U.S. Department of the Interior

U.S. Geological Survey

Techniques and Methods 4–F4

Version 1.1, July 2016 

Advanced Methods for Modeling Water-Levels and 

Estimating Drawdowns with SeriesSEE, an 

Excel Add-In
Cover art: Screen showing SeriesSEE toolbar and example 

workbook that was created with SeriesSEE
Advanced Methods for Modeling WaterLevels and Estimating Drawdowns with 

SeriesSEE, an Excel Add-In

By Keith Halford, C. Amanda Garcia, Joe Fenelon, and Benjamin Mirus

Techniques and Methods 4–F4

Version 1.1, July 2016 

U.S. Department of the Interior

U.S. Geological Survey

U. S. Department of Energy, National Nuclear Security Administration, 

Environmental Restoration Program, Underground Test Area Project 
U.S. Department of the Interior

KEN SALAZAR, Secretary

U.S. Geological Survey

Marcia K. McNutt, Director

U.S. Geological Survey, Reston, Virginia 

First release: 2012, online 

Revised: July 2016 (ver. 1.1), online 

For more information on the USGS—the Federal source for science about the Earth, its natural and living 

resources, natural hazards, and the environment—visit http://www.usgs.gov or call 1–888–ASK–USGS.

For an overview of USGS information products, including maps, imagery, and publications, 

visit http://store.usgs.gov.

Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the 

U.S. Government.

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

reproduce any copyrighted materials contained within this report.

Suggested citation:

Halford, K., Garcia, C.A., Fenelon, J., and Mirus, B., 2012, Advanced methods for modeling water-levels and 

estimating drawdowns with SeriesSEE, an Excel add-In, (ver. 1.1, July, 2016): U.S. Geological Survey Techniques and 

Methods 4–F4, 28 p., http://dx.doi.org/10.3133/tm4F4. 

ISSN 2331-7047 (online) 
iii

Preface

This report documents a spreadsheet add-in for viewing time series and modeling water 

levels that was developed in Microsoft® Excel 2010. Use of trade names does not constitute 

endorsement by the U.S. Geological Survey (USGS). The spreadsheet add-in has been tested 

for accuracy by using multiple datasets. If users find or suspect errors, please contact the USGS. 

Every effort has been made by the USGS or the United States Government to ensure the 

spreadsheet add-in is error free. Even so, errors possibly exist in the spreadsheet add-in. The 

distribution of the spreadsheet add-in does not constitute any warranty by the USGS, and no 

responsibility is assumed by the USGS in connection therewith. 
iv

Acknowledgments

The report was prepared in cooperation with the U.S. Department of Energy, National 

Nuclear Security Administration, Nevada Site Office, Office of Environmental Management 

under Interagency Agreement DE-AI52-12NA30865. 

The tide-challenged authors are indebted to Devin Galloway for clarifying the ebb and flow 

of tides. 
v

Contents

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

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

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

Environmental Fluctuations......................................................................................................................... 2

Barometric Effects .............................................................................................................................. 2

Tidal Effects .......................................................................................................................................... 3

Background Water Levels ................................................................................................................. 3

Water-Level Modeling ................................................................................................................................. 4

Water-Level Model Components ...................................................................................................... 5

Moving Average ......................................................................................................................... 5

Theis Transform .......................................................................................................................... 7

Computed Tides .......................................................................................................................... 8

Step Change ................................................................................................................................ 9

Pneumatic Lag ............................................................................................................................ 9

Gamma Transform .................................................................................................................... 11

Calibration........................................................................................................................................... 12

Drawdown Estimation ...................................................................................................................... 13

SeriesSEE .................................................................................................................................................... 15

Data Requirements ........................................................................................................................... 16

Supporting Utilities ........................................................................................................................... 16

Water-Level Modeling ...................................................................................................................... 18

Applications of Water-Level Modeling ................................................................................................... 19

Hypothetical Example ....................................................................................................................... 20

Pahute Mesa Example ...................................................................................................................... 24

Water-Level Modeling Strategies ........................................................................................................... 25

Summary and Conclusions ....................................................................................................................... 25

References .................................................................................................................................................. 26

Appendix A. SeriesSEE add-in ................................................................................................................. 29

Appendix B. Source Codes for SeriesSEE ............................................................................................. 29

Appendix C. Verification of Analytical Solutions .................................................................................. 29

Appendix D. Hypothetical Test of Theis Transforms ............................................................................ 29

Appendix E. Pahute Mesa Example ........................................................................................................ 29
vi

Figures

1. Graphs showing daily precipitation, groundwater levels, barometric change, and 

earth tide at Air Force Plant 6, Marietta, Georgia, April 22 to May 28, 2004 ..................... 3

2. Graphs showing input series of barometric pressure, input series of background 

water level, and computed gravity tide ................................................................................... 4

3. Graphs showing two time series with different collection frequencies and 

sampling times ............................................................................................................................. 5

4. Graphs showing input series and four additional water-level model components that 

were created by averaging in periods of 0.5, 1, 2, and 4 days ............................................. 6

5. Graphs showing theis transform of a pumping schedule to water-level changes at 

radial distances between 1,250 and 10,000 feet from a pumping well for a fixed 

transmissivity and storage coefficient .................................................................................... 7

6. Schematics of one-dimensional, confined aquifer and an areally extensive, thick 

unsaturated zone that experience similar step-changes to a time-varying 

specified-head boundary such as a river or barometric-pressure difference ................. 9

7. Graphs showing average daily barometric pressure and simulated air pressure 

at the water table ...................................................................................................................... 10

8. Graphs showing an infiltration schedule and water-level rises simulated with 

gamma transforms that were defined by six pairs of shape (n) and scale (k) 

parameters ................................................................................................................................. 11

9. Graphs showing estimated drawdown from summing Theis transforms and 

subtracting residuals ................................................................................................................ 13

10. Graphs showing discharge from pumping wells ER-20-8 upper and ER-20-8 lower, 

estimated drawdowns, residuals, RMS errors, and signal-to-noise ratios in 

observation wells ER-EC-12 shallow and ER-EC-6 deep..................................................... 14

11. Screen showing SeriesSEE toolbar and example workbook that was created with 

SeriesSEE.................................................................................................................................... 15

12. Screen showing format of headers and values for creating a viewer file with 

SeriesSEE.................................................................................................................................... 16

13. Graphs showing shifting series to a common reference with the offset utility.................. 18

14. Screen showing table of contents and an explanation page in the help system for 

SeriesSEE ................................................................................................................................... 19

15. Map showing background wells, observation wells, pumping well, and selected 

fault structures at Pahute Mesa Nevada National Security Site...................................... 21

16. Illustration showing hydraulic conductivity distribution of a subset of a hypothetical 

aquifer system that has been bisected by a fault, showing well locations and 

labeled quadrants ..................................................................................................................... 22

17. Graphs showing barometric pressure, background water levels, and water levels 

with known drawdowns in hypothetical well O3 ................................................................. 23

18. Graphs showing known drawdowns (MODFLOW), drawdowns estimated from 

“measured” water levels, and drawdowns estimated directly from MODFLOW 

results in well O3 ....................................................................................................................... 23

19. Graphs showing measured water levels, synthetic water levels, Theis transforms, 

and estimated drawdowns in well ER-20-7 from pumping ER-20-8 main upper and 

lower zones, Pahute Mesa, Nevada National Security Site .............................................. 24
vii

Tables

1. Water-level model components......................................................................................... 5

2. Abbreviations and descriptions of tides that are computed in SeriesSEE ........................... 9

3. Summary of estimable parameters and parameter groups for water-level 

modeling components ...................................................................................................12

4. Summary of available tools in SeriesSEE ................................................................................. 17

5. Summary of verification tests for analytical models in the FORTRAN program 

WLmodel ..................................................................................................................................... 18

6. Site information and completion depths for wells at Pahute Mesa, Nevada National 

Security Site that were used in hypothetical example and field investigation ............. 20
viii

Conversion Factors and Datums

Multiply By To obtain

Length

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Volume

gallon (gal) 3.785 liter (L) 

Flow rate

gallon per minute (gal/min) 0.06309 liter per second (L/s)

Transmissivity*

foot squared per day (ft2

/d) 0.09290 meter squared per day (m2/d) 

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

(NAVD 88).

Horizontal coordinate information is referenced to the North American Datum of 1983 (NAD 83).

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

*Transmissivity: The standard unit for transmissivity is cubic foot per day per square foot times 

foot of aquifer thickness [(ft3

/d)/ft2]ft. In this report, the mathematically reduced form, foot 

squared per day (ft2/d), is used for convenience.
Advanced Methods for Modeling Water-Levels and 

Estimating Drawdowns with SeriesSEE, an Excel Add-In

By Keith Halford, C. Amanda Garcia, Joe Fenelon, and Benjamin Mirus

Abstract

Water-level modeling is used for multiple-well aquifer 

tests to reliably differentiate pumping responses from natural 

water-level changes in wells, or “environmental fluctuations.” 

Synthetic water levels are created during water-level modeling and represent the summation of multiple component 

fluctuations, including those caused by environmental forcing 

and pumping. Pumping signals are modeled by transforming 

step-wise pumping records into water-level changes by using 

superimposed Theis functions. Water-levels can be modeled 

robustly with this Theis-transform approach because environmental fluctuations and pumping signals are simulated 

simultaneously. Water-level modeling with Theis transforms 

has been implemented in the program SeriesSEE, which is a 

Microsoft® Excel add-in. Moving average, Theis, pneumaticlag, and gamma functions transform time series of measured 

values into water-level model components in SeriesSEE. Earth 

tides and step transforms are additional computed water-level 

model components. Water-level models are calibrated by minimizing a sum-of-squares objective function where singular 

value decomposition and Tikhonov regularization stabilize 

results. Drawdown estimates from a water-level model are the 

summation of all Theis transforms minus residual differences 

between synthetic and measured water levels. The accuracy 

of drawdown estimates is limited primarily by noise in the 

data sets, not the Theis-transform approach. Drawdowns much 

smaller than environmental fluctuations have been detected 

across major fault structures, at distances of more than 1 mile 

from the pumping well, and with limited pre-pumping and 

recovery data at sites across the United States. In addition to 

water-level modeling, utilities exist in SeriesSEE for viewing, 

cleaning, manipulating, and analyzing time-series data. 

