Journal of Environmental Quality 31:1370-1379 (2002)
© 2002 American Society of Agronomy, Crop Science Society of America, and Soil Science Society of America
TECHNICAL REPORTS
Waste Management
Uncertainty Analysis of the Water Balance Technique for Measuring Seepage from Animal Waste Lagoons
J. M. Ham*
Department of Agronomy, Throckmorton Hall, Kansas State Univ., Manhattan, KS 66506
* Corresponding author (snafu{at}ksu.edu)
Received for publication October 1, 2001.
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ABSTRACT
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Water balance measurements can be used to estimate seepage rates from animal waste lagoons and earthen storages. This method requires detailed measurements of depth changes and cumulative evaporation during 5- to 10-d periods. Quantifying the uncertainty surrounding the measurements is crucial if data from seepage tests are used to determine if lagoons are meeting engineering specifications and operating within regulatory guidelines. Uncertainty analyses, using a 95% confidence interval, were applied to field data collected during studies of animal waste lagoons in Kansas and Oklahoma. Changes in depth were measured with float-based recorders and evaporation was estimated from meteorological observations. Results showed that rate changes in depth could be measured to within ±0.28 mm d-1 or better when wind speeds at the start and end of the test were less than 4 m s-1. Uncertainty in evaporation was the most significant factor affecting the seepage estimate, and surface temperature and relative humidity were the main sources of imprecision in the evaporation calculations. Evaporation could be estimated to within 10 to 20%, with the largest uncertainty occurring during windy conditions. Uncertainty in the calculated seepage rate increased as evaporation increased. When evaporation rates are low (e.g., <4 mm d-1), seepage can be estimated to within ±0.5 mm d-1 with 95% confidence. A precision of ±0.25 mm d-1 is possible when research-grade instruments are deployed under favorable weather conditions. A measurement duration of 5 d is adequate for most water balance tests. In many cases, precision of the water balance technique will be sufficient in determining if a working lagoon is within regulatory guidelines.
Abbreviations: AFO, animal feeding operation • DOY, day of year
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INTRODUCTION
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EARTHEN LAGOONS are used at many animal-feeding operations (AFOs) to store and treat manure waste. The effluent in the lagoons contains nutrients, salts, pathogens, and other chemicals that could affect ground water quality if waste is not properly contained. Most states have design standards for lagoons that limit seepage to specified values (Ham and DeSutter, 2000; Parker et al., 1999b). These regulations are used mainly to design the thickness and permeability of the compacted-soil liner. Typically, soil samples of the proposed liner material or intact cores of the completed liner are subjected to laboratory permeability tests (e.g., ASTM D 5084-90, 1991). These analyses provide an estimate of the coefficient of permeability that, when used in a Darcy's Law flow equation, can provide an estimate of the lagoon seepage rate for a given liner thickness and waste depth. Permeability data are useful during the design process and for quality control, but are poor predictors of the actual seepage rate from working lagoons. Researchers have shown that many factors affect seepage on a whole-lagoon basis that are not accounted for by laboratory soil analyses (Daniel, 1984; Rowe et al., 1995; Ham, 2002). Processes like erosion of side slopes can increase seepage, while clogging from organic sludge can reduce flow (Maule et al., 2000). A compete review of these processes can be found in the literature (Hills, 1976; McCurdy and McSweeney, 1993; Ham and DeSutter, 1999, 2000; Maule et al., 2000).
Evaluating seepage and liner performance at existing lagoons requires measurement techniques that account for the multiple "real-world" factors affecting seepage. Research has shown that whole-lagoon seepage can be estimated at working AFOs using short-term (e.g., 5-d) water balance tests (Ham, 1999). Seepage is calculated from detailed measurements of changes in lagoon depth and cumulative evaporation. These techniques could prove useful to both regulators and producers as a way to evaluate liner performance in situ. Ham (2002) measured the water balance of 20 lagoons and found the average seepage rate was 1.1 mm d-1. Glanville et al. (1999) performed similar studies on lagoons in Iowa. Ham (2002) also showed that seepage estimates could be combined with information on lagoon geometry to estimate the apparent permeability of the compacted liner while the lagoon was in use.
