Introduction

The quality and chemical budget of surface waters and groundwater depend on the source of water, and on the extent and timing of the interaction between precipitation, lake waters and groundwater. Those interactions have been studied previously in the Trout Lake Watershed in Wisconsin (Fig. 1) as part of the Long Term Ecological Research (LTER) and Water, Energy, and Biochemical Budget (WEBB) programs. Elder et al. (1992) and Magnuson et al. (1984) summarized those research programs. The Long Term Ecological Research (LTER) Network is a collaborative effort involving more than 1,800 scientists and students investigating ecological processes over long temporal and broad spatial scales. The Network promotes synthesis and comparative research across sites and ecosystems and among other related national and international research programs. The National Science Foundation established the LTER program in 1980 to support research on long-term ecological phenomena in the United States. The 26 LTER sites represent diverse ecosystems and research emphases. More detailed information can be found in LTER (2011).

Fig. 1
figure 1

Geographical location of the study area: a location of the State of Wisconsin (WI) in the United States; b location of Trout Lake Watershed in Wisconsin and of relevant sites in the watershed. LTER Long Term Ecological Research, ALAC Allequash site, MD Middle site, HRD Hardwood site

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The US Geological Survey initiated the Water, Energy, and Biogeochemical Budgets (WEBB) program in 1991 to understand the processes controlling water, energy, and biogeochemical fluxes over a range of temporal and spatial scales and to understand the interactions of these processes, including the effect of atmospheric and climatic variables. Five small research watersheds were selected, in part because they had existing long-term research data sets on which the WEBB program could build, and in part to be geographically and ecologically diverse and represent a range of hydrologic and climatic conditions. More detail can be found in WEBB (2011).

Estimation of the time lag between precipitation and groundwater level response is important because it can help to determine the source of water in cases of several potential sources. Furthermore, the magnitude and timing of fluctuations in precipitation and groundwater level may affect water quality, ecological function and use (Kelly 2001). In other words, the quality and ecological function of Trout Lake watershed aquifers depends on whether the source of water is lake discharge or direct precipitation recharge. Several previous studies have analyzed the relationship between precipitation and groundwater response at time scales on the order of days. Changnon et al. (1988) used autoregressive integrated moving average modeling to identify a time lag of 0–2 months between monthly precipitation and groundwater-level time series in Illinois. Kelly (2001) found a 2-day lag time between river stage changes and changes in stage at a flood plain wetland, and a 1-day lag between the river stage changes and groundwater level fluctuations in flood plain wetland.

Several studies within the LTER and the WEBB programs dealt with the interactions between precipitation, lake levels and groundwater at seasonal and interannual time scales. Krabbenhoft et al. (1990) used stable isotopes and a numerical model to estimate groundwater inflow and outflow to Sparkling Lake, one of the lakes in the Trout Lake watershed. Anderson and Cheng (1993) found that both short-term and long-term transient effects influence the hydrological and chemical budgets of Crystal Lake. Cheng and Anderson (1994) found that groundwater fluxes are more variable for lakes located higher in the Trout Lake watershed. Kim et al. (2000) studied the effect that seasonal precipitation and lake levels fluctuations in Crystal and Big Muskellunge Lakes had on the vertical mixing in the 50-m thick sandy aquifer between these two lakes. Dripps (2003) estimated groundwater recharge in the Trout Lake Basin using time series analysis techniques, analytical and numerical models. A complete list of references on the Trout Lake watershed is available in USGS (2011).