Introduction

Multiple-well, aquifer testing provides the most direct, 

integrated assessment of bulk hydraulic properties within complex geologic systems (Bohling and others, 2003; Sepúlveda, 

2006; Yeh and Lee, 2007; Walton, 2008). The aquifer volume investigated with multi-well aquifer tests increases with 

increasing distance at which drawdown, or the pumping 

signal, can be detected (Risser and Bird, 2003; Halford and 

Yobbi, 2006). Drawdown analyses at distances of more than 

1 mile (mi) often fail because environmental water-level fluctuations typically overwhelm the pumping signal. Barometric 

change, tidal forces, surface-water stage changes, or other 

external stresses induce these natural water-level changes in 

wells, which collectively are referred to here as “environmental fluctuations.” 

Barometric change and tidal forces can induce water-level 

fluctuations in a well greater than 1 foot (ft) during periods 

of less than a few days (Fenelon, 2000). Daily barometric 

changes alone typically exceed 0.3 ft where aquifers are 

confined or the unsaturated zone is thicker than 500 ft (Weeks, 

1979; Merritt, 2004). Episodic recharge events can cause 

water-level rises that exceed 1 ft (O’Reilly, 1998). Climatic 

variations in recharge can induce long-term rising trends 

of more than 3 feet per year that affect detection of small 

pumping signals (Elliott and Fenelon, 2010; Fenelon, 2000). 

Drawdowns can be a fraction of the environmental fluctuations 

in distant observation wells that are more than a mile from a 

pumping well. 

Environmental fluctuations have been modeled previously 

to differentiate natural water-level changes from pumping 

responses. Barometric and tidal effects typically are modeled 

independently and removed from water-level records (Erskine, 

1991; Rasmussen and Crawford, 1997; Toll and Rasmussen, 

2007). These approaches do not remove regional trends, such 

as long-term recharge, and are difficult to automate because all 

significant stresses that affect water levels other than pumping 

are not simulated simultaneously. 

Water levels from background wells can be used to 

explicitly model water-level changes from recharge responses, 

surface-water stage changes, or any other external stress 

(Halford, 2006; Criss and Criss, 2011). A background well 

monitors water levels that are affected by tidal potential-rock 

interaction, imperfect barometric coupling, and all other 

stresses, excluding analyzed pumping, that affect water 

levels in observation wells. The need for antecedent data and 

background water levels has long been recognized (Stallman, 

1971), but these trends and corrections typically have been 

estimated qualitatively. 
2 Advanced Methods for Modeling Water-Levels and Estimating Drawdowns with SeriesSEE, an Excel Add-In

Environmental fluctuations can be simulated as synthetic 

water levels, which represent the summation of multiple time 

series of barometric-pressure change, tidal potential, and background water levels, if available (Halford, 2006). Synthetic 

water levels are fitted to measured water levels for a period 

just prior to pumping, which should be more than three times 

greater than the period affected by pumping (Halford, 2006). 

Amplitude and phase of each time series are adjusted to minimize differences between synthetic and measured water levels. 

These synthetic water levels are projected into the pumping 

period, and drawdown is the difference between synthetic and 

measured water levels. This approach is referred to here as the 

“projection approach” to water-level modeling. The projection approach becomes unreliable where most of the analyzed 

period is affected by pumping. 

Simultaneous modeling of environmental fluctuations 

and pumping signals overcomes the limitations of long-term 

extrapolation by using the projection approach. Environmental 

fluctuations can be defined during the entire period of record, 

which includes pumping and prolonged recovery periods. 

Variable pumping rates, as defined by a schedule of step 

changes, can be transformed to pumping signals by superimposing multiple Theis functions (Theis, 1935). Simultaneous 

simulation of all significant stresses affecting water-level 

changes is discussed as the “Theis-transform approach” to 

water-level modeling. 

These water-level modeling approaches have been implemented in the program SeriesSEE, which is a Microsoft®

Excel add-in. Water levels to be modeled, component fluctuations, and period of analysis are defined interactively and 

viewed in workbooks that are created by SeriesSEE. Water 

levels are modeled with a FORTRAN program that is called 

from Excel. Differences between synthetic and measured 

water levels are minimized with PEST (Doherty, 2010a and 

2010b). Water-level models are calibrated rapidly because 

PEST files are created and executed seamlessly. 

Water-level modeling with SeriesSEE differs from existing 

applications that filter environmental fluctuations or simulate 

pumping (Toll and Rasmussen, 2007; Harp and Vesselinov, 

2011). This is because models of environmental fluctuations, 

Theis transforms, and parameter estimation are integrated in 

SeriesSEE. BETCO (barometric and earth tide correction) and 

similar programs simulate barometric and tidal water-level 

fluctuations but not regional trends and pumping effects (Toll 

and Rasmussen, 2007). Theis transforms have been applied 

previously in other water-level models, but environmental 

fluctuations were simulated with linear trends (Harp and Vesselinov, 2011). 

Purpose and Scope

The purpose of this report is to document the approach 

used in SeriesSEE. This is the supporting software for modeling water levels that respond to environmental fluctuations and 

pumping. Water levels are modeled so pumping signals can be 

differentiated from environmental fluctuations. A method for 

fitting these water-level models to measured series by adjusting the selected parameters of each component is reported. 

The spreadsheet add-in is compatible with Microsoft® Excel 

2010 (version 14.0) or higher. Use of the spreadsheet add-in 

requires basic knowledge of Excel. Use and applicability of 

this software is documented in this report. The hydrologic 

concepts and methods used in the data processing also are 

described briefly.

Environmental Fluctuations

Environmental fluctuations in measured water levels, or 

natural water-level changes, can be modeled by using pertinent time series, such as barometric pressure, tidal potential, 

background water levels, and stream stage. These time series 

represent potential components used to create synthetic water 

levels in a water-level model. Relevant components can be 

selected where a relation is expected with the water-level 

record. For example, water-level fluctuations in well b4mwh 

appear to be related to earth tide, barometric pressure fluctuations, recharge, and pumping (fig. 1). Simulating these 

environmental fluctuations in well b4mwh requires that earth 

tide, barometric pressure, and background water level (wells 

rw204 and sct4) components are included so that synthetic 

water levels can replicate measured water levels. 

Barometric Effects

Barometric pressure induced water-level fluctuations are 

greatest in deep, confined aquifers where the rock matrix 

absorbs most of the atmospheric load (Merritt, 2004). Fluctuations increase because pressure instantly affects water 

levels in wells, whereas a stiff rock matrix transfers little of 

the increased atmospheric load to the confined water column. 

Atmospherically induced water-level fluctuations typically 

are less than 0.2 ft during a day. Large barometric-pressure 

changes from regional storms can cause water-level fluctuations of more than 1 ft during a week.

Barometric changes also measurably affect water levels 

in unconfined aquifers (Weeks, 1979). Pressure changes do 

not propagate instantaneously through the unsaturated zone 

because air is highly compressible. The relatively low pneumatic diffusivity of the unsaturated zone creates substantial 

phase lags between atmospheric and water-level changes. 

Unconfined water-level fluctuations can approach the magnitude of confined water-level fluctuations where the depth 

to water exceeds 500 ft. This is because atmospheric loading 

through the wellbore is not balanced by diffusion through the 

unsaturated zone. 
Environmental Fluctuations 3

Tidal Effects

Tidal forces distort the crust of the earth, which creates 

water-level fluctuations in mid-continent wells (Bredehoeft, 

1967; Marine, 1975; Hanson and Owen, 1982; Narasimhan 

and others, 1984). Earth tides periodically deform (dilate and 

compress) the skeleton of the aquifer system, changing the 

porosity and causing measurable water-level fluctuations of as 

much as 0.1 ft or more in wells penetrating aquifers with small 

storage coefficients (fig. 1). Coupling between the mechanical 

deformation and the fluid filling the secondary porosity amplifies water-level response in wells hydraulically connected to 

the secondary-porosity features, such as fractures or faults. 

The presence of secondary porosity typically renders the 

formation more compliant to imposed stresses, depending on 

orientation of the fractures or faults with respect to the principal component directions of the imposed stress. The theoretical crustal strain tensors that result from the two principal 

lunar daily and semidiurnal tides are largely horizontal and 

orthogonal to one another. Subvertical fractures with azimuths 

oriented perpendicular to the strain tensor for a particular 

tide tend to amplify the strain and, thereby, the water-level 

response (Bower, 1983). 

The diurnal rise and fall of ocean levels are the most common manifestation of varying gravitational forces and are 

referred to as ocean tides. Ocean tides affect coastal groundwater levels through direct head changes in an aquifer or as 

loads applied through a confining unit (Merritt, 2004). Oceantide effects are better approximated with a nearby tidal gage 

than calculated tides because wind and coastal geometry also 

affect ocean tides in addition to direct gravitational forcing.

Background Water Levels

Recharge events, regional pumping, and change in surfacewater stage are identifiable stresses that typically affect large 

areas but are not predicted easily with independent time series 

such as barometric change and tidal potential. Recharge events 

and regional pumping stresses can create similar water-level 

Figure 1. Daily precipitation, groundwater levels, barometric change, and earth tide at Air 

Force Plant 6, Marietta, Georgia, April 22 to May 28, 2004. 

0

1

2

3

0

1

2

Water level change, in feet

Daily precipitation,

in inches

Well b4mwh

Well rw204

Well sct4

Barometric Pressure

Earth Tide

4/22 4/29 5/6 5/13 5/20 5/27

2004 Modified from Halford, 2006
4 Advanced Methods for Modeling Water-Levels and Estimating Drawdowns with SeriesSEE, an Excel Add-In

changes in multiple wells over areas of many square miles. 

Change in surface-water stages locally affects groundwater 

levels and can be measured directly. Water levels in wells 

sufficiently removed from an aquifer test can simulate these 

regional stresses, local changes in surface-water stages, and 

any other unidentified pervasive stresses. Water levels in 

these remote wells are referred to as background water levels 

(Halford, 2006).

Background water levels can be more effective correctors 

than independent barometric and tidal time series even where 

only barometric and tidal stresses are significant (Halford, 

2006). Barometric forcing through the unsaturated zone lags 

behind water-level changes because of the small permeability 

of unsaturated rock relative to an open well (Weeks, 1979). 

The complex relation between barometric pressure and water 

level in a well is explained poorly with barometric efficiency 

where the unsaturated zone is thick. Background water levels 

from another well of similar construction better approximate 

this relation. Likewise, rock properties and fracture orientation 

in an aquifer control tidal water-level fluctuations as much as 

tidal forcing. Water levels from background wells can better 

approximate the rock-tide interaction than theoretical tidal 

components alone. Independent barometric and tidal time 

series frequently remain necessary because of differences in 

rock properties, fracture orientation, and well completions 

around measured and background wells. 

