2. Data Compilation and Equation Development

Figure 3. Typical data collection site. From German [2000]. [larger image]

[5] New equations developed for estimating changes in heat energy stored in a column of wetland surface water are presented. Existing hydrometeorological data [German, 2000] in the Everglades wetland areas of southern Florida were compiled before these equations were developed.

2.1. Data Compilation

[6] The hydrometeorological data compiled for this analysis were taken from German [2000] (Figure 1 and Table 1), who estimated evapotranspiration in the Everglades using a Bowen ratio approach [Bowen, 1926]. Data collected at sites included the surface water depth, rainfall, wind speed and direction, incoming solar radiation, net radiation, soil heat flux, air temperature, relative humidity, vapor pressure gradient, air temperature gradient, and water temperature at both the water surface and at depth. An example of a typical data collection site is shown in Figure 3.

[7] At each site, sensor measurements were made every 30 s and averaged and stored on site at 15 or 30 min intervals [German, 2000]. Measurements were stored at 15 min intervals at sites 4–9. Thirty minute means or totals (for rainfall) were computed for sites 4–9 so that analysis and equations could be applied at each location using data with a consistent time step. Data used most frequently in this paper are measurements of air and surface water temperature, and surface water depth. German [2000] measured surface water temperature (1) directly below the water surface, and (2) at depth near the land-water interface adjacent to decaying peat and vegetation at the bottom of the surface water column. The height above land surface where air temperature was measured ranged from about 1 to 4 m (Table 1), and depended on the vegetation height.

[8] German [2000] describes in detail (1) the criteria for selecting monitoring sites, (2) data processing and screening, and (3) site maintenance; these procedures are summarized here. Factors considered in site selection included plant community, duration of water inundation, security and logistics. Each site was located at the center of a circle of relatively uniform vegetative cover with a radius of at least 100 times the height of the air temperature/relative humidity sensor. Data processing included screening tests based on range limits and visual inspections to eliminate data collected by sensors that were clearly malfunctioning. Sites were visited every 4 to 6 weeks for inspection and maintenance. It is important to note the accuracy of air and water temperature measurements used in this study. Accuracy of these measurements may be about ±0.4% of the reading (Omega Engineering Inc., oral communication, 2005). The mean temperatures measured at all nine sites were about 23.3° and 25.1°C, respectively, for air and water [German, 2000] which translates into an average error of about ±0.1°C. An error of ±0.2°C is possible for air and water temperature changes, since these are the difference between two consecutive readings.

2.2. Equation Development

[9] Several equations were considered to estimate changes in stored heat energy in a column of wetland surface water. The goals were to (1) capture (reasonably well) the observed variability in stored heat energy changes at 30 min and daily time intervals, and (2) require only readily available data.

[10] Estimating stored heat energy changes at short time intervals (30 min and daily) was desirable because high-frequency estimates are needed to make local and regional energy budget estimates of evapotranspiration. For example, 30 min and daily fluxes of stored heat energy can be used to estimate evapotranspiration with the Priestley-Taylor [Priestley and Taylor, 1972] and Penman [1948] equations. Also, high-frequency results can be temporally up-scaled to fit the needs of most evapotranspiration estimators. For example, 30 min fluxes of stored heat energy can be upscaled to weekly and monthly values for energy budget based estimators of evapotranspiration.

[11] Ideally, stored heat energy is calculated from changes in the vertically averaged water column temperature measured by thermistor or thermocouple strings. Unfortunately, historical water temperature data are limited. Historical air temperature data, however, are readily available and, for shallow layers of surface water, air temperature changes are the primary driver of water temperature changes. For example, air temperature changes explain most of the variability in water temperature changes in Everglades wetland areas. The similarity of changes in mean daily air and surface water temperatures at site 3 in 1997 demonstrates this point (Figure 4). These temperature changes were computed as the difference between two consecutive mean vertical and daily surface water temperatures. Although air temperature changes seem to explain most of the variability in water temperature changes, some differences are apparent (Figure 4). The differences could be caused by heat exchange processes other than air temperature affecting water temperature, including water management activities, evaporative cooling, rainfall, and perhaps surface water and groundwater interactions.

Figure 4. Changes in mean daily and vertical water and air temperatures at site 3 in the Everglades during 1997. Site location is shown in Figure 1. [larger image]

[12] The goal of capturing the variability of changes in stored heat energy at short-time intervals (less than 1 hour) presented additional challenges, particularly the ability to account for the thermal memory of the wetland surface water. Thermal memory is a length of time that an individual air temperature change will impact future water temperature changes. Air temperature changes within the surface water’s thermal memory are dampened, phase shifted, and superimposed to produce the current water temperature change (Figure 5). Air temperature changes that occurred more recently within the surface water’s thermal memory impact the current water temperature change more strongly than air temperature changes that occurred later within the surface water’s thermal memory. Notably, the concept of thermal memory differs from the concept of thermal inertia, which is the square root of the product of the media thermal conductivity, density and specific heat capacity. Thermal inertia is the ability of a material to conduct and store heat, whereas thermal memory only represents a length of time. Although not previously applied to heat exchange hydrologic problems, the general convolution integral can account for the thermal memory of surface water when computing changes in mean vertical surface water temperature.

