 6). 

Domain-based fracture-correlated lineaments 

were determined by defining fracture families for each 

of the two quadrangles. A square-cell sampling grid, 

or domain, was developed so that no cell would have 

less than five sample points. The optimal grid cell 

measured 3,300 x 3,300 m. Fracture families for each 

cell were derived from spatial analysis of mapped 

fracture data for each of the two quadrangles by 

plotting frequency-azimuth (rose) diagrams in the 

Structural Data Integrated System Analyzer software 

(DAISY 2.19) by Francesco Salvini, Dipartimento 

12 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
METHODS AND APPROACH 13

Figure 4. Simplified bedrock geologic map of the Pinardville quadrangle, New Hampshire (Location of quadrangle shown in 

figure 1.)
14 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire

Figure 5. Simplified bedrock geologic map of the Windham quadrangle, New Hampshire (Location of quadrangle shown in 

figure 1.)
METHODS AND APPROACH 15Figure 6. Fracture-correlated lineaments in the Pinardville and Windham quadrangles, New Hampshire.
di Scienze Geologiche, Universita degli Studi di 

“Roma Tre”, Rome, Italy (computer software, 2000). 

The DAISY software uses a Gaussian curve-fitting 

routine for determining peaks in directional data 

(Salvini and others, 1999) that was first described by 

Wise and others (1985). The rose diagrams included 

strike data for steeply dipping fractures (dips greater 

than 45°, after Mabee and Hardcastle, 1997). In this 

method, primary fracture families on the rose 

diagrams were defined as the highest peak, and 

secondary fracture families were defined as normalized peaks between 30 and 99 percent of the highest 

peak. Lineaments that are in or pass through a given 

cell, and are parallel to the fracture family for that cell, 

are identified as fracture correlated. Parallelism is 

defined by lineament trends that match primary and 

secondary fracture families within one standard 

deviation as determined by the Gaussian curve-fitting 

routine. The term “fracture correlated” in the domain 

method follows the definition by Mabee and others 

(1994) in that the correlated lineaments cannot be used 

to unconditionally identify fractures on the ground, but 

merely to imply that fractures are likely in the given 

area. Results from the domain analysis indicate that, 

by lineament length, 30.5 percent of the lineaments in 

the two quadrangles correlate with fracture trends, and 

14.6 percent correlate with the primary fracture trend. 

These filtered, domain-based fracture-correlated 

lineaments were tested in the regression model. 

Discrete-analysis-based fracture-correlated 

lineaments were determined by use of a more rigorous 

and conservative method than domain analysis (fig. 6). 

Domain analysis assumes that a statistically identified 

fracture family is applied to an entire area or domain, 

and, thus, only can be used to filter groups of 

lineaments that match the fracture trend in that 

domain. The domain approach does not address spatial 

variability in the domain and spatial variability is 

inherent in quadrangle-scale fracture-trend analysis 

(Walsh and Clark, 2000). This discrete-analysis 

method examines each lineament individually, or 

discretely, by comparing the strike of each lineament 

in a data set with the strike of nearby steeply dipping 

planar features or the trend of linear features in a 

structural geology data set. The discrete method 

identified lineaments that correlated with many types 

of structures (including fractures) measured during 

geologic mapping. The discrete method identified 

individual lineaments that matched the strike or trend 

within ±5° (after Mabee and others, 1994) of the 

structures observed in outcrops up to 1,000 ft (305 m) 

from a lineament. Lineaments that match these 

structures are called “structure-correlated lineaments,” 

and may be parallel to ductile or brittle features identified during quadrangle-scale geologic mapping. A 

subset of these lineaments includes the brittle fracturecorrelated lineaments, and this subset was tested for 

inclusion in the regression model. Results of the 

discrete method indicate that, by lineament length, 

30.9 percent of the lineaments were identified as 

fracture correlated. 

Creation of a Verification Data Set 

Twenty percent of the well data were randomly 

selected statewide and set aside as a verification data 

set. Sequestered data are used for verification of 

testing results of the model developed by use of the 

primary data set. The primary data set contained data 

from 16,302 wells. The verification data set contained 

data from 4,050 wells. 

Development of the Statewide Regression 

Model 

Multivariate regression was selected as the 

primary analytical technique because it is the most 

efficient tool for predicting a single variable (such as 

the natural log of well yield) with many potentially 

independent variables. Multivariate-regression 

analysis, unlike simple bivariate analyses, examines 

the added contribution of each parameter in explaining 

the variance of the dependent variable. The regression 

model was structured to account for drillers targeting 

well yields to specific demands. For a group of wells 

drilled to meet similar demands, such as domestic 

needs, the result is a minimized range of yields. 

If well depths were random, or if a uniform depth was 

drilled at each well site, there would be a greater 

range, or variance, among the well yields. 

Because domestic wells make up most of the 

well-yield data set, the demand criteria by 

homeowners needs to be factored into the analysis. 

Domestic needs generally are met by a yield of a few 

(usually less than 10) gallons per minute, and the wide 

range of yields is decreased by targeting similar yields. 

At a potentially high-yield site, drilling typically is 

stopped at a shallower depth than at an average site, 

16 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
which effectively reduces the number of deep highyielding wells in that population. At low-yield sites, 

drilling is continued to a greater depth to obtain a 

desired yield, which consequently reduced the number 

of low-yielding shallow wells. Applying an ordinary 

least squares model to the relation between yield and 

depth would result in a model that was biased 

downward; that is, the coefficient of the depth would 

be lower than its true value in an unbiased data set. 

Assuming that drillers are targeting yield, the 

variable “DEPTH” was “endogenous” (not independent of yield but developed with yield) and correlated 

with the residual of the yield regression. A regression 

of yield on depth violates the regression assumption 

that predictor variables and the error term are independent. To overcome this problem, the statistical 

technique of using instrumental variables (Davidson 

and MacKinnon, 1993) was applied in the New 

Hampshire Bedrock Aquifer Assessment. By this 

technique, the model evaluates the effects of demand. 

The bias, caused by the endogenous depth variable, 

can be corrected if there are variables (instrumental 

variables) that are correlated with yield demand but 

uncorrelated with random variation in the physical 

yield-depth relation. Initially, an attempt was made to 

include water use in the model because it is related to 

demand requirements. The water-use variables tested 

were the indicator variables that identify if the well is a 

commercial well or a public supply well. Other factors, 

likely related to well-construction techniques, such as 

well diameter, however, cause higher well yields in 

public supply and commercial wells than yields in 

domestic wells and water-use variables were, thus, not 

useful. Other more useful instrumental variables were 

(1) year drilled, (2) well driller, and (3) median 

household income based on the 1990 census (U.S. 

Census Bureau digital data, 1990). These variables are 

correlated with desired yield but are uncorrelated with 

random variation in the physical yield-depth relation. 

Well data from the Pinardville quadrangle 

(Lawrence Drew, U.S. Geological Survey, oral 

commun., 2000) indicated that the yields of newly 

constructed wells, measured by the drillers, increased 

with time because of an increase in demand for water. 

Local drillers confirmed that they gradually are 

drilling new wells deeper to meet this increased 

demand. Year of drilling can be used as a surrogate for 

increased demand, and it was introduced into the 

regression model as an instrumental variable. 

A two-stage regression model with instrumental 

variables was used to control the depth variable. The 

principle is depicted in figure 7. In this example, only 

year of drilling is used as an instrumental variable. 

Conceptually, it is included in the first stage of the 

model, which predicts lnDEPTH, the natural log of 

depth. The predicted lnDEPTH then is used in the 

second stage as a predictor. By excluding year from 

the second stage and substituting the predicted 

lnDEPTH from the first stage, a depth coefficient 

describing the physical relation of depth to yield, 

rather than the demand relation, is computed (fig. 7). 

The independent variables in the model are accounted 

for and the differences from year to year define the 

lnYIELD to lnDEPTH relation, providing a coefficient 

for depth that represents the physical relation. 

A statistical analysis of well depth by drilling 

company, for the 25 major drillers in New Hampshire, 

indicated that 21 drillers tend to drill deeper or 

shallower than average. As a result, drilling company 

also was used as an instrumental variable that is 

correlated with desired yield but uncorrelated with 

random variation in the physical yield/depth relation. 

The spatial distribution of median annual 

household income, determined from the TIGER data 

for the 1990 census (U.S. Census Bureau, 1990), also 

was used as an instrumental variable (fig. 8). 

Generally, wells in areas of high-median income are 

assumed to be drilled deeper to meet a somewhat 

higher demand for water than in areas of low-median 

Figure 7. Use of the instrumental variable “YEAR,” and 

a two-stage regression model to separate demanddriven relations from physical relations between well 

depth (lnDEPTH) and yield (lnYIELD) used in this study 

of New Hampshire. 
18 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire

Figure 8. Median household income for New Hampshire using the 1990 U.S. Bureau of Census data. Median income was 

used as a surrogate demand variable (instrumental variable) in the model.
income. Where census polygons lacked data, because 

of no population in 1990, the median county income 

was assigned to the income polygon. (This assignment 

was done to cover all possible wells, including those 

wells drilled post 1990.) Each well was assigned a 

median income based on the polygon in which the 

well was located and this income was used as another 

instrumental variable. 

Residuals from the two-staged regression model 

are computed as the difference between actual 

lnYIELD and predicted lnYIELD from the regression 

using actual lnDEPTH rather than predicted lnDEPTH. 

This process was used to essentially estimate the 

variance of a population of wells that would exist if 

yield had not been a function of demand. Details on 

how the model was built are given in appendix A. A 

list of variables considered and tested for inclusion in 

the model are given in appendix B. 

FACTORS RELATED TO WELL YIELD 

Numerous variables were tested for inclusion in 

the regression model. The strategy for adding or 

deleting variables during the model construction is 

outlined in appendix A. All variables included have 

met a 95-percent confidence level. 

