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 Temporal scaling of the hydraulic head time series, h(t), was found in a previous analysis of hourly measured head data. This issue is further investigated in this paper with nonstationary spectral analyses and numerical simulations. The results show that temporal scaling may indeed exist in h(t), which fluctuates like a fractional Brownian motion in most aquifers. On the basis of a linear reservoir model with a white noise recharge input, we show that the variance and covariance of h(t) are functions of time: The head variance increases with time and approaches a constant limit as time progresses, while the covariance decreases with the separation time interval for a fixed time and approaches the typical exponential covariance as time increases. The spectra of the simulated h(t) using a one-dimensional transient groundwater flow model with a white noise recharge in both homogeneous and heterogeneous aquifers are shown to be proportional to f-β, where f is frequency and β ∼ 1.84 (or H = 0.42). Heterogeneity in the hydraulic conductivity may affect the fractal dimension of h(t) in highly permeable aquifers but not in the low permeable aquifer simulated in this study.
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 Groundwater flow in an aquifer is a dynamic system which is usually recharged by precipitation and discharges through evapotranspiration (ET) and as base flow to rivers and creeks. Fluctuations of the hydraulic head (water level) are dynamic responses of an aquifer to the changes in recharge and discharge, and thus contain significant amounts of information about the nature of recharge and discharge processes. Valuable insights about groundwater recharge and discharge may be obtained by studying fluctuations of water levels in monitoring wells. Compared with groundwater recharge, ET, and base flow, water levels in monitoring wells are relatively easy to measure. Extensive data of groundwater levels sampled at various time periods of many years or decades are often available. In the United States, for example, groundwater levels in over 25,000 observation wells have been monitored by the U.S. Geological Survey sometime since the last century.
 Natural fluctuations in groundwater levels have been investigated traditionally with deterministic approaches [e.g., Jacob, 1943; Dagan, 1964; Pinder et al., 1969] and lately with stochastic methods. One of the important stochastic methods, the spectral method, has been applied to study temporal and spatial variations of groundwater quantity and quality [e.g., Gelhar, 1974; Bakr et al., 1978; Gelhar and Axness, 1983; Duffy et al., 1984; Duffy and Gelhar, 1985, 1986; Jin and Duffy, 1994]. A critical assumption in many studies with stochastic approaches, including stationary spectral analyses, is that the spatial and temporal variations of the hydraulic head and contaminant concentration are wide-sense stationary [e.g., Dagan, 1989; Gelhar, 1993]. Nonstationary effects, e.g., flow in a bounded domain and transport of a finite contaminant plume, need to be considered in many applications. One of the effective ways to deal with nonstationarity is so-called “evolutionary spectra,” which have been used since 1950s in time series analysis [Cramer, 1951; Priestley, 1981] and applied to groundwater problems recently [Li and McLaughlin, 1991, 1995]. This nonstationary spectral method is adopted in this paper to study temporal scaling of the hydraulic head fluctuations due to natural groundwater recharge and discharge.
 Their findings are further investigated in this study by theoretical analyses with a nonstationary spectral method and by numerical simulations of transient groundwater flow under a hydrogeological condition similar to that at the Walnut Creek site, Iowa, using a one-dimensional transient groundwater flow model with a white noise recharge process. The hydraulic heads and discharge (base flow) to a creek in both homogeneous and heterogeneous aquifers are simulated. The theoretical and simulation results support the finding of Zhang and Schilling  that the hydraulic head fluctuations due to natural recharge and discharge may be a temporal fractal and an aquifer may act as a fractal filter which takes stationary recharge inputs and produces fractal head fluctuations and base flow variations. In the following we will present first the measured data and their analysis, then theoretical derivations and numerical simulations, and finally some conclusions.
