U.S. Department of the Interior

U.S. Geological Survey

Open-File Report 2020–1007

Prepared in cooperation with the National Oceanic and Atmospheric Administration’s National 

Integrated Drought Information System

Spectral Analysis to Quantify the Response of Groundwater 

Levels to Precipitation—Northwestern United States
Cover: U.S. Geological Survey groundwater site 473442118162201 (also known as the 

“Davenport well”), Lincoln County, Washington. [Shown in the picture are the well 

head and the equipment used for transmitting real-time data. Photograph by 

Kimberly Cesal, U.S. Geological Survey.]
Prepared in cooperation with National Oceanic and Atmospheric Administration’s 

National Integrated Drought Information System

Spectral Analysis to Quantify the Response of 

Groundwater Levels to Precipitation—Northwestern 

United States 

Andrew J. Long and Christopher P. Konrad

Open-File Report 2020–1007

U.S. Department of the Interior

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

DAVID BERNHARDT, Secretary

U.S. Geological Survey

James F. Reilly II, Director

U.S. Geological Survey, Reston, Virginia: 2020

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

https://www.usgs.gov/ or call 1–888–ASK–USGS (1–888–275–8747). 

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

visit https://store.usgs.gov/.

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

endorsement by the U.S. Government.

Although this information product, for the most part, is in the public domain, it also may

contain copyrighted materials as noted in the text. Permission to reproduce copyrighted items

must be secured from the copyright owner.

Suggested citation:

Long, A.J., and Konrad, C.P., 2020, Spectral analysis to quantify the response of groundwater levels to 

precipitation—Northwestern United States: U.S. Geological Survey Open-File Report 2020–1007, 18 p., 

https://doi.org/10.3133/ofr20201007. 

ISSN 2331-1258 (online)
iii

Contents

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

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

Methods......................................................................................................................................................... 4

Precipitation Analysis for the Example Site ................................................................................................ 5

Spectral Analysis ................................................................................................................................... 5

Cumulative Departure from the Mean.................................................................................................... 8

Groundwater Analysis for the Example Site ............................................................................................... 8

Spectral Analysis ................................................................................................................................... 9

Cumulative Departure from the Mean.................................................................................................. 11

Lag in Groundwater Response .................................................................................................................... 13

Synthesis of Results for all Sites ................................................................................................................. 14

Possible Alternative Methods ...................................................................................................................... 15

Summary and Conclusions.......................................................................................................................... 16

References Cited......................................................................................................................................... 17

Figures

Figure 1. Map showing study area with monitoring wells included in groundwater-level analysis................ 2

Figure 2. Graph showing frequency spectrum of precipitation for site WA-4734, Davenport, Washington .. 6

Figure 3. Graph showing filtered precipitation record for site WA-4734, with all frequencies removed 

except those at 10 and 20 cycles per decade................................................................................................ 6

Figure 4. Graph showing cumulative departure from the mean , computed on a daily basis, for 

precipitation at site WA-4734, Davenport, Washington.................................................................................. 8

Figure 5. Graph showing groundwater level for site WA-4734 and monthly precipitation at National 

Oceanic and Atmospheric Administration station USC00452007 .................................................................. 9

Figure 6. Graph showing frequency spectrum of groundwater level for site WA-4734, Davenport, 

Washington.................................................................................................................................................. 10

Figure 7. Filtered groundwater-level record for site WA-4734, with all frequencies removed except those at 

10 and 20 cycles per decade....................................................................................................................... 10

Figure 8. Graph showing measured groundwater level and cumulative departure from the mean for 

precipitation for site WA-4734, Davenport, Washington .............................................................................. 12

Figure 9. Graph showing frequency spectrum of the cumulative departure from mean precipitation at site 

WA-4734, Davenport, Washington .............................................................................................................. 12

Figure 10. Graph showing illustrative graph showing typical annual cycles of precipitation and groundwater 

level for site WA-4734.................................................................................................................................. 14

Tables 

Table 1. Sites described and analyzed. ....................................................................................................... 3

Table 2. Results of data analysis for each site. ............................................................................................ 7
iv

Conversion Factors

U.S. customary units to International System of Units

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

inch (in.) 25.4 millimeter (mm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

square mile (mi2) 259.0 hectare (ha)

square mile (mi2) 2.590 square kilometer (km2) 

Flow rate

inch per year (in/yr) 25.4 millimeter per year (mm/yr)

International System of Units to U.S. customary units 

Multiply By To obtain

Length

kilometer (km) 0.6214 mile (mi)

kilometer (km) 0.5400 mile, nautical (nmi) 

Datums 

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 (NAVD 88) or the National Geodetic 

Vertical Datum of 1929 (NGVD 29). 

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.
1 

Spectral Analysis to Quantify the Response of 

Groundwater Levels to Precipitation—Northwestern 

United States 

Andrew J. Long and Christopher P. Konrad

Abstract

Persistent atmospheric patterns that lead to wet and dry seasons and droughts over 

periods of months to decades and longer-term climate change over periods of decades to 

millennia affect groundwater resources. Changes in groundwater storage and the resulting 

groundwater discharge from most aquifers are relatively slow and steady compared to the 

variability of daily precipitation. The response of groundwater levels to precipitation can be 

complex because of a combination of processes that include evapotranspiration, surface runoff, 

and infiltration of net recharge from precipitation through the vadose zone. Typically, this

response is delayed and results in a change in groundwater storage reflected in a time series of 

groundwater levels. Understanding the relations between variations in precipitation and changes 

in groundwater storage is essential to water resources planning. The objectives of this study were 

to (1) characterize the relation between precipitation and responses in groundwater levels at 