Water-Level Modeling

Water-level modeling assumes that measured water-level 

fluctuations can be approximated by summing multiple-component fluctuations (Halford, 2006). Input series of barometric 

pressure, input series of background water levels, and computed earth tides explain most environmental fluctuations 

(fig. 2). Pumping signals are simulated with multiple Theis 

solutions that transform pumping schedules to water-level 

fluctuations. 

Water-level model components are summed to create a 

synthetic water level. A synthetic water level at time, t, is 

determined:

SWL(t) 0

1

n

i

i=

= + C WLMC ∑ (1)

where 

C0 is an offset (L) that allows mean values of 

synthetic water levels to match mean 

values of measured water levels, 

n is the number of water-level model (WLM) 

components, and 

WLMCi is the ith WLM component in units of the 

modeled water level.

Water-level model results are denoted with the word 

synthetic rather than simulated to differentiate between waterlevel and groundwater-flow model results. 

Figure 2. Input series of barometric pressure, input series of background water level, and computed gravity tide. Water-level change, in feet

2011

0

0.5

1.0

5/1 5/8 5/15 5/22 5/29

Barometric Pressure

Background

Gravity Tide
Water-Level Modeling 5

Water-Level Model Components

Input series are measured water levels, barometric pressures, or pumping schedules that are transformed to represent 

water-level change. All input series are assumed to be continuous between each discrete measurement where continuity can 

be piecewise linear or stepwise. Water levels and barometric 

pressures typically are used as piecewise linear functions. 

Pumping schedules typically are used as stepwise functions. 

All input series are transformed into WLM components that 

are smooth, differentiable functions. 

WLM components are created from input series with one 

of six transforms. The parameters that define each transform 

generically are referred to as coefficients because characteristics and terminology are not consistent among transforms 

(table 1). Moving averages are most frequently used to transform interpolated time series of barometric pressure and background water levels into WLM components. Pumping schedules are transformed into water-level fluctuations with Theis 

transforms. Earth tides are computed for a given observation 

well location (Harrison, 1971). Transducer displacement, as a 

result of resetting a transducer in a well, is simulated with the 

step transform following a user-specified time. Lag and attenuation of barometric-pressure changes between land surface and 

water table are simulated with the pneumatic-lag transform. 

Water-level rises from infiltration events are simulated with 

the gamma transform. 

WLM components are smooth functions because values 

are interpolated linearly between consecutive data pairs or 

transformed from stepwise data to a smooth function. Interpolation or transformation allows data to be collected at variable 

intervals within a time series. Collection frequencies can differ 

among time series and do not need to be synchronized because 

interpolation or transformation synchronizes comparisons 

(fig. 3).

Moving Average 

Fluctuations of different frequencies exist in input series 

such as barometric changes and background water levels. 

Barometric changes exhibit diurnal, weekly, and seasonal 

fluctuations that differ in amplitude and frequency. Frequencydependent differences in water-level fluctuations also exist 

between wells because of differences in well construction and 

aquifer properties. Diurnal water-level fluctuations will be less 

where communication between well and aquifer is impeded 

and wellbore storage is increased. Poorly developed wells with 

large casing diameters and short screens damp high-frequency 

water-level fluctuations. Aquifers with large storage coefficients and small transmissivity values also will damp waterlevel fluctuations. 

Table 1. Water-level model (WLM) components. 

[— is not applicable]

WLM

component

Time

series

Coefficient

1 2 3 4 5

Moving average Any series Multiplier Phase Averaging period — —

Theis transform Pumping scheduleTransmissivity Storage coefficientRadial distance Flow-rate

conversion

—

Tide Computed Multiplier Phase Latitude Longitude Altitude

Step — Time Offset — — —

Pneumatic lag a Barometric pressureKAIR SAIR Thickness of unsaturated zone

— —

Gamma1 Infiltration Multiplier k n Time conversion Multiplication 

series

a Hydraulic properties of the Pneumatic-lag transform, KAIR & SAIR, are with respect to air. KAIR is hydraulic conductivity of air and is about 60 times 

greater than KWATER.SAIR is average air-filled porosity divided by mean air pressure.

1 The k and n terms represent scale and shape parameters, respectively in the Gamma Probability Distribution Function.

0

0.05

0.10

0:00 6:00 12:00

Water level change, in feet 

Time, in hour:minute

Series 1

Series 2

Figure 3. Two time series with different collection frequencies 

and sampling times.
6 Advanced Methods for Modeling Water-Levels and Estimating Drawdowns with SeriesSEE, an Excel Add-In

Input series frequently are composed of multiple signals 

of different frequencies. These different frequencies can be 

separated into multiple WLM components with multiple 

moving averages of the input series (fig. 4). Water levels can 

be averaged over periods of hours to days where duration of 

averaging periods and the number of WLM components are 

arbitrary quantities. More than a half dozen WLM components 

frequently are created from a single input series because a 

broad range of averaging periods are more likely to simulate 

the environmental fluctuations. An excess of WLM components generally does not degrade results. High-frequency signals are approximated indirectly by summing multiple WLM 

components with ranges of averaging periods. The original 

input series and WLM component are one and the same where 

an averaging period of 0 is specified (table 1).

The moving-average transform is applied to ith WLM component at time, t:

( ) WLMC aV t i ii i = +φ (2)

where

ai is the amplitude multiplier of the ith

component in units of the modeled water 

level divided by units of the ith component,

Φi is the phase-shift of the ith component (t), and 

Vi(t+Φi) is the value of the moving average of ith

input series at time t+ Φi in units of ith

component.

Amplitude (a) and phase (Φ) are estimated in equation 

2 to minimize differences between synthetic and measured 

water-levels.

Moving averages are centered about the evaluation time, t, 

where averaging periods are defined by time, not the number 

of measurements. For example, a 12-hr, moving average at 

the time when sampling increased from hourly to 15-minute 

measurements would average 31 values. Six values were measured prior to the evaluation time, another value was measured 

at the evaluation time, and 24 values were measured after the 

evaluation time.

Figure 4. Input series and four additional water-level model components that were created 

by averaging in periods of 0.5, 1, 2, and 4 days (d). 

Input series

Average 0.5 d

Average 1.0 d

Average 2.0 d

Average 4.0 d

0.2

0.4

0.6

Water-level change, in feet

2010

09/03 09/06 09/09 09/12

0
Water-Level Modeling 7

Theis Transform

Pumping schedules are converted into water-level 

responses with a simple model: the Theis (1935) solution. 

Water-level changes or drawdown, s, from pumping-rate 

changes are simulated:

    = = =

2

( ) 4 4 4 i π π T T T t   ∆

WLMC s W u W Q Q r S

(3)

where 

Q is the flow rate (L3/t), 

T is the transmissivity (L2/t), 

W(u) is the exponential integral solution, 

u is dimensionless time, 

r is the radius (L), 

S is the storage coefficient (dimensionless), and 

Δt is the elapsed time since the flow rate 

changed (t).

Multiple Theis solutions are superimposed in time to 

simulate water-level responses to variable pumping schedules 

(fig. 5). The effects of multiple pumping wells also can be 

simulated by superposition in space (Harp and Vesselinov, 

2011). Each pumping well with its unique pumping schedule 

and radial distance is simulated with a WLM component in 

SeriesSEE. Pumping signals are discussed here as drawdowns, 

regardless of pumping rate, because discrete drawdown and 

recovery periods do not exist when variable pumping schedules are simulated.

Superimposed Theis solutions serve as transform functions, where step-wise pumping records are translated into 

approximate water-level responses at observation wells. Logtransforms of transmissivity (T) and storage coefficient (S) 

are estimated in equation 3 to minimize differences between 

synthetic and measured water-levels. Estimates of T and S can 

characterize correctly the hydraulic properties of an aquifer 

if assumptions of the Theis solution are honored. These same 

Figure 5. Theis transform of a pumping schedule to water-level changes at radial distances between 1,250 

and 10,000 feet from a pumping well for a fixed transmissivity and storage coefficient.

04/29 05/06 05/13 05/20 05/27

2010

Water-level change, in feet

–1.0

–0.5

0

Radius = 1,250 feet

Radius = 2,500 feet

Radius = 5,000 feet

Radius = 10,000 feet

0

200

400

Discharge, in gallons per minute

Pumping Schedule

 Transmissivity = 25,000 feet squared per day

 Storage Coefficient = 0.005 dimensionless
8 Advanced Methods for Modeling Water-Levels and Estimating Drawdowns with SeriesSEE, an Excel Add-In

parameters primarily are fitting terms with little physical 

significance in hydrogeologically complex aquifer systems 

because assumptions of the Theis solution are violated. This 

component of the water-level model is referred to as a “Theis 

transform,” here, and applies to the pumping schedule of a 

single well.

Hydrogeologic complexity and uncertainty are addressed 

by applying multiple Theis transforms to a single pumping schedule. Relatively fast and slow elements of pumping 

signals propagate through complex aquifer systems. These fast 

and slow elements are approximated by Theis transforms with 

relatively high and low hydraulic diffusivities, respectively.

Computed Tides

The tides are displacements of the particles in a celestial 

body caused by the forces of attraction in a neighboring body. 

The terrestrial tides on Earth consist of the atmospheric tides, 

the earth tides, and the ocean tides and are related to the lunar 

and solar cycles (Defant, 1958). Simulated tidal forcing and 

body tides of a solid Earth (oceanless) produced by the moon 

and sun are computed from gravitational and astronomical 

theory for a specified point on the Earth for a specified time by 

using the Harrison (1971) model. Changes in the solid Earth 

caused by the ocean tides are not considered here. Many of the 

model parameters, and thus the computed tidal components, 

are functions of time based on the ephemerides, which are 

computed in the model but are not included here explicitly.

The earth tides result as the crust undergoes volumetric 

strains, ƐV , due to variations in tide-generating forces:

( ) 1

3 V θθ λλ rr ε εεε = ++ (4)

where, ƐƟƟ, Ɛλλ, and Ɛrr (positive downwards) represent the 

principal components of the strain-tide tensor with respect to 

polar north, east, and radial, respectively. Most of the stress 

close to the Earth’s surface is plane stress, and the resultant 

strain tide is predominately an areal strain, ƐA (Melchior, 1966:

( ) 1

2 A θθ λλ ε εε = + (5)

The areal strain produced by earth tides is computed from 

theoretical considerations (Harrison, 1971, 1985; Beaumont 

and Berger, 1975; Berger and Beaumont, 1976) by using the 

tidal potential, V (L2/t2), as formulated by Bartels (1957, 1985) 

and computed by Harrison (1971):

The areal strain tide component is formulated as a scaled 

function of the tidal potential (Munk and McDonald, 1960; 

Melchior, 1966, Bredehoeft, 1967):

( ) 2 6 A

V h l

rg

ε = − (7)

where 

h ̄ and l ̄ are Love numbers at the Earth’s surface, and

g is the gravitational acceleration (L/t2).