As with all measurements, there is a degree of uncertainty surrounding seepage data. Therefore, the precision and resolution of the technique must be quantified so stakeholders can assess the utility of the results. This is especially relevant when lagoons are being tested to verify that they are operating within regulatory guidelines. The objectives of this report are to discuss measurement uncertainties surrounding the water balance technique of Ham (1999)( 2002) and estimate the accuracy of the method under a range of environmental conditions. Operating guidelines for improving measurement precision are proposed. While the discussion centers on anaerobic lagoons at AFOs, results could apply to water balance studies of any earthen basin or pond containing liquid effluent.
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WATER BALANCE ESTIMATES OF SEEPAGE
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The water balance technique for measuring seepage from a pond or lagoon is based on conservation of mass. If there are no inputs or outputs into a contained body of waste (no pumping, precipitation, or runoff), then seepage can be determined from measurement of changes in liquid depth and cumulative evaporation. Several researchers used this technique in the early days of lagoon research (e.g., Davis et al., 1973). Historically, water balance tests were carried out over long periods (e.g., 30 d) because relatively imprecise methods were used to measure depth and evaporation. That is, imprecision in the measuring devices was offset by increasing the measurement period so cumulative depth changes and evaporation were large. The method of Ham (1999)( 2002) uses the same water balance principle, but employs the latest electronic technology and micrometeorology techniques, allowing tests of shorter duration and greater precision. Briefly, seepage is determined from measurements of evaporation and changes in depth when all other waste inputs and outputs are precluded or quantified. In most cases, data are collected for 5 to 10 d and the seepage rate, S (mm d-1) is calculated as:
 | [1] |
where 
P is total precipitation and runoff from the side embankments (mm); E is cumulative evaporation (mm); Dt1 and Dt2 are the relative depths of the lagoon at the beginning and end of the test (mm), respectively; and t2 - t1 is the duration of the test (d). In practice, data are often analyzed when precipitation is negligible (P = 0). Use of Eq. [1] requires data collected at different time scales. For example, Dt1 and Dt2 are single point-in-time measurements while E is calculated by summing evaporation calculated from short-time-interval (e.g., hourly) environmental measurements. Thus, the duration of a test, t2 - t1, has an effect on the uncertainty surrounding S.
Evaporation is measured using the bulk transfer method as described by Eq. [3] in Ham (1999). Evaporation is computed every 30 min as:
 | [2] |
where Ts and Ta are temperatures (°C) of the waste surface and air, respectively; es(Ts) and es(Ta) are saturation vapor pressures at the temperatures of the waste surface and air (kPa); RH is relative humidity of the air; U is wind speed (m s-1) at the reference height (e.g., 1 m); Ce is the bulk transfer coefficient (dimensionless); Rd is the gas constant (287.04 J kg-1 K-1); and 622 is from the ratio of the molecular weights of water and dry air. Examination of Eq. [2] shows that calculating E requires measurement of four environmental variables (Ta, Ts, RH, and U) and an approximation of Ce. The determination of Ce for lagoons was described in Ham (1999) and is assumed constant at 2.8 x 10-3. Saturation vapor pressure, es(Ts) and es(Ta), can be calculated using the formula of Murray (1967):
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where Tx (°C) represents Ts or Ta. Ham and DeSutter (1999) measured the saturation vapor pressure curve of lagoon effluent and found it was not distinguishable from water. Parker et al. (1999a) discuss how the physical and chemical properties of cattle feedyard effluent may affect evaporation. Surface temperature, Ts, is measured with infrared transducers (IRTs). However, readings from the transducers must be adjusted to correct for the effect of surface emissivity and the effect of downwelling radiation from the atmosphere (Ham and Senock, 1992). Given the direct, uncorrected reading from an infrared transducer (Ti), the actual temperature of the surface (i.e., skin temperature) is calculated as:
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where 
is the emissivity of the surface (0.96 for water), 
is the Stefan–Boltzmann constant (5.67 x 10-8 W m-2 K-4), and B represents the effect of surrounding radiation and characteristics of the infrared transducer filter. An approximation of B can be obtained from the Stefan–Boltzmann equation:
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where k is the emissivity of the atmosphere. Equation [5] assumes that Ta approximates the thermodynamic temperature of the atmosphere contributing to B. The emissivity of the atmosphere (k) can be approximated using the equation of Brutsaert (1975):
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As will be shown later, uncertainty in S is dependent on the magnitude and errors of the input variables (i.e., Dt, Ta, Ts, RH, U, Ce) and the amplification or attenuation of these errors in Eq. [1] through [6].