Most previous studies used field measurements and time domain methods, including physically based modeling. Physically based models require extensive, multidimensional and distributed field data to provide boundary and initial conditions and data for model verification, and significant computational effort. Frequency domain methods such as coherency analysis and the estimation of phase lag between hydrologic time series, provide an alternative, efficient method of analysis. Kuo et al. (1990) used squared coherency and phase functions to study the relationship between CO2 emission on the Earth and global temperature; the estimated phase lag was found to be 5 months. Tsonis et al. (2005) studied the coherence between ENSO and global temperature. Also Namdar Ghanbari and Bravo (2008) analyzed the coherence between large-scale signals, regional climate and Great Lakes levels, and Namdar Ghanbari et al. (2009) analyzed the coherence between teleconnections, local climate and lake-ice-cover duration. In addition, Namdar Ghanbari (2007) found significant coherencies between precipitation and groundwater level for aquifers in northern forested areas of Wisconsin. Reliable estimation of the interaction between precipitation, lakes and groundwater, and groundwater recharge can improve predictions made by physically based models and decisions made by water resources managers.

The objectives of this study were: (1) to estimate the phase lag between precipitation and groundwater level using spectral methods and 2 years of daily data, (2) to analyze the interactions between precipitation, lake levels and groundwater in the Trout Lake Watershed using spectral methods and almost two decades of quarterly data. The phase lag function between precipitation and groundwater is used to estimate recharge travel time.

Area of study and data

The Trout Lake Basin is in the Northern Highland area of north-central Wisconsin (Fig. 1). This site is included in the North Temperate Lakes LTER network sponsored by the National Science Foundation and the WEBB of the US Geological Survey program. The site includes five open lakes and two bog lakes. The area is a sandy outwash plain consisting of 30–50 m of unconsolidated sand and coarser till overlying Precambrian igneous bedrock (Walker and Bullen 2000). The soils are mainly thin forest soils with high organic content in the uppermost layer. The site is a typical glacial lake district that is usually seen in the upper Midwest and Canada, but certain individual characteristics of this area are different from other similar lake areas. Most of the lakes in the area are seepage lakes with no surface water inlet or outlets; therefore, precipitation and groundwater are two important sources of recharge in the lake budgets.

The data used in coherency analysis between precipitation, groundwater level and lake level consist of 19 years of quarterly data, from 1985 to 2003. Quarterly precipitation was calculated from daily records obtained at two stations near the Trout Lake Basin, namely Minocqua Dam and Woodruff Airport stations which are located 10 and 7 miles to the south of the study area, respectively. Figure 2a shows the arithmetic average of precipitation records at the two stations. Figure 2b shows groundwater level in a network of 12 wells in the area. Some wells have lower or higher magnitude fluctuations but all the wells show similar oscillatory behavior in terms of interannual frequencies. One well (well K2) was therefore selected to represent the groundwater level fluctuation in the area around Big Muskellunge Lake (Fig. 2c). The Big Muskellunge Lake was also selected to represent the lake-water-level time series in the analysis (Fig. 2d).

Fig. 2
figure 2

Time series used in coherency analysis: a Quarterly precipitation time series estimated as the average of Minocqua Dam and Woodruff Airport data, Minocqua Dam and Woodruff Airport stations are located ten and seven miles to the south of the study area ; b Quarterly groundwater-level time series in the whole well network. There are twelve wells and the data for each is shown in a different color; c Quarterly groundwater-level time series at well K2; d Quarterly Big Muskellunge Lake water level time series. In all cases the data have been standardized

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The data used in the estimation of phase lag between precipitation and groundwater level includes hourly groundwater levels measured by Dripps (2003) with periods of record varying from 274 to 408 days. The arithmetic average of daily precipitation, at the two stations mentioned previously, was used as the precipitation time series. Groundwater levels at three sites in the area were selected (Allequash, Middle, and Hardwood sites), and one well per site was studied. All wells are shallow with groundwater depth not exceeding 10 feet. The soil in all three sites is mainly sandy outwash consisting of sand and coarse till and a thin layer of forest soil on top, which is rich in organic material. Saturated hydraulic conductivity varies by less than an order of magnitude among all the sites. Middle site has the highest saturated hydraulic conductivity, Allequash site has the lowest hydraulic conductivity, and Hardwood site has a hydraulic conductivity between the highest and lowest values. Field capacity varies from the lowest to highest in Middle site, Allequash site and Hardwood site, respectively. The vegetation type in Allequash, Middle, and Hardwood sites include mixed forest, conifer, and hardwood respectively. These sites were selected in order to have different vegetation type for each site and clear groundwater response to precipitation events due to the shallow groundwater level.