Model results show that bedrock well yields are 

related to a number of physiographic, geologic, 

socioeconomic, and various other factors. These 

factors are best illustrated by examining the regression 

model equation, which relates independent variables 

to yield. Testing of the model provides confidence in 

the relations. The model then can be used to forecast 

and create maps of yield probability. 

Regression Model Equation 

There are a total of 66 variables used to predict 

lnDEPTH in the first stage of the model, of which 43 

are included in the second stage of the model along 

with the predicted lnDEPTH variable (table 2). Of 

these 43 variables, all but the two water-use variables 

(commercial- and public-supply wells) and the method 

of construction variable are site characteristics. Many 

of the variables also are dichotomous-indicator 

variables and are zero where a given condition is not 

met. For this reason, at any one location, there is a 

maximum of 12 site-dependent variables (with any 

indicator variables equal to 1) and a minimum of 

5 site-dependent variables that can apply. For all 

indicator variables, the magnitudes of the coefficient 

can be directly compared to one another. Indicator 

variables with large coefficients (in either a positive or 

negative direction) have greater effects on lnYIELD 

than do indicator variables with smaller coefficients. 

The coefficients of the continuous variables, such as 

elevation and slope, cannot be directly compared to 

one another in this manner. 

The instrumental variable (two-stage) approach 

affects the calculated regression coefficient for the 

natural log of depth (lnDEPTH). The coefficient is 

positive (0.53) indicating that total well yield increases 

with total well depth, but incremental yield (yield per 

foot of well depth) decreases with well depth. This 

result is in accordance with the physical relation, 

where water-bearing fractures generally decrease in 

size and number with depth (U.S. Geological Survey, 

1984, p. 304). The demand-driven coefficient for 

lnDEPTH (-0.88) was negative and could not possibly 

represent the physical relation. 

The coefficient of slope of the land surface 

(variable SLOPE, a unitless percent) is negative 

(-0.014), which indicates steep slopes tend to have low 

yields. Similarly, the land-surface elevation at each 

well (ELEV24, in feet above sea level) is derived from 

1:24,000- or 1:25,000-scale maps with a negative 

coefficient of -0.00012, indicating that high elevations 

tend to have low yields. The curvature of the land 

surface determined from the 1:250,000-scale regional 

DEM (CURV250) has a negative coefficient of -2.15, 

indicating that, at a regional scale, concave downward 

surfaces are associated with low yields and concave 

upward surfaces are associated with higher-thanaverage yields. 

Distance to the nearest waterbody, determined 

using the 1:24,000-scale Digital Line Graph (DLG), 

was negatively related to lnYIELD, with a coefficient 

of -0.00012. This relation indicates that wells further 

away from waterbodies tend to have lower than 

average yields, although other variables in the regression equation must be taken into consideration for 

estimating yields. The drainage area to the well, 

expressed as the natural log of the number of 

1:24,000-scale DEM cells upgradient of the well 

(including the well cell, lnACCUM), has a positive 

coefficient of 0.024. This result indicates that large 

upgradient drainage areas are associated with higherthan-average yields, although other variables in the 

equation must be taken into consideration. 

FACTORS RELATED TO WELL YIELD 19
Table 2. Regression equation used by this study 

[Geologic units from Lyons and others (1997), Walsh and Clark (1999), and Thomas R. Armstrong and William C. Burton (U.S. Geological Survey, written 

commun., 2000); *, dichotomous indicator variable; DEM, Digital elevation model; DLG, Digital line graph] 

Coefficient Variable Explanation of variable name 

model mean square error = 1.84 

lnYIELD Dependent variable the natural log of yield, in gallons per minute 

= 0.53 lnDEPTH Natural log of well depth, in feet 

Topographic variables including proximity to surface water 

− .014 SLOPE Percent slope from 1:24,000-scale DEM 

− .00012 ELEV24 Elevation derived from 1:24,000-scale map, in feet above sea level 

− 2.15 CURV250 Curvature from 1:250,000-scale DEM 

− .00012 DISTH2O Distance to water, in feet, determined from the 1:24,000-scale DLG 

+ .024 lnACCUM Natural log of the number of 1:24,000-scale DEM grid cells uphill of the well 

+ .81 L7DRAW3 *Valley bottoms (1:24,000-scale DEM) lithologic group 7 (Massabesic Gneiss Complex and 

Breakfast Hill Granite) 

Major lithologic group 

− .27 LITHO2 *Major lithologic group 2, well-foliated igneous rocks 

Lineaments 

+ 1.23 D180BUF *Domain fracture-correlated lineaments from high-altitude 1:80,000-scale photographs 

+ .37 ALFRACP *Discrete fracture-correlated lineaments in plutons from four platforms 

+ .44 PAC15 *Statewide lineaments in plutons from angle category 15 (N30 oW to N40 oW) 

Geologic units from the Pinardville and Windham quadrangles 

− .14 SOB *Berwick Formation 

+ .61 SOBW *Well-bedded Berwick Formation 

+ .36 ZMLG *Layered paragneiss and orthogneiss 

+ 1.55 ZMA *Amphibolite gneiss 

+ .86 PZMG *Layered Migmatite 

− .44 PDG *Damon Pond Granite 

+ .41 DSG *Spaulding Tonalite 

+ .44 AXMBUF *Within 2,000 ft of the axis of the Massabesic Gneiss Complex anticline 

Geologic units from the State map 

+ .18 RKTYP101 *Perry Mountain Formation (Sp) 

− 1.32 RKTYP102 *Undivided Perry Mountain & Rangeley Formations (Spr) 

+ .36 RKTYP103 *Eliot Formation (SOe) 

− 1.14 RKTYP105 *Perry Mountain Formation member (Spvx) 

+ .62 RKTYP107 *Kittery Formation (SOk) 

− 1.13 RKTYP112 *Rangeley Formation member (Srup) 

− .36 RKTYP120 *Massabesic Gneiss Complex (Zmz) 

− .20 RKTYP121 *Member of Berwick Formation (SObc) 

− .52 RKTYP122 *Calef member of Eliot Formation (SOec) 

− .37 RKTYP125 *Ammonoosic Volcanics member (Oal) 

+ 0.48 RKTYP139 *Granite, granodiorite, trondhjemite (Oo1-3A) 

+ 1.66 RKTYP174 *Rye Complex (OZrz) 

+ .34 RKTYP45 *Two Mica Granite (D1m) 

+ 1.96 RKTYP53 *Metamorphosed gabbro, diorite, and basalt dikes (DS9) 

+ .22 RKTYP57 *Concord Granite (Dc1m) 

20 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
Table 2. Regression equation used by this study—Continued 

Explanation of variable name Coefficient Variable model mean square error = 1.84 

- 1.28 RKTYP58 

- .15 RKTYP66 

- 1.77 RKTYP72 

- .30 RKTYP83 

+ .66 RKTYP90 

+ .89 RKTYP91 

+ .61 RKTYP92 

+ .49 CABLE 

+ .38 COMERCL 

+ .93 PUBLIC 

− .80 

Geologic units from the State map--Cont. 

*Gile Mountain Formation (Dg)


*Undivided Littleton Formation (Dl)


*Littleton Formation member with volcanic lentils (Dlv)


*Winnipesaukee Tonalite (DW3A)


*Undivided Frontenac Formation undivided (Sfr)


*Frontenac Formation member (Sfrb)


*Frontenac Formation member (Srfc)


Well characteristics 

*Well drilled with a cable tool rig 

*Commercial supply well 

*Public supply well (note: domestic wells are those not identified as either public or commercial) 

End of equation 

Swales or valley bottoms (L7DRAW3) were 

identified using the 1:24,000-scale DEM. In major 

lithologic unit 7 (LITHO7) (Massabesic Gneiss 

Complex and Breakfast Hill Granite) a positive coefficient of 0.81 indicates that these sites tend to have 

higher-than-average well yields. 

The variable identifying well sites within 100 ft 

of lineaments that were identified using the highaltitude (1:80,000-scale) aerial photography, and also 

correlated with the primary fracture direction (regional 

analysis) (D180BUF), had a large positive coefficient 

of 1.23. This result indicates that wells near these 

lineaments tend to have high well yields. 

Similar to the description above, the indicator 

variable identifying well sites in plutons (lithologic 

groups 1, 2, and 7) within 100 ft of lineaments 

(ALFRACP), which were correlated with observed 

fractures in outcrops, measured within 1,000 ft of these 

lineaments, and identified by use of (1) Landsat; (2) 

high-altitude aerial photography; (3) color infra-red 

photography; or (4) topographic maps, had a positive 

coefficient of 0.37. This result indicates an association 

of these lineaments to higher-than-average well yields. 

The variable identifying well sites in plutons 

within 100 ft of lineaments trending N35o

W ±5 

degrees (PAC15) that were identified using Landsat, or 

high-altitude or low-altitude aerial photography, had a 

positive coefficient of 0.44. PAC15 sites are associated 

with high yields. Statewide, lineaments in this orientation seem to represent a group of fractures with higherthan-average yields, possibly the result of openness of 

these fractures or a similar method of formation. The 

degree of openness and hydrologic connectivity to 

horizontal fractures may be greater in the plutons than 

in metasedimentary rocks. 

Seven detailed geologic map units (Zma, SOb, 

SObw, Zmlg, PZmg, Pdg, and Dsg), identified during 

the detailed geologic mapping of the Pinardville and 

Windham quadrangles, were significant in the regression model. Well yields were either higher or lower 

than average depending on the geologic unit. 