Areal strain tide is computed by using  ̄h = 0.638 and 

l ̄ = 0.088 and is expressed in parts per billion strain (dimensionless). The resulting areal ‘dry’ (in the absence of saturating 

fluid) tidal dilatation at the Earth’s surface, Δt can be expressed 

(Bredehoeft, 1967):

1 2

1 t A

v

v ε

  − ∆ =     −

(8)

where v is Poisson’s ratio.

The gravity tide oriented downwards normal to the Earth’s 

ellipsoid, gN, is computed (Harrison, 1971):

= − V V

gN r r θ

∂ ∂

∂ ∂ δ (9)

where 

Ɵ is the geocentric polar angle of the observation point (radians), and 

δ is the difference between the geodetic and 

geocentric latitudes. 

For example, δ attains a value of about 3.37 × 10-3 radians 

at 45° latitude. Gravity tide is expressed in terms of microgals 

(L/t2). 

The tilt tide in a plane tangent to the Earth’s ellipsoid along 

a specified azimuth oriented with respect to 0° N, γT is computed (Harrison, 1971):

  1 1 cos sin

sin T

V V V

g r r r

γ α α

θ θ λ

 ∂ ∂   ∂      ∂ ∂ ∂  = +δ + (10)

where 

λ is the terrestrial east longitude of the 

observation point (radians) and 

α is the specified azimuth of tilt (radians). 

Tilt tide is expressed in nanoradians. 

2 2 2 22 2

3 3

3cos 1 5cos 3cos 3cos 1

22 2

m mm s

sm s

GMr z zz z r GSr V

RR R

  −− −  

=+ +       

(6)

where 

G is the Newtonian constant of gravitation (L3 / M1-t2), 

M and S are the masses of the moon and sun, respectively (M), 

r is the distance between the center of the Earth and the observation point on the Earth’s surface (L), 

Rm and Rs are the distances of the moon and sun, respectively, from the Earth’s center (L), and 

zm and zs are the zenith angles of the moon and sun, respectively (radians).
Water-Level Modeling 9

Dry, gravity, and tilt tides (Table 2) result from changes in 

gravitational forces as the relative positions of the sun, moon, 

and earth change (Harrison, 1971). These theoretical earth 

tides are computed functions that only require the location of 

an observation well. 

Adjustable WLM components are created by multiplying 

computed dry, gravity, or tilt tide (table 2) by an amplitude. 

Zenith angles primarily are specified by longitude and time 

as referenced to Greenwich Mean Time. A phase shift can be 

applied to the zenith angles through the specified time. Amplitude (a) and phase (Φ) are estimated to minimize differences 

between synthetic and measured water-levels. 

Step Change

Step changes in water-level records are introduced when a 

transducer is disturbed or replaced. Transducer submergence 

can change if the hanger position is moved. Replacing a transducer is likely to change submergence because the devices 

can differ and cable stretch can occur. A step-change WLM 

component is necessary because shifts of less than 0.03 ft are 

detectable in WLM results. 

A step change in the water-level measurement is simulated 

as follows: 

for 

0 for 

i i STEP

i STEP

WLMC h t t

WLMC t t

= ∆ ≥

= < (11)

where

Δhi is the step change of the ith component and t

is the time. The step change is estimated 

in equation 11 to minimize differences 

between synthetic and measured 

water-levels.

Pneumatic Lag 

The pneumatic lag between barometric-pressure changes at 

land surface and the water table can be simulated with a onedimensional diffusion equation instead of being approximated 

with multiple moving averages. This alternative approach is 

advantageous for estimating the hydraulic properties of the 

unsaturated zone and precludes using multiple moving averages of barometric pressure. The propagation of barometric 

changes through the unsaturated zone is solved analytically 

by using equivalent solutions for surface-water/groundwater 

interaction (Rorabaugh, 1964; Barlow and Moench, 1998). 

Stage changes of a fully penetrating river that perturb 

groundwater levels behave similarly to barometric pressure 

changes that perturb air pressures in the unsaturated zone 

(fig. 6). This assumes that pressure changes are small relative 

to the mean air-pressure so air density and specific storage 

are affected minimally. Barometric changes typically are less 

than 2 ft while mean air-pressure ranges between 26 and 34 

ft (Merritt, 2004; Fenelon, 2005). Boundary conditions for a 

one-dimensional, confined aquifer are equivalent to boundary conditions of an areally extensive, thick unsaturated zone. 

The water table is an impermeable boundary because air-filled 

pores cease to exist. 

Table 2. Abbreviations and descriptions of tides that are computed in SeriesSEE.

Tide DESCRIPTION Units Equation

DRY Areal strain tide parts per billion 8

GRAVITY Normal to the Earth ellipsoid microgals 9

TILT Plane tangent to the Earth ellipsoid nanoradians 10

Figure 6. Schematics of one-dimensional, confined aquifer and an areally extensive, 

thick unsaturated zone that experience similar step-changes to a time-varying specifiedhead boundary such as a river or barometric-pressure difference.
10 Advanced Methods for Modeling Water-Levels and Estimating Drawdowns with SeriesSEE, an Excel Add-In

Equivalent hydraulic conductivity and specific storage of 

the unsaturated zone differ from the confined aquifer solution 

because the pores are filled with air rather than water. Equivalent hydraulic conductivity is air permeability divided by the 

viscosity of air and is about 60 times greater than saturated 

hydraulic conductivity because the ratio of water-to-air viscosity ranges from 70 to 40 for temperatures between 10 and 

30°C. Air permeability is affected negligibly by changes in 

barometric pressure (Baehr and Hult, 1991). Specific storage 

of the unsaturated zone is the air-filled porosity divided by the 

mean air pressure. 

Pressure change at a given depth in the unsaturated zone 

from a step-change in pressure at land surface is simulated as 

follows:

WLMCi = Δp − 2∑ cos � m −

∞

m=1

−1m+1

� m − 1

2

1

2

e

−�2

 m−

2 ΔtKAIR

SAIR a2 1

2 1− z

a (12)

where 

Δp is the step change in air pressure at land surface (L),

m is an index, 

Δt is elapsed time since the step change (t),

KAIR is the air permeability divided by viscosity of air (L/t),

SAIR is air-filled porosity divided by the mean air-pressure (1/L), and 

a is the thickness of the unsaturated zone (L).

Multiple step changes are superimposed in time to simulate air-pressure changes at the 

water table by using barometric-pressure changes at land surface (fig. 7). 

Water-table changes are assumed equal and opposite of 

air-pressure changes at the water table. Log-transforms of KAIR

and SAIR are estimated in equation 12 to minimize differences 

between synthetic and measured water-levels. If the objective 

of a water-level model is to estimate hydraulic properties of 

the unsaturated zone by using equation 11, then multiple moving averages of barometric pressure cannot be used as WLM 

components. 

Figure 7. Average daily barometric pressure and simulated air pressure at the water table.

Air pressure, in feet of water

2009 2010

26

27

28

Dec Jan Feb

Land Surface

Water Table

Property Value Units

K AIR 200 feet per day

S AIR 0.0007 1 per feet

Depth to water (a) 2,000 feet
Water-Level Modeling 11

Gamma Transform

The gamma transform was adapted from a Water-Balance/

Transfer Function (WBTF) model that simulates recharge 

to the water table from precipitation (O’Reilly, 2004). The 

gamma transform retains the transfer function from the 

WBTF model that translates a discrete pulse of infiltration 

below the root zone to recharge at the water table. The delay 

between infiltration and recharge at the water table increases 

as unsaturated-zone thickness increases. Recharge pulses also 

are attenuated and prolonged as unsaturated-zone thickness 

increases. The WBTF model was selected because the transfer 

function simulates these characteristics (O’Reilly, 2004). 

Water-level rise, rather than recharge, is simulated with the 

gamma transform. Water-level rise equals recharge divided by 

specific yield, where the aquifer is unconfined, and consequently has a greater magnitude than recharge (fig. 8). 

Water-table rise from each infiltration event is simulated as 

follows: 1

( )

t n k

i i

e t WLMC a I

kn k

∆− −

  ∆ =   Γ   (13)

where 

ai is the amplitude multiplier of the ith component, 

I is amount of infiltration during an event (L), 

Δt is elapsed time since the infiltration event(t),

k is a scale parameter (t), 

n is a shape parameter (dimensionless) , and

Γ(n) is the gamma function, (dimensionless), which 

is equivalent to (n – 1) for integer values of n

(Potter and Goldberg, 1987, p. 111). 

Multiple step changes are superimposed in time to simulate 

water-table fluctuations from infiltration events below land 

surface (O’Reilly, 2004).

Figure 8. An infiltration schedule and water-level rises simulated with gamma transforms that 

were defined by six pairs of shape (n) and scale (k) parameters. Infiltration, in feet per day Water-level rise, in feet per day

0

0.02

0.04

0

0.1

0

0.1

Aug Sep Oct Nov

2010

n = 1; k = 2

n = 3; k = 2

n = 5; k = 2

n = 2; k = 1

n = 2; k = 2

n = 2; k = 5
12 Advanced Methods for Modeling Water-Levels and Estimating Drawdowns with SeriesSEE, an Excel Add-In

Physical significances have been attributed to the fitting 

parameters ai, k, and n (O’Reilly, 2004). The amplitude multiplier (ai

) converts recharge to water-level rise and should be 

proportional to the inverse of the storage coefficient. The scale 

parameter (k) controls the average delay time imposed by the 

unsaturated zone (Dooge, 1959). The shape parameter (n) has 

been characterized as “the number of linear reservoirs necessary to represent the unsaturated zone” by O’Reilly (2004). 

These explanations are interesting, but estimated values of ai, 

k, and n should be interpreted with great skepticism, if at all. 

Superimposed gamma transforms translate step-wise precipitation or infiltration records into approximate water-level 

responses at observation wells. Amplitude (a) and the logtransform of the scale parameter (k) are estimated in equation 

13 to minimize differences between synthetic and measured 

water-levels. The shape parameter (n) is assigned and is not 

estimated. Multiple gamma transforms should be used with 

different values of n if the effect of n is investigated. 

Calibration

Water-level models must be calibrated to reliably differentiate small pumping responses from environmental fluctuations. Efficient and effective calibration requires a quantitative 

measure of model misfit so model parameters can be estimated automatically as is done with the parameter estimation 

software PEST (Doherty, 2010a, 2010b). Differences between 

synthetic and measured water levels, or residuals, define the 

goodness-of-fit and are summed in the measurement objective 

function: 

( )

2

1

( ) ( )

nobs

MEAS j j

j

x SWL x MWL =Φ= − ∑ (14)

where 

x is the vector of parameters being estimated, 

nobs is the number of observations compared, 

SWL(x)j is the jth synthetic water level, and 

MWLj is the jth measured water level.