Following the approach of Ham (1999), instrumentation to estimate seepage using Eq. [1] through [6] includes three main components: a meteorological raft or buoy positioned near the center of the lagoon, a water level recorder mounted inside a stilling well, and a bank station that houses the electronic data collection equipment and a tipping-bucket rain gauge (Fig. 1)
. The meteorological buoy supports a cup anemometer, an infrared thermometer (to sense surface temperature), and a relative humidity and air temperature probe, all positioned 1 m above the waste surface. Depth changes are measured with either float-based water level recorders that use high-resolution linear displacement transducers or with submersible pressure probes. All instruments are sampled every 10 s and stored as 30- or 60-min averages using computerized data acquisition equipment. At the end of a water balance test (e.g., 5 to 10 d), S is calculated using Eq. [1]. The terms P, Dt1, and Dt2 are measured directly. However, E is determined by first calculating E for each 30-min period from the environmental measurements using Eq. [2] through [6]. These data are then summed over the entire test period (t2 - t1) to estimate E.
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UNCERTAINTY ANALYSIS
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Theory
Equation [1] suggests that uncertainty surrounding calculation of S is dependent on errors in P, Dt2 - Dt1, and E. Typically, S is calculated during periods when P is zero; thus, only errors attributed to Dt and E will be considered here. The uncertainty of any calculated result is determined using a root-sum-square (RSS) formula that considers the combined effect of uncertainty in each input variable (Coleman and Steele, 1989). Furthermore, imprecision in each input variable is the sum of precision and bias errors. Detailed discussions of these concepts are given in Coleman and Steele (1989). The uncertainty surrounding S, hereafter called 
S (mm d-1), can be calculated as:
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where Dt2 and Dt1 are the uncertainties of the depth measurements at the end and beginning of the test, respectively, and E is the uncertainty in cumulative E, all in mm. Normally, each uncertainty term on the right hand side of Eq. [7] is multiplied by a sensitivity coefficient, the partial derivative of the result with respect to the measured variable. However, in the case of Eq. [1] these sensitivities are unity and do not appear. Also, the RSS formula assumes that errors associated with each input are independent (i.e., not correlated). Because Dt1 and Dt2 are measured by the same float recorder, this assumption would appear to be violated. However, the magnitudes of Dt1 and Dt2 mainly are governed by wind effects (i.e., waves), factors that may be drastically different at t1 and t2. Thus, the imprecision of the two depth readings will be treated as independent variables.
The uncertainty in E is calculated by first predicting the instantaneous uncertainty of E on an hourly basis as:
where terms Ta, Ti, RH, U, and Ce are the uncertainties in the five input variables, and the partial derivatives are the sensitivity coefficients. All of the these variables, with the exception of Ce, are measured on an hourly basis. Once hourly estimates of E are available, they are summed over the duration of the water balance test to estimate cumulative uncertainty in E:
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The uncertainty in the depth measurements coupled with the results from Eq. [9] are then used in Eq. [7] to predict S. Because S is dependent on environmental conditions during the test, uncertainty will be unique for each water balance experiment.
The uncertainty analysis used here does not consider several sources of error that could affect the accuracy of the E calculation. There may be situations when evaporation is affected by factors not included in Eq. [2]. Model errors of this type may occur, for example, if nearby buildings, tree lines, or local topography affect wind flow and alter the characteristics of the surface boundary layer. Also, sampling errors could occur if the environment at the location of the meteorological buoy (i.e., center of lagoon) is different than conditions near the edge of the lagoon. Webster and Sherman (1995) address the difficulties of estimating evaporation from small water bodies.