seasonal to decadal scales and (2) develop methods that are transferable on a continental scale to 

any groundwater-level record. Spectral analysis was applied to daily precipitation and 

groundwater levels for eleven monitoring wells in the northwestern Unites States with records 

ranging in length from 5.9 to 23.9 years. The analysis provided a quantitative characterization for 

each monitoring well that met both objectives and indicated that maximum and minimum 

precipitation rates generally occurred in December and August, respectively. Maximum 

groundwater levels and storage occurred from February to August, and minimum values 

occurred from January to December. The lag in the annual peak response of groundwater to peak 

precipitation ranged from 2.2 to 8.8 months, with a median value of 5.3 months. Groundwater 

responses to wet and dry seasons were evident in the relatively high amplitude frequencies of 10 

and 20 cycles per decade (cpdec). A high amplitude frequency at 1 cpdec represents a drought 

cycle that resulted in larger groundwater level changes than typical seasonal water level 

fluctuations. 

Introduction

Long-term records of groundwater levels are critical for assessing the effects of drought 

on water resources. The response of groundwater levels to precipitation occurs more slowly than 

the response of surface water (levels and flows) and is less sensitive to short-term variation in 

weather. Persistent atmospheric patterns, including wet and dry seasons and droughts, occurring 

over periods of months to decades, and longer-term climate change, occurring over periods of 
2 

decades to millennia, affect groundwater resources. For example, an aquifer might provide a 

reliable water supply during a drought when surface water availability is reduced. Thus, the 

aquifer may serve as a buffer to reduce the temporal variability of the overall water supply. At 

the end of an extended drought, however, surface water may recover quickly in response to 

increased precipitation, while the recovery of groundwater will likely lag. Understanding the 

relations between precipitation and groundwater storage is essential to water resources planning. 

Time series of groundwater-level measurements are the primary datasets, or records, used to 

monitor changes in groundwater storage. The objectives of this study were to (1) characterize the 

relation between precipitation and responses in groundwater levels at seasonal to decadal scales

and (2) to develop methods that are transferable on a continental scale to any groundwater-level 

record of sufficient length. Groundwater-level response to precipitation was analyzed for eleven 

monitoring wells in the northwestern Unites States (fig. 1; table 1), which are part of the Climate 

Response Network (CRN) of the U.S. Geological Survey (2019a). The CRN consists of wells 

that are useful for monitoring groundwater responses to climate variability, because by design, 

they are minimally affected by human activities, such as nearby groundwater withdrawals and 

irrigation recharge. 

Figure 1. Map showing study area with monitoring wells included in groundwater-level analysis (red dots). 
3 

Table 1. Sites described and analyzed. 

[A site is a location that includes one groundwater monitoring well. Data obtained from U.S. Geological Survey 

(2019b). Site name: WA (Washington), ID (Idaho), and OR (Oregon) indicate the state the site is located in. 

Abbreviations: ft, feet; NAVD88, North American Vertical Datum of 1988; NGVD29, National Geodetic Vertical 

Datum of 1929; USGS, U.S. Geological Survey] 

 Groundwater period of record

Site 

name

USGS 

Groundwater 

station No.

Other

identifier 

Principal 

aquifer

Local 

aquifer

Land surface 

altitude, in ft

Well depth, in ft

Begin date (yearmonth-day)

End date (yearmonth-day)

Years of record

WA_4734 473442118162201 Davenport

Columbia Plateau 

basaltic-rock 

aquifers

Wanapum

Basalt 

Formation

2,3711 117 2003-

09-03

2017-

11-02 14.2

WA_4619 461935118081501 Whetstone

Columbia Plateau 

basaltic-rock 

aquifers

Grande 

Ronde Basalt 

Formation

1,4031 173 2011-

11-22

2017-

11-02 5.9

WA_4740 474011117072901 Spokane Valley Columbia Plateau 

basin-fill aquifers

Scabland 

Flood 

Deposits

2,0591 129 2011-

12-21

2017-

11-02 5.9

WA_4650 465033122570202 Scatter Creek Puget Sound 

aquifer system

Unclassified 

Overburden 2211 82 2003-

09-30

2017-

11-02 14.1

ID_4754 475439116503401 53N 04W 

28CAB1

Pacific Northwest 

basin-fill aquifers Outwash 2,4321 449 2004-

07-14

2017-

11-02 13.3

ID_4327 432700112470801

02N 31E 

35DCC1

USGS 1

Snake River Plain 

basaltic-rock 

aquifers

Snake River 

Group 5,0222 630 1999-

06-20

2017-

11-02 18.4

OR_4300 430029121552101 30.00S/07.00E06AAA01

Pacific Northwest 

volcanic-rock 

aquifers

Mazama 

Pumice 4,7952 205 2000-

09-12

2016-

10-13 16.1

OR_4529 452912122312801 01.00S/02.00E14ABC01

Willamette 

Lowland basin-fill 

aquifers

Troutdale 

Formation 2292 59 1998-

10-02

2017-

11-01 19.1

OR_4520 452033122195901 02S/04E29DAD Other aquifers Sandy River 

Mudstone 7122 177 2001-

09-07

2017-

11-01 16.2

OR_4422 442242121405501 14S/09E08ABA

Pacific Northwest 

volcanic-rock 

aquifers

Volcanic 

Rocks 3,3802 403 1993-

11-24

2017-

11-01 23.9

OR_43443 434400121275801 21S/11E19CCC 

Pacific Northwest 

basin-fill aquifers Valley Fill 4,2202 100 1999-05-15

2016-

10-26 17.5

1

NAVD 88

2

NGVD 29

3

Discontinued in 2016 because of discontinued funding

Long periods of record for groundwater levels, in response to precipitation, allow 

analysis of climatic variation over seasonal to interdecadal scales. Seasonal cycles in this report 

refer to those that occur within a period of 1 year (yr). Of the 17 monitoring wells in 

Washington, Oregon, and Idaho that are part of the CRN, 6 were not suitable for this analysis 

because the periods of record were less than 5 yr. The remaining 11 CRN wells were included in 
4 

this study. Periods of daily record available for these 11 wells ranged from 5.9 to 23.9 yr, and all 

but 2 wells had more than 13 yr of record (table 1). 