2. Measured Hydraulic Head Fluctuations and Their Spectra
 As part of a monitoring project established in April 1995 by the Iowa Geological Survey Bureau at the Walnut Creek watershed, Iowa [Zhang and Schilling, 2004, Figure 1], seven groundwater monitoring wells (four shallow wells, i.e., 6B, 5B, 4B, and 7B, and three deep wells, i.e., 6A, 5A, and 4A) were installed along the transect A-A' from an upland to Walnut Creek at the valley bottom (Figure 1). The distance to the creek is 105 m from well 6A and 6B, 226 m from 5A and 5B, 324 m from 4A and 4B, and 395 m from 7B. Hourly water level measurements were made in all seven wells for more than 4 years from January 1996 to March 2000 [Zhang and Schilling, 2004, Figure 3]. A total of more than 35,000 head values were measured for each well. This high-resolution, long-term data set was analyzed by Zhang and Schilling . More detailed site description is given by Zhang and Schilling .
 The sample power spectra of the measured hydraulic head, Shh, at each of the seven wells were estimated and plotted by Zhang and Schilling [2004, Figure 4]. In order to compare with the numerical simulation results obtained in this paper with a one-dimensional transient groundwater flow model, we averaged the two head time series in each pair of shallow and deep well at three locations (well 4A and 4B, 5A and 5B, and 6A and 6B) and estimated the spectra of the averaged heads (Figure 2). The head spectra (Shh) of the averaged heads at the four locations behave similarly to that of seven individual wells, i.e., decrease linearly with frequency (?) in the log-log scale or Shh(?) ? 1/?β, which indicates scaling of the hydraulic head variations from 1 to 35,000 hours or 1458 days. In other words, the temporal variations of the water level at the monitoring wells have power law variograms ?(t) = C0t2H, where C0 is a constant, t is time, and H is the Hurst coefficient. Straight lines are well fitted to each of the four sample spectra in Figure 2 with the least square method, and the values of their slopes (β) are listed in Table 1 along with the fractal dimension, defined as D = (5 - β)/2, and Hurst coefficient given by H = 2 - D [Turcotte, 1992; Carr, 1995]. It is seen in Table 1 that the values of β at the four wells vary between 2.02 and 2.14, and the corresponding D values range between 1.43 and 1.49 and H values range between 0.51 and 0.57. The spectrum of the mean heads of all seven wells is fitted with a straight line of slope β = 2.07, or D = 1.47 or H = 0.53 (Table 1).
Table 1. Estimated Spectral Coefficient β, Hurst Coefficient H, and Fractal Dimension D of the Measured and Simulated Hydraulic Heads in the Walnut Creek Watershed
Measured at Walnut Creek
Simulated in a Homogeneous Aquifer
Simulated in a Heterogeneous Aquifer
Base flow I
Base flow II
 Also presented in Figure 2 is the spectrum of the daily base flow in the Walnut Creek. It is seen in Figure 2f that the power spectrum of the base flow may be broken into two segments with a transition time or break point at f = 0.057 day-1 or t = 18 days. Two straight lines are fitted to the base flow spectrum by the least squares method. The low-frequency segment I was fitted well by a straight line with β = 1.30, D = 1.85, or H = 0.15 and the high-frequency segment II with β = 2.88, D = 1.06, or H = 0.94 (Table 1). Similar breakpoints in the base flow spectra in four rivers in Iowa were found by Zhang and Schilling . Such a breakpoint or transition time was also found in the global temperature record [Tsonis et al., 1998]. With this transition time, Tsonis et al.  found that a time of 20 months separates climate processes (e.g., La Niña) that promote a trend in the past from processes (e.g., El Niño) that reverse this tendency. Tsonis et al. [1998, p. 2821] emphasized, “This characteristic scale has important implications one of which might be that the El Niño/La Niña cycle may act as a mechanism countering the tendency of shorter time scale events to organize a positive or a negative temperature trend.” The breakpoint in the base flow may bear similar significance. The high-frequency variations of the base flow may be caused by individual rainfall events ranging from a few hours to a few days, which are persistent and positively correlated (H > 0.5), and the low-frequency variations may be the results of the seasonal and annual changes in rainfalls which are antipersistent and negatively correlated (H < 0.5).