Figure 5. Changes in 30 min mean air and water temperature at site 3 in the Everglades. Site location is shown in Figure 1. [larger image]

2.2.1. Convolution Integral

[13] The convolution integral [Dodge, 1959; Besbes and DeMarsily, 1984; Morel-Seytoux, 1984; Wu et al., 1997; Weiler et al., 2003; Long and Putnam, 2004; O’Reilly, 2004] uses a time series forcing function and transfer function to calculate a time series response function. For this paper, the physics of heat energy exchange between the atmosphere and surface water is encapsulated as both a time-varying and nonlinear transfer function. The transfer function is time varying because the amount of heat energy exchanged between the atmosphere and surface water depends not only on the temperature gradient close the water surface, but also the surface-water depth, which can change within the water’s thermal memory. The nonlinearity of the transfer function is a consequence of it’s derivation from heat transfer equations, which will be discussed later.

[14] Numerous studies have used the convolution integrals. For example, Besbes and DeMarsily [1984] used the convolution of a time series of infiltrated water (forcing function) with a linear transfer function to compute a net recharge time series (response function) for an unconfined aquifer in northern France. Long and Putnam [2004] used the convolution of a time series of the stable isotope of oxygen (¹⁸O) for recharge water (forcing function) with a nonlinear transfer function (representing the effect of conduit and intermediate groundwater flow) to compute a time series of ¹⁸O (response function) at a discharge point. In this paper, the convolution integral is presented in a format similar to that of Long and Putnam [2004] and takes the form

(2) D

where y(t) is the computed time series of changes in mean vertical surface water temperature (the response function), h(t - ) is the time-varying and nonlinear transfer function derived and described herein, c() is the time series of measured changes in air temperature (the forcing function), and d is the derivative of time.

[15] Using the discrete form of the convolution integral, changes in the response function (water temperature) were computed as the linear superposition of a thermal memory of dampened and phase-shifted individual changes in the forcing function (air temperature). The discrete form of the convolution integral for this heat exchange problem is

(3) D

where y, h, and c are the discrete forms of the continuous functions in equation (2); i is the integer time step for computing changes in the response function (surface water temperature); and j is the integer time step discretizing the surface water’s thermal memory. The variable IMEM + 1 is the number of historical time steps discretizing the surface water’s thermal memory. Not present is an equation symbol for the water’s thermal memory; instead, the thermal memory is the time length spanning the summation of the individual products of h and c from j = 0 to IMEM. Note that the transfer function, h_i-j, needs to be derived.

2.2.2. Transfer Function

[16] The transfer function describes the temporal response of the response function to a unit change in the forcing function and, as such, encapsulated the physics of heat exchange between the atmosphere and surface water. Assume that the derivative with respect to time of the difference between the equilibrium [Edinger et al., 1968] and actual water temperatures is proportional to the difference between equilibrium and actual water temperatures and inversely proportional to surface water depth. The mathematical formula becomes

'

(4) D

where T^e_w is equilibrium water temperature [T], T_w is actual water temperature [T], dt is the derivative of time [t], k_ex is the thermal exchange coefficient [Lt^-1], and D is the surface water depth [L]. The units [T], [t] and [L] represent the units of temperature, time and length, respectively. Equilibrium water temperature is defined as the temperature at which no net heat exchange occurs in the area of interest. This condition is met if the sum of various heat exchange processes, such as air temperature, sensible heat flux, solar radiation, and evaporative cooling totals zero. The k_ex variable is a proportionality constant that describes the rate at which water temperature responds to heat exchange processes [Edinger et al., 1968]. In some cases treating k_ex as a constant may oversimplify the actual system dynamics because k_ex likely depends on processes such as the temperature gradient near the water surface, local turbulence, and radiation.

[17] Solving the ordinary differential equation (4) (see Appendix A) yields

'

(5) D

where T_a is the rate of change in air temperature [Tt-1]. The rate of change in water temperature with time is determined by differentiating equation (5) with respect to time, which results in

.

(6) D

The discrete form of this derivative introduces a coefficient, (unitless), which relaxes the assumption that a specific air temperature change causes an equivalent water temperature change. The coefficient represents the fraction of air temperature change that eventually causes an equivalent water temperature change, and is less than or equal to 1.0. This mathematical formula becomes

.