Methods of analysis

The application of spectral methods assumes stationary processes and therefore all the precipitation, lake level and groundwater-level time series data were linearly detrended and normalized before applying spectral analysis. The normalization was done by subtracting the mean from the time series and dividing by the standard deviation of the time series. Squared coherency is the frequency domain analogue of correlation. Squared coherency was estimated in this study following Jenkins and Watts (1968) and Bloomfield (1976). The squared coherency function shows the frequency bands where the two time series are either coherent or non-coherent. Squared coherency function varies between 0 and 1; a value of 0 indicates no coherency and a value of 1 indicates perfect coherency exists between the two time series.

Coherence analysis, or cross-spectral analysis, may be used to identify variations that have similar spectral properties (high power in the same spectral frequency bands). This method can be thought of splitting the time series into two components: high (H) and low (L) frequencies: X t = X (H)t  + X (L)t ; and Y t = Y (H)t  + Y (L)t . It is of interest to know whether the slow components of X and Y vary together in some way (Von Storch and Zwiers 1999).

The cross-spectrum is defined from the covariance function Cxy where ω is frequency and τ is a small shift in time:

$$ {\Gamma _{ {{\rm{xy}}}}}\left( \omega \right) = \sum\limits_{{\tau = - \infty }}^{\infty } {{C_{{{\rm{xy}}}}}\:\exp } \left\{ { - 2\pi \,\,i\tau \omega } \right\},\,\:\,\,\quad \omega \in \left[ { - 1/2,.,1/2} \right] $$
(1)

This is a complex function where the power is:

$$ {A_{{{\rm{xy}}}}}{\left( \omega \right)^{2}} = {\hbox{Re}}{\left( {{\Gamma _{ {{\rm{xy}}}}}\left( \omega \right)} \right)^{2}} + {\hbox{Im}}{\left( {{\Gamma _{ {{\rm{xy}}}}}\left( \omega \right)} \right)^{2}} $$
(2)

where A is the power spectrum, cross spectrum shown by Г is a complex function with imaginary and real parts, Im stands for the imaginary and Re stands for real. The phase is:

$$ {\Phi _{ {{\rm{xy}}}}}\left( \omega \right) = {\tan ^{{ - 1}}}\left( {\frac{{{\text{Im}}\left( {{\Gamma _{ {{\rm{xy}}}}}\left( \omega \right)} \right)}}{{{\hbox{Re}}\left( {{\Gamma _{ {{\rm{xy}}}}}\left( \omega \right)} \right)}}} \right) $$
(3)

where Ф is the phase function and “tan” stands for tangent.

A cross-spectrum for two similar processes, but with one shifted in time with respect to the other [X(t) and X(t + τ)], gives the same power spectrum as for the same analysis applied to two identical time series, X(t), but instead of a phase difference of zero, the phase is linear in frequency with a slope proportional to the phase shift:

$$ {\Phi _{ {{\rm{xy}}}}}\left( \omega \right) = 2\pi \tau \omega $$
(4)

The coherence spectrum is analogous to the conventional correlation coefficient and is defined as:

$$ {\kappa _{ {{\rm{xy}}}}}\left( \omega \right) = \frac{{{A_{{{\rm{xy}}}}}{{\left( \omega \right)}^{2}}}}{{{\Gamma _{ {{\rm{xx}}}}}\left( \omega \right){\Gamma _{ {{\rm{yy}}}}}\left( \omega \right)}} $$
(5)

where K stands for the coherence spectrum between x and y variables for different frequencies (ω). Estimated coherencies are considered significant at the 95% level of confidence when they are larger than the critical value T derived from the upper 5% points of the F-distribution on (2, d–2) degrees of freedom:

$$ T = \frac{{2F}}{{d - 2 + 2F}} $$
(6)

where d is the degrees of freedom associated with the univariate spectrum estimates. The critical T value is derived from the upper 5% of the F-distribution.