3. Nonstationary Spectral Analyses
 As it is shown in Figure 2e, the mean head of all seven wells fluctuates similarly to the water level at individual wells. We thus carry out our analysis for the mean head with the simple linear reservoir model
where R(t) is the recharge rate which is a function of time, a is outflow constant, h(t) is the average water level in the aquifer, hB is the water level in the creek, SY is the specific yield, and h0 is the initial water level. The parameters a, SY, hB, and h0 are considered to be constant, and only R (and thus h) is assumed to be a random process.
 Let the random fluctuations of R and h about their respective constant means be represented by R' and h'; the following perturbation equation can be easily derived:
This model was developed by Gelhar  to derive the spectral relationship between h' and R'. The spectral method he used is only valid for stationary processes and thus is not applicable to the nonstationary fractal head fluctuations observed at the Walnut Creek site, although the recharge may be stationary [Zhang and Schilling, 2004]. Therefore the nonstationary evolutionary spectral method [Priestley, 1981; Li and McLaughlin, 1991, 1995] is used to derive the head covariance and spectrum in the following.
 The stationary recharge process R' can be expressed in term of its Fourier-Stieljes integral,
where i = , ? is angular frequency, and ? = 2pf, dZR(?) is the random Fourier increments of R'. Similarly, the nonstationary process h' may be expressed in term of a generalized Fourier-Stieljes integral over the stationary increment dZR(?) since equation (2) is linear
where ?hR(t, ?) is an unknown complex-valued transfer function and ?*hR(t, ?) is the complex conjugate of ?hR(t, ?). Replacing R' and h' in (2) with (3) and (4), respectively, we obtain
Multiplying (5) by its complex conjugate, taking expectation, and using the orthogonal property of the random Fourier increment dZR, we have
Equation (6) is an ordinary differential equation whose solution is given by
The (auto) covariance Chh of the head fluctuation h' is given by
where SRR is the spectrum of the recharge fluctuations. The head covariance in (8) may be evaluated for any known recharge spectrum. Zhang and Schilling  indicated that an aquifer system may behave like a fractal filter, producing temporal scaling in the hydraulic head in response to either a white noise or fractal recharge process depending on the value of the aquifer response time (tc). The value of tc is usually large for most aquifers since tc is related to the aquifer transmissivity (T) and the dimension (L) by
where a = 3T/L2 [Gelhar and Wilson, 1974] and T « L2 in most aquifers. They further indicated that an aquifer may take a white noise input signal and output a fractal process when tc » 1. The value of tc for the Walnut Creek watershed they estimated is about 760 days, where the values of SY, T, and L are 0.25, 17.1 m2/d, and 395 m, respectively. This value is large enough to warrant a white noise recharge process in the Walnut Creek watershed. We thus assume the recharge is a white noise process in this analysis in order to compare our theoretical results with the observed.
 The spectrum for a white noise process is given by SRR = sR2?R/p [Gelhar, 1993, p. 34], where sR2 and ?R are the variance and correlation length of recharge fluctuations, respectively. Using this spectrum and replacing ?hR in (8) with (7), the head covariance is derived as
and t' is the dimensionless separation time interval and t' is the dimensionless time. The head variance can be obtained from equation (10a) by setting t' = 0, i.e.,
which is a function of time. In other words, the head fluctuations is nonstationary and its variance increases with time and approaches its constant asymptotic limit of CR as t' ? 8 (Figure 3a), meaning that the head fluctuations approach stationary as time progresses. Equation (11) can be approximated as
when t' « 1 or the head variance is proportional to time. According to Turcotte , a necessary condition that a time series be a fractal is that its variance has a power law dependence on time, i.e., s2(t) ? t2H. We show in (12) that the head fluctuation is a temporal fractal and more specifically a Brownian motion (H = 0.5) when t' « 1 or tc = SY/a » 1. As we mentioned earlier, for most aquifers T « L2 or tc » 1, and thus the head fluctuations may be a Brownian motion. This conclusion will be further verified with the head spectrum presented later. Unlike a stationary covariance which is a function of t' only, the head covariance in (9) for the nonstationary process h(t) is a function of not only t' but also t' (Figure 3b). For a fixed t', Chh decreases from its variance (which depends on t') to zero as t' increases. The smaller the t', the smaller variance and more quickly Chh decreases to zero. The head covariance approaches the stationary exponential covariance CRe-t' as t' ? 8 (Figure 3b). In other words, the random process h(t) is nonstationary at early time and becomes stationary at late time.