In this report, a “site” is defined as a location with a groundwater monitoring well. To 

describe the approach used in the study, one site is selected to demonstrate the methods, 

assumptions, and interpretations that were applied to all sites, for which results are then

summarized. Site WA-4734, located in Lincoln County in eastern Washington, is used for

demonstration (fig. 1; table 1). The well is 117 feet (ft) deep and open to the Wanapum Basalt 

Formation, which is a local aquifer contained within a principal aquifer named the Columbia 

Plateau basaltic-rock aquifers (fig. 1), one of many designated principal aquifers in the United 

States (U.S. Geological Survey, 2018). The Wanapum Basalt Formation has an areal extent of 

about 25,000 square miles (mi2

) within the Columbia River basin and ranges in thickness from 

less than 10 to 1,200 ft (Kahle and others, 2009). At site WA-4734, the Wanapum Basalt

Formation is exposed at land surface (2,371 ft above NAVD 88) and is about 400 ft thick. The 

formation comprises multiple basalt layers, also known as “basalt flows,” with permeable zones 

at the contacts between basalt layers. At this site, eight primary permeable zones are present 

within the Wanapum Basalt Formation and range in thickness from about 10 to 50 ft (Myers, 

1972). The well is open to the uppermost permeable zone, which consists of broken and 

honeycombed basalt overlain by a basalt layer about 100 ft thick. The basalt layers generally 

have low permeability, but groundwater may flow vertically through fractures in these units and 

provide possible hydraulic connection between the land surface and the permeable zones 

(Vaccaro and others, 2015). 

Methods

Spectral analysis was used to characterize the dominant frequencies in the precipitation 

and groundwater-level records. The discrete Fourier transform (DFT) was executed in 

MATLAB® (MathWorks, 2019) to compute the frequency spectra of precipitation and 

groundwater-level (hydrologic) records by using methods described by Frigo and Johnson (1998) 

and FFTW (2018). The DFT was used to quantify the magnitude of variation in the hydrologic 

records for frequencies ranging from 0.25 to 45 cycles per decade (cpdec) and thereby 

characterize seasonal cycles, as well as possible longer-term climatic cycles. The DFT breaks 

down a time-series record into a set of sinusoidal functions, each with a specific amplitude and 

frequency (periodicity) that, when summed, reproduce the time-series record. Each frequency in 

the spectrum defines a single sinusoid, and any component with a particularly high amplitude 

indicates strong variation in the record at that frequency. A general description of spectral 

analysis, including the DFT, is provided in Jenkins and Watts (1968), Smith (2003), and other 

references on time-series analysis. 

The cumulative departure from the mean (CDM) for daily precipitation was used to 

identify the timing (years and months) of wet and dry periods for the precipitation record. The 

CDM is calculated for each day as the difference between the cumulative sum of the difference 

between daily precipitation and mean daily precipitation. The DFT was applied to the CDM for 

precipitation, which provided information complimentary to the application of the DFT to the 

measured precipitation record.
5 

Precipitation Analysis for the Example Site

Spectral Analysis 

A record of estimated daily precipitation was obtained from DAYMET (Thornton and 

others, 2018) for the 1-kilometer (km) × 1-km grid cell containing each site for 1980‒2016 (37-

yr period). Advantages of using DAYMET are that precipitation estimates are available across 

the United States, with no missing daily values. For this analysis, we assumed that precipitation 

at a site provides an index, or proxy, for the timing and variation of groundwater recharge, 

However, precipitation stored as snowpack was considered in interpretation of results. The mean 

precipitation (1980‒2016) from DAYMET for site WA-4734 was 16.3 inches per year (in/yr).

As a comparison, the mean precipitation (1980‒2016) measured daily by the National Oceanic 

and Atmospheric Administration (NOAA) at a location 7.6 miles to the northeast had a mean 

value of 14.0 in/yr (station number USC00452007, available at https://www.ncdc.noaa.gov/).

About 1 percent of the daily values were missing from the measured record. Although site WA4734 had a long-term precipitation station within 10 miles, this is not the case for most sites, and 

therefore, DAYMET precipitation was used for all sites. 

The frequency spectrum for DAYMET precipitation at this site has a dominant (high 

amplitude) frequency component at 10 cpdec and a subdominant frequency at 20 cpdec (fig. 2), 

which equate to sinusoid wavelengths (λ) of 1.0 and 0.5 yr, respectively. Because 1 year spans 

one complete cycle of the seasons, dominant wavelengths that are factors of 1.0 yr, such as 0.5 or 

0.25, indicate strong seasonality. For example, a sinusoid with λ = 1 yr contains one upward peak 

and one downward peak in a 1-yr period, which might represent a wet and a dry season, 

respectively. A sinusoid with λ = 0.5 yr has two upward and two downward peaks that might 

represent other characteristics of seasonality. Therefore, to characterize seasonality, the inverse 

DFT was used to reconstruct the precipitation record as a function of time (time domain) from 

only the two dominant frequencies of 10 and 20 cpdec (frequency domain). This reconstruction 

is referred to as the filtered precipitation record because all but the two dominant frequencies 

were removed (fig. 3). 
6 

Figure 2. Graph showing frequency spectrum of precipitation for site WA-4734, Davenport, Washington. 