Coherence measures the linear dependency of the oscillatory components in two detrended signals. Furthermore, phase estimates can provide temporal (lead/lag) relationships between two variables.

The term groundwater level refers to measured static-water levels in observation wells. At the LTER site vertical hydraulic gradients are small, so water level in wells represents a good estimate of the water-table position. The physical process of groundwater recharge is complex, involving wetting fronts, redistribution, finger flow, pressure waves, and other physical phenomena. In the context of this study, recharge travel time means the time lag between a precipitation event and a rise in the water level in a well, without regard for process-based modelling. Groundwater level fluctuation is caused by precipitation recharge—or lack thereof—and discharge, which in turn could be caused by flow to nearby lakes and evapotranspiration. Discharge occurs mainly in the absence of precipitation. Therefore the simplifying assumption that groundwater level fluctuation is caused primarily by precipitation recharge, or lack thereof, was used. The phase lag in days is an indication of the groundwater-level response to a precipitation event. The phase lag between precipitation and groundwater-level response was estimated at the frequency bands where the precipitation and groundwater level are coherent at or above 95% confidence level.

In the coherency and phase analyses, precipitation is taken as the leading time series. A positive phase lag indicates that the second time series (groundwater level) follows the first one (precipitation). In this study, the lag between the two time series is calculated for each frequency band by taking an average of phase function for that specific frequency band. The assumption made in this averaging process is that phase function is linear for the frequency band of significance. A cause–effect relationship is supported by the physics that governs the relation between precipitation and groundwater level and by the coherence and phase analysis.

Figure 3a shows the squared coherency between precipitation and groundwater level at the Allequash site. The peaks above the horizontal line show the frequency bands with squared coherency at or above 95% confidence level. Figure 3b shows a phase function plot for the two time series and illustrates the phase lag calculation. To demonstrate the phase lag computation between precipitation and groundwater level, consider the frequency band of (6.5–10.5)–1 cycles/day. That frequency band is shown approximately as 0.1–0.15 cycles/day in Fig. 3b, where the numbers in the figure are the reciprocal of those between parentheses in the text. The time lag between precipitation and groundwater level for this frequency band is calculated by taking the slope of this roughly linear segment of the phase function, i.e., P (radians)/[2π F (cycles/day)], where P is the phase change and F is frequency change in Fig. 3b. The value of F in this case is 0.05 (cycles/day) and the value of P is 1.55 radians; therefore the time lag between precipitation and groundwater level is estimated at approximately 5 days. The time lags between precipitation and groundwater level for the other two frequency bands shown in Fig. 3 are given in Table 1.

Fig. 3
figure 3

Precipitation–groundwater-level coherency and phase functions for the a–b Allequash, c–d Hardwood and e–f Middle sites. Sampling interval = 1 day. a–b Illustrates the phase lag estimation for the (6.5–10.5)–1 cycles/day frequency band in the Allequash site; the estimated phase lag in that case is approximately 5 days. The horizontal line (a, c, e) shows the 95% confidence level

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Table 1 Phase lags for significant frequency bands between precipitation and groundwater level in Trout Lake Basin. Sampling interval   is 1 day
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Results

Coherency between precipitation, groundwater level and lake level

Figure 4a–c and Table 2 illustrate the results from coherency analysis between precipitation, groundwater level and lake level. A horizontal line in the squared coherency plots shows the estimated 95% confidence level. The values that fall above this line are considered statistically significant.