(7) D

Thus the transfer function takes the form

'

(8) D

where t is the elapsed time within the surface water’s thermal memory. The transfer function clearly is time variant because of dependence on surface water depth, D_{i – j}, which changes within the thermal memory of the wetland surface water. The thermal exchange coefficient, k_ex, does not make the transfer function time variant because this coefficient is a regression-defined constant.

[18] The summation shown in equation (3) was carried out over an assumed finite thermal memory of the surface water. To avoid discretization errors, the transfer function was normalized by dividing by the total area under the discrete transfer function of finite memory. For various values of k_ex and D_{i – j} (Figure 6), a normalized form of the transfer function, equation (8), is indicative of exponential decay, suggesting air temperature changes that occurred more recently in the water’s thermal memory have greater impact on current water temperature changes than air temperature changes that occurred later within the water’s thermal memory. Increasing the thermal exchange coefficient, k_ex, reduces the impact of recent air temperature changes on the current water temperature change, and increases the impact of later air temperature changes on the current water temperature change. Likewise, an increased surface water depth, D_{i – j}, reduces the impact of recent air temperature changes on the current water temperature change but has little effect on the impact of later air temperature changes on the current water temperature change. The final discrete form of the transfer function and convolution integral, used to compute mean vertical water temperature changes, and ultimately, changes in heat energy stored in a column of wetland surface water, takes the form

Figure 6. Transfer function for various values of the thermal exchange coefficient (kex) and surface water depth (D). [larger image]

(9) D

[19] It is important to note that the distance between data collection locations may affect the accuracy of equation (9). For example, equation (9) is applied in the wetland areas of the Everglades (Figure 1), where air temperature measurements always were made less than 3 m away from the location where predicted mean vertical water temperature changes were desired. Using air temperatures measured several thousand meters away from the target location for predicting water temperature changes may increase the error statistics.

2.2.3. Parameter Estimation

[20] Regression was performed using UCODE [Poeter and Hill, 1998] to estimate parameter values for the transfer function that minimized the errors between mean vertical water temperature changes computed with the convolution integral and those measured in the field. Accurately computed water temperature changes are required to obtain accurate estimates of changes in heat energy stored in a column of wetland surface water. For example, an error of 0.1°C in water temperature change for a 30 cm deep water column during a 30 min time period results in an error of about 70 W m^-2 in stored heat energy flux. This stored heat energy error can be substantial, considering mean 30 min net radiation ranged from -50 to 500 W m^-2 within the Everglades wetland areas. Negative values of net radiation were common during the night, when incoming short-wave solar radiation was zero. The regression-defined parameter values served to minimize the error in water temperature changes computed with the convolution integral, and provided further insight regarding the variability of heat exchange processes acting within the wetland system.

[21] Regression minimized the objective function, S(b), [Hill, 1998] that was quantitatively defined as

(10) D

where b is a vector containing values of each of the parameters being regression estimated, nt is the number of water temperature changes measured in the field, is the ith measured water temperature change, is the equivalent of the ith measured water temperature change computed by the convolution integral (a function of b). Measured water temperature changes, , in S(b) were derived from the German [2000] data as follows. Thirty minute mean water temperatures measured at the top and base of the water column were averaged to estimate the mean 30 min and vertical water column temperatures. The difference between two consecutive mean vertical water column temperatures was used as the mean vertical water column temperature change ().

[22] Field data requirements for equation (9) included time series of air temperature changes (T_a) and water depths (D). These time series data were taken from the German [2000] data set. Field data were not available for the thermal exchange (k_ex) and alpha () coefficients, therefore these coefficients were estimated with regression to minimize equation (10). A conservative overestimate of the surface water’s thermal memory also was used in order to solve equation (9) because it prevented errors in convolution-computed water temperature changes caused by truncation of the transfer function. For example, using 0.25 days for the surface water’s thermal memory truncates relatively large values of the transfer function (Figure 6). This truncation creates errors in convolution-computed water temperature changes (equation (9)) because air temperature changes occurring prior to 0.25 days are not multiplied by the transfer function and summed to compute the current water temperature changes. In contrast, over estimating the surface water’s thermal memory only truncates very small values of the transfer function (Figure 6) which minimizes truncation errors. Thus a period of 12 days was used as a conservative overestimate of the surface water’s thermal memory, based on the decay of the transfer function (Figure 6) for values of k_ex and D expected in this study. This 12 day thermal memory was used at all sites every year.

[23] The regression process was initiated by perturbing k_ex and by 1% from their initial values to compute parameter sensitivities. These sensitivities were used to solve for estimates of k_ex and that minimized S(b). The detailed forms of S(b), sensitivity, and regression equations are presented by Poeter and Hill [1998] and thus are not repeated here. S(b) was considered minimized when the difference between two successive solutions for k_ex and was less than the parameter tolerance criteria (equal to 0.01). Within this parameter tolerance, further regression iterations did not substantially reduce errors between changes in mean vertical water temperature computed with the convolution integral and those measured in the field.