Hector R. Bravo and Feng Jiang
University of Wisconsin-Milwaukee, P.O. Box 784, Milwaukee WI 53201-0784, USA
(E-mail: hrbravo@uwm.edu)
Randall J. Hunt
U.S. Geological Survey, Middleton, Wisconsin, USA
Abstract: The estimation of model parameters under transient conditions has the potential of revealing important dynamic characteristics of a system and the danger of a prohibitive computational cost. This paper presents the application of frequency domain analysis to select appropriate time scales for the estimation of parameters in a groundwater flow and heat transport model of a wetland system. It was verified, both at the annual cycle and daily cycle levels, that the periodicity in heat transport is clearer than periodicity in groundwater flow. We anticipate that the methodology will have transferability to other temperate wetlands where large annual and daily temperature fluctuations occur.
Harmonic analysis of land surface temperature and groundwater temperature data showed that most of the variances are explained in terms of the annual harmonic cycle. That analysis showed a remarkable consistency, at the yearly cycle level, between land surface temperature and groundwater temperature at all depths. Frequency domain analysis for harmonic components higher than the first (or yearly one) showed that the significance of frequency relations decreases as one considers temperature records at deeper depths.
Keywords: Frequency domain analysis, parameter estimation, groundwater flow and heat transport models, wetland systems
The estimation of model parameters under transient conditions has the potential of revealing important dynamic characteristics of a system and the danger of a prohibitive computational cost. Bravo et al. (2000) faced this problem while inverting head and temperature data to estimate inflows to a wetland system located near Wilton, Wisconsin and the hydraulic conductivity of the underlying aquifer; that was the motivation of the present study.
Bravo et al. (2000) implemented a method to estimate flux across the water table and hydraulic conductivity by coupling a 3D groundwater flow and heat transport model with automatic calibration. The joint simulation overcomes restrictive hypotheses inherent in methods such as temperature profile modeling and areal two-dimensional water balance modeling (Hunt et al., 1996). However, the handling of a free surface and heat transport required two-step modeling, namely a free surface flow model and a fixed-region flow and heat transport model. The procedure became involved for transient flow periods and it was necessary to determine time scales that would allow extracting information conveyed by the flow and thermal fields while maintaining reasonable computational cost. This paper presents the use of frequency domain analysis to select appropriate time scales in that study.
Hunt et al. [1996], describe the natural and constructed wetlands located in Southwestern Wisconsin. A peat/fluvial sediment layer and a sandstone layer approximately100 m thick characterize the wetland considered here, and a temperature field that varies seasonally and daily within a few meters below the land surface. Figure 1 illustrates the typical flow pattern in a cross section of the system as calculated by the model, with the flow converging to the river. The associated simulated temperature field is also shown. Figure 2 shows hourly water level data collected in 1996 at site F2 in the constructed wetland. Figure 3.a shows evidence of an annual cyclical component in groundwater temperature at the same site and Figure 3.b illustrates variations in hourly groundwater temperature data collected in 1996.
The present analysis included univariate analysis of water level, groundwater temperature and land surface temperature, and bivariate analysis of the relation between land surface and groundwater temperature data. The univariate analysis consisted in the estimation of autocorrelation functions (acf), sample spectral density, and harmonic analysis. As might be expected in a temperate climate, inspection of temperature data revealed periodicity with a period of around 1 year.
The bivariate analysis focused on the residuals obtained by subtracting from the measured data the mean value and the annual cycle in order to satisfy the assumption of stationarity. Calculations in the frequency domain included the smoothed sample spectrum of each series, cross correlation functions of the bivariate series, smoothed sample cross spectrum of the bivariate series, cross amplitude spectrum and squared coherency, gain function and phase from the univariate and bivariate spectra [The Numerical Algorithms Group, 1999].
Univariate analysis of water level and
temperature data
Between 1990 and 1996 water level was measured with various sampling frequencies. The available 1996 hourly water level record was not long enough to quantify an annual cycle but contained enough information about smaller-period cycles. One period of continuous and simultaneous water level and groundwater temperature measurements was identified between May 22, 1996 and July 22, 1996. Inspection of water level data in Figure 2 reveals a hint of a growing-season cycle pattern, a daily cycle, and six rainfall events in the 61-day period of interest. Furthermore, one can find periods when the water table is nearly steady, such as in early October 1996. There are no clearly visible “steady” periods in the temperature record.
Figure 4.a shows the acf of water level and Figure 4.c shows the corresponding estimated spectral density. Figures 4.b and 4.d show the corresponding results for groundwater temperature at a depth of 1.2 ft. The acf of groundwater temperature is broader than that of water table level. The spectral density of groundwater temperature (Fig. 4.d) shows a clear peak at 1 cycle/day and decays quickly for higher frequencies. The spectral density of water table level decreases more gradually and shows no clear peak at 1 cycle/day. Because the daily cycle of heat transport is always important while the daily groundwater flow cycle may be unimportant during certain periods, a few daily cycles of temperature data (while the water table remains nearly steady) may contain enough information to constrain the calibration.