Fig. 4
figure 4

Coherency analysis results: a Squared coherency between precipitation and groundwater level; b Squared coherency between precipitation and lake level; c Squared coherency between groundwater level and lake level. Sampling interval is 0.25 year. Precipitation time series is an average between Minocqua Dam and Woodruff Airport stations precipitation time series. Groundwater-level time series is from groundwater well K2 indicated in Fig. 1b, and lake level time series is from Big Muskellunge Lake given in Fig. 1b. The horizontal line (a–c) shows the 95% confidence level

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Table 2 Squared coherency and significant frequency bands between precipitation, groundwater level and lake level in Trout Lake basin. Sampling interval = 0.25 year
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Figure 4a and the first row in Table 2 show the results of coherence analysis between precipitation and groundwater level. The figure and the numerical values in the table show that precipitation and groundwater level are significantly coherent in the frequency bands of (2–8)–1, (1.4–1.6)–1, (0.9–1.1)–1, and (0.67–0.7)–1 cycles/year.

Figure 4b and the second row in Table 2 shows the results of coherence analysis between precipitation and lake level. The figure and the numerical values in the table show that precipitation and lake level are significantly coherent in the frequency bands of (2.2–8)–1, (1.3–1.6)-1, (0.9–1.0)–1, and (0.6–0.7)–1 cycles/year.

Figure 4c and the third rows in Table 2 show the results of coherence analysis between groundwater level and lake levels. The figure and the table show that groundwater level and lake levels are significantly coherent in the frequency bands of (>2.4)–1 and (0.9–1.3)–1 cycles/year.

Phase lag between precipitation and groundwater level

The numerical results for phase lag between precipitation and groundwater-level response are given in Table 1 for the chosen three pairs of precipitation and groundwater-level time series. Figure 3a and b illustrates the estimation of phase lag for the Allequash site. Recall that the phase lag between precipitation and groundwater-level response at the Allequash site for the frequency band of (6.5–10.5)–1 cycles/day is estimated to be 5 days, meaning that it takes the precipitation water 5 days to affect the water table.

Figure 3c–f shows the coherency and phase lag between the other two pair of precipitation and groundwater-level time series. In all cases precipitation leads groundwater-level response by phase values up to 5 days. Significant frequency bands in Table 1 are in cycles per day and the corresponding squared coherency in each site is reported. Phase lag is reported for the frequency bands with coherence at or above 95% confidence level.

Discussion

Interaction between precipitation, lake levels and groundwater

The effect of precipitation on both lake level and groundwater level is clearly demonstrated by the similarity between the precipitation–lake-level coherency function and the precipitation–groundwater-level coherency function. Both coherency functions show significant peaks at interannual frequencies and at seasonal frequencies. The lake level and groundwater-level spectra, shown in Fig. 5, clearly display interannual components, suggesting that lakes and groundwater act like low pass filters.

Fig. 5
figure 5

Spectral estimation of groundwater level and lake level quarterly data: a Groundwater level spectrum (well K2); b Lake-level spectrum (Big Mukelunge Lake). The estimated background noise and associated 90, 95, and 99% confidence levels are shown by solid, dot-dash, doted, and dashed smooth curves in the figures, respectively. Annual and interannual signals are significant at 99% confidence level

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The interaction between groundwater and lakes was analyzed based on the groundwater–lake-level coherency function. That function shows a significant, broad peak at interannual frequencies, and no significant peak at seasonal frequencies. This behavior suggests a preponderance of long memory in the interaction between groundwater and lakes. The results of coherency analysis between precipitation, groundwater-level and lake-level time series clearly show that precipitation is a source of recharge for both the shallow groundwater aquifer and lakes in Trout Lake Area. It suggests that the phase lag between daily precipitation and groundwater fluctuations can provide a reasonable estimate of the recharge water travel time through the vadose zone. Seasonal and interannual variability in precipitation affects groundwater and lake levels at similar time scales. Disregarding the interannual time scale of variability can cause a considerable error in the results of any groundwater–lake-level model.