Although the sum-of-squares error serves as the measurement objective function, root-mean-square (RMS) error,

( )MEAS x RMS

nobs

Φ = (15)

is reported because RMS is easily compared to measurements.

Residuals are not weighted in the measurement objective 

function because suspect measured water levels should be 

discarded rather than assigned a low weight. Each measured 

water level is assumed equally important so all water levels 

are weighted equally. Uniform weighting causes differences 

between synthetic and measured water levels to equally affect 

the measurement objective function (eq. 14). 

Stable parameter-estimation results are ensured with 

selective parameter transformation and regularization. Logtransforms of hydraulic properties are estimated in the Theis, 

pneumatic lag, and gamma transforms to scale parameters and 

precluded negative hydraulic properties (table 3). Regularization avoids estimating insensitive parameters and guides 

estimates toward preferred values. Parameter estimates have 

little to no significance because the parameter values generally 

are not interpreted. Drawdown estimates are interpreted and 

are the ultimate water-level model result. 

Parameter estimation for water-level modeling is unconditionally stable because singular-value decomposition (SVD) 

regularization is used (Doherty and Hunt, 2010). Insensitive or 

highly correlated parameters are not estimated and remain at 

their assigned values if eliminated by SVD regularization. 

Tikhonov regularization guides estimates to preferred conditions (Doherty, 2010a, 2010b). Regularization observations 

are added to define preferred relations between parameters 

(Doherty and Johnston, 2003). Homogeneity within each of 

the three parameter groups of amplitude, phase, and hydraulic 

property was the preferred relation that was enforced with 

Tikhonov regularization (table 3). 

The balance between fitting measurement and regularization observations is controlled by the sum-of-squares measurement error, PHIMLIM, in PEST (Doherty, 2010a, 2010b). 

An expected RMS error defines PHIMLIM, which equals the 

square of the expected RMS error times the number of measured water levels (nobs). The expected RMS error defaults to 

0.003 (L) in SeriesSEE, but can be changed by the user. 

Table 3. Summary of estimable parameters and parameter groups for water-level modeling (WLM) components. 

[— is not applicable]

WLM

component

Coefficient

1

Parameter

group

Coefficient

2

Parameter

group

Moving Average a Amplitude ɸ Phase

Theis Transform T Hydraulic Property S Hydraulic Property

Tide a Amplitude ɸ Phase

Step — — a Amplitude

Pneumatic Lag KAIR Hydraulic Property SAIR Hydraulic Property

Gamma a Amplitude k Hydraulic Property
Water-Level Modeling 13

Drawdown Estimation

Drawdown estimates from a water-level model are the 

difference between measured water levels and synthetic water 

levels without the Theis transforms. Alternatively, drawdowns 

can be computed directly by summing all Theis transforms 

and subtracting residuals (fig. 9). The summation of all Theis 

transforms is the direct estimate of the pumping signal. 

Residuals represent all unexplained water-level fluctuations. 

These fluctuations should be random residuals during nonpumping periods, but can contain unexplained components 

of the pumping signal during pumping and recovery periods. 

This method of estimating drawdowns is called the Theistransform approach. 

A limited, application of water-level modeling, the projection approach, was developed prior to the Theis-transform 

approach (Halford, 2006). Synthetic water levels were developed and calibrated during a period prior to pumping with the 

projection approach. Calibrated, synthetic water levels were 

then projected forward during pumping and recovery. Drawdown was the difference between projected synthetic values 

and measured values. This approach ensures that environmental fluctuations and the pumping signal are uncorrelated 

because pumping is not simulated during model calibration to 

antecedent water levels. 

The projection approach is limited primarily because 

regional water-level trends are simulated poorly. Excluding 

pumping and recovery periods from WLM calibration eliminated much of the regional trends from the calibration period. 

This drawback weakened the projection approach and limited 

the usefulness of background well information, particularly 

where pumping and recovery periods were greater than the 

antecedent data period. 

The Theis-transform approach is a more robust application of water-level modeling because environmental fluctuations and pumping signal are simulated during pumping and 

recovery in addition to antecedent water levels. This allows for 

calibration of synthetic water-levels to all measured data. The 

effects of pumping on measured water levels are approximated 

by using a simple approach, Theis transforms, so that simulations are quick. Efficiency and speed are mandatory because 

water levels are modeled independently in every observation 

well. These requirements preclude numerical groundwaterflow models or any other laborious approach for translating 

pumping schedules to water-level responses. 

Drawdown detection with the Theis-transform approach 

becomes ambiguous when the signal-to-noise ratio is low or 

where environmental fluctuations and pumping signals can be 

correlated. Signal and noise are defined herein as the maximum drawdown in a well during an aquifer test and the RMS 

Figure 9. Estimated drawdown from summing Theis transforms and subtracting residuals. Fast and slow Theis transforms represent the 

relatively fast and slow elements of pumping signals that propagate through a complex aquifer system. Water level change, in feet

2010

09/09 09/16 09/23 09/30

-0.2

-0.1

0

0.1

Estimated drawdown

Fast Theis transform

Slow Theis transform

Residuals
14 Advanced Methods for Modeling Water-Levels and Estimating Drawdowns with SeriesSEE, an Excel Add-In

error, respectively. Drawdown has been detected definitively 

where the signal-to-noise ratio was greater than 10 and correlation was unlikely. Correlation is unlikely where sharply 

defined pumping signals (saw-tooth shape) exist or considerable recovery has been observed (fig. 10, ER-EC-6 deep, 

r = 6,800 ft). Correlation between environmental fluctuations 

and the pumping signal is possible where observed drawdown 

can be approximated by a linear trend during all or part of the 

period of analysis (fig. 10, ER-EC-12 shallow, r = 8,900 ft). 

The potential for correlation increases as hydraulic diffusivity 

decreases, distance between observation and pumping well 

increases, or recovery diminishes. 

Figure 10. Discharge from pumping wells ER-20-8 upper and ER-20-8 lower, estimated drawdowns, residuals, RMS errors, and signalto-noise ratios in observation wells ER-EC-12 shallow and ER-EC-6 deep. Discharge, in gallons per minute

2011

Water level change, in feet

0

50

100

150

200

-0.15

-0.10

-0.05

0

0.05

-0.05

0

0.05

Apr May Jun Jul Aug Sep Oct

RMS = 0.006 foot

Signal-to-Noise = 3

Residuals

Estimated drawdowns

RMS = 0.006 foot

Signal-to-Noise = 22

ER-20-8 upper

ER-20-8 lower

ER-EC-12 shallow

ER-EC-6 deep
SeriesSEE 15

SeriesSEE

SeriesSEE is a Microsoft® Excel add-in for viewing, cleaning, manipulating, and analyzing time-series data where waterlevel modeling is a primary analysis tool. SeriesSEE creates a 

viewer file from a data workbook that can contain more than 

16,000 series. The maximum number of series that can be 

viewed simultaneously is limited to twelve. Time series are 

displayed on two charts where all data are shown in one chart, 

and a magnified subset is shown in the other chart (fig. 11). 

Borehole geophysical logs also can be viewed, cleaned, 

manipulated, and analyzed with SeriesSEE, where the two 

charts are displayed top-to-bottom, rather than left-to-right. 

SeriesSEE software, installation instructions, and help for all 

tools can be downloaded in the zipped file, which is described 

in appendix A. 

All source code that was developed for SeriesSEE can be 

downloaded freely (appendix B). All utilities, except WLM, 

are processed exclusively with VBA code in the SeriesSEE 

add-in or supporting add-in files named SSmodule_*.SerSee. 

Source codes for these files are in the VBA folder of appendix B and are named SSmodule_*.xlsm. Water levels to be 

modeled, input series, and period of analysis are defined with 

VBA routines. WLM components are transformed (table 1) 

and water levels are simulated with the FORTRAN program 

WLmodel, which reads ASCII files written by VBA programs. 

Differences between synthetic and measured water levels are 

minimized with PEST (Doherty, 2010a, 2010b). A copy of 

Figure 11. SeriesSEE toolbar and example workbook that was created with SeriesSEE. 
16 Advanced Methods for Modeling Water-Levels and Estimating Drawdowns with SeriesSEE, an Excel Add-In

PEST exists in the SeriesSEE installation files, but also can be 

downloaded independently from http://www.pesthomepage.

org/. The VBA utility WLM writes the PEST control file,

*.pst, as multiple, commented input files, which are concatenated and stripped of comments with the FORTRAN program 

NoComment. Source codes and documentation of WLmodel

and NoComment are in the FORTRAN folder of appendix B. 

Data Requirements

Data must be arranged as a continuous series of headers and values where all headers are in a single row (fig. 12). 

Multiple time columns can be specified, which allows for 

specification of series with different or irregular sampling 

intervals. All series are independent, so time columns need not 

be synchronous. Multiple data series can share a common time 

column (fig. 12, See columns C, D, and E), but the shared time 

column must be the first time column to the left of the data 

series. 

A Viewer file is created by selecting a cell in the 

block of data to be analyzed and pressing the button 

(fig. 11). The entire data block is copied from the user’s original file into the viewer file by default. All equations within the 

block of data are converted to values in the viewer file, which 

breaks all linkages to the user’s original workbook. Original 

data and formulas are not altered in the user’s original file 

because all SeriesSEE operations act on a copy of the data in 

the viewer file. 

Supporting Utilities

SeriesSEE features more than 20 supporting utilities in 

addition to the viewer creation and water-level modeling utilities already discussed (table 4). Many utilities exist to provide 

data-handling capabilities that can be used prior to water-level 

modeling. Related utilities are grouped and labeled as Clean 

Data, Analysis, Tools, Import, Export, Adjust, and Chart Tools 

(table 4). 

Time-series data generally must be cleaned before analyzing. Cleaning removes erroneous measurements, converts 

units, reconciles continuous and periodic measurements, 

and removes step changes from transducer disturbances. 

All changes between the original and cleaned series can be 

recorded with explanations for each data change if the track 

utility is active. Changes and explanations are recorded to an 

auxiliary workbook that also contains the original and revised 

series. Utilities in the clean data and analysis groups perform 

these tasks (table 4). 

Simple analysis and inspection of series are supported by 

utilities in the analysis group (table 4). New series can be created by adding, subtracting, multiplying, or dividing one series 

by another with the utility. Measurement frequencies of the two series can differ because of interpolation. 