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METHODS
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The uncertainty analysis, as defined in the preceeding section, was applied to five case studies from a group of 20 seepage experiments conducted at commercial AFOs between 1997 and 2000 (Ham, 1999, 2002; Ham and DeSutter, 1999, 2000). The lagoons selected for analysis included two cattle feedlot lagoons, one dairy lagoon, and two swine waste lagoons (Table 1)
. These sites were selected because they provided a range of lagoon types, sizes, depths, and locations. All lagoons had compacted-soil liners between 0.3 and 0.46 m thick with the exception of Swine Lagoon 2, which had a 1-mm-thick plastic liner. Studies were chosen to span different seasons of the year, thus providing a range of evaporation rates. The water balance of each lagoon was measured and calculated using the approach of Ham (1999) as described here in Eq. [1] through [6] and Fig. 1. Changes in depth were measured with float-based recorders housed in 0.2-m-diam. stilling wells (Ham and DeSutter, 1999; Fig. 3). Linear displacement transducers on the recorders had full-scale ranges of 635 mm and resolutions of 0.16 mm (LXPA25; Unimeasure, Corvallis, OR). Air temperature and humidity were measured with a shielded HMP45C probe (Campbell Scientific, Logan, UT), and wind speed was measured with a cup anemometer (Wind Sentry; RM Young, Traverse City, MI). An infrared transducer with a 15-degree field of view was used to measure the temperature of the waste surface (Everest Interscience, Logan, UT). Signals from the instruments were sampled every 10 s and averaged over 30- or 60-min periods using battery-powered data acquisition equipment (CR10X; Campbell Scientific).
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Table 1. Water balance analyses of five animal waste lagoons. Uncertainties for depth change, evaporation, and seepage represent 95% confidence intervals as calculated using the uncertainty formulas defined in the text. The precision of input variables were assigned values listed as Case 1 in Table 2.
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Table 2. The uncertainty (95% confidence interval) assigned to input variables used to compute the uncertainty in the evaporation and seepage rates. Two scenarios are described: a "typical" case with a 5-d study duration and the manufacturer's stated precision of each instrument, and a "best" case with a 10-d study duration using research-grade instruments where some variables are measured with greater precision.
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Fig. 3. Effect of wind speed on the standard deviation (SD) of measurements from a float-based water level recorder. The SD was computed every 30 min from samples collected every 10 s. Data were collected for 45 d at Swine Lagoon 2 (see Table 1).
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Uncertainty analysis was applied to the five case studies in Table 1 by first setting the uncertainty surrounding the input variables. The uncertainty terms (Dt, Ta, RH, etc.) in the RSS formulas can represent random errors or bias, components that are sometimes evaluated separately in what Coleman and Steele (1989) call a "detailed uncertainty analysis." However, when estimating the uncertainty in E, random errors will tend to cancel out as the duration of the test increases. Evaporation may be overestimated on one day but underestimated on another such that the net effect of the errors is reduced by the end of the test. Conversely, the uncertainties listed in Table 2
, with the exception of Dt, are assumed to represent bias limits. Unfortunately, biases do not cancel out as the test duration increases, but can accumulate and fluctuate over time. These errors could be caused by improper sensor calibration, deployment blunders, or the use of improper or oversimplified mathematical models in Eq. [2] through [6]. For example, the bulk transfer coefficient, Ce, in Eq. [3] may not be constant, but may in fact be a more complex function that varies spatially and temporally (Webster and Sherman, 1995). Thus, the uncertainty assigned to Ce in Table 2 is associated with variance in Ce not accounted for in Eq. [2]. The uncertainty in Dt1 and Dt2 represents both random and systematic errors, because depth is a point-in-time measurement at the beginning and end of the test. As will be shown, wind speed and wave action strongly affect Dt.
Two different levels of input uncertainty were evaluated (Table 2). The case termed "typical" represents the uncertainty specified by the manufacturer of the instruments with some adjustment based on field experience. The case termed "best" represents the best precision possible if sensors used to measure Ti and RH underwent improved laboratory calibration and were deployed by an experienced research team. Durations of the typical and best scenarios were assumed to be 5 and 10 d, respectively. The procedure for computing S for each of the five case studies was as follows: (i) Using data from the site in question, E was computed for each 30-min or hourly period using Eq. [8], (ii) E was calculated using Eq. [9], and (iii) results were combined with Dt1 and Dt2 into Eq. [7] to compute S. Equation [8] was solved numerically using Mathcad 2000 (MathSoft, 2000).