The findings of this study in the context of other studies on the same groundwater–lake system are discussed in the following. Anderson and Cheng (1993) found that the 10-year record of chlorophyll concentration in Crystal Lake did not show a correlation with groundwater input. They concluded that the trends in lake chemical budget that occurred within their record were influenced more by changes in precipitation input than groundwater inputs. Anderson and Cheng’s (1993) study focused on Crystal Lake, situated in the same groundwater–lake system as Big Muskellunge Lake (analyzed herein), but located in the upper portion of the watershed. Big Muskellunge Lake is a flow-through seepage lake, located in intermediate position between the groundwater divide and the discharge area. Cheng and Anderson (1994) found that groundwater fluxes into and out of the lake located in a lower position were higher and more stable than lakes further upgradient. Thus the significant, broad peak found in this study at interannual (rather than seasonal) frequencies in the groundwater–lake-level coherency function is consistent with Anderson and Cheng’s (1993) and Cheng and Anderson’s (1994) studies for the same groundwater–lake system.

Recharge travel time

Groundwater fluctuation is partially caused by precipitation events acting as a recharge source. Recharge occurs with some lag that could range from hours to days and even months. Recharge-water travel time is the time that water takes from the ground surface until it affects the water table. The recharge travel time at several measurement sites in the Trout Lake watershed was estimated using frequency domain methods. The results were consistent across several sites such as the AC Allequash site, the Middle site and the Hardwood site. In all cases, analyzed precipitation leads groundwater level by up to 5 days. For example, at the Allequash site the squared coherency between precipitation and groundwater level function showed a significant peak in the (6.5–10.5 days)–1 frequency band, and the corresponding phase lag was estimated as 5 days. In other words, groundwater level is significantly correlated with precipitation at frequencies of the order of 1 week, and groundwater level lags precipitation by 5 days. This argument was verified with time domain calculations because the cross correlation function between precipitation and groundwater showed a peak for a 5-day lag, as shown in Fig. 6.

Fig. 6
figure 6

Cross correlation coefficient between daily precipitation and groundwater level at Allequash site

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Previous studies have found lags of the order of days between precipitation and groundwater level. For example Kelly (2001) compared concurrent measurements of river stage, rainfall, groundwater level, and wetland stage for two Missouri River flood-plain wetlands, in order to characterize the spatial and temporal relations between the measured variables, and to determine the source of water to each wetland. Kelly (2001) plotted figures consisting of pairs of hydrographs such as the hydrographs of Missouri River stage and wetland stage, and the hydrographs of Missouri River stage and groundwater level. A 2-day lag in the first figure and a 1-day lag in the second figure produced close similarity between the hydrographs.

The lag for the wells considered (Allequash, Middle, and Hardwood) was less than 5 days as the outwash cover per sediment type for the Trout Lake basin is relatively uniform. One could expect that watersheds with more spatial heterogeneity in soil type would show more temporal variability.

Conclusions

The recharge travel time at several measurement sites was estimated through the coherency and phase lag functions between precipitation and groundwater level. The results consistently showed that precipitation leads groundwater level by up to 5 days in all cases. In other words the recharge travel time, the time that takes the precipitation water to affect the water table, is less than or equal to 5 days in all cases under analysis.

The effects of precipitation on lake level and groundwater level display strong similarities. Both the precipitation –lake level and the precipitation–groundwater-level coherency functions show significant peaks at interannual and seasonal frequencies. The groundwater –lake-level coherency function shows a significant, broad peak at interannual frequencies, and no significant peak at seasonal frequencies, demonstrating a preponderance of long memory in groundwater–lake interaction.

Time-domain methods such as physically based modeling, have traditionally been used to study the recharge travel time and the precipitation–groundwater–lake interactions. In this article, the innovative and efficient use of spectral methods was demonstrated to estimate the time scales that are significant in those interactions and estimate the recharge travel time, which is extracted from the daily data time series under analysis. The recharge travel time and the relevant time scales calculated in this study using spectral methods are consistent with values estimated in previous studies.