Major lithologic unit 2 (LITHO2), foliated 

plutons, had a negative coefficient of -0.27 indicating 

that these areas tend to have low yields. Twenty-two of 

the 154 geologic map units, depicted on the Bedrock 

Geologic Map of New Hampshire (Lyons and others, 

1997), also were statistically significant at the 

95-percent confidence level. These mapped units have 

either a positive or negative effect depending on the 

unit. For example, mapped units in major lithologic 

category 2, LITHO2 (fig. 3), also are significant in the 

regression model (RKTYP139, RKTYP174, 

RKTYP45, RKTYP53, RKTYP57, and RKTYP83, 

table 2). These represent local refinements to the 

generalized rock groupings. The effect of these units in 

LITHO2 is essentially to either add or subtract from 

the overall negative coefficient of major LITHO2 unit. 
In addition to site characteristics, certain well 

characteristics also were determined to be statistically 

significant. CABLE, which identifies whether or not a 

well was drilled using a cable tool rig, has a positive 

relation to well yield. High yields among these drilled 

wells may be the result of the drilling technique in 

which the bit is pounded into the earth, perhaps 

opening up or enhancing fractures. Also, there may be 

less clogging of fractures by bedrock cuttings when 

the well is drilled with a cable-tool rig. 

Wells drilled for commercial or public water 

supply (COMERCL and PUBLIC) also are associated 

with high yields. These yields may be related to well 

construction characteristics, such as well diameter, 

that may increase the well yield at a given site to meet 

the demand. The fact that these wells tend to be deeper 

than private wells already is accounted for in the first 

stage of the regression. 

Other major variables, which individually were 

significant but dropped out as new variables were 

introduced, include (1) the distance of a well to any 

lineament in a pluton, (2) thickness of overburden, and 

(3) whether a well penetrates through stratified drift. 

With the introduction of variables, such as distance to 

surface-water bodies (DISTH2O) and upgradient 

drainage area (lnACCUM), these three variables 

dropped out of the model because of shared variance. 

Variables DISTH2O and lnACCUM were statistically 

more significant than those values that were dropped. 

The problem of shared variance is one of exclusion. If 

two variables share variance and both are included in 

the model and both remain significant, then there is no 

modeling problem. If, however, variables are excluded 

from the model because they fail to meet the 

95-percent confidence level because of shared 

variance, then there is the potential to exclude 

variables that should be included. This is an unavoidable consequence of any regression analysis. The 

model is stochastic or statistical, not a processoriented geologic, and cannot account for all physical 

processes involved, which may be best explained by 

some excluded variable. 

Regression Model Testing 

The statewide analysis of bedrock yields by the 

USGS involved developing a complex regression 

model. Large data sets were used to examine geospatial data. The resultant model then was subjected to six 

statistical tests (appendix C), and the results are 

summarized in the following paragraphs: 

1. First, a test was conducted to determine if the use of 

instrumental variables was necessary. A test was 

applied to evaluate the use of the instrumentalvariable technique to eliminate the effect of the 

inherent relation between well yield and depth. In 

applying this test, lnDEPTH was endogenous and 

a two-stage regression model using instrumental 

variables was appropriate. 

2. Second, the appropriateness of the form of the 

model was tested. A test of the model specification revealed that nine variables are heteroscedastic. That is, the distribution of the model 

residuals varies depending on the value of these 

nine variables. The effects of heteroscedastic 

variables on model probabilities is discussed in 

appendix C. The form of the model was 

appropriate. 

3. A test for normality of the residuals indicated that 

the graphical distribution is nearly normally 

distributed; however, at the 95-percent 

confidence level, the residuals are not normally 

distributed. For practical purposes, the model 

was considered appropriate for estimating 

probabilities of equaling or exceeding a given 

yield. 

4. Declustering of the data (statistically deleting some 

of the excess data where there is an over 

abundance) was not necessary on the basis of a 

spatial test of the residuals for the Pinardville 

quadrangle. Pinardville is the quadrangle with 

the highest density of well data in the State. 

Results of the spatial test did not show spatial 

significance, which means that the data are not 

related to those of neighboring wells in the most 

clustered area of the map, and would not be a 

potential source of bias. 

5. A verification test of model results was done. A 

summary Chow test (Chow, 1960) was used to 

compare the verification and primary data sets. 

The results indicate that the data sets are statistically distinguishable from one another by the 

model; they do not behave identically. However, 

a single variable, PZmg, accounts for most of the 

difference between the verification data set and 

the primary data-set (with coefficients of 

opposite sign between the two data sets). 

6. A sixth test, unique to the use of instrumental 

variables, was a test of finite sample bias. The 

test results showed no indication of significant 

finite-sample bias. 

22 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
Alternative models expressing yield and depth 

were considered. The final form of the selected model 

was considered appropriate because of the following 

reasons: 

1. The model adjusts for the severe endogenous bias 

associated with depth. 

2. The residuals are, for all practical purposes, log 

normally distributed indicating that the decision 

to express yield and depth in log space was 

appropriate. 

3. The errors are multiplicative (higher errors at higher 

yields) further indicating that a model with 

lnYIELD as the dependent variable (as opposed 

to yield) was appropriate. 

4. The specification tests demonstrate that the 

functional form was appropriate for the continuous variables. 

5. The functional form of the model was not an issue 

because only dichotomous variables were identified with the specification test. Heteroscedasticity was found for some dichotomous variables, 

but this can be considered when interpreting the 

results. 

Yield-Probability Forecasting 

The probability of obtaining a given yield can 

be forecast statewide for a given well depth. A 

statewide GRID of probabilities of equaling or 

exceeding 40 gal/min when a well depth of 400 ft is 

drilled is presented in figure 9. A well depth of 400 ft 

was used because this has become a common depth at 

which a driller stops drilling if the desired yield has 

not been reached. 

Maps of Yield Probability 

To apply the results of the regression model, a 

map is needed. Maps (figs. 9, 10, and 11) clearly show 

areas of various yield probabilities. Users then can use 

probability criteria to select areas where other 

techniques, such as geophysical techniques, can be 

applied to further refine site selection for drilling in 

the bedrock aquifer. Results of tests of geophysical 

techniques for ground-water exploration in the 

fractured-bedrock aquifer of New Hampshire are 

presented in a companion report by Degnan and others 

(2001). 

The regression model developed during this 

study was used to produce yield-probability maps, 

statewide (fig. 9) and for the Pinardville and Windham 

quadrangles, N.H. (plates 1 and 2). The maps were 

created by the following procedures. A grid of points, 

with a 98-ft (30-m) spacing, was created. For each 

point, all predictive parameters were compiled and 

model results applied. For each point, a probability of 

equaling or exceeding a given yield was calculated. 

The grid of points then was converted to a raster GRID 

(using ARC/INFO GRID software) and plotted on a 

map. The probability at each GRID cell was 

determined by using equations 1 and 2 and the 

predicted value of lnYIELD that was computed by the 

regression equation. The procedure used to estimate 

cell probability, expressed as a percentage, is 

Cell Probability = 100 * (1 – PROBNORM (Z)), (1) 

where 

PROBNORM (SAS computer software, 1996) is a 

function that returns the probability that an 

observation from a standard normal distribution is less than Z, 

Z is a function that defines the point where the 

conditional yield (to be equaled or 

exceeded) is relative to the predicted distributions of yields for each cell, and 

Z is normalized by the standard deviation of 

predicted yields for each cell. 

Z = (ln ( ) C ) – Y

ˆ ⁄ (MSE + BETAVAR ) , (2) 

where 

C is the conditional yield to be equaled or 

exceeded, 

Y

ˆ is the predicted natural log of yield for the 

cell (lnYIELD), 

MSE is the model mean square error, and 

BETAVAR	 is the variance associated with the coefficients. (This involves the covariance 

matrix and is dependent on the magnitude 

of the predictor variable). 

The map data are displayed in shades of color 

on the basis of the value of probability (figs. 9, 10A, 

and 11A, and plates 1 and 2). The effect of topography 

is reflected in the probability zones. Low probabilities 

FACTORS RELATED TO WELL YIELD 23
Figure 9. Model results of the probability of obtaining 40 gallons of water per minute from a 400-foot-deep 

well in New Hampshire. 

24 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
FACTORS RELATED TO WELL YIELD 25

Figure 10. Model results of the probability of obtaining 40 gallons of water per minute or more from a 400-foot-deep well for the Pinardville quadrangle, 

New Hampshire, with (A) detailed mapping variables included and (B) only the statewide variables included 
26 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 

Figure 11. Model results of the probability of obtaining 40 gallons of water per minute or more from a 400-foot-deep well for the Windham quadrangle, 

New Hampshire, with (A) detailed mapping variables included and (B) only the statewide variables included. 
on the steep slopes are represented by dark shades of 

orange. Lineaments and topographic swales 

(determined from the 1:24,000-scale DEM) are clearly 

highlighted in white or green. This result is especially 

true of the lineaments identified using the highaltitude aerial photography that are correlated with the 

primary fracture direction. These lineaments define 

most of the locations with the highest probability 

areas, which indicates that the regression model is 

sensitive to photolinear variables despite measurement 

error and shared variance. Results of this analysis 

indicate that using a filtered subset of the lineaments 

(based on lineament orientation and correlation with 

nearby fracture orientations) to identify potential highprobability areas is feasible elsewhere in New 

Hampshire. 

A broad swath of high probabilities associated 

with the anticlinal axis of the Massabesic Gneiss 

Complex (figs. 4 and 10A), across the southeastern 

quadrant of the map, is prominent. Likewise, the 

positive (light colors) and negative effect (dark 

orange) of the various detailed mapped units identified 

in the model are clearly evident. For example, the 

bedrock geologic units of the amphibolite gneiss 

(Zma), layered migmatite (PZmg), and Spaulding 

Tonalite (Dsg) (fig. 4) all are highlighted as zones of 

high-yield probabilities (fig. 10A and plate 1). 

Comparison of Regional Variables with 

Statewide Variables 

To measure the added value of field-based 

methods for well-yield-probability forecasting, the 

regression model was run with and without the 

variables developed from the detailed geologic 

mapping and fracture-correlated lineaments (Zma, 

SOb, SObw, Zmlg, PZmg, Pdg, Dsg, AXMBUF, 

D180BUF, AND ALFRACP (table 2)). Four 

probability maps were produced (figs. 10 and 11) for 

the Pinardville and Windham quadrangles, one with 

and one without the added variables for each 

quadrangle. These maps show the benefit of 

conducting additional detailed geologic mapping. 