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RESULTS AND DISCUSSION
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The results of the water balance experiments will be presented first, followed by an uncertainty analysis of Dt, E, and the effect of test duration. Finally, the combined effect of these factors on S will be summarized.
Water Balance Results
Environmental conditions during the water balance tests ranged from wintry, near-freezing conditions at Cattle Lagoon 1 to warm, summer-like conditions at Swine Lagoon 1 (Table 3)
. The experiments at Cattle Lagoon 2 and the Dairy Lagoon had more moderate weather. Large differences in environmental conditions led to big differences in the water balances among sites. Rate changes in depth, the combined effect of seepage and evaporation, ranged from a low of 11.7 mm d-1 at Cattle Lagoon 1 to a high of 44.9 mm d-1 at Swine Lagoon 1 (Table 1). Correspondingly, evaporation at Swine Lagoon 1 was 8.1 mm d-1, which was 4.5 times greater than at Cattle Lagoon 1. Figure 2
shows the time course of depth changes and E at four of the study sites; divergence between depth change and E represents seepage. Figure 2a, the wintertime case at Cattle Lagoon 1, shows a period where evaporation apparently was greater than changes in depth. This can sometimes occur when seepage is very small and minute errors in the calculation of E make it appear as if the lagoon is gaining water through the liner. The freezing conditions during this test may have contributed to this effect. However, the test at Cattle Lagoon 1 was executed for an additional 10 d (not shown) and clear separation between evaporation and depth change emerged. Figures 2b–d represent more desirable cases because E was consistently less than the total depth change.
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Table 3. Environmental conditions during the water balance studies. Included are maximum and minimum air temperatures (Ta), maximum and minimum waste surface temperatures (Ts), average vapor pressure deficit (VPD), and average wind speed (U) as measured over each 5-d test period. A description of each lagoon is provided in Table 1.
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Fig. 2. Change in depth and evaporation from four animal waste lagoons during water balance experiments. A description of the lagoons is given in Table 1.
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Despite the large differences in E and total depth change, the calculated seepage rate among the four soil-lined lagoons only ranged between 0.6 and 1.0 mm d-1 (Table 1). Ham (1999) showed that site-to-site variability in seepage among a population of lagoons was small and that results were log-normally distributed. At the plastic-lined lagoon (Swine Lagoon 2) where seepage was assumed to be zero, the apparent calculated value of S was 0.12 mm d-1. This result in itself was a measure of the error surrounding the water balance test because the "true" answer (i.e., S = 0) was known. An error of 0.12 mm d-1 is small and suggests that S can probably be measured to within a fraction of 1 mm d-1. The following sections will quantify this uncertainty in detail.
Uncertainty in Depth Measurements
Calculation of S (Eq. [1]) requires precise measurements of depth change over the study period (i.e., Dt2 and Dt1). Figure 2 shows how very small changes in depth can be discerned using the high-resolution float-based recorders of Ham and DeSutter (1999). These recorders use linear displacement transducers that are designed primarily for robotics and precision machining. Thus, the accuracy of the measurements is very high when connected to quality data acquisition equipment (e.g., ±0.3% full scale range). A bigger concern is sampling error caused by wind and wave effects (i.e., dynamic bias or drift). The effect of waves is apparent during the water balance experiment at Swine Lagoon 1 (Fig. 2d) where the depth measurements appear "noisy" during windy conditions. Wave height in a small water body is dependent on wind speed, wind duration, and fetch (i.e., distance between the leading edge of the lagoon and the waste level recorder). Figure 3
shows the standard deviation (SD) of the waste level data as a function of wind speed at a 2.5-ha lagoon in southwestern Kansas. The SD was computed over 30-min periods from samples collected every 10 s from an anemometer positioned 1 m above the waste surface. Variance in the recorder output increased exponentially with increasing wind speed. Similar results were found at other locations with the power of the exponential ranging from 3.7 to 4.1. This relationship is very similar to the theoretical relationship between wave height and wind speed. Data suggest that the effect of waves tends to increase rapidly when wind speeds exceed 4 to 5 m s-1.