 It is evident that the spectrum of the head fluctuations is also a function of time since Chh is. The Wigner spectrum [Priestley, 1981], which is defined as the Fourier transform of Chh, regarded as a function of t' with t' fixed,
can be derived as
where SRR = sR2?R/p, d(?) is the Dirac delta function. The first term inside the brackets in (14) results from an exponential covariance, and the second term depends on time but is zero where ? ? 0. Considering that the head measurements are always collected at finite frequencies, i.e., d(?) = 0 when ? ? 0, it is seen that the spectrum of the nonstationary head fluctuations is the same with the one for an ordinary exponential covariance [Gelhar, 1974], i.e.,
It is important to point out that when tc » 1 or a/SY « 1, Shh(?) ? ?-2, meaning that head time series becomes a Brownian motion with H = 0.5, which is consistent with that derived from the covariance. As we mentioned early, the condition tc » 1 is satisfied in the Walnut Creek watershed and in most aquifers since L2 » T, and thus equation (15) may be applicable to most field-measured water level data.
4. Numerical Simulations
 Groundwater flow in an unconfined aquifer along a hillslope from the topographic high (groundwater divide) to a stream, similar to the setting at the Walnut Creek Watershed (Figure 1), can be approximated with the one-dimensional transient groundwater flow equation under the Dupuit's assumption
where K is the hydraulic conductivity, h(x, t) is the hydraulic head above the bottom of the aquifer, R(t) is a time-dependent recharge (or evapotranspiration) rate, and SY is the constant specific yield. The left boundary is set as a constant head boundary of 15 m at a river, and the right boundary is taken as a nonflux boundary at a groundwater divide. The length (L) of the aquifer from the divide to the creek is 400 m, which is discretized into 256 cells with the size of each cell, dx = 1.56 m. Equation (16) was solved with the block-centered finite difference method with the strongly implicit procedure solver implemented in MODFLOW 2000. The initial water table is obtained by running a steady flow model with average daily recharge. The transient groundwater flow is simulated for 4 years or 1460 days, and the time step is taken to be 1 day. A time series of 1460 white noise daily recharge rates was first generated with a mean recharge rate of 0.0009 m/d and a standard deviation of 0.05 m/d. The generated rates (Figure 4) have both positive and negative values representing daily recharge and evapotranspiration, respectively. The spectrum of the generated recharge rates (Figure 5) fluctuates around a horizontal line, indicating that the generated recharge time series is indeed a white noise process.
4.1. Simulation in a Homogeneous Aquifer
 The hydraulic conductivity (K) is taken be the constant value of 0.1 m/d for the homogeneous aquifer. The simulated hydraulic heads at the locations of wells 4, 5, 6, and 7 vary with time similarly (Figure 6). As expected, the water levels in the wells at higher elevation (e.g., well 7) are always higher than those at lower elevation (e.g., well 6). The power spectra of the simulated water levels at the four wells (Figure 7) are very similar to those based on the measured data (Figure 2) and show distinct slopes in the log-log plot, indicating temporal scaling in the water level time series. The aquifer simulated does act as a fractal filter, which takes a white noise recharge process and outputs a fractal head fluctuations as discovered by Zhang and Schilling  in the measured water levels. The value of tc for the homogeneous aquifer simulated is about 8889 days or a/SY = 1.12 × 10-4, since S, T, and L values are 0.25, 1.5 m2/d, and 400 m, respectively. For such a large value of tc, the spectrum of the simulated heads in this aquifer is a fractal under the white noise recharge inputs based on (15).