Smoother series can be created from noisy series with moving 

averages or LOWESS (LOcally Weighted Scatterplot Smoothing), which is a nonparametric method of fitting a curved 

line to data (Helsel and Hirsch, 1992, p. 288–291). Potential 

correlations among multiple series of disparate scales can be 

inspected by normalizing these series to a common scale with 

the utility. 

Water-level modeling and other analyses can be expedited 

and improved by data reduction where there has been oversampling. Data can be reduced by averaging within periods 

such that 1-minute data are reduced to 1-hour averages with 

the utility. Continuous records of flow rates with 

many thousands of measurements can be reduced accurately to 

a few dozen step changes with the utility. Simplified pumping schedules increase the efficiency and speed at 

which drawdowns can be simulated in WLMs. Geophysical 

logs are approximated with a simple polyline using the PolyFit 

utility, , which can eliminate extraneous fluctuations and constrain the polyline to monotonic increases. Utilities in the tools group perform these tasks (table 4). 

Time series can be imported from ASCII files and database 

tables to a SeriesSEE data table with utilities in the import 

group (table 4). Multiple data-logger files are read interactively with the utility to create a single SeriesSEE data table. Database tables with site identifiers, times, and 

water levels grouped into three columns can be reformatted to 

a SeriesSEE data table with the utility. 

Figure 12. Format of headers and values for creating a viewer file with SeriesSEE.
SeriesSEE 17

Table 4. Summary of available tools in SeriesSEE.

Group Utility Description Name

Create Create Viewer file by selecting a cell in a block of data in an original source file, which is 

copied to the viewer file. All equations are converted to values in the Viewer file. View

Clean Data Bad data conditionally can be commented and/or eliminated. Conditional

Bad data in a single series can be commented and/or eliminated graphically. Points

Data gaps from the cleaning process can be filled by linear interpolation, loaded with a 

dummy value, eliminated altogether, or gaps can be created for alignment. GapFill

Shift data segments. Estimate shift with simple water-level models that use a few guide 

series. Alternatively, shifts can be assigned from other estimates. Align

Data reduction by averaging where oversampled. Average

Float series to tape downs without changing slope of transducer data. Float

Force an explanation to be appended to each data change in an auxillary workbook that also 

contains the original and revised series. Track

Analysis

Create new series by addition, subtraction, multiplication, and division of existing series. 

Second series interpolated to times in the first series. Series can also be smoothed with a 

moving average or LOWESS curve.

Compare

Series can be normalized to common scales. Rescale

Removes derived series that are created by Compare or Rescale. Remove

Tools Data reduction tool where selected series are binned by time periods or depth intervals to 

compute statistics. Subtotal

Reduces pumping rates to a simple schedule. SimpleQ

Geophysical logs are approximated with a simple polyline. PolyFit

Model water levels interactively in a new workbook, where water levels are simulated with 

a FORTRAN program and differences are minimized with PEST. WLM

Import Series from data-logger files are read interactively and concatenated in a SeriesSEE format. GetLogger

Split 3 columns of site identifiers, time, and water levels into SeriesSEE input where a new 

series is identified at each change in site identifier. Split

Export Output from tracking workbooks to selected ASCII formats. ASCII

Export individual series with options to create drawdown observations. Drawdown observations require shifting, binning, and truncating to a time window. Series

Data are copied to a new workbook and reduced to a user-specified period. Window

Adjust Individual, selected, or all series can be shifted such that the average, minimum, maximum, 

or first value will equal zero. Offset

Chart 

Tools Refresh the list of available series after manually adding or deleting series on the data page. Refresh

Create temporary hyperlinks between visible series and charted data in the Viewer file. HyperData

Magnify subareas of plot. First click adds a rectangle. Second click re-scales both axes to 

rectangle area. Third click restores plot. BoxFocus

Inform Controls and usage of SeriesSEE are explained. Help

Display ad copy about SeriesSEE. About
18 Advanced Methods for Modeling Water-Levels and Estimating Drawdowns with SeriesSEE, an Excel Add-In

Series can be viewed and inspected at scales as fine as 

discrete measurements with utilities in the “adjust” and 

“chart tools” groups (table 4). Series can be shifted so that 

all measurements fluctuate about a common reference with 

the utility, which eases comparisons among series 

(fig. 13). Subareas of charts in SeriesSEE viewer and auxiliary 

files can be magnified interactively with the 

utility. Discrete measurements can be selected graphically and 

connected to the cell with the numerical value in the Viewer 

file with the utility, which creates temporary 

hyperlinks between charted points and the cell with the plotted 

value.

Each SeriesSEE utility is fully documented in the help 

system, which can be called with the utility or from 

context sensitive help calls in each utility (appendix A). Each 

group, utility, form, and auxiliary workbook is explained 

briefly, and step-by-step instructions (fig. 14). Complex 

utilities such as water-level modeling are documented with 

multiple pages that explain each form and action. 

Water-Level Modeling

Water levels are modeled interactively with the 

utility in SeriesSEE. Water levels to be modeled, input series, 

period of analysis, and WLM components are defined through 

the use of data-entry forms. A new workbook for modeling 

water levels is created with user-specified information from 

these forms. Fitting periods and WLM components can be 

modified in the WLM workbook. 

Analytical models that transform WLM components in the 

FORTRAN program WLmodel have been verified (table 5). 

The analytical models for moving average and step transforms 

were verified against intrinsic functions in Excel. The analytical models for Theis, tide, pneumatic lag, and gamma transforms were verified against solutions that were computed with 

published programs. Source problems, programs, and comparisons between WLmodel output and published programs 

are documented fully in appendix C. 

Differences between synthetic and measured water levels 

are minimized with PEST. Parameter estimates, transformed 

WLM components, synthetic water levels, and differences 

are imported automatically into the WLM workbook after 

PEST finishes. Model fit is defined by RMS error and evaluated graphically. Parameters are estimated and WLM results 

are evaluated iteratively until the user deems the fit to be 

adequate. 

Figure 13. Shifting series to a common reference with the offset utility.

2011

Offset series

Original series

Water level change, in feet

-0.5

0

0.5

0

1,000

2,000

3,000

5/1 5/8 5/15

W_EC-6-S.FT

W_ue20bh1.FT

B-EC-11.FT

Table 5. Summary of verification tests for analytical models in 

the FORTRAN program WLmodel.

WLM Component SeriesSEE Label Verification Source

Moving Average SERIES Excel function

Theis Transform THEIS Barlow and Moench, 1999

Tide TIDE Harrison, 1971

Step STEP Excel function

Pneumatic Lag AIR-LAG Barlow and Moench, 1998

Gamma GAMMA O'Reilly, 2004
Applications of Water-Level Modeling 19

Drawdowns and transformed WLM components are 

returned to the SeriesSEE viewer once the user accepts a 

WLM, where drawdowns are the sum of all Theis transforms 

minus differences between synthetic and measured water 

levels. Drawdowns and transformed WLM components are 

selected individually, so the number of returned series can 

range from 0 to all WLM components. The WLM workbook 

can be archived as a macro-free workbook with re-activation 

capabilities. 

Applications of Water-Level Modeling

Water-level modeling applications of SeriesSEE are demonstrated with a hypothetical example and a field investigation 

at Pahute Mesa, Nevada National Security Site (NNSS). The 

hypothetical example emulated the complex hydrogeology 

beneath Pahute Mesa so that known drawdowns could be 

simulated in a complex aquifer system. Limitations of the 

Theis-transform approach were investigated with these known 

drawdowns. Environmental noise, which was the record of 

water levels in background well ER EC-6 shallow (table 6), 

was added to known drawdowns. The field investigation demonstrated that drawdowns much smaller than environmental 

Figure 14. Table of contents and an explanation page in the help system for SeriesSEE. 

fluctuations can be detected across a major fault structure 

more than 1 mile from the pumping well. Explanations, data 

sets, and ancillary software for the hypothetical example and 

field investigation are in appendixes D and E, respectively. 

Water-level modeling was developed and tested with data 

from Pahute Mesa, NNSS, (fig. 15) because detection of 

distant drawdowns is imperative and complicated by more 

than 2,000 ft of unsaturated zone. Migration of radionuclides 

from underground testing of nuclear devices drives the need 

to quantify groundwater flow and transport beneath Pahute 

Mesa (Laczniak and others, 1996). The great depth to water 

and accessibility limit the number of wells, which typically 

penetrate a mile of volcanic rock and are more than 1-mi 

apart (Fenelon and others, 2010). Environmental water-level 

fluctuations are substantial beneath Pahute Mesa because of 

the thick unsaturated zone and high hydraulic diffusivity of the 

volcanic rocks. 

The aquifer system beneath Pahute Mesa comprises layered sequences of volcanic rocks that have been faulted into 

distinct structural blocks (Warren and others, 2000). Rhyolitic 

lavas or welded ash-flow tuffs such as in the Benham and 

Topopah Springs aquifers, respectively, comprise aquifers. 

Bedded and non-welded, zeolitized tuffs typically comprise 

confining units (Blankennagel and Weir, 1973; Prothro and 

Drellack, 1997; Bechtel Nevada, 2002). More than a half 
20 Advanced Methods for Modeling Water-Levels and Estimating Drawdowns with SeriesSEE, an Excel Add-In

dozen faults with offsets in excess of 500 ft have been mapped 

previously in Pahute Mesa (McKee and others, 2001), and 

additional faults are mapped with each new well (for example, 

National Security Technologies, LLC, 2010). 

Hypothetical Example

The reliability of differentiating environmental fluctuations and pumping responses with water-level models was 

tested with a hypothetical aquifer system. Drawdown from a 

hypothetical aquifer test was simulated where the hydrogeologic complexity and distribution of hydraulic properties were 

assigned. The hypothetical aquifer system is comprised of 

ash-fall tuff, bedded tuff, welded tuff, and lava units that are 

flat-lying, laterally isotropic, and homogeneous (fig. 16). A 

fault 1,500 ft east of the pumping well, P1, bisects the aquifer 

system, vertically displaces hydrogeologic units 1,000 ft, and 

alters hydraulic properties around the structure. 

The hypothetical aquifer system was simulated with a 

three-dimensional MODFLOW model (Harbaugh, 2005). 

The model domain was discretized laterally into 135 columns 

of 135 rows with a variably spaced grid (fig. 16). Cell sizes 

ranged in width from 10 ft by the pumping well to 40,000 ft 

at the model edges. Model edges were about 200,000 ft away 

from the pumping well, P1, and were simulated as no-flow 

boundaries. The model grid extended vertically from an impervious base at sea level to the water table at 4,200 ft above sea 

level. Vertical discretization was uniform, with 200-ft thick 

layers except for a 1-ft thick layer at the water table. The 

thickness differed so that the storage coefficient and specific 

storage were equivalent, and it allowed specific yield to be 

assigned directly in a layer. Changes in saturated thickness of 

the aquifer were not simulated because maximum drawdown 

at the water table was small relative to the total thickness. 