The dark zones in the northwestern quadrant of 

plate 1 and figures 10A and 10B are associated with 

the steep slopes and concave downward curvature of 

the Uncanoonuc Mountains. Likewise, valleys and 

swales are evident as light zones in the Massabesic 

Gneissic Complex in the southeastern half of the plate. 

Lineaments trending N35oW in the igneous rocks 

(plutons) also are shown as distinct lines. Fracturecorrelated lineaments (trending in various directions), 

the zone of horizontal jointing associated with the 

anticlinal axis of the Massabesic Gneiss Complex, and 

the effects of five detailed geologic units only are 

evident on the map that included the detailed mapping 

variables (fig. 10A). 

Well-yield probability maps with and without 

the detailed mapping variables can be compared 

quantitatively (table 3). At any selected probability 

value, more sites are identified with the inclusion of 

the detailed mapping. For example, with a 20-percent 

probability of obtaining 40 gal/min from a 400-ft-deep 

well, more than 3 times as many sites could be located 

by including the variables from the detailed mapping. 

As the criterion is restricted to higher percentage 

probabilities, the benefit of using detailed mapping to 

identify areas increases. For example, at and above 

45-percent probability, the model with the excluded 

detailed geologic mapping variables does not identify 

any sites, whereas the model with the variables 

included continues to identify sites (table 3). The 

amount of detail in the probability maps also can be 

summarized in terms of the variance (or standard 

deviation) of predicted probability values. For the 

Pinardville quadrangle, the variance increased from 

12.43 to 40.56 with the inclusion of the detailed 

variables and the standard deviation increased from 

3.53 to 6.37. These wide ranges in estimated probabilities indicate that more high-probability sites are 

identified by including the detailed mapping variables. 

A similar comparison, as described in the 

previous paragraph, can be made in the Windham 

quadrangle (fig. 11). The effects of the statewide 

topographic variables are evident in figures 11A and 

11B. The dark orange zones with highly irregular 

shapes on the maps reflect the negative effects of hills. 

A dark orange zone also indicates the low probability 

associated with the Ayer granodiorite, part of LITHO2 

(fig. 3), along the southern boundary of the quadrangle 

(figs. 11A and B, plate 2) near the center. Similarly, 

the light zones associated with RKTYPE45 (D1m, the 

Two Mica Granite of northern and southeastern New 

Hampshire) are evident on figures 5 (unit Dg) and 

11A. Proximity to water is shown as light shades of 

color on both maps. The well-bedded subunit of the 

Berwick Formation (SObw) has high probabilities 

(lighter-shaded zone of the northwest corner 

(fig. 11A), which is not identified in the statewide 

map (fig. 11B). 

FACTORS RELATED TO WELL YIELD 27
Table 3. Percentage of area identified as having a 

probability of obtaining 40 gallons per minute or more from a 

400-foot-well depth with detailed geologic mapping variables 

included and excluded for the Pinardville quadrangle 

Probability of obtaining Percentage of quadrangle meeting 

40 gallons per minute or probability criteria with detailed 

more with 400-foot-well geologic mapping variables 

depth, 

in percent Included Excluded 

Pinardville quadrangle, New Hampshire 

20 7.94 2.33 

25 3.93 .80 

30 2.22 .10 

35 1.33 .02 

40 0.72 .01 

45 .39 0 

50 .21 0 

55 .11 0 

60 .06 0 

65 .02 0 

70 .013 0 

75 .004 0 

80 .002 0 

Various lineaments associated with the igneous 

rock show up as linear patterns of increased 

probability on both quadrangles (figs. 10A and 11A, 

plates 1 and 2). The highest probabilities identified in 

linear pattern on figures 10A and 11A, and plates 1 and 

2 are from lineaments identified using the high-altitude 

aerial photography; these lineaments are correlated 

with the primary fracture direction (regardless of 

bedrock type). 

A quantitative comparison of probability maps 

for the Windham quadrangle is given in table 4. More 

sites are identified at selected probability criteria with 

the detailed mapping. For this quadrangle, however, 

the benefits of the detailed geologic mapping seems to 

be even greater than for the Pinardville quadrangle. 

For example, 23 times as many sites are located with a 

20-percent probability of obtaining 40 gal/min with a 

400-ft-deep well, when the detailed geologic mapping 

variables were included. At or above 25-percent 

probability, the model, that excludes variables, fails to 

identify any sites. The model with the variables 

included continues to identify sites (table 4). The 

amount of detail in the probability maps also can be 

summarized in terms of the variance (or standard 

deviation) among predicted probability values. For the 

Windham quadrangle, the variance increased from 

3.64 to 26.74, with the inclusion of the detailed 

variables, and the standard deviation increased from 

1.91 to 5.17. 

Table 4. Percentage of area identified as having a 

probability of obtaining 40 gallons per minute or more from a 

400-foot-well depth with detailed geologic mapping variables 

included and excluded for the Windham quadrangle 

Probability of obtaining Percentage of quadrangle meeting 

40 gallons per minute or probability criteria with detailed 

more with 400-foot-well geologic mapping variables 

depth, 

in percent Included Excluded 

Windham quadrangle, New Hampshire 

20 10.23 0.44 

25 6.28 0 

30 1.33 0 

35 .62 0 

40 .40 0 

45 .18 0 

50 .065 0 

SUMMARY AND CONCLUSIONS 

The New Hampshire Bedrock Aquifer Assessment, done in cooperation with the New Hampshire 

Department of Environmental Services, was designed 

to provide information that can be used by communities, industry, professional consultants, and other 

interests to evaluate the ground-water development 

potential of the fractured-bedrock aquifer. This assessment was done at three scales—statewide, regional, 

and local—to uncover relations that potentially will 

increase the probability of successfully locating highyield water supplies in the fractured-bedrock aquifer 

throughout New Hampshire. The statewide scale was 

designed as a reconnaissance-level investigation of 

bedrock well yield in relation to bedrock type, 

lineament characteristics, topography, and other wellsite and construction characteristics. The regional 

scale adds field-based geologic-data collection, 

additional lineament data, and lineament filtering to 

assess what effect quadrangle-scale data acquisition 

has on the resulting well-yield relations. Local investigations were done to assess the effectiveness of 

geophysical tools for identifying the location and 

orientation of fracture zones. Results of the statewideand regional-scale investigations are described in this 

report. Results of the local investigations are presented 

in a companion report. 

In investigating the factors that relate to highyield bedrock-aquifer sites, many possible variables 

were examined. Statewide data sets were compiled 

from available sources or were created for this study. 

The primary information that was analyzed included 
well information, bedrock lithology, surficial geology, 

lineaments, topography, and various derivatives of 

information in those data sets. 

Additionally, more detailed geologic, fracture, 

and lineament data were collected for the Pinardville 

and Windham, New Hampshire quadrangles. These 

quadrangles were selected because they represent 

differing geohydrologic settings and have the largest 

number of georeferenced bedrock wells in the State. 

Geologic mapping at the 1:24,000 scale provided an 

accurate distribution of lithologies and data on brittle 

fracture and ductile-structure orientation of bedrock 

for correlation with lineaments. The purpose of this 

regional-scale investigation was to determine the 

degree to which predictive well-yield relations, 

developed as part of the statewide reconnaissance 

investigation, can be improved with quadrangle-scale 

mapping. 

Detailed information on water wells constructed 

in the State since 1984 is collected and maintained in a 

computer database by NHDES. Many of these wells 

have been field-inventoried to obtain accurate 

locations. Basic well data are reported by drillers and 

include the following six characteristics used in this 

study: well yield, well depth, well use, method of 

construction, date drilled, and depth to bedrock (or 

length of casing). The well data set compiled by 

NHDES contains complete records, including 

geographic coordinates for 20,308 of the total number 

of wells (over 70,000) reported since 1984. A 

randomly selected subset (20 percent or 4,050 wells) 

of the available well data were reserved for model 

evaluation. The availability of this large data set 

provided an opportunity for statistical analysis of 

variables related to bedrock well yields. Well yields in 

the database ranged from zero to greater than 

500 gal/min. 

Multivariate regression was selected as the 

primary method of analysis because it is the most 

efficient statistical tool for predicting a single variable 

(such as the natural log of well yield) with many 

potentially independent variables. The dependent 

variable that was analyzed in this study was the natural 

log of the reported well yield. Well yield, however, is 

in part a function of demand for water by the well 

owner. This restriction introduces a major problem in 

using well yield as the dependent variable in a 

multivariate-regression analysis. Generally, a well 

only is drilled as deep as is needed to meet a specific 

yield. Thus, the reported yield is not necessarily the 

maximum potential yield for that site. Because well 

yield is partially a function of demand, an innovative 

technique that involves the use of instrumental 

variables was used. Instrumental variables that are 

correlated to demand-driven yield but uncorrelated 

with random variation in the physical yield-depth 

relation are (1) year drilled, (2) well driller, and 

(3) median household income based on the 1990 

census. 

Results of the regression model show the 

following: 

1. The effect of the instrumental variable approach is 

apparent in the calculated regression coefficient 

for the natural log of depth (lnDEPTH). The 

regression coefficient is positive (0.53) indicating 

that total well yield increases with total well 

depth, but incremental yield (yield per foot of 

well depth) decreases with well depth. 

The previously demand-driven negative coefficient (-0.88) was corrected by a coefficient that 

represents the physical relation. Presumably at 

some depth (hundreds to thousands of feet) this 

relation ceases because of lithostatic pressure. 

2. Slope of the land surface determined from the 

1:24,000 scale DEM, has a negative coefficient 

implying that yields tend to decrease as slopes 

increase. Similarly, the elevation of the land 

surface at each well, derived from 1:24,000 or 

1:25,000-scale maps, has a negative coefficient 

of -0.00012, indicating that yields tend to 

decrease as elevation (in feet above sea level) 

increases. The curvature of the land surface 

determined from the 1:250,000-scale regional 

DEM has a negative coefficient of –2.15, 

indicating that, at a regional scale, wells on 

hilltops tend to have low yields and wells in 

valleys tend to have higher-than-average yields. 

