Vadose Zone Journal 2:186-192 (2003)
© 2003 Soil Science Society of America
A Time Domain Transmission Method for Determining the Dependence of the Dielectric Permittivity on Volumetric Water Content
An Application to Municipal Landfills
K. Masbrucha and
T. P. A. Ferré*,b
a Department of Environmental Management, City of Tucson, 100 N. Stone Ave., Suite 215, P.O. Box 27210, Tucson, AZ 85726-7210
b Department of Hydrology and Water Resources, University of Arizona, Bldg. 11, Rm. 246, 1133 E. North Campus Drive, P.O. Box 210011, Tucson, AZ 85721-0011
* Corresponding author (ty{at}hwr.arizona.edu)
Received 5 August 2002.
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ABSTRACT
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A method for determining the dependence of the dielectric permittivity of municipal refuse on its volumetric water content is developed and tested. The method is based on time domain transmission (TDT) measurements collected with an automated network analyzer (ANA). Measurements in variably saturated sand were first made to test the method's ability to measure the apparent dielectric permittivity. The method was then extended to samples of municipal refuse to determine whether a single calibration can be used to describe the relationship between the dielectric permittivity and the volumetric water content of refuse. The results show that the relationship between the dielectric permittivity and the volumetric water content within the refuse shows significant spatial variability. The results do show, however, that a calibration based on multiple samples collected throughout the landfill can be used to measure the volumetric water content with a root mean square volumetric water content measurement error of <0.04 cm3 cm-3. Deviations from the common calibration are not explained by variations in bulk density or porosity among samples. It is recommended that a calibration be derived from samples collected throughout a municipal landfill before a water content monitoring method based on dielectric permittivity measurements is used.
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INTRODUCTION
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UNTIL RECENTLY, abandoned municipal landfills were considered undesirable sites for development. However, this perception is changing with continued urban growth and increased land costs. Some active landfills currently employ a new technology, enhanced landfill biodegradation, to reduce landfill-operating costs, remove landfill gas, and maximize landfill capacity (Reinhart and Townsend, 1997). This approach has been suggested to speed the rate of degradation and settling of abandoned landfills to allow for more rapid development. Enhanced landfill biodegradation requires the judicious addition of water to control and remove the heat generated during biodegradation, while not impeding air flow throughout the refuse. As a result, an effective method of water content measurement is necessary to ensure rapid, safe mass removal throughout a landfill (Pacey et al., 1999).
Time domain reflectometry (TDR) has been shown to be a reliable and nondestructive method for inferring the volumetric water content based on measurements of the dielectric permittivity of a wide range of porous media ranging from soils (Topp et al., 1980) to crushed concrete (van der Aa and Boer, 1997). Li and Zeiss (2001) conducted a systematic study of the dependence of the response of TDR to components of municipal refuse and found that, with independent knowledge of the porosity, the water content could be determined accurately. However, there is no published report of the expected uncertainty of TDR-determined water contents in municipal refuse. Another electromagnetic method, borehole ground penetrating radar, can be used to measure water contents over larger areas than TDR (e.g., Binley et al., 2001). This method allows for water content monitoring on a scale that is more practical for landfill operations. However, the use of either of these methods requires an improved understanding of the relationship between the dielectric permittivity and the volumetric water content of municipal refuse.
The purposes of this investigation were (i) to develop a rapid and effective method for determining the dependence of the dielectric permittivity of municipal refuse on its volumetric water content, (ii) to compare the dependence of the dielectric permittivity of refuse on its water content with standard relationships used in soils, (iii) to determine whether the relationship determined for refuse varies strongly within a municipal landfill, and (iv) based on these results, to determine whether electromagnetic methods can be used to monitor water contents and water content changes in municipal landfills either with or without site-specific calibration.
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THEORY
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Electromagnetic geophysical methods, such as TDR, TDT, and ground penetrating radar (GPR), provide rapid and nondestructive means of measuring the volumetric water content of porous materials. Time domain methods generally rely on direct measurement of the unitless bulk apparent dielectric permittivity, Ka, of a material based on the travel time of an electromagnetic wave through the medium. The bulk apparent dielectric permittivity of a soil sample can be uniquely correlated with the volumetric water content because the dielectric permittivity of water is much higher than that of the other soil components (e.g., Topp et al., 1980).
In transmission mode methods, the total travel time of an electromagnetic wave through a medium is measured. Generally, the total travel time, tT (s), includes the travel time through the medium under examination and through connecting cables. Additionally, depending on the length of the sample under investigation, a section of the rods may extend beyond the medium. This gap could be anywhere along the rods. But, for our experimental conditions, the medium generally settled to the base of the measurement system, leaving a gap at the top of the column. Under these conditions, the electromagnetic wave will also travel through air after passing through the medium for a time, tair. The total one-way travel time is then:
 | [1] |
The velocity of propagation through the medium, vmedium (m s-1), can be defined from tmedium and the length of the transmission line through the sample, Lmedium (m), as:
 | [2] |
Ka can be calculated based on the ratio of the speed of light in a vacuum, c, (3 x 108 m s-1) to the velocity of propagation through the sample as:
 | [3] |
The travel time in the connecting cables can be accounted for by measuring the travel times through the column filled with the medium, tT, and through an empty, air-filled column, tT-air. Assuming that the speed of propagation in air is the same as that in a vacuum, this total travel time can be expressed as the sum of the travel time through the air-filled column and the travel time through the connecting cables:
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The travel time through the connecting cables is then:
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Combining Eq. [1] through [5] gives an expression for the dielectric permittivity based on a pair of measured travel times tT-air and tT.