[Site information detailed in table 1.]

Figure 3. Graph showing filtered precipitation record for site WA-4734 (Davenport, Washington), with all 

frequencies removed except those at 10 and 20 cycles per decade. [Site information detailed in table 1. 

The record was shifted to a mean of zero.] 

Because the sinusoids composing the filtered precipitation record have wavelengths that 

are factors of 1 yr (λ = 0.5 and 1.0), the high and low spectral amplitude values occur at the same 

time during each year. These values represent the most likely time of occurrence of extremes that 

reoccur on an annual basis and provide a way to characterize seasonality. The filtered 

precipitation record for WA-4734 indicates that the maximum and minimum values (5.9×10-3
7 

and -7.3×10-3) occur on December 10 and August 17, respectively, indicating a time difference 

from maximum to minimum (annual period of decrease) of about 8.2 months (table 2). The 

period from maximum dry to maximum wet (annual period of increase) occurs over 3.8 months 

(table 2). 

Table 2. Results of data analysis for each site.

[Period of annual increase: Time between the annual minimum to the annual maximum. Period of annual 

decrease: Time between the annual maximum to the annual minimum. Lag in water level response: Time period 

from annual maximum precipitation to annual maximum groundwater level. Abbreviations: cpdec, cycles per 

decade; ––, could not be determined from the analysis or not applicable]

Precipitation analysis Groundwater level analysis

Site name

Annual maximum (daymonth)

Annual minimum (daymonth)

Period of annual 

increase, in months

Period of annual 

decrease, in months

Dominant seasonal 

frequencies, in cpdec

Relative dominance of 

seasonal frequencies

Average annual 

precipitation, in inches

Annual maximum (daymonth)

Annual minimum (daymonth)

Period of annual 

increase, in months

Period of annual 

decrease, in months

Dominant seasonal 

frequencies, in cpdec

Relative dominance of 

seasonal frequencies

Lag in water level 

response, in months

WA_4734 10-Dec 17-

Aug 3.8 8.2 10, 

20 Strong 16.3 22-Apr 5-Dec 4.5 7.510, 20 Strong 4.4

WA_4740 12-Dec 18-

Aug 3.8 8.2 10, 

20 Strong 21.9 23-

May 6-Feb 3.5 8.5

10, 20 Strong 5.3

WA_4619 10-Dec 10-

Aug 4.0 8.0 10, 

20 Strong 18.7 3-May 20-

Dec 4.4 7.6

10, 20

Moderate 4.7

WA_4650 13-Dec 30-Jul 4.5 7.5 10, 

20 Strong 54.6 19-Feb 9-Oct 4.4 7.610, 20 Strong 2.2

ID_4754 12-Dec 16-

Aug 3.9 8.1 10, 

20 Strong 31.1 3-Aug 31-Jan 6.1 5.910

Moderate 7.7

ID_4327 9-May 21-

Aug 8.6 3.4 10, 

20

Weak 10.4 1-Feb 4-Aug 6.0 6.0

10

Moderate 8.8

OR_4529 13-Dec 5-Aug 4.3 7.7 10, 

20 Strong 51.9 3-Apr 1-Oct 6.0 6.010 Strong 3.6

OR_4520 12-Dec 5-Aug 4.2 7.8 10, 

20 Strong 65.5 18-

May 

16-

Nov 6.0 6.0

10, 20 Strong 5.2

OR_4422 20-Dec 9-Aug 4.4 7.6 10, 

20 Strong 46.3 19-Jul 15-Jan 6.1 5.910

Moderate 6.9

OR_4344 21-Dec 21-

Aug 4.0 8.0 10, 

20 Strong 18.1 13-

Aug 10-Feb 6.1 5.9

10 Strong 7.7

OR_4300 27-Dec 11-

Aug 4.5 7.5 10, 

20 Strong 36.5 – – – – None – – 

Median 12-Dec 11-

Aug 4.2 7.8 – – 31.1 10-

May 2-Sep 6.0 6.0 – – 5.3

The 20 cpdec frequency (λ = 0.5 yr) results in a small precipitation increase in the filtered 

record at about 0.3 yr into the middle of the overall period of decreasing precipitation (fig. 3). 

This increase in the filtered precipitation record may not represent an actual increase in 

precipitation during spring but probably is because of omitted frequencies. The physically 

meaningful effect of the 20 cpdec frequency is that, when combined with the 10 cpdec frequency 

(λ = 1 yr), it has the effect of lengthening the period of annual decreasing precipitation. Without 

the 20 cpdec frequency, the periods of annual increase and decrease each would be 6 months. If 

other dominant frequencies in multiples of 10 were present, those might indicate additional 

aspects of seasonality. Dominant frequencies not in multiples of 10 might indicate additional 

climatic cyclicity. 
8 

Cumulative Departure from the Mean

Dominant frequencies less than 10 cpdec (λ > 1 yr) are not apparent in the frequency 

spectrum for precipitation (fig. 2), so longer term climate variation, such as recurrent drought, 

are not evident from the spectrum. However, the CDM for the precipitation record indicates two

interannual wet periods and two interannual dry periods for site WA-4734 (fig. 4). Upward and 

downward slopes in the CDM represent wet and dry periods, respectively, and major upward and 

downward trends occur on roughly decadal scales. Also evident in figure 4 are annual 

fluctuations that represent seasonality, and higher frequency fluctuations at the sub-month scale. 

Therefore, the CDM provides useful precipitation information at a wide range of time scales. 

Figure 4. Graph showing cumulative departure from the mean, computed on a daily basis, for precipitation 

at site WA-4734, Davenport, Washington. [Site information detailed in table 1.]