The data indicate that valleys have more 

saturated fractures than hilltops. 

3. The distance to the nearest waterbody, determined 

using the 1:24,000 scale Digital Line Graph data 

(DLG), is negatively related to lnYIELD. This 

means that wells farther away from waterbodies 

tend to have lower yields than those closest to 

waterbodies. 

4. The drainage area to the well, expressed as the 

natural log of the number of 1:24,000-scale DEM 

cells upgradient of the well (including the well 

SUMMARY AND CONCLUSIONS 29
cell) (lnACCUM) has a positive coefficient of 

0.024. This value indicates that wells with large 

upgradient drainage areas tend to have higherthan-average yields. 

5. Swales or valley bottoms, identified using the 

1:24,000 scale DEM, in the Massabesic Gneiss 

Complex and Breakfast Hill Granite, tend to have 

higher-than-average yields. 

6. Major lithologic unit 2, foliated plutons, tended to 

have lower than average yields. 

7. Sites within 100 ft of lineaments, identified using 

the high altitude (1:80,000-scale) aerial photography that are correlated with the primary 

fracture direction (regional analysis), have a large 

positive coefficient of 1.23. This value indicates 

a strong association between these lineaments 

and higher-than-average well yields. 

8. The quadrangle-scale well sites in plutons within 

100 ft of fracture-correlated lineaments (discrete 

analysis), defined as a subset of all lineaments 

identified using Landsat, high-altitude aerial 

photography, color infra-red photography, or 

topographic maps, have higher-than-average 

yields. Perhaps sheeting (near-horizontal) 

fractures in plutons create more hydrologic 

connections between wells and adjacent 

lineaments. 

9. Sites in plutons within 100 ft of lineaments trending 

N35oW ±5 degrees, identified using Landsat, or 

high- or low-altitude aerial photography, have a 

positive coefficient of 0.44, indicating higherthan-average yields. 

10. Seven geologic map units (Zma, SOb, SObw, 

Zmlg, PZmg, Pdg, and Dsg) identified by 

detailed geologic mapping of the Pinardville and 

Windham quadrangles, had a relation to well 

yield. Well yields were statistically above or 

below average depending on the geologic unit. 

11. Twenty-two geologic map units on the Bedrock 

Geologic Map of New Hampshire also had a 

relation to well yield. These had either a positive 

or negative relation to yield depending on the 

type of unit. 

12. In addition to the above site characteristics, certain 

well-construction characteristics also were 

determined to be statistically significant. Wells 

drilled using a cable-tool rig were positively 

related to well yield. 

13. Wells drilled for commercial or public-supply 

purposes also tended to have higher-than-average 

yields. 

A statistical relation predicting well yield was 

developed to produce yield-probability maps. Plates 

show the yield probabilities for the Pinardville and 

Windham quadrangles. Probabilities are color coded 

on these maps. The results of the yield-probability 

maps indicate the following: 

1. The effect of topography is reflected in the 

probabilities. Probabilities of high yields 

decrease on the steep slopes. 

2. Specific categories of lineaments result in linear 

patterns of high probabilities on the maps. This is 

especially true of the lineaments identified 

using the high-altitude aerial photography 

(1:80,000 scale) that are correlated with the 

primary fracture direction. These lineaments 

define most of the locations with the highest 

probabilities in the Pinardville and Windham 

quadrangles. As a result of this analysis, the use 

of a filtered subset of lineaments appears to be a 

promising technique for identifying highprobability areas elsewhere in New Hampshire. 

3. A broad swath of high probabilities is associated 

with the anticlinal axis of the Massabesic Gneiss 

Complex across the southeastern quadrant of the 

Pinardville quadrangle. Likewise, the positive 

and negative effect of the variables representing 

various detailed lithologies, identified in the 

model, are evident. For example, the bedrock 

geologic units of the amphibolite gneiss (Zma), 

layered migmatite (PZmg), and Spaulding 

Tonalite (Dsg) are all represented as zones of 

high-yield probabilities. 

4. To measure the added value of using field-based 

methods for well-yield-probability forecasting, 

probabilities were estimated with and without the 

variables developed from the detailed geologic 

mapping and fracture-correlated lineaments. The 

probability maps provide a visual comparison for 

evaluating the benefit of including additional 

detailed geologic mapping variables. The results 

of including these variables for each quadrangle 

are as follows: 

30 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
4A. In the Pinardville quadrangle, more sites are 

identified at selected probability criteria with 

detailed mapping variables. For example, using 

the information from the detailed mapping, more 

than 3 times as many sites are associated with a 

20-percent probability of obtaining 40 gal/min 

from a 400-ft-deep well. As probability criteria 

are restricted to percentages higher than 

20 percent, the advantage of the detailed 

mapping in identifying additional areas of highyield increases. At and above 45-percent 

probability, the model using just the statewide 

variables fails to identify any sites, whereas the 

model using the detailed mapping variables 

continues to identify sites. 

4B. Similar results were found for the Windham 

quadrangle where more sites were identified 

using the detailed mapping variables. For this 

quadrangle, however, the increased percentage of 

area with a high probability of obtaining 

40 gal/min or more with detailed geologic 

mapping variables included, is even greater than 

for the Pinardville quadrangle. In areas with a 

20-percent or more probability of obtaining 

40 gal/min from a 400-ft-deep well, roughly 

23 times as many sites are identified using the 

information from the detailed mapping. At or 

above 25-percent probability, the model with just 

the statewide variables does not identify any 

sites, whereas the model with the detailed 

mapping variables continues to identify sites. 

The amount of detail in the probability maps 

also can be expressed in terms of the variance (or 

standard deviation) among predicted probabilities. For 

the Pinardville quadrangle, the variance describing the 

range of predicted probability values increased with 

the inclusion of the detailed mapping variables, from 

12.43 to 40.56, and the standard deviation increased 

from 3.53 to 6.37. For the Windham quadrangle, the 

variance increased from 3.64 to 26.74, and the 

standard deviation increased from 1.91 to 5.17. These 

wide ranges in probabilities indicate that more highprobability sites can be identified by including the 

detailed mapping. Results from both quadrangles 

clearly demonstrate the advantage of using detailed 

geologic mapping to identify potential new groundwater supplies in the fractured-bedrock aquifer of 

New Hampshire. 

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34 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
35 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
36 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
APPENDIX A


APPENDIX 35 
APPENDIX A. REGRESSION MODEL 

CONSTRUCTION 

The overall approach in deciding which 

variables to include in the model was as follows: 

1. Ordinary least-squares regression was used to 

provide an initial estimate of which variables are 

significant predictors of well depth for the first 

stage of the model. Significant predictors of the 

natural log of depth of the well then could be 

identified. 

2. These predictor variables then were used as the 

starting point for building the two-stage model. 

This process favors the inclusion of variables that 

affect well depth and yield. 

3. Predictor variables were removed from the twostage model if they were insignificant as predictors of the natural log of yield. 

4. The two-stage model was built by comparing the 

residuals with the remaining unused variables. 

Guided by the relation, new variables were 

introduced into the two-stage model. The major 

groups of lithologies, which cover all of New 

Hampshire, had priority over individual map 

units to make a more inclusive model. 

5. Regional variables were tested in the model, which 

created a nested model. Statewide mapped 

bedrock units were regionally subdivided into 

more detailed categories. The more general 

statewide variables remained active in those 

regions (Pinardville and Windham quadrangles) 

where detailed mapping was conducted. 

6. Additional variables tested in the model included 

uphill drainage area, proximity to surface water, 

driller (as an instrumental variable), construction 

method, and median household income (also as 

an instrumental variable). 

36 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
APPENDIX B


APPENDIX 37
Appendix B. List of variables tested for the regression model 

[DEM, Digital elevation model; DLG, Digital line graph] 

* Indicator Variables Definition of variable variable 

DEPENDENT VARIABLES 

YIELD Yield, in gallons per minute 

lnYIELD Natural log of yield 

YIELDFT Yield/length, in gallons per minute per foot of open hole 

PREDICTOR VARIABLES 

lnDEPTH Natural log of the well depth, in feet 

OVER Overburden depth to bedrock 

OPEN Length of open hole 

CSN Casing length 

IPLUTN Dichotomous variable identifying wells drilled into plutons * 

Seven major lithologic units 

LITHO1 Lithologic unit 1: medium to coarse-grained (nonfoliated to weakly foliated igneous rock) * 

LITHO2 Lithologic unit 2: well-foliated, coarse-grained igneous rock * 

LITHO3 Lithologic unit 3: fine-grained, tightly bonded metasedimentary rock * 

LITHO4 Lithologic unit 4: fine to medium-grained granular, metasedimentary rock; * 

LITHO5 Lithologic unit 5: graphitic micaceous metapelite, locally interbedded with calcareous or * 

siliceous rock 

LITHO6 Lithologic unit 6: fine and coarse-grained, volcanic rock and associated metapelite * 

LITHO7 Lithologic unit 7: Massabesic Gneiss Complex and Breakfast Hill Granite * 

All bedrock units, with well data, are shown on the 1997 State Bedrock Geologic Map (Lyons and others, 1977) 

Bedrock unit 

RKTYP2 –Cjb * 

RKTYP3 Dir * 

RKTYP4 Dif * 

RKTYP5 K1a * 

RKTYP6 K1bx * 

RKTYP7 K1r * 

RKTYP8 K2 * 

RKTYP9 K7C * 

RKTYP14 Kc1b * 

RKTYP21 J1r * 

RKTYP22 J1x * 

RKTYP23 J7h * 

RKTYP25 J4hx * 

RKTYP27 J5 * 

RKTYP28 Sfrv * 

RKTYP29 Oalx * 

RKTYP30 J7x * 

RKTYP31 J8 * 

38 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
[DEM, Digital elevation model; DLG, Digital line graph]