Hydraulic properties typical of volcanic units were 

assigned to the hypothetical aquifer system. Ash-fall tuff, 

bedded tuff, welded tuff, and lava were assigned hydraulic conductivities of 0.001, 0.1, 3, and 50 ft/d, respectively. 

Horizontal-to-vertical anisotropy of one was assigned to all 

units. A uniform value of 0.02 was assigned for specific yield. 

The specific storage of all hydrogeologic units was 2×10–6 1/ft. 

Hypothetical aquifer-test results were simulated and 

analyzed during a 3-month period that was divided into five 

stress periods. The antecedent, pumping, recovery, pumping, and recovery periods were 21, 10, 10, 10, and 40 days, 

respectively. Pumping rates were 500 gpm during the second 

and fourth stress periods. Flow and drawdown in pumping and 

observation wells were simulated and sampled with the MultiNode Well (MNW) package (Harbaugh, 2005). Flow to the 

pumping well was distributed proportionally to cell transmissivities by the MNW package. 

Water levels with a “known” pumping signal and environmental fluctuations (noise) shown in figure 17 for well O3 

were created by adding simulated drawdowns from MODFLOW to measured water levels in well EREC-6 shallow 

(fig. 17). Simulated drawdowns from MODFLOW in well O3, 

which is 7,800 ft from well P1, were interpolated in time to 

match measured water levels in well EREC-6 shallow. Simulated drawdowns from MODFLOW and simulated drawdowns 

with environmental noise added are in appendix D in the file .\

WLMs\00_Hypo+Meas2SeriesSEE.xlsx. 

Drawdowns were estimated by modeling “measured” water 

levels in well O3. Environmental fluctuations were simulated 

with computed tides, barometric pressure and background 

water levels in wells PM-3 and UE-20n 1 (fig. 17). Pumping 

effects were simulated with a Theis transform of the hypothetical pumping schedule. The water-level model was calibrated 

during the period from November 18, 2010, to March 6, 2011. 

Table 6. Site information and completion depths for wells at Pahute Mesa, Nevada National Security Site that were used in 

hypothetical example and field investigation.

Well name: Names are listed in alphabetical order. Bold part of name is well site as shown on Figure 15.

U.S. Geological Survey site identification number: Unique 15-digit number identifying well.

Latitude/Longitude: Latitude and longitude coordinates, referenced to North American Datum of 1927.

Land-surface altitude: Altitude, referenced to National Geodetic Vertical Datum of 1929.

Open intervals: Depth, in feet below land surface, of the top and bottom of open annulus.

Well Name U.S. Geological Survey site 

identification number

Latitude (degrees, minutes, 

seconds)

Longitude (degrees, minutes, seconds)Land-surface altitude

(feet)

Open intervals

ER-20-5 #1 371312116283801 37°13'12.2" 116°28'37.8" 6,242 2,249–2,655

ER-20-6 #3 371533116251801 37°15'33.1" 116°25'17.5" 6,466 2,436–2,807

ER-EC-6 shallow 371120116294805 37°11'19.6" 116°29'48.1" 5,604 1,606–1,948

ER-EC-11 main 371151116294102 37°11'51.2" 116°29'41.1" 5,656 3,196–3,385

3,590–4,148

PM-3-1 371421116333703 37°14'20.7" 116°33'36.6" 5,823 1,872–2,192

UE-20n 1 371425116251902 37°14'25.1" 116°25'19.0" 6,461 2,308–2,834

ER-20-7 371247116284502 37°12'47.0" 116°28'44.8" 6,209 2,292–2,924

ER-20-8 main 371135116282601 37°11'35.1" 116°28'26.3" 5,848 2,440–2,940

3,070–3,442
Applications of Water-Level Modeling 21

West Silent Canyon structural zone

Purse faultPurse fault

East Greeley fault

LineamentLineament

Canyon

Thirsty Purse faultPurse fault

West

West Boxcar fault

Boxcar faultBoxcar fault

West Greeley

fault

ER-20-7 fault

Northern Timber Mountain moat structural zone

ER-20-8 fault

Area 20 caldera structural margin

Timber Mountain caldera complex structural margin

Area 20 Area 19

Area 18

Nevada Test and

Training Range

Nevada National

Security Site

ER-20-5-1ER-20-5-1

ER-EC-11ER-EC-11

ER-EC-6

PM-3

UE-20n 1UE-20n 1

ER-20-6 #3ER-20-6 #3

ER-20-8

ER-20-7

116°32' 116°28' 116°24'

37°16'

37°12'

37°08'

Base from U.S. Geological Survey digital data 1:100,000 1978–88

High-resolution aerial imagery for the United States copyright 2007 i-cubed

Universal Transverse Mercator Projection, Zone 11, NAD83 

0 1 2 3 MILES

0 1 2 3 4 KILOMETERS

Geologic structures modified from U.S. Department of Energy (2009) 

and Sigmund L. Drellack, National Security Technologies, LLC, 

written commun. (2011)

EXPLANATION

The Bench area

Fault

Pumping well

Observation well

Las

Vegas

N E V A D A

Nevada

National

Security Site

Carson

City

Figure 15. Background wells, observation wells, pumping well, and selected fault structures at Pahute Mesa Nevada National Security 

Site. 
22 Advanced Methods for Modeling Water-Levels and Estimating Drawdowns with SeriesSEE, an Excel Add-In

Drawdowns that were estimated from “measured” water 

levels in well O3 agreed with known drawdowns within the 

noise of the data set (fig. 18). A maximum drawdown of 

0.18 ft was estimated which was identical to the known maximum. The RMS error of differences between synthetic and 

measured water levels was 0.013 ft. The RMS error of differences between synthetic and known drawdowns was 0.015 ft. 

Drawdowns alternatively were estimated in well O3 by 

modeling the original MODFLOW results with Theis transforms. No other WLM components were considered because 

environmental fluctuations did not exist in the original 

MODFLOW results. This alternative water-level model also 

was calibrated during the period from November 18, 2010, to 

March 6, 2011. 

Drawdowns that were estimated directly from MODFLOW 

results could be replicated almost perfectly with Theis transforms. Differences between MODFLOW results and a single 

Theis transform could be reduced to a RMS error of less than 

0.006 ft. RMS error declined to less than 0.0006 ft with the 

addition of a second Theis transform (fig. 18). Deviations of 

less than 0.001 ft approach the accuracy of the numerical solution of the hypothetical aquifer test. 

The simplicity of Theis transforms did not introduce error 

because MODFLOW results could be replicated near perfectly 

with Theis transforms. Differences between known drawdowns and drawdowns that were estimated from “measured” 

water levels differed because of noise in the measured input 

series. 

The hypothetical model and SeriesSEE input were created 

with HypoFrame, which is a workbook for simulating hypothetical aquifer tests and creating water levels with known 

pumping signals and environmental noise. Hypothetical 

aquifer systems must have flat-lying geologic units of uniform 

thickness and laterally isotropic, homogeneous hydraulic conductivity. A hypothetical aquifer system can be subdivided into 

four quadrants by two intersecting faults. Rock sequences in 

each quadrant can be displaced vertically within each quadrant. The HypoFrame workbook and documentation are in 

appendix D.

Figure 16. Hydraulic conductivity distribution of a subset of a hypothetical aquifer system that has been bisected by a fault, showing 

well locations and labeled quadrants (upper left, UL; upper right, UR; lower right, LR; lower left, LL). 
Applications of Water-Level Modeling 23

Figure 17. Barometric pressure, background water levels, and water levels with known drawdowns in hypothetical well O3. Water level change, in feet

0

1

2

-0.2

0

Nov-10 Dec-10 Jan-11 Feb-11 Mar-11

Pumping

Period

Known drawdown (MODFLOW)

Barometer UE-20n 1

Background well UE-20n 1

Background well PM-3

"Measured" well O3

Pumping

Period

Drawdown, in feet

-0.3

-0.2

-0.1

0

0.1

Nov-10 Dec-10 Jan-11 Feb-11 Mar-11

Known (MODFLOW)

Estimated from "measured" water levels

Estimated from MODFLOW results with 

Theis transforms

Figure 18. Known drawdowns (MODFLOW), drawdowns estimated from “measured” water levels, and drawdowns estimated directly 

from MODFLOW results in well O3. 
24 Advanced Methods for Modeling Water-Levels and Estimating Drawdowns with SeriesSEE, an Excel Add-In

Pahute Mesa Example

Water-level modeling was tested in a complex hydrogeologic system by estimating drawdown from two aquifer tests 

beneath Pahute Mesa (Halford and others, 2011). The upper 

and lower zones of well ER-20-8 main produced water from 

the Tiva Canyon and Topopah Spring aquifers sequentially 

between June 16, 2011, and August 8, 2011. Each well was 

pumped a total of 20 d, where pumping periods were evenly 

divided between well development and a constant-rate test 

(fig. 19).Drawdown from pumping both zones was estimated 

in observation well ER-20-7, which is screened in the Topopah 

Spring aquifer. Pumping and observation wells are 1.4 mi 

apart and penetrate different structural blocks (fig. 15). 

Drawdown in well ER-20-7 was estimated with multiple 

Theis transforms in the water-level model. Environmental 

fluctuations were simulated with computed tides, barometric 

pressure, and background water levels from well UE-20bh-1 

(fig. 15). Pumping effects were simulated with two Theis 

transforms for each of the two pumping schedules (fig. 19). 

The fitting period was from April 20, 2011, to November 11, 

2011. Synthetic water levels matched measured water levels 

with a RMS error of 0.004 ft. 

Drawdown in well ER-20-7 also was estimated with an 

identical water-level model, except that WLM components 

with background water levels were negated. Synthetic water 

levels matched measured water levels with a RMS error of 

0.027 ft during the same fitting period from April 20, 2011, to 

November 11, 2011 (fig. 19). Each drawdown estimate was 

the difference between a synthetic water level without Theis 

transforms and a measured water level. 

Poor drawdown estimates from the water-level model without background water levels demonstrates the need to simulate as much of the environmental fluctuations as possible. 

Antecedent conditions were simulated poorly where estimated 

drawdowns should be zero. Estimated drawdowns unambiguously were wrong during October and November when net 

water-level rises from pumping were estimated (fig. 19). 