 Unlike the values of β, D, and H based on the measured h(t) which varies slightly from well to well (Table 1), the same values of β(1.84), D(1.58), and H(0.42) were obtained for the simulated water levels at all four wells as well as their mean head (Table 1). The discrepancy between the measured and simulated results may be caused by, for example, the simple one-dimensional model and/or nonuniform recharge. In reality, groundwater flow occurs in two or three dimensions where the Dupuit's assumption is not valid [e.g., Boufadel, 2000], and the recharge rates may be correlated over space and time. We are currently working on theoretical derivations and numerical simulations in which some of these issues are considered.
 The simulated discharge flux to the creek or base flow rates (Figure 6b) were calculated using Qb(t) = K?h/?x where ?h is the head difference between the water level in the creek and the head at the cell next to the creek. The power spectrum of the simulated Qb exhibits a distinct slope (Figure 7f) which is much gentler or less steep than that of the simulated Shh. The simulated base flow spectrum was well fitted with a straight line with β = 1.22, which is equivalent to D = 1.89 and H = 0.11 (Table 1), clearly indicating scaling in the simulated base flow. The large fractal dimension or small Hurst coefficient means rapid fluctuations or less correlation in the temporal variations of base flow since the homogeneous aquifer responds to a white noise recharge quickly and produces an output of base flow which also fluctuates quickly. The spectrum of the simulated base flow shows no slope change, and the value of D (1.89) is close to the D value (1.85) of the first segment of the estimated base flow. That the breakpoint in the spectrum of the measured base flow (Figure 2f) is not observed in the simulated data (Figure 7f) may be simply because the water level in the creek (hB) is taken to be constant while in reality hB varies with time and the spectrum of hB may be different from that of h(t). Base flow should be affected by both the hydraulic head variations in the aquifer and the water level changes at the creek. They may vary quite differently with time and thus have different spectra. The fact that the slope of the simulated base flow spectrum (β = 1.22) is closer to that of the observed base flow spectrum at low frequency (β = 1.30) may indicate that the variation of the observed base flow at low frequency (segment I in Figure 7f) may be related the water level fluctuations in the aquifer while that at high-frequency (segment II) may be due to the water level changes in the creek.
4.2. Simulation in a Heterogeneous Aquifer
 For the heterogeneous case, the log hydraulic conductivity (Y = ln K) field is assumed to be normally distributed and is generated with a geometric mean of 1.01 m/d and an exponential variogram model
where s is the separation distance, and sY2 and ?Y are the variance and correlation length of Y and are taken to be 2.40 and 5 m, respectively. The variogram of the generated ln K field matches well with the theoretical model given in (17) (Figure 8).
 The simulated hydraulic heads in the heterogeneous aquifer at the locations of wells 7, 5, 4, and 6 vary with time similarly (Figure 9). Like the observed and simulated for the homogeneous case, the power spectra of the simulated h(t) at all four wells in the heterogeneous aquifer are shown to have distinct slopes (Figure 10). Furthermore, the values of D (1.83) in the heterogeneous case are almost the same as those in the homogeneous aquifer (D = 1.84). In other words, it seems that the aquifer heterogeneity has little effect on temporal scaling. We think that this is because of the small effective K value (0.01 m/d) we used in our simulation. On the basis of equation (15), the head spectrum becomes independent of K if the value of a/SY is relatively small as compared with ?. The value of a/SY is 1.13 × 10-4 1/d (K = 0.1 m/d) for the homogeneous aquifer and 1.13 × 10-5 1/d (K = 0.01 m/d) for the heterogeneous one. Both these values are much smaller than the minimum values of ? (>1.0 × 10-3) presented in Figures 7 and 10. It is expected that the heterogeneity will affect the temporal scaling in a more permeable aquifer, as it is easy to see in equation (15) that Shh may change from being proportional to ?-2 when a/SY « ? to being the same process as the input recharge process or independent of ? when a/SY » ?.