The DFT fits sinusoids to a hydrologic record, and these sinusoids are stationary because 

these functions do not have an overall upward or downward trend. Therefore, the DFT is most 

appropriate and meaningful if the record also is a stationary time series, and a useful application 

of the CDM is for assessing the stationarity of a time series. If the mean of the CDM is nearly 

zero, the underlying time series is nearly stationary because departures from the mean are 

balanced between positive and negative. For site WA-4734, we consider the precipitation record

to be nearly stationary because the CDM mean is about 1.7 inches, which is small in comparison 

to its total range of values (26.6). 

Groundwater Analysis for the Example Site

The period of record analyzed for site WA-4734 is October 2003–October 2017 (14-yr 

period), with a groundwater level ranging from 2,321 to 2,342 ft above NAVD 88 and median 

value of 2,333 ft (fig. 5). During this period, seasonal water level fluctuations ranged from 2 to 

14 ft, and decadal fluctuations were as high as 15‒20 ft. Groundwater-level records for all sites 

were obtained from the National Water Information System (NWIS; U.S. Geological Survey, 

2019b). All groundwater stations and available data are searchable and retrievable, respectively, 

in NWIS by the station number shown in table 1. 
9 

Figure 5. Graph showing groundwater level for site WA-4734 and monthly precipitation at National 

Oceanic and Atmospheric Administration station USC00452007. [Site information for site WA-4734 detailed 

in table 1.]

Spectral Analysis

The frequency spectrum for groundwater level for site WA-4734 has dominant (high 

amplitude) frequencies in the bandwidth of 1‒4 cpdec, which correspond to wavelengths of 10.0, 

5.0, 3.4 and 2.6 yr (fig. 6). These represent interannual fluctuations in the record. For example, 

the 11-yr period 2005‒15 began and ended with anomalous declines in groundwater levels, when 

seasonal recharge was insufficient for recovery to typical annual maximum levels (~2335 ft) (fig. 

5). Both of these declines were followed by a rapid recovery (2006 and 2016‒17). Most of this 

area was in drought during 2005 and 2014‒15 (U.S. Drought Monitor, 2019) indicating that the 

declining groundwater levels were likely a result of low recharge. These two droughts were 

separated by about 1 decade, and the high amplitude at 1 cpdec largely represents the timing of 

the two droughts, which resulted in larger groundwater level changes than typical seasonal water 

level fluctuations (fig. 5). The meaning of the frequency bandwidth of 2‒4 cpdec is less obvious 

but represents a combination of climatic and hydrologic factors that may be partly related to 

drought. For example, these frequencies may reflect differences in the severity of the two 

droughts, as well as the rapid post-drought groundwater recoveries in contrast to the slower

groundwater recession rates during droughts. Also, the groundwater level peaks gradually 

declined for the periods 2007‒10 and 2011‒14 (fig. 5), a pattern that likely is represented by 

some combination of the 2‒4 cpdec bandwidth. 
10 

Figure 6. Graph showing frequency spectrum of groundwater level for site WA-4734, Davenport, 

Washington. [Site information detailed in table 1.] 

Within the bandwidth of potential seasonality (10‒40 cpdec), there is a dominant and 

subdominant frequency peak at 10 and 20 cpdec, respectively (fig. 6), which is similar to the 

precipitation spectrum. These are the two highest peaks within this bandwidth; frequencies less 

than 10 cpdec represent longer hydrologic cycles that include drought. Therefore, to characterize 

the groundwater seasonality, the inverse DFT was used to reconstruct the groundwater-level 

record from only these two seasonal frequencies, similarly to the approach applied to 

precipitation. This reconstruction is referred to as the filtered groundwater-level record and 

indicates that the highest groundwater levels generally occurred in mid-April, and lowest levels

about 7.5 months later in early December (fig. 7; table 2). Therefore, periods of annual 

increasing and decreasing groundwater levels occurred over 4.5 and 7.5 months, respectively. 

Figure 7. Filtered groundwater-level record for site WA-4734 (Davenport, Washington), with all 

frequencies removed except those at 10 and 20 cycles per decade. [Site information detailed in table 1. 

The record was shifted to a mean of zero.] 
11 

It is noteworthy that the seasonal fluctuation in the filtered record (fig. 7) is much smaller 

than the seasonal fluctuation in the measured groundwater record (fig. 5). The reason is that the 

filtered groundwater-level record includes only the water-level change corresponding to exactly

10 and 20 cpdec. However, the 10 cpdec frequency peak is centered on a bandwidth of relatively 

high amplitudes (9.7‒10.3 cpdec) that contributes to seasonal fluctuations in the groundwaterlevel record, and the 20 cpdec frequency is centered on another bandwidth of similar effect 

(19.7‒20.3 cpdec; fig. 6). To determine the occurrence of annual high and low groundwater 

levels, only the highest amplitude frequency component within each of these bandwidths was 

necessary. Therefore, by ignoring the full bandwidths that capture seasonal water-level 

fluctuations, the amplitudes of the seasonal fluctuations in the filtered record were reduced, but 

the temporal characterization of the seasonal fluctuation was achieved. If the full bandwidths of 

the two frequency peaks were used to construct the filtered record, this would provide an 

estimate of the magnitude of temporal variation resulting from those seasonal bandwidths. 

However, the specific upper and lower bounds of the bandwidths would have to be selected 

carefully to obtain a useful estimate. 