Figure 19. Measured water levels, synthetic water levels, Theis transforms, and estimated drawdowns in well ER-20-7 from pumping 

ER-20-8 main upper and lower zones, Pahute Mesa, Nevada National Security Site. Water level change, in feet Drawdown, in feet Discharge, in gallonsper minute

-0.2

-0.1

0

0.1

0

0.1

0.2

0.3

0

100

200

2011

Apr May Jun Jul Aug Sep Oct Nov Dec

No background well

Background well

Measured

20-8 upper

20-8 lower
Summary and Conclusions 25

Water-Level Modeling Strategies 

Estimating drawdowns that have been obscured by environmental fluctuations is the primary goal of the water-level 

modeling approach. This approach is most effective and efficient where many WLM components are specified and fitting 

periods are great. This approach has been summarized, sometimes derisively, as the flak-gun, fishing-with-dynamite, and 

kitchen-sink approaches. All phrases accurately depict testing 

many WLM components simultaneously. Unique contributions 

from each WLM component remain unknown, but pumping 

signals are not correlated with environmental fluctuations. The 

flak-gun approach was adopted here. 

The flak-gun approach uses WLM components that could 

have been excluded. This is not a problem because mechanisms exist to negate WLM components. Amplitudes tending to zero will negate a WLM component. Multiple WLM 

components also can negate one another by summing to zero. 

Likewise, Theis transforms also are negated by a large transmissivity or storage coefficient value where pumping signals 

are below detection or absent. Negated WLM components 

aesthetically are lacking, but do not affect results. Systematic 

investigation of WLM components is possible with SeriesSEE, 

but has not been automated. 

The flak-gun approach has many advantages, especially 

when estimating drawdowns in dozens of wells. Reporting is 

easier because the same input series and WLM components 

were used in all of the water-level models. Water-level models 

calibrate quickly after analyzing the first or second well 

because WLM components are defined with fair initial estimates of amplitude and phase. The flak-gun approach can fail 

when the fitting period decreases and correlation becomes possible between pumping signals and environmental fluctuations. 

Correlation between weak pumping signals and environmental fluctuations is possible and requires further investigation. Nebulous drawdown estimates can be investigated with 

multiple water-level models where water levels initially are 

simulated without Theis transforms. An alternative water-level 

model is created by adding a Theis transform to the initial 

water-level model. The initial transmissivity and storage coefficient should create a small but measureable maximum deflection in the added Theis transform. Drawdowns likely were not 

detected if the RMS error cannot be reduced by more than 30 

percent. 

Input series of greater duration potentially can degrade 

with time as pressure transducers fail. For example, multiple 

input series could be good for the first four months, while one 

input series degrades during the last two months. Degradation likely will be apparent in the WLM residuals as scatter 

increases. Identifying the onset of failure in a specific input 

series requires modeling water levels during subsets of the 

fitting period. Degrading input series can be investigated 

manually with SeriesSEE, but an automated tool would be a 

better approach. 

Summary and Conclusions

Pumping responses can be differentiated reliably from 

environmental fluctuations with water-level modeling. Waterlevel modeling approximates measured water-level fluctuations by summing multiple component fluctuations. Environmental fluctuations primarily are composed of barometric 

and background water-level input series and computed tide 

components. Pumping signals are modeled by superimposing 

multiple Theis transforms, where step-wise pumping records 

of flow are transformed into water-level changes. The summation of all component fluctuations is a synthetic water-level 

series. 

Water-levels can be modeled robustly with the Theis-transform approach because environmental fluctuations and pumping signals are simulated simultaneously. Long-term trends are 

well simulated because environmental fluctuations are defined 

with entire periods of record. Fitting periods are extended 

greatly where pumping and recovery affect a majority of the 

record. Multiple Theis responses with different hydraulic diffusivities are summed to approximate lithologic variability. 

Water-level modeling with Theis transforms has been 

implemented in the program SeriesSEE, which is a Microsoft®

Excel add-in. Water levels to be modeled, input series, period 

of analysis, and water-level model components are defined 

interactively and viewed in workbooks that are created by 

SeriesSEE. Water levels are modeled with a FORTRAN program that is called from Excel. Differences between synthetic 

and measured water levels are minimized with PEST. 

Water-level model components are transformations of input 

series. Moving average, Theis, pneumatic-lag, and gamma 

transforms are available transforms in SeriesSEE. Moving 

averages most frequently transform input series of barometric 

pressure and background water levels. Pumping schedules are 

transformed into water-level fluctuations with Theis transforms. Pneumatic-lag transforms barometric pressure changes 

at land surface to lagged and attenuated responses at the water 

table. Water-level rises from infiltration events are simulated 

with the gamma transform. Earth tides and step transforms are 

purely computed quantities that do not require input series. 

Many utilities exist in SeriesSEE for viewing, cleaning, 

manipulating, and analyzing time-series data in addition to 

water-level modeling. Supporting utilities exist because data 

handling frequently consumes more time and effort than 

water-level modeling. Each SeriesSEE utility is documented 

with a brief explanation and step-by-step instructions that are 

accessed through context sensitive help. 

Water-level models must be calibrated to reliably differentiate small pumping responses from environmental fluctuations. Differences between synthetic and measured water levels define goodness-of-fit. Sum-of squares of differences are 

minimized by PEST where singular value decomposition and 

Tikhonov regularization are used to assure stable results, not 

to inform estimated parameter values. Preferred homogeneity 

within amplitude, phase, and hydraulic property parameters is 

enforced with Tikhonov regularization. 
26 Advanced Methods for Modeling Water-Levels and Estimating Drawdowns with SeriesSEE, an Excel Add-In

Drawdown estimates from a water-level model are the 

summation of all Theis transforms minus residuals. The summation of all Theis transforms is the direct estimate of the 

pumping signal. Residuals represent all unexplained waterlevel fluctuations. These fluctuations should be random residuals during non-pumping periods, but can contain unexplained 

components of the pumping signal during pumping and 

recovery periods. 

The simplicity of Theis transforms did not introduce 

error because results from a hydrogeologically complex 

MODFLOW model could be replicated near perfectly with 

Theis transforms. Differences between known drawdowns 

and drawdowns that were estimated from “measured” water 

levels differed because of noise in the measured input series. 

Estimated drawdowns are affected minimally by the Theistransform approach relative to the inaccuracies that result from 

noise in the data sets. 

Drawdowns much smaller than environmental fluctuations 

have been detected across a major fault structure more than 

1 mile from the pumping well beneath Pahute Mesa, Nevada 

National Security Site. A maximum drawdown of 0.1 ft was 

estimated in well ER-20-7 during an 8-month period of analysis. Drawdown estimates in well ER-20-7 were consistent 

with a plausible pattern of drawdowns at all observation wells. 

Drawdowns could not have been detected without water-level 

modeling as implemented in SeriesSEE. 

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Appendix E. Pahute Mesa Example 29

Appendix A. SeriesSEE add-in 

The SeriesSEE add-in, example data sets, and installation instructions in the zipped file, AppendixA_SeriesSEE.v.1.20.zip, 

can be accessed and downloaded at http://dx.doi.org/10.3133/tm4-F4. The SeriesSEE add-in, supporting modules, templates, and 

compiled FORTRAN codes are in the subfolder AddIN. Examples of geophysical log, data logger input, other time series, and 

water-level modeling data sets are in the subfolders Example_BOREHOLE, Example_LOGGER, Example_TIME, and Example_WLM, respectively. An Adobe PDF version of the help files, SeriesSEE.V1.00_Explain.pdf, is in the root directory because 

compressed help files that are on servers can be disabled, http://support.microsoft.com/kb/896358. Contents of all subdirectories 

are reported in README file in the root directory of the unzipped AppendixA_SeriesSEE.v.1.20.zip file. 

Appendix B. Source Codes for SeriesSEE 

Source code for SeriesSEE exists as FORTRAN, XML, and VBA codes in the zipped file, AppendixB_Codes-SeriesSEE.

v1.20.zip, which can be accessed and downloaded at http://dx.doi.org/10.3133/tm4-F4. The FORTRAN codes NoComment and 

WLmodel support PEST and solve water-level models, respectively, and are in the FORTRAN subfolder. All VBA code are 

in the SeriesSee.V*.xlsm and SSmodule_*.xlsm files in the VBA subfolder. The XML that defines SeriesSEE commands and 

buttons in the Excel ribbon are in the XML subfolder. Contents of all subdirectories are reported in a README file in the root 

directory of the unzipped AppendixB_Codes-SeriesSEE.v1.20.zip file.

Appendix C. Verification of Analytical Solutions 

Analytical solutions that were computed with the FORTRAN program WLmodel and published results of the same solutions in the zipped file, AppendixC_Verification.zip, can be accessed and downloaded at http://dx.doi.org/10.3133/tm4-F4. The 

analytical models for pneumatic lag, gamma, moving average, step, Theis, and tide are verified against known solutions in the 

subfolders AirLAG, Gamma, MovingAverage, Step, Theis, and Tide, respectively. Contents of all subdirectories are reported in 

a README file in the root directory of the unzipped AppendixC_Verification.zip file. 

Appendix D. Hypothetical Test of Theis Transforms

The Excel program, HypoFrame, measured water levels, measured barometric changes, and reported water-level models 

in the zipped file, AppendixD_HypotheticalAquifer.zip, can be accessed and downloaded at http://dx.doi.org/10.3133/tm4-F4.

HypoFrame is a workbook for simulating hypothetical aquifer tests and creating water levels with known pumping signals and 

environmental fluctuations. The premise and usage of HypoFrame are documented in the compressed help file 00_HypoFrameHELP.chm. Measured water levels and barometric changes that serve as environmental fluctuation sources and background 

water levels are in the file 00_Meas+Back-for-Analysis.xlsx. Reported water-level models and tools for viewing parameter correlation are in the subfolder WLMs. 

Appendix E. Pahute Mesa Example

Measured water levels, measured barometric changes, pumping signals, and reported water-level models in the zipped file, 

AppendixE_PahuteMesaExample.zip, can be downloaded at http://dx.doi.org/10.3133/tm4-F4. The zip file contains the pumping 

response in well ER-20-7 from the ER-20-8 main upper and lower aquifer tests.
For additional information, contact: 

Director, Nevada Water Science Center 

U.S. Geological Survey 

2730 N. Deer Run Rd. 

Carson City, NV 89701 

http://nevada.usgs.gov/ 

Halford and others—Advanced Methods for Modeling Water-Levels and Estimating Drawdowns with SeriesSEE, an Excel Add-In—Techniques and Methods 4–F4, ver. 1.1, July 2016

ISSN 2328-7047 (online) 

http://dx.doi.org/10.3133/tm4F4