Appendix B. List of variables tested for the regression model—Continued 

Variables Definition of variable * Indicator 

variable 

Bedrock unit--Continued 

RKTYP32 J9A * 

RKTYP35 Jc1b * 

RKTYP37 Jo1b * 

RKTYP38 Jo1h * 

RKTYP39 P1m * 

RKTYP40 Oalb * 

RKTYP43 D1b * 

RKTYP44 Oalg * 

RKTYP45 D1m * 

RKTYP49 Oh1-2h * 

RKTYP50 D3Bb * 

RKTYP52 Sfrx * 

RKTYP53 DS9 * 

RKTYP54 DSlr * 

RKTYP55 Db2b * 

RKTYP57 Dc1m * 

RKTYP58 Dg * 

RKTYP59 Dgc * 

RKTYP60 Oh2-9A * 

RKTYP61 Dgm * 

RKTYP62 Dgv * 

RKTYP63 Di * 

RKTYP64 Dih * 

RKTYP65 Dk2x * 

RKTYP66 Dl * 

RKTYP67 Dlc * 

RKTYP69 Oo4Ch * 

RKTYP70 Dll * 

RKTYP71 Dlu * 

RKTYP72 Dlv * 

RKTYP74 Oalf * 

RKTYP75 Oals * 

RKTYP77 Ds1-6 * 

RKTYP78 Ds6-9B * 

RKTYP79 D3Ab * 

RKTYP80 Oaux * 

RKTYP81 PM1m * 

RKTYP82 K4x * 

RKTYP83 Dw3A * 

APPENDIX 39
[DEM, Digital elevation model; DLG, Digital line graph]

Appendix B. List of variables tested for the regression model—Continued 

Variables Definition of variable * Indicator 

variable 

Bedrock unit--Continued 

RKTYP86 Sa2x * 

RKTYP87 Sc * 

RKTYP88 Sf * 

RKTYP89 Sfc * 

RKTYP90 Sfr * 

RKTYP91 Sfrb * 

RKTYP92 Sfrc * 

RKTYP93 Sfrg * 

RKTYP95 Sg * 

RKTYP96 SObg * 

RKTYP98 Smsf * 

RKTYP99 SOb * 

RKTYP100 Dc3Am * 

RKTYP101 Sp * 

RKTYP102 Spr * 

RKTYP103 SOe * 

RKTYP104 Spvs * 

RKTYP105 Spvx * 

RKTYP106 Sr * 

RKTYP107 SOk * 

RKTYP108 Src * 

RKTYP109 Srl * 

RKTYP110 Srlp * 

RKTYP111 Sru * 

RKTYP112 Srup * 

RKTYP114 Ssf * 

RKTYP115 Ssfb * 

RKTYP118 De9 * 

RKTYP119 Ssfx * 

RKTYP120 Zmz * 

RKTYP121 SObc * 

RKTYP123 Sn2-3A * 

RKTYP124 Sn1x * 

RKTYP128 Oaus * 

RKTYP136 Oh2h * 

RKTYP139 Oo1-3A * 

RKTYP140 Oo1-3B * 

RKTYP142 Oo1b * 

RKTYP144 Oo1h * 

40 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
, v p

Appendix B. List of variables tested for the regression model—Continued 

[DEM Digital ele ation model; DLG, Digital line gra h] 

* Indicator Variables Definition of variable variable 

Bedrock unit--Continued 

RKTYP145 Oo2-3A * 

RKTYP146 Oo2b * 

RKTYP147 Oo2bx * 

RKTYP148 Oo2h * 

RKTYP149 Oo3B * 

RKTYP150 Oo3B-6 * 

RKTYP152 Oo4-7h * 

RKTYP159 Opv * 

RKTYP160 Oq * 

RKTYP161 O-Cd * 

RKTYP162 O-Cdp * 

RKTYP163 O-Czl * 

RKTYP164 O-Czu * 

RKTYP173 OZrb * 

RKTYP174 OZrz * 

Topographic variables 

SLOPE Percent slope derived from 1:24,000-scale DEM 

SECDRV Curvature derived from 1:24,000-scale DEM 

SLOPE250 Slope derived from 1:250,000-scale DEM 

CURV250 Curvature derived from 1:250,000-scale DEM 

HILLTOP Derived from 1:24,000-scale DEM * 

HILLTOP2 Derived from 1:250,000-scale DEM * 

DRAW2 Valley bottom or swale derived from 1:250,000-scale DEM * 

DRAW3 Valley bottom or swale derived from 1:24,000-scale DEM * 

HILLSIDE Derived from 1:24,000-scale DEM * 

HILLSID2 Derived from 1:250,000-scale DEM * 

ELEV24 Elevation derived from 24:000-scale map or DEM 

lnACCUM Natural log of the number of DEM grid cell uphill of the well 

ACCUM The number of DEM grid cell uphill of the well 

DISTH20 Distance to surface water, in feet determined from the 1:24,000-scale DLG 

LNDSTH20 Natural log of the distance to surface water 

SDCODE Stratified drift * 

Proximity to lineaments (within 100 feet) 

DSTLO Proximity to low-altitude lineaments * 

DSTLSAT Proximity to Landsat lineaments * 

DSTHI Proximity to high-altitude lineaments * 

DSTALL3 Proximity to all three * 

APPENDIX 41
[DEM, Digital elevation model; DLG, Digital line graph]

Appendix B. List of variables tested for the regression model—Continued 

* Indicator Variables Definition of variable variable 

Interactions between major lithologic groupings and proximity to lineaments 

L1HI L2HI L3HI L4HI L5HI L6HI * 

L7HI L1LO L2LO L3LO L4LO L5LO L6LO L7LO * 

L1LSAT L2LSAT L3LSAT L4LSAT L5LSAT * 

L6LSAT L7LSAT R29HI * 

Interactions between plutons and lineaments 

PLUTHI * 

PLUTLO * 

PLUTLSAT * 

PLUTALL3 * 

Interactions between metasediments and lineaments 

METAHI * 

METALO * 

METALSAT * 

METALL3 * 

Interaction between plutons, lineaments, and 10-degree orientation categories 

PAC1-PAC18 * 

Interaction between metasediments, lineaments, and 10-degree orientation categories 

MAC1-MAC18 * 

Interactions between rock units and proximity to lineaments 

R29LO R29LSAT R35HI R35LO R35LSAT R38HI R38LO R38LSAT R39HI R39LO * 

R39LSAT R45HI R45LO R45LSAT R55HI R55LSAT R57HI R57LO R57LSAT * 

R66HI R66LO R66LSAT R71HI R71LO R71LSAT R77HI R77LO R77LSAT R81HI * 

R81LO R83HI R83LO R83LSAT R86HI R86LO R86LSAT R97HI R97LO * 

R97LSAT R101HI R101LO R101LSAT R102HI R102LO R102LSAT R103HI R103LO * 

R103LSAT R107HI R107LO R107LSAT R114HI R114LO R114LSAT R118HI R118LO * 

R118LSAT R120HI R120LO R120LSAT R121HI R121LO R121LSAT R122HI * 

R122LO R122LSAT ... * 

Interactions between major lithologic groupings and topographic variables 

L1SDRV L2SDRV L3SDRV L4SDRV L5SDRV L6SDRV L7SDRV * 

L1HILL L2HILL L3HILL * 

L4HILL L5HILL L6HILL L7HILL L1HLL2 L2HLL2 L3HLL2 L4HLL2 L5HLL2 L6HLL2 L7HLL2 * 

L1DRAW L2DRAW L3DRAW L4DRAW L5DRAW L6DRAW L7DRAW * 

L1DRAW3 L2DRAW3 L3DRAW3 L4DRAW3 L5DRAW3 L6DRAW3 L7DRAW3 * 

Water use 

COMERCL Commercial supply well * 

PUBLIC Public supply well * 

Construction method 

CABLE Well drilled with a cable tool rig * 

42 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
[DEM, Digital elevation model; DLG, Digital line graph]

Appendix B. List of variables tested for the regression model—Continued 

* Indicator Variables Definition of variable variable 

Demand instruments 

YEAR Year well drilled * 

INCOME Median household income from 1990 census data * 

DRILLER1-DRILLER25 Drillers in the State with the most number of wells drilled * 

REGIONAL VARIABLES 

Lineaments (The following lineament variables restricted to just plutons or just metasediments) 

TOPOBUF Within 100 feet of a lineament identified by the use of the topographic contours * 

CIRBUF Within 100 feet of a lineament identified by the use of color infrared photographs * 

Fracture-correlated lineaments - DOMAIN ANALYSIS (primary fracture set) 

D180BUF Within 100 feet of a 80,000 lineament -- domain analysis * 

D120BUF Within 100 feet of a 20,000 lineament -- domain analysis * 

D1LSTBUF Within 100 feet of a Landsat lineament -- domain analysis * 

D1TPOBUF Within 100 feet of a topographic lineament -- domain analysis * 

D1CIRBUF Within 100 feet of a color infrared lineament -- domain analysis * 

Fracture-correlated lineaments - DOMAIN ANALYSIS (any peak greater than 30 percent of primary 

peak) 

D380BUF Within 100 feet of a 80,000 lineament -- domain analysis * 

D320BUF Within 100 feet of a 20,000 lineament -- domain analysis * 

D3LSTBUF Within 100 feet of a Landsat lineament -- domain analysis * 

D3TPOBUF Within 100 feet of a topographic lineament -- domain analysis * 

D3CIRBUF Within 100 feet of a color infrared lineament -- domain analysis * 

Fracture-correlated lineaments - DOMAIN ANALYSIS (any peak greater than 50 percent of primary peak) 

D580BUF Within 100 feet of a 80,000 lineament -- domain analysis * 

D520BUF Within 100 feet of a 20,000 lineament -- domain analysis * 

D5LSTBUF Within 100 feet of a Landsat lineament -- domain analysis * 

D5TPOBUF Within 100 feet of a topographic lineament -- domain analysis * 

D5CIRBUF Within 100 feet of a color infrared lineament -- domain analysis * 

Fracture-correlated lineaments - 1,000 foot buffer technique 

K80FRACT High-altitude lineament * 

K20FRACT Low-altitude lineament * 

LSTFRACT Landsat lineament * 

TPOFRACT Topographic map lineament * 

CIRFRACT Color infrared lineament * 

ALLFRACT High-altitude, Landsat, topographic, or color infrared lineament * 

APPENDIX 43
 Appendix B. List of variables tested for the regression model—Continued 