Cumulative Departure from the Mean

The CDM for precipitation has some synchrony to the measured groundwater-level 

record (fig. 8). The CDM for precipitation is similar to the measured groundwater-level record, 

particularly regarding seasonal fluctuations. Vaccaro and others (2015) compared the CDM for 

annual precipitation at NOAA station USC00452007 to the groundwater-level record for WA4734 and also noticed similarities. The similarity suggests that, in a general sense, groundwater 

levels represent the cumulative effects of precipitation. Limitations in interpretation of the CDM 

for hydrologic analysis were discussed by Weber and Stewart (2004); application of the CDM in 

this report avoids these limitations by focusing on the CDM’s spectral characteristics for 

understanding groundwater response periodicity, rather than as a proxy for actual groundwater 

storage or levels. 

The smoothing effect of the CDM for precipitation provides useful information over a 

wide range of time scales that otherwise would be difficult to obtain by examining the measured 

daily precipitation record directly. However, computing the CDM for groundwater levels was not 

useful. Because groundwater processes controlling the response of water levels to recharge from 

precipitation inherently act as a filter to attenuate and smooth the response, a computed CDM for 

groundwater level represents an additional smoothing of the water-level response that obscures 

critical information related to these controlling processes. Therefore, for the purposes of this 

study, the CDM for groundwater level was not used in the subsequent analyses. 
12 

Figure 8. Graph showing measured groundwater level (red) and cumulative departure from the mean

(CDM) for precipitation (blue) for site WA-4734, Davenport, Washington. [Site information detailed in table 

1.]

The similarity of the measured groundwater-level record to the CDM for precipitation 

(fig. 8) suggests that their respective frequency spectra also might be similar. Therefore, the DFT 

was applied to the CDM for precipitation (fig. 9). This frequency spectrum has characteristics 

similar to the spectrum of the groundwater-level record, which is evident by comparison of fig. 9 

to fig. 6. Both spectra have dominant frequencies of 10 and 20 cpdec, as well as dominant low 

frequencies less than 5 cpdec. The similarity of these two spectra might increase if the 

groundwater-level record (14-yr period) was similar in length to that of the precipitation record 

(37 yr). 

Figure 9. Graph showing frequency spectrum of the cumulative departure from mean precipitation at site 

WA-4734, Davenport, Washington. [Site information detailed in table 1.]
13 

The frequency spectrum of the CDM for precipitation has dominant frequencies at 10 and 

20 cpdec (fig. 9), similarly to that of the original precipitation record (fig. 2). However, a 

characteristic in the frequency spectrum for the CDM for precipitation that is absent from the 

frequency spectrum of the original precipitation record is the dominance of low frequencies, with 

a peak at 0.56 cpdec, which corresponds to a wavelength of 18 yr (fig. 9) and indicates

interannual or interdecadal cycles that were not detectable from spectral analysis of the original 

precipitation record (fig. 2). Therefore, the analysis shown in figure 9 preserved useful 

information from the original precipitation record (seasonal cycling), but also provided 

information regarding interannual and interdecadal cycling. 

The interannual cycling of groundwater levels was previously determined to have 

wavelengths ranging from 2.6 to 10 yr (fig. 6). This range corresponds to that of the CDM for 

precipitation (3–100 yr), except that λ > 10 yr are not dominant in the groundwater spectrum, 

because the period of record was only 14 yr. This suggests, however, that groundwater levels 

respond to interannual precipitation cycling, but quantification of this response through spectral 

analysis, such as estimating response lag times, is not possible without a longer groundwater 

record. 

Lag in Groundwater Response

Several factors affect the response of groundwater levels to precipitation, including those

related to evapotranspiration, runoff, vadose zone infiltration, and aquifer hydraulics. Winter 

snow accumulation, sublimation, and spring snowmelt are additional factors in many climates, 

including the climate at site WA-4734. Comparison of the seasonality characteristics of 

precipitation to those of groundwater for this site indicates that the lag time from the annual 

precipitation peak (early December) to the annual groundwater peak (mid-April) is 4.4 months, 

which likely would be shortened if not for other factors, most notably snow accumulation. The 

lag in the annual peak response of groundwater following the annual peak precipitation was 

computed as the time difference between these two peaks in the filtered records, as quantified by 

the DFT analysis. Figure 10 illustrates the hydrologic characteristics of site WA-4734 as 

determined by the analysis: the times of occurrence of the highest and lowest hydrologic values, 

the lag time from peak precipitation to peak groundwater level, and the periods of annual 

increasing and decreasing hydrologic values. 
14 

Figure 10. Graph showing illustrative graph showing typical annual cycles of precipitation and groundwater 

(GW) level for site WA-4734. [The time lag from the peak in precipitation to the peak in groundwater level is 

4.4 months.] 

Synthesis of Results for all Sites

All sites were analyzed by the same methods as those described for the example site 

(WA-4734), and results are summarized in table 2. Dominant seasonal frequencies were those at 

10 and 20 cpdec, which were evident in the precipitation spectra for all sites. For groundwater, 

five sites had dominant frequencies at 10 and 20 cpdec, and five sites had dominant frequencies 

at 10 cpdec but not at 20 cpdec (table 2). One site (OR-4300) had dominant frequencies only at 

the lowest end of the spectrum, with no indication of dominant seasonal frequencies. The relative 

dominance of seasonal frequencies, as a comparison between all sites, was qualitatively assigned 

as “strong,” “moderate,” or “weak.” Ten sites were characterized as having strong seasonal 

precipitation. The one site with weak seasonal frequencies for precipitation (ID-4327) had the 

lowest average precipitation of all sites (10.4 in/yr). For groundwater levels, six sites where 

characterized as having strong seasonality, and four were characterized as having moderate

seasonality (table 2). 