The hourly groundwater temperature data between March 7, 1995 and December 15, 1996 (Figure 3.b) was used to quantify the yearly cycle. Daily average temperatures were calculated from the hourly records. Daily air temperature at a nearby weather station was used to represent land surface temperature. Calculated acfs for land surface temperature and groundwater temperature showed the relevance of the yearly cycle in both data series. The referred data set was therefore used to carry out a harmonic analysis for the land surface temperature and for groundwater temperature at site F2, at different depths below the land surface. Figures 5.a and 5.b and Table 1 summarize the results of that analysis. One can see that, as the depth increases, the amplitude R decreases in an exponential fashion while the phase (or lag) f increases. The yearly cycle of temperature at the 1.2 ft depth lags the air temperature cycle by 12 days, and the 3.2 ft depth temperature is lagged an additional 17 days. The values of r2 indicate that most of the total variance is explained by the first harmonic (or yearly) component. The relevance of the yearly cycle is clear from these results.
Bivariate analysis of temperature data
The bivariate analysis elucidated how the relation between groundwater temperature and the land surface temperature varies with depth. Figure 5.c shows the sample cross correlation functions between land surface temperature and groundwater temperature at depths equal to 1.2 ft and 3.2 ft. At the 1.2 ft depth, the groundwater temperature residual shows a clear peak for a lag equal to 1 day, which seems to justify the neglect of the unsaturated zone and the use of a fixed-region flow and heat transport model. On the other hand, at a depth of 3.2 ft the cross correlation function becomes flat, with a weak peak at lag of 5 days.
The calculation of additional frequency relations between land surface temperature and groundwater temperature at different depths, for harmonics beyond the first (or yearly) one confirmed that the importance of conduction relative to advection decreased with increasing depths. The correlation between the frequency components of the data series (squared coherency) was significant at 1.2 ft and negligible at 3.2 ft. The phase remained bounded at the lower depth and grew continuously with frequency at higher depths. Both the contribution of different harmonics to the correlation at lag zero (cross amplitude spectrum) and the gain function (or ratio between amplitudes of the frequency components) decrease in a fairly uniform fashion at lower depth and very rapidly at higher depth.
It was verified, at the annual cycle and daily cycle levels, that the periodicity in heat transport is clearer than that in groundwater flow. Field data and models that account for heat transport unsteadiness while the flow remains steady at those time scales can thus provide valuable information for parameter estimation. We expect that the procedure will have transferability to other temperate wetlands with large annual and daily temperature fluctuations.
The implicit assumption of steady flow in temperature profile modeling is not flawless, but the use of temperature profiles measured throughout the year and steady water flow assumption constitute a sensible and practical modeling compromise.
Harmonic analysis of groundwater temperature data showed that most of its variance is explained in terms of the annual (or first harmonic) cycle. That analysis showed a remarkable consistency, at the yearly cycle level, between land surface temperature and groundwater temperature at all depths. For frequency components other than the yearly cycle the significance of frequency relations decreases as one considers temperature records at deeper depths. In other words, the relative importance of heat conduction from and to the land surface decreases rapidly with depth.
This study was partly funded by the Sate of Wisconsin Groundwater Coordinating Council.
Bravo, H., F. Jiang and R. Hunt, “Using Groundwater Temperature Data to Constrain Parameter Estimation in a Groundwater Flow Model of a Wetland System”, submitted to Water Resources Research, 2000.
Hunt, R.J., D.P. Krabbenhoft, M.P. Anderson, “Groundwater inflow measurements in wetland systems”, Water Resources Res., 32 (3), pp. 495-507, 1996.
Hunt, R., J. F. Walker and D.P. Krabbenhoft, “Characterizing Hydrology and the Importance of Ground-Water Discharge in Natural and Constructed Wetlands”, Wetlands, 19(2).
The Numerical Algorithms Group Ltd, “Computer-based documentation for the NAG Library”, Oxford UK, 1999.
Table 1 Typical results of harmonic analysis of temperature data
|
Depth (ft) |
Amplitude R (°C) |
Sample mean m (°C) |
Phase f (day of the year) |
Sample variance ST(°C)2 |
r2 |
|
0 |
15.87 |
6.79 |
110 |
150.73 |
0.835 |
|
1.2 |
10.21 |
8.15 |
122 |
54.54 |
0.956 |
|
3.2 |
6.90 |
8.22 |
139 |
24.73 |
0.963 |
|
5.5 |
5.76 |
8.48 |
146 |
17.20 |
0.966 |
Fig.1 Typical simulated flow and temperature fields in synthetic section model. Velocity vectors converge to the river and are perpendicular to contour lines of hydraulic head h. Contour lines of temperature T shows significant variation near the surface.
Fig. 2 Hourly head data collected during the 1996-growing season in Well F2 in the constructed wetland.
Fig. 3 (a) Typical groundwater temperature measurements showing evidence of an annual cycle component; b) Detail of hourly temperature data collected between May 22 and October 11, 1996.
(a) (b)
(c) (d)
Fig. 4 (a) Autocorrelation function of hourly water level data; (b) autocorrelation function of hourly groundwater temperature at a depth of 1.2 ft.; (c) spectral density of hourly water level; (d) spectral density of hourly groundwater temperature at a depth of 1.2 ft.
Fig. 5 (a) Measured air temperature and groundwater temperature at site F2, at depths of 1.2 ft. and 3.2 ft.; (b) First harmonic component of data in Figure 3.a; (c) Cross correlation functions between air temperature residuals and groundwater temperature residuals.