 In general, the simulation results presented above support our observations with the field measured data and our theoretical derivations. Temporal scaling may indeed exist in the hydraulic head time series. An aquifer may act as a fractal filter in most cases, and it takes a white noise recharge rate and produces fractal head fluctuations and base flow variations. The time series of h(t) is nonstationary and a fractal process since its variance follows a power law with ever-increasing variance and never approaches a sill. Our theoretical results are obtained with a nonstationary spectral method. Different approaches have been suggested by others. Neuman  and di Federico and Neuman  showed that any random field with stationary increments can be viewed as an infinite hierarchy of mutually uncorrelated stationary process. These stationary processes have covariance functions that increase as a power of scale and variograms that define stationary fields associated with constant variance. They can be obtained by means of cutting off lower integral scale and higher integral scale in space domain or cutting off lower frequency and higher frequency in time domain. Molz et al.  also mentioned that nonstationary functions in a finite domain can be well approximated by a stationary function with correlation existing at a larger scale than the domain size. All these theories make it possible to study nonstationary fractal processes with existing stochastic models which are normally developed for stationary process if cutoffs are appropriately used.
 The theoretical results obtained in this paper are based on a simple reservoir model where the spatial variations of the water level and of physical properties of the aquifer, e.g., SY and T, are neglected. These assumptions may introduce some bias in the results since heterogeneous conductivity may filter recharge and make recharge from head data more difficult to assess. The heterogeneity of an aquifer may increase effective response time and smooth hydraulic head process with a decrease of variance compared with the situation in which an aquifer is homogeneous, and a much smooth hydraulic process will be expected. These factors have been considered in our current research.
 Theoretical analyses using a nonstationary spectral method with a linear reservoir model and numerical simulations of one-dimensional transient groundwater flow in a homogeneous and heterogeneous aquifer were carried out to investigate the temporal variations of the hydraulic head (h) and discharge (base flow) to a creek due to a white noise recharge process. The results support temporal scaling of the h fluctuations and the base flow variations, found in a previous study based on the hourly h data measured over a 4–year period at seven monitoring wells in the Walnut Creek watershed in Iowa. The following conclusions can be drawn from this study:
 1. Our measured data and theoretical and simulation results show that an aquifer may indeed act as a fractal filter which takes a random nonfractal signal (a white noise recharge process in this study) and produces a fractal hydraulic head and base flow time series.
 2. On the basis of a simplified linear reservoir model with a white noise recharge input, the hydraulic head variance and covariance are shown to be functions of time. The head variance increases with time and approaches a constant limit as time progresses, while the covariance decreases with the separation time interval for a fixed time and approaches the typical exponential covariance as time increases.
 3. The hydraulic head fluctuation is shown to be a nonstationary Brownian motion with the Hurst coefficient (H) of 0.5 when the aquifer response time tc » 1. The condition tc » 1 is usually satisfied since tc = SYL2/(3T) and SYL2 » T in most aquifers, and thus this conclusion may apply to most field-measured water level data.
 4. The spectrum of the simulated h fluctuations using a one-dimensional transient groundwater flow model with a white noise recharge in both a homogeneous and heterogeneous aquifer is shown to be proportional to f-β where f is frequency and β ∼ 1.84 (or H = 0.42).
 Heterogeneity in the hydraulic conductivity had little effect on the fractal dimension of the head time series in the low permeable aquifer simulated in this study. This may not be the case in a highly permeable aquifer.
 We express our thanks to two anonymous reviewers, whose constructive comments greatly improved this paper. This work was partially supported by a research grant from the Iowa Department of Natural Resources.