The time of occurrence of annual maximum and minimum values for precipitation and 

groundwater levels were determined for all sites (table 2). For precipitation, the annual 

maximum values were primarily in December for 10 sites and May for 1 site (ID-4327), whereas 

the annual minimum values were in late July or August for all sites. The annual period of 

decrease, or time from maximum to minimum precipitation, ranged from 7.5 to 8.2 months for 

ten sites, with one site far outside of this range (3.4 months for site ID-4327). For groundwater, 

the occurrence of annual maximum and minimum values had much larger ranges than those for

precipitation: maximum values ranged from February to August, and minimum values ranged 

from January to December. The annual period of decrease for groundwater also had a wider 
15

range than that of precipitation, ranging from 5.9 to 8.5 months, with a median value of 6.0

months.

The lag in the annual peak response of groundwater following the annual peak

precipitation is influenced by several factors, as previously discussed, and may vary widely 

across the study area, ranging from 2.2 to 8.8 months, with a median value of 5.3 months (table 

2). In some cases, this lag results in groundwater levels that are highest during dry seasons, as 

evidenced by three sites that had annual groundwater maximums at about the same time that 

precipitation was at a minimum (ID-4754, OR-4422, and OR-4344). This information is useful

for management of groundwater and surface water resources on a seasonal basis. 

The first objective of this study was to characterize the relation between precipitation and 

responses in groundwater levels at seasonal to decadal scales. The first objective was met, in that 

seasonal cycles were characterized well, and it was concluded that groundwater levels respond to 

interannual precipitation cycling. However, quantification of the interannual response through 

spectral analysis was not possible without longer groundwater records. Therefore, while spectral 

analysis indicated that the peak groundwater response is delayed by several months after peak 

precipitation, it did not provide an estimate of the full length of time that the effects of

precipitation on groundwater may last. For example, Long and Mahler (2013) estimated that this 

system memory generally was on the order of years to decades and was a substantial factor in the 

long-term groundwater response. Additional methods are available and could provide more 

detailed assessments of how groundwater storage responds to precipitation over a wide range of 

time scales. The following section briefly describes these methods.

The second objective was to develop methods that are transferable on a continental scale 

to any groundwater-level record. Because this method was applied to 11 wells that are widely 

distributed across the northwestern United States and are located in a variety of aquifer settings, 

the method is likely to be effective for any groundwater system that responds primarily to 

precipitation. 

Possible Alternative Methods

Spectral analysis was useful in estimating the time lag of seasonal groundwater response 

to seasonal precipitation cycles and for general comparison of interannual and interdecadal 

cycles in precipitation and groundwater level. However, a useful next step in time-series analysis 

that would quantifiably characterize the response of groundwater levels to precipitation over all 

relevant time scales (seasonal, interannual, and interdecadal) is time-series modeling. For 

example, Jakeman and Hornberger (1993), Von Asmush and others (2002), and Long and 

Mahler (2013) described time-series models to simulate groundwater and surface-water 

responses to precipitation by estimation of an impulse-response function (IRF). The IRF

succinctly represents the full spectrum of long- and short-term responses. By examining the IRF, 

these authors estimated that the response of groundwater levels to a single precipitation event can 

last for decades, in some cases. The IRF also defines the peak response and recession curve of 

groundwater levels. Another advantage of modeling is that a quantitative measure of the 

goodness of fit between measured and simulated water levels can be computed, providing an 

indication of the accuracy of the estimated response times and groundwater storage changes over 

those response times. The Rainfall-Response Aquifer and Watershed Flow Model 

(RRAWFLOW) could be used for this purpose and is available without cost (Long, 2015, 2019). 
16

Summary and Conclusions

Understanding the relations between variations in precipitation and changes in 

groundwater storage as indicated by changes in groundwater levels is essential to water resources 

planning. The objectives of this study were to (1) characterize the relation between precipitation 

and responses in groundwater levels at seasonal to decadal scales and (2) to develop methods 

that are transferable on a continental scale to any groundwater-level record. Eleven groundwaterlevel monitoring wells were selected for this study, with periods of record ranging from 5.9 to 

23.9 years (yr). For each well location, estimated precipitation records of 37 yr in length were 

analyzed. 

For analyzing precipitation records, spectral analysis was useful for characterizing 

seasonality, and examining the cumulative departure from the mean (CDM) was useful for 

assessing precipitation cycles over short- and long-term time scales. The smoothing effect of the 

CDM provides useful information that otherwise would be difficult to obtain by examining the 

measured daily precipitation record directly. Spectral analysis of the precipitation CDM was 

useful for quantifiably characterizing seasonal, interannual, and interdecadal cycling. Therefore, 

spectral analysis of the precipitation record and of the CDM for precipitation is recommended, 

but the latter might provide all necessary information. Maximum and minimum precipitation 

rates generally occurred in December and August, respectively, in the study area. The period 

from annual minimum-to-maximum precipitation generally was shorter (<6 months) than period 

of annual maximum-to-minimum precipitation (>6 months). 

For analyzing groundwater-level records, spectral analysis was useful for relating the 

seasonality of precipitation to groundwater responses and provided a general characterization of 

interannual groundwater-level cycles. Because groundwater moves slowly, allowing aquifers to 

store water for long periods, the groundwater-level record may represent the cumulative effects 

of precipitation over long time scales. Therefore, the groundwater-level record may resemble the 

CDM for precipitation for the same location. For the sites analyzed, maximum groundwater 

levels occurred from February to August, and minimum values occurred from January to 

December. The magnitudes of groundwater level responses to episodic droughts in comparison 

to those of seasonal cycles can be assessed from the higher amplitudes of the frequency spectrum 

at frequencies less than about 4 cycles per decade (cpdec) relative to smaller amplitudes for

frequencies at 10 and 20 cpdec.