Variables Bedrock 

unit Definition of variable * Indicator variable 

Geologic units mapped in the Pinardville and Windham quadrangles, New Hampshire 

DG Dg Binary granite and granitic pegmatite * 

DMG Dmg Muscovite granite * 

DP Dp Pegmatite * 

DSG Dsg Spaulding granite * 

MD Md Diabase and lamprophyre dike 

MDP Mdp Porphyritic diabase dike * 

PDMG PDmg Migmatite gneiss * 

PZMG PZmg Layered migmatite and restite * 

PDG Pdg Damon Pond granite * 

PPG Ppg Granite and pegmatite * 

SOB SOb Biotite-plagioclase-quartz granofels and schist and calc-silicate rock * 

SOBQ SObq Quartzite, biotite-plagioclase-quartz granofels, and calc-silicate rock * 

SOBW SObw Well bedded biotite-plagioclase-quartz granofels and schist and calc-silicate rock * 

SAG Sag Porphyritic granite to granodiorite * 

SAGD Sagd Granodiorite * 

SAGDG Sagdg Garnetiferous granodiorite * 

SR Sr Light-gray biotite-muscovite schist * 

SRB Srb Biotite schist and granofels * 

SRLB Srlb Layered biotite granofels * 

SRMG Srmg Migmatite schist and gneiss * 

SRQ Srq Rusty quartzite and schist * 

SRR Srr Rusty-biotite-muscovite schist * 

ZMA Zma Amphibolite gneiss * 

ZMLG Zmlg Layered paragneiss and orthogneiss * 

CNTXBUF Within 100 feet of a mapped contact * 

AXMBUF Within 100 feet of a mapped fold axis * 

DIKBUF Within 100 feet of a mapped dike * 

FAUBUF Within 100 feet of a mapped fault * 

ISOBUF Within 100 feet of a mapped isograd * 

AXMBUF Within 2,000 feet of the axis of the Massabesic Gneiss Complex anticline * 

REFERENCE CITED 

Lyons, J.B., Bothner, W.A., Moench, R.H., and Thompson, J.B., Jr., 1997, Bedrock geologic map of New Hampshire: 

U.S. Geological Survey State Geologic Map, 2 sheets, scale 1: 250,000 and 1:500,000. 

44 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
APPENDIX C


APPENDIX 45
APPENDIX C. 

TESTS 

REGRESSION MODEL 

The statewide analysis of bedrock yields by this 

project involved developing a complex statistical 

model. Large data sets were used to examine geospacial data, and the resultant model was subjected to a 

variety of statistical tests. 

Test for an Endogenous Variable 

A test was applied to evaluate the use of the 

instrumental variable technique to eliminate the effect 

of the inherent relation between well yield and depth. 

The use of instrumental variables is not a preferred 

method of estimation if the predictor variables are not 

correlated with the error term. Thus, in order to 

determine the appropriateness of this method, a statistical test of the predictor variable, natural logarithm of 

depth, was required to determine if it is an endogenous 

variable. The test used is described by Hausman 

(1978) and evaluates the null hypothesis that the 

regressors and the residual are independent. Under the 

null hypothesis, coefficient estimates obtained by 

either ordinary least squares (OLS), or by instrumental 

variables, are consistent and unbiased in large 

samples. Also, under the null hypothesis, the method 

of OLS produces more precise coefficient estimates 

than do instrumental variables. The test consists of 

obtaining two sets of estimates of the coefficients, one 

estimated by OLS and the other estimated using 

instrumental variables, and determining if the difference in the coefficient estimates is significantly 

different from zero. 

An “omitted variables” form of the test is 

described by Godfrey (1988, p.194-195). The original 

regression equation is augmented with additional 

explanatory variables, which consist of the residuals of 

the potentially endogenous variable (lnDEPTH) 

determined from an OLS regression of lnDEPTH on 

the original set of instrumental variables not under 

investigation. The augmented equation is estimated 

using the instrumental-variable technique with instruments consisting of the original set, plus the 

potentially endogenous variables. If the coefficients 

associated with the augmented variables are jointly 

significant, as determined by a likelihood-ratio-test 

statistic distributed as chisquare with degrees of 

freedom equal to the number of augmented variables 

(1), then the null hypothesis that the potentially 

endogenous variables are exogenous and can be 

rejected. 

Applying the test described in the previous 

paragraph, lnDEPTH is shown to be endogenous, and 

the use of a two-stage model with instrumental 

variables is appropriate. Results of the test yield a chisquare test statistic of 521 with one degree of freedom, 

and a significance of 0.00. As the significance is less 

than 0.05, the null hypothesis that lnDEPTH is 

exogenous was rejected. 

Specification Test 

A “well-specified” model (Godfrey, 1988) is 

one where the functional form is correct and the 

residuals are homoscedastic (for example, the distribution of the residuals are similar throughout the range 

of predicted values). The coefficient estimates and the 

associated estimated standard errors are unbiased. 

Concurrently, predictions from the model will be 

unbiased and the standard formula for evaluating 

prediction accuracy will be correct. 

There are many tests that can be used to validate 

model specification. Godfrey (1988, chap. 4) presents 

a comprehensive description of specification tests. The 

test used (White, 1982; p. 1-25) for the regression 

model addresses the issue of specification if there is no 

explicit alternative model under consideration. The 

test determines if squared values of residuals are 

correlated with instrumental variables. The test is 

sensitive to misspecification in the regression equation 

and heteroscedasticity in the data and will likely 

trigger a false positive indication of misspecification, 

particularly if there are a lot of instrumental variables 

used. 

The test is significant for the model, as 

determined by its F statistic, indicating that there is 

evidence of heteroscedasticity. The F statistic is 194.5 

with a p-value less than 0.0001. The null hypothesis 

that there is no misspecification is rejected. As most 

predictors are dichotomous, the likely cause of the 

rejection is heteroscedasticity. All nine predictors of 

lnYIELD, which were identified as significant by the 

test (table C1), are dichotomous. In other words, there 

are significantly different variances between populations where the indicator variables are 1 and where 

they are 0. The weight column in table C1 represents 

the ratio of the variance of the residuals, where the 

variable is zero, divided by the variance of the 

residuals, where the variable is one. 

46 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
Table C1. Significant predictors in the specification test of 

the model used in this study 

[All variables are defined in table 2] 

Variable Coefficient Significance Weight ratio 

SOb 0.45 0.0001 0.82 

Zmlg -.45 .0199 .95 

PZmg -1.13 .0180 1.49 

RKTYP103 -.40 .0291 1.22 

RKTYP107 -.64 .0101 1.39 

RKTYP120 .69 .0001 .79 

RKTYP57 -.25 .0165 1.16 

CABLE -1.06 .0001 2.33 

LITHO2 .15 .0203 1.00 

The practical implications of the tests described 

previously are that estimated probabilities will be 

somewhat over or underestimated for sites in the 

following areas. For those sites where the weight ratio 

is less than one (SOb, Zmlg, and RKTYP120), the 

probability of obtaining a given yield tends to be 

underestimated for probabilities less than 50 percent 

and overestimated for probabilities greater than 

50 percent. For those sites where the weight is greater 

than one (PZmg, RKTYP103, RKTYP107, and 

RKTYP57), probabilities tend to be overestimated for 

probabilities less than 50 percent and underestimated 

for probabilities greater than 50 percent. The weight 

for LITHO2 is close to 1 and the probability estimates 

should be consistent if LITHO2 is 0 or 1. There also is 

much less variance among wells drilled by cable tool 

than for wells drilled using other techniques. In 

predicting probabilities, however, it is assumed that the 

site is not to be drilled by cable-tool construction, and 

an adjustment of the probabilities is not necessary. 

Residual Statistical Distribution 

Testing for normality of the residuals is 

important with a regression model. The cumulative 

distribution function of the residuals was determined 

by use of SAS/INSIGHT software. Confidence bands, 

at the 95-percent confidence level, also were 

determined using SAS/INSIGHT. The Kolmogorov 

statistic D, the maximum vertical distance between the 

two distribution functions, was then used to test the 

null hypothesis that the population distribution is 

normally distributed. In the normality test, the null 

hypothesis is rejected because at some point, the 

normal distribution falls outside the confidence band 

(fig. C1). 

Although the Kolmogorov statistic D test 

indicates that the residuals are not normally distributed, 

at the 95-percent confidence level, the graphical distribution (fig. C1 (A and B)) is near normal. Thus, for 

practical purposes, the model is considered appropriate 

for estimating probabilities equaling or exceeding a 

given yield. 

Figure C1. Distribution of model residuals for New Hampshire shown as a (A) bell curve and (B) cumulative curve. 

Kolmogorov statistic D is 0.0209 with a probability greater than D of less than 0.01. 
Residual Spatial Distribution and Spatial 

Analysis 

The spatial distribution of residuals was 

evaluated to see if there is a need for further regionalization of the model. Statewide maps of the residuals 

were made to see if positive or negative residuals 

(color coded) are grouped in regions. A non-uniform 

coverage of well distribution can cause bias in a 

misspecified regression model, and there are areas in 

southern New Hampshire with significantly higher 

densities of data than in the rest of the State. The 

regression model, however, accommodates this by 

using predictor variables that describe the well-site 

characteristics. Areas with large amounts of data in 

southern New Hampshire potentially could dominate 

the equation. If this happened, then a pattern should be 

apparent on statewide plots of the residuals maps. 

Areas with few data would show a bias positively or 

negatively, and this was not observed in the map. 

Instead, an even distribution of positive and negative 

residuals generally were displayed. 