Although the effect of wave chop is a concern, data can be averaged over longer periods to reduce the apparent "noise" in the signal. A bigger concern is bias in the depth measurements caused by wind "pushing" water toward the downwind side of the lagoon. Sample calculations show that shearing stress could cause the waste depth to increase by 1 to 2 mm on the downwind edge of a large lagoon (White and Denmead, 1989). Conversely, the same effect could cause an underestimate of depth if the recorder was near the upwind edge. Figure 4
examines wind-induced fluctuations in depth measurements for a 3-d period at Swine Lagoon 2. Because the lagoon had a plastic liner, changes in depth should have equaled E (Fig. 4a). Thus, deviations between the recorder output and the calculated evaporation line can be considered error. Fluctuations in the depth measurement ranged between ±0.5 and ±2 mm (Fig. 4b), and were correlated with wind speed (Fig. 4c). Recorder output was the most stable on Day of Year (DOY) 266 when wind speeds were about 2 m s-1.
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Fig. 4. Fluctuations in recorded waste depth during a 3-d period at a plastic-lined swine waste lagoon. Because seepage was zero, the change in depth should equal evaporation. Shown are (a) change in depth as detected by the waste level recorder (30-min averages) and the evaporation rate predicted by Eq. [3], (b) deviations between the recorded waste depth and modeled evaporation, and (c) wind speed and wind direction.
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Figures 3 and 4 suggest that Dt2 and Dt1 can be determined to within ±1 mm when wind speeds are less than 4 m s-1. Therefore, the official start and end of a water balance test should coincide with periods of low wind speed to minimize errors caused by wave action shearing stress. Typically, wind speeds are lowest in the predawn period, thus 0600 h might be an opportune time of day to start and stop a test. Errors caused by shearing stress could be reduced by positioning the recorder near the center of the lagoon. Unfortunately, this is often difficult to achieve in the field. An uncertainty of 1 mm was assigned to Dt1 and Dt2 for all five case studies in Table 1. The imprecision in the change in depth measurement (Dt2 - Dt1) can be computed using Eq. [7], excluding the E term. Thus, the assigned uncertainty in the depth change measurements for all case studies was the square root of 2, or ±1.4 mm (Table 1).
Uncertainty in Evaporation
Uncertainty in evaporation is the "Achilles Heel" of any water balance experiment. Thus, the smaller the evaporation rate the better. Using the "typical" input uncertainties described in Table 2, the uncertainty in evaporation ranged from 0.3 mm d-1 at Cattle Lagoon 1 to 1.1 mm d-1 at Swine Lagoon 1 (Table 1). On average, the predicted uncertainty surrounding E was ±16%. When the precision of the inputs was increased to the "best" case (Table 2), the average uncertainty in E was ±11% (not shown). Figure 5
shows the fraction of uncertainty contributed by each input variable in the evaporation equation. Measurements of Ti and RH were the largest sources of imprecision, often accounting for more than 70% of the total. Wind speed and Ce had a lesser effect, while errors attributed to Ta were almost negligible. Surface temperature and RH are used to compute the vapor pressure difference between the waste surface and the air, the driving force underlying evaporation. It follows that any bias in these terms could lead to severe overestimates or underestimates of E in Eq. [3].
The combined effect of U and Ce, terms describing aerodynamic conditions, were similar in magnitude to the effect of Ti and RH. Almost all the partial derivatives in Eq. [8] are multiples of U, Ce, or both. Figure 6
shows the relationship between E(t) and hourly estimates of E at Swine Lagoon 1. Results appear to show a family of curves with different slopes, each depicting a linear relationship between E and E. Each apparent line of points are data from a different day of the test, each having a unique range of wind speeds. Uncertainty in E was greatest (approximately 20%) on the last day of the test, DOY 162, when average wind speeds were 6 to 8 m s-1. Much less uncertainty (approximately 10 to 15%) was calculated for the first three days, DOY 157 to 159, when wind speeds were less than 5 m s-1. In general, levels of uncertainty surrounding E(t) were bracketed between 10 and 20%, a trend that was found at the other study sites as well.