Comparing the frequency spectrum of groundwater levels to that of the precipitation 

record provides an estimate of the typical lag time required for groundwater levels to reach a

peak response that follows the annual precipitation peak. This method also provides the typical 

times of year that precipitation and groundwater levels reach their highest and lowest values. Lag

times for this study ranged from 2.2 to 8.8 months, with a median value of 5.3 months.

Advantages of this method are that (1) it provides important metrics to characterize an aquifer’s 

response to precipitation, (2) it requires only daily precipitation and daily groundwater levels, 

and (3) it is not labor intensive and does not require model calibration. 

Time-series modeling is a related method that also uses daily precipitation and 

groundwater levels but would more fully characterize the response of groundwater levels to 

precipitation. Groundwater at a single location can respond to precipitation on a wide range of 

time scales, and these responses may be superimposed in the measured water-level record. Timeseries modeling can be used to characterize these different responses; for example, over monthly,

seasonal, interannual, and interdecadal time scales.
17

References Cited

Frigo, M., and Johnson, S.G., 1998, FFTW—An adaptive software architecture for the FFT: 

Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal 

Processing v. 3. p. 1381–1384, DOI:10.1109/ICASSP.1998.681704.

FFTW, 2018, FFTW (version 3.3.8): FFTW, accessed Jan 2019, at http://www.fftw.org.

Jakeman, A.J., and Hornberger, G.M., 1993, How much complexity is warranted in a rainfallrunoff model?: Water Resources Research, v. 29, no. 8, p. 2637–2649. 

Jenkins, G.M., and Watts, D.G., 1968, Spectral analysis and its applications: San Francisco, 

California, Holden-Day, 525 p.

Kahle, S.C., Olsen, T.D., and Morgan, D.S., 2009, Geologic setting and hydrogeologic units of 

the Columbia Plateau Regional Aquifer System, Washington, Oregon, and Idaho: U.S. 

Geological Survey Scientific Investigations Map 3088. [Also available at

https://pubs.er.usgs.gov/publication/sim3088.]

Long, A.J., 2015, RRAWFLOW—Rainfall-Response Aquifer and Watershed Flow Model 

(v1.15): Geoscientific Model Development, v. 8, no. 3, p. 865–880. 

Long, A.J., 2019, RRAWFLOW—Rainfall-response aquifer and watershed flow model: U.S. 

Geological Survey, Dakota Water Science Center. [Also available at

https://www.usgs.gov/centers/dakota-water/science/rrawflow-rainfall-response-aquifer-andwatershed-flow-model.]

Long, A.J., and Mahler, B.J., 2013, Prediction, time variance, and classification of hydraulic 

response to recharge in two karst aquifers: Hydrology and Earth System Sciences, v. 17, no. 1, 

p. 281–294.

MathWorks, 2019, MATLAB for artificial intelligence: MathWorks, web, accessed Jan 2019, at 

https://www.mathworks.com/. 

Myers, D.A., 1972, Test observation well near Davenport, Washington—Description and 

preliminary results: U.S. Geological Survey Open-File Report 72-265, 23 p. [Also available at 

https://doi.org/10.3133/ofr72265.]

Smith, S.W., 2003, Digital signal processing—A practical guide for engineers and scientists: 

Newnes, Amsterdam, 650 p., ISBN: 0-750767-44-X.

Thornton, P.E., Thornton, M.M., Mayer, B.W., Wei, Y., Devarakonda, R., Vose, R.S., and Cook, 

R.B., 2018, Daymet: Daily Surface Weather Data on a 1-km Grid for North America, Version 

3: Oak Ridge, Tennessee, accessed Jan 2019, at https://doi.org/10.3334/ORNLDAAC/1328.

U.S. Geological Survey, 2018, Map of the principal aquifers of the United States: U.S. 

Geological Survey database, accessed Jan 2019, at

https://water.usgs.gov/ogw/aquifer/map.html.

U.S. Geological Survey, 2019a, Climate response network: U.S. Geological Survey, 

Groundwater Watch, web. [Also available at 

https://groundwaterwatch.usgs.gov/CRNHome.asp.]

U.S. Geological Survey, 2019b, National water information system: U.S. Geological Survey, 

web, accessed Jan 2019, at https://waterdata.usgs.gov/nwis.

U.S. Drought Monitor, 2019, Time Series: U.S. Drought Monitor, accessed May 23, 2019, at

https://droughtmonitor.unl.edu/Data/Timeseries.aspx.
18

Vaccaro, J.J., Kahle, S.C., Ely, D.M., Burns, E.R., Snyder, D.T., Haynes, J.V., Olsen, T.D., 

Welch, W.B., and Morgan, D.S., 2015, Groundwater availability of the Columbia Plateau 

Regional Aquifer System, Washington, Oregon, and Idaho: U.S. Geological Survey 

Professional Paper 1817, 87 p. [Also available at https://doi.org/10.3133/pp1817.]

Von Asmuth, J.R., Bierkens, M.F.P., and Maas, K., 2002, Transfer function-noise modeling in 

continuous time using predefined impulse response functions: Water Resources Research, 

v. 38, no. 12, p. 23-1–23-12.
Publishing support provided by the U.S. Geological Survey

Science Publishing Network, Tacoma Publishing Service Center

For more information concerning the research in this report, contact the

Director, Washington Water Science Center

U.S. Geological Survey

934 Broadway, Suite 300

Tacoma, Washington 98402

https://www.usgs.gov/centers/wa-water
Long and Konrad—Spectral Analysis to Quantify the Response of Groundwater Levels to Precipitation—Northwestern United States—Open-File Report 2020–1007

ISSN 2331-1258 (online)

https://doi.org/10.3133/ofr20201007