Locally, spatial distributions also were 

evaluated in the Pinardville quadrangle where 

numerous wells are clustered. There is a similarity of 

well yields with medium and low yields (less than 

40 gal/min) for wells aligned in a direction northnortheast to one another among and in specific 

geologic units (Drew and others, 1999). The hypothesis of the model described in this report, however, is 

that spatial features inherent in the predictor variables 

explain systematic spatial features of the yield data. If 

this hypothesis is correct, then the residuals from the 

regression model will be independent. If it is not 

correct, then standard errors of the regression coefficients will be biased downward and predictions of 

well yield at a given site could be improved by 

incorporating known yield data at neighboring sites. 

This result can be tested by examining the spatial 

distribution of the model residuals. 

Declustering of the data was not necessary, on 

the basis of a spatial test of the residuals for the 

Pinardville 7.5-minute quadrangle (fig. C2). Pinardville is the quadrangle with the highest density of data 

in the State. Results of the test did not show spatial 

significance, therefore, corrections to the model for 

this potential bias are not considered necessary. 

Estimates of spatial correlation for model 

residuals in the Pinardville 7.5-minute quadrangle are 

shown in figure C2. Correlations are computed by 

binning the separation distances between pairs of 

wells. The results for binning separation distances into 

100 cells across the quadrangle are shown in figure 

C2. The results of the estimates show little evidence of 

spatial correlation or of any directional preference in 

residual correlations. All correlations are small even at 

fine scales such as a 1,000-cell analysis (not shown). 

Numerical correlations for the center of the 

separation space are listed in table C2. For the 100-cell 

analysis, correlations are all statistically insignificant. 

Each correlation in the table is based on at least 

1,332 pairs. There are few correlations that exceed the 

significance threshold (or absolute value of the 

correlation greater than 1 over the square root of n, 

where n is the number of paired residuals used to 

compute the correlation for a given cell—the 

minimum n is 337). There is no recognizable pattern to 

the correlogram (fig. C2), and the statistical significance of certain correlations appears random. A 1,000-

cell analysis (not shown) has fewer observations per 

cell than a 100-cell range. Again, none of these results 

showed evidence of a spatial pattern. The site characteristics that are used as input to the regression model, 

including the additional regional variables, account for 

the spatial relation observed between well yields; 

therefore, the spatial correlation of the model residuals 

is not a concern. 

Comparison of Verification and Primary 

Data Sets 

There are a number of tests available for 

comparing the results of a primary data set with those 

of a verification data set. Chow (1960) provides a test 

to determine if there is a significant difference in the 

coefficient estimates of two independent regressions. 

Two sets of data combined into a common data set. A 

test model is then constructed in which each coefficient in the model is estimated with a companion 

coefficient that is multiplied by a verification data-set 

indicator. For example, the effect of depth in the test 

model would be represented as 

(b dI + ) ln DEPTH , (3) 

48 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
APPENDIX 49

Figure C2. A correlogram showing the spatial correlation of model residuals that have a low-resolution binning (100 cells across) for 

the Pinardville quadrangle, New Hampshire. A spatial correlation pattern is not apparent.

EXPLANATION

Separation ranges for low-resolution binning (100 cells across the quadrangle)

 -0.72 to -0.47

 -0.46 to -0.23

 -0.22 to -0.02

Table C2. Numerical correlations of the model residuals for the Pinardville quadrangle, New Hampshire, for distance 

separations less than 1 kilometer

[The correlations in greater detail for x and y spatial separations less than 1 kilometer (km); * = correlations that exceed the significance threshold (absolute 

value of the correlation greater than 1 over the square root of n, where n is the number of paired residuals used to compute the correlation for a given cell; 

the minimum n is 337)]

Vertical distance separation 

(km)

 Horizontal separation 

(km)

-0.91 -0.71 -0.50 -0.30 -0.10 0.10 0.30 0.50 0.71 0.91

0.98 0.00 -0.07 0.01 -0.05 0.08 0.01 0.02 0.07 0.01 -0.03

.88 .07 -.02 .00 .06 -.05 .06 -.03 .04 -.01 -.02

.77 -.04 -.04 .08 -.01 -.02 .03 .01 .00 .03 -.07

.67 -.05 -.04 .06 .00 -.04 .03 .05 .05 .08 -.04

.57 .03 .04 .02 -.00 .05 -.01 .04 -.02 -.05 .00

.46 .02 -.06 -.06 -.06 -.03 -.07 .01 .01 -.03 .09

.36 -.00 .01 .06 -.11 * -.01 -.01 .05 .03 -.02 -.00

.26 .04 .12 * .02 .07 -.05 .00 .02 .01 -.05 -.01

.15 -.06 .01 -.03 -.03 -.01 .02 -.00 -.00 -.03 -.04

.05 .03 -.02 .03 .01 -.00 .07 * .04 -.03 .09 .16 *
where 

b is the depth coefficient, 

d is the depth companion coefficient, 

I is the indicator variable for the verification 

data set (I equals 1 if the observation is 

from the verification data set and equals 

zero otherwise), and 

lnDEPTH is the regressor natural log (ln) of depth. 

The test rejects the null hypothesis that the 

results are consistent across data sets if the companion 

coefficients are jointly significant. This rejection is 

determined by a likelihood-ratio statistic, in which 

distributed chi-square with degrees of freedom are 

equal to the number of companion coefficients 

estimated. Coefficients for only two of the predictor 

variables were found to be jointly significant at the 

95-percent confidence level when applying the above 

test to the pooled data set. These two variables are 

PZmg (layered migmatite of the Massabesic Gneiss 

Complex), and RKTYP72 (Dlv, Littleton formation 

with volcanic lentils). With 44 variables, and a 

95-percent confidence level, two or three variables 

should test significant randomly. 

Using a summary Chow test, however, to 

compare the two subsamples, indicates that there is a 

statistically significant difference between the verification and the primary data sets. The computed Chow 

test statistic is 64.46, with 45 degrees of freedom, and 

a significance level of 0.030. The null hypothesis that 

the subsamples are the same is rejected because the 

significance level is less than 0.05. The variable, 

which accounts for the largest difference between the 

sequestered data and the primary data set is PZmg 

(with coefficients of opposite sign between the two 

data sets). Eliminating this variable from the model 

would decrease the Chow statistic to 39.90 with a 

significance of 0.65, in which case the null hypothesis 

that the subsamples are the same would be accepted. 

Finite Bias Test 

The model also is tested for finite-sample bias. 

Problems may be present with instrumental-variables 

estimation if the relation between the instruments and 

the endogenous variable are weak (Bound and others, 

1995). According to Bound, weak instruments may 

lead to bias in the coefficients even if there is a large 

sample such as we have in this study. The bias is in the 

same direction as the bias under ordinary least squares 

(for example, the coefficient for lnDEPTH would be 

less than its true value). To test for this bias, an F 

statistic for the significance of the “excluded” instruments, the year, income, and driller variables is 

computed. This statistic is equal to 

2 F = ((n – k) ⁄ q)(R2 – Rr ) ⁄ (1 – R2 ) , (4) 

where 

n is the number of observations (16,302), 

k is the number of instruments for the 

unrestricted regression (for example, the 

total number of instruments) (70) 

q	 is the number of “excluded” instruments 

(for example, the number of instruments 

not used in the second stage of the model) 

(23), 

R2	 is the r-square for the unrestricted regression (the R-square for the first-stage 

regression of depth on all the instruments 

including the public supply and commercial indicator variables and all the other 

exogenous variables that appear in the 

yield equation) (0.202259), and 

R2 is the r-square for the restricted regression r 

(the R-square for a regression of depth on 

all the exogenous variables in the yield 

equation but not the excluded variables) 

(0.148000). 

Given these values, the value for the F statistic 

is 48.00. According to Bound and others (1995), 48.00 

implies a finite sample bias of an instrumental 

variables coefficient estimate relative to the OLS bias 

of 0.00 (Bound and others, 1995, table A.1, k = 2 and 

t

2

/k (= F) = 100). Thus, the test shows no indication of 

significant finite-sample bias. 

Overall Statement 

The model performs well given these test. As 

expected, the variable lnDEPTH was endogenous. 

Predicted probabilities may be over or underestimated 

in specific locations associated with certain heteroscedastic variables. However, the form of the model is 

appropriate. The residuals are near normally distributed, and spatial declustering of the data is not 

necessary. Also, the verification data set (with the 

possible exception of variable PZmg) indicates that 

the data set used to calibrate the model is composed of 

representative data. Lastly, there is no indication of 

significant finite-sample bias. 

50 Factors Related to Well Yield in the Fractured-Bedrock Aquifer of New Hampshire 
REFERENCES CITED 

Bound, John, Jaeger, D.A., and Baker, R.M., 1995, 

Problems with instrumental variables estimation when 

the correlation between the instruments and the 

endogenous explanatory variable is weak: Journal of 

the American Statistical Association, v. 90, no. 430, 

p. 443-50. 

Chow, G.C., 1960, Tests of equality between sets of coefficients in two linear regressions: Econometrica, v. 28, 

p. 591-605. 

Drew, L.J., Karlinger, M.R., Armstrong, T.R., and 

Moore, R.B., 1999, Relations between igneous and 

metamorphic rock, fracture patterns, and ground-water 

yield from the variography of water-well yields, 

Pinardville quadrangle, New Hampshire: Natural 

Resources Research, v. 8, no. 2, p. 137-152. 

Godfrey, L.G., 1988, Misspecification tests in econometrics: The Lagrange Multiplier Principle and other 

approaches: New York, Cambridge University Press, 

252 p. 

Hausman, J., 1978, Specification tests in econometrics: 

Econometrica, v. 46, p. 1,251-1,271. 

White, H., 1982, Maximum likelihood estimation of 

misspecified models: Econometrica, 50, p. 1-25. 

This product was prepared by Ann Marie Squillacci (Layout and Design), Debra H. Foster (Editing), and 

Anita Cotton (Illustrations) in the U.S. Geological Survey New Hampshire/Vermont District, Pembroke, 

New Hampshire. 

APPENDIX 51