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Fig. 6. Uncertainty in the hourly estimates of evaporation plotted against evaporation rate. Data are from an experiment at Swine Lagoon 1 (Table 1) conducted between Day of Year (DOY) 157 and 161, 2000. Variation among measurement days was caused by differences in average daily wind speed.
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Effect of Study Duration
Lengthening the duration of a water balance experiment reduces the effect of random errors and increases resolution. Figure 7 shows how the apparent calculated seepage rate (Eq. [1]) changes following the start of an experiment. Results at any elapsed time represent S values that would have been calculated if the experiment had been stopped hypothetically at that moment. The elapsed time is essentially t2 in Eq. [1]. In all cases, the variance in the calculations is very high at the start of the experiment and becomes more stable over time. This pattern was not caused by changes in actual seepage rate, but shows how the effect of random errors in the input variables were moderated with more sampling. This is much like computing a running mean (integration) of a steady signal contaminated with random noise. Results from Cattle Lagoon 2 and the Dairy Lagoon site were stable after about three days and became almost constant on the fifth day (Fig. 7b,c). At Swine Lagoon 1, S also exhibited very little upward or downward trend after 3 d (Fig. 7d). However, the effect of random errors is more apparent at Swine Lagoon 1 because E was larger and conditions were very windy compared with the other sites. A bigger concern is the pattern observed at Cattle Lagoon 1, where there was a noticeable upward trend in S over time (Fig. 7a). This pattern may indicate that more data were needed or that some biophysical factor was causing a change in the actual seepage rate over time. Air temperature during the test was fluctuating between -3 and +5°C, which may have caused errors in water level recorders if ice formed on the floats and stilling wells. As mentioned previously, data were collected for more than 15 d at Cattle Lagoon 1 and the calculated seepage became stable. These data are shown to demonstrate how plots like Fig. 7 are useful tools for testing data quality and ensuring adequate study duration. In the cases presented here, as well as the 20 studies in Ham (2002), five days usually was adequate for seepage determination if weather conditions were favorable.
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Fig. 7. The apparent seepage rate as calculated at different times following the start of a water balance test. Individual plots correspond to the same sites depicted in Fig. 2 and Table 1.
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Total Uncertainty in the Seepage Calculation
The uncertainty of any seepage calculation (S) is dependent on the precision and bias limits of the inputs as well as environmental conditions during the test. When the input uncertainties were set at the "typical" levels (Table 2), S for the five case studies ranged from ±0.4 mm d-1 at Cattle Lagoon 1 to ±1.1 mm d-1 at Swine Lagoon 1 (Table 1). The uncertainty in S was correlated with E. In comparison, when the input uncertainties were set to the "best" case (Table 2), S was reduced ±0.2 mm d-1 at Cattle Lagoon 1 and ±0.8 mm d-1 at Swine Lagoon 1. On average, increasing the precision of Ti and RH and extending the study duration to 10 d caused a 33% reduction in S. There was a linear relationship between S and evaporation rate for both the typical and best scenarios (Fig. 8)
. At higher rates of evaporation (e.g., Swine Lagoon 1), S was greater than S itself—an undesirable trait (Table 1). Only when evaporation was less than about 4 mm d-1 was S consistently less than S.
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Fig. 8. Uncertainty in the overall seepage estimate as a function of average evaporation rate. Data were integrated over measurement periods for each of the water balance tests described in Table 1. The uncertainty of the input variables was set using two scenarios, typical and best, as described in Table 2.
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The relationship between S and E shown in Fig. 8 may prove useful when planning water balance experiments at lagoons. If the objective is to determine if a basin is seeping less than some specified value, then the required level of precision can be established a priori. Once an acceptable level of uncertainty has been negotiated among the interested stakeholders, the relationships in Fig. 8 could be used to identify the time of year and environmental conditions when E and the calculated values of S will meet these requirements. For many states in the central Great Plains, evaporation rates less than 4 mm d-1 are observed in the winter, or during unusually cool weather in late fall and early spring. Perhaps nothing will ensure the precision of seepage test results more than restricting measurement periods to certain seasons of the year when evaporation rates are small (e.g., 
5 mm d-1).
Given that S is correlated with evaporation, it might be advantageous to estimate S using only nighttime data [i.e., when E(t) is small]. Nighttime data could be partitioned from the continuous 5- or 10-d dataset, and Dt2 - Dt1 and E could be computed for each night of the test. Daily results could be summed over all measurement days to compute a nighttime water balance and the corresponding uncertainties. For the five case studies in Table 1, nighttime evaporation accounted for 25 to 40% of E. According to Fig. 8, a corresponding decrease in S might be predicted. Unfortunately, uncertainty from the depth measurements, Dt1 and Dt2, is much larger when using this approach because depth must be estimated 10 times during a 5-d experiment (e.g., sunset and sunrise each day). Sample calculations for Swine Lagoon 1 showed that S using only nighttime data was ±1.24 mm d-1, slightly larger than the ±1.1 mm d-1 value in the original analysis (Table 1). A nighttime-based approach may improve the precision of the water balance if more emphasis is placed on measurement of depth change. For example, multiple waste-level recorders could be installed in the lagoon to reduce Dt.
At some locations, it may be possible to estimate S when the surface of the lagoon is frozen. Changes in depth still can be measured in these cases using pressure transducers (e.g., PTX1830; Druck, New Fairfield, CT). Sublimation and evaporation from the surface is negligible under these conditions, so seepage can be determined solely from the rate change in depth. Thus, in northern latitudes, measurements of S under frozen conditions may provide the highest precision. More water balance experiments on frozen lagoons are needed to quantify uncertainty for this special case.
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CONCLUSION
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Results clearly show that water balance measurements of lagoon seepage should be conducted when evaporation rates are less than 4 to 5 mm d-1 (late fall, winter, early spring). The resolution of the seepage estimate was high, but the uncertainty (i.e., imprecision) was often near ±0.5 mm d-1, even during favorable measurement conditions. Analysis showed the highest possible precision is about ±0.3 mm d-1 when research-grade instruments are employed, evaporation is less than 3 mm d-1, and the duration of the test is 10 d or longer. The levels of uncertainty stated here represent the 95% confidence interval on the result, not the expected error rate. Using the Cattle Lagoon 2 study as an example (Table 1), one could state that the best estimate of S was 1.0 mm d-1 and that there is only a 1 in 20 chance that the actual seepage was greater than 1.5 mm d-1 or less than 0.5 mm d-1. Thus, the uncertainty analysis is inherently conservative.
The precision of water balance measurements could be optimized by following some simple guidelines:
- Measurements of depth changes and evaporation should follow published scientific methods (e.g., Ham, 1999).
- All instruments used in the test should be calibrated by a third party within 12 mo prior to testing.
- Evaporation should be less than 5 mm d-1.
- The duration of the test must be 5 d or longer.
- Depth measurements at the start and end of the test should only be taken when wind speeds are less than 4 m s-1.
- A test should be discontinued if cumulative precipitation exceeds 0.5 mm.
- Methods used to ensure that waste was not entering the lagoon during the test should be documented.
If very high rates of precision are required, more attention should be given to measuring RH and Ti, and the duration should be extended to approximately 10 d. These measures, coupled with additional requirements specified by the regulatory agency, could make the water balance technique a useful tool for determining if existing lagoons are operating within regulatory guidelines. Finally, the uncertainty analysis shown here is not overly complex and could be performed whenever a water balance test is conducted. Seepage results could be reported with site-specific estimates of precision, a feature that could possibly avoid litigation at some later date. A more beneficial approach would be to run the uncertainty analysis on a daily basis while a water balance test is underway. The duration of the test could be extended until the desired level of precision is achieved. In summary, water balance techniques will remain an important tool for determining if existing lagoons are operating within design specifications, because no other technique can estimate whole-lagoon seepage rates or the in situ, areal performance of a compacted-soil liner.
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NOTES
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Contribution no. 02-120-J from the Kansas Agric. Experiment Station, Manhattan, KS.
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[Abstract]
[Full Text]
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ABSTRACT
INTRODUCTION