 | [6] |
If the medium does not fill the column length entirely, then the travel time through the air above the medium will be:
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If the medium does fill the column length entirely, then there is no travel time through air, so the tair term is removed. In either case, Eq. [6] and [7] simplify to:
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Finally, this reduces to:
 | [9] |
where tT-diff is the difference in the measured travel times, tT and tT-air. The travel time through the medium, tmedium, is:
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For measurements made in a vertical column, the volumetric water content can be related to the mass of the sample before any water is added Mi (g), the oven-dried sample mass Mdry (g), the mass of water added 
M (g), the sample length Lmedium, the cross-sectional area of the column A (m2), and the density of water 
water (g cm-3), as:
 | [11] |
Many authors (e.g., Ledieu et al., 1986; Alharthi and Lange, 1987, Herkelrath et al, 1991, and Hook and Livingston, 1995) have found a linear dependence of the square root of the dielectric permittivity (or the travel time) on the volumetric water content. This can be expressed as:
 | [12] |
Because of its direct relationship to mixing models, this form is generally seen as more informative than the polynomial form first presented by Topp et al. (1980) and adopted by Li and Zeiss (2001).
Combining Eq. [9], [11], and [12] gives an expression relating the change in travel time with the change in mass of the column as:
 | [13] |
In this study, we construct plots of tT-diff vs. M for variably saturated sand and refuse. These plots are then used to determine the constants, a and b, in Eq. [12].
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EXPERIMENTAL METHODS
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An upward infiltration method was developed to correlate the velocity of propagation of an electromagnetic signal to the volumetric water content of municipal refuse samples. Ferré et al. (1996) explained why TDR measurements made with water contents that vary along a TDR probe can be used to calibrate a correlation between the square root of the dielectric permittivity and the volumetric water content. Young et al. (1997) showed this experimentally. The method developed here is similar to that shown by (Young et al., 1997), but this method is based on travel time measurements made in transmission mode rather than in reflection mode. Reflection mode measurements, made with TDR instruments, are made with metallic probes that terminate within a sample. Transmission measurements are made using metallic rods that pass through the entire sample. There are a number of advantages to making measurements in transmission mode. First, uncertainties related to the behavior of the electromagnetic wave at the ends of TDR rods can give rise to the need for probe-specific calibration for TDR measurements. In contrast, TDT measurements eliminate these fringing field effects. Second, TDT waveform analysis is simpler than TDR waveform analysis because only one point on the waveform must be identified. Furthermore, in TDR analysis, identification of the reflection from the ends of the rods can be complicated due to multiple reflections from the beginning of the rods. This complication is not a factor for TDT analysis. Despite its advantages, the TDT method currently requires more expensive instruments than TDR-based methods. Widespread, practical application of the following method will require the development of less expensive TDT instruments.
All measurements were made in specially designed columns (Fig. 1) within a climate-controlled laboratory. The columns were designed to allow for transmission measurements through samples during controlled upward infiltration. The columns were fabricated with a 26.0-cm-long section of 10.2-cm-diameter schedule 40 chlorinated polyvinyl chloride (CPVC) pipe. Two 30.0-cm-long stainless-steel rods (0.48-cm diam.) were centered inside the column, approximately 3.0 cm apart, and held in place by a poured, epoxy base. The rod separation was chosen based on numerical analysis following that of Knight et al. (1997) and based on direct measurements in columns of varying diameter to ensure that there was no influence of the column walls on the dielectric permittivity measurements. The rods were connected to a 50-
coaxial cable that extended through the column wall and was embedded in the epoxy base; this cable was connected to the reflection port of the ANA (labeled "reflection cable" in Fig. 2). A second cable ("transmission cable" in Fig. 2) connected the transmission port of the ANA to the top of the rods. The cable was connected to the rods using a strip connector. Plastic tubing (0.64-cm o.d.) passed through the column wall and was attached to a ring of T-connectors. Three T-connectors extended above the epoxy base to allow uniform addition of water at the base of the sample. Water entered the column via gravity feed from a volumetric cylinder. No water overflowed the top of the column.
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Fig. 1. Side view of the experimental column. Note that "reflection coaxial connector to ANA" refers to the label on an automated network analyzer (ANA). The reflection port is used for signal input for transmission measurements together with a transmission port to deliver the transmitted signal to the instrument.
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Measurements were made using a Hewlett Packard (HP) 8752C ANA (Hewlett Packard, Palo Alto, California) in the transmission mode. The time domain response was constructed with onboard Fast Fourier Transform hardware from 401 single frequency measurements equally spaced between 0.3 and 1.5 GHz. Measurement parameters were set and data were collected via connection with a desktop computer. The total travel time can be determined from any of the time domain waveforms as the time of intersection of two straight lines, one fitted to the early-time constant amplitude section of the waveform and a second fitted to the rising limb of the waveform (Fig. 3).
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Fig. 3. The time at the point of intersection of the horizontal and sloped dashed lines that have been fitted to the waveform (solid line) defines the time of first arriving energy of the time domain transmission (TDT) pulse, 18.93 ns.
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Before adding refuse to the column, the mass of the empty column was measured with a Sartorius BP 3100 S digital scale (Sartorius, Goettingen, Germany) with a repeatability of 0.0002 g. Then, the phase and amplitude of the signal transmitted through the empty column, with no refuse present, were measured. Following this initial, air-filled column measurement, the dependence of the dielectric permittivity of variably saturated sand on its volumetric water content was determined and compared with published results. This step was completed as a validation of the method. Following validation, the method was applied to refuse samples.
Refuse samples were collected from a municipal solid waste site in Tucson, AZ that was operated between 1960 and 1967. The solid waste is composed primarily of brick and wood construction debris and residential waste. Soil boring data shows that the landfill ranges in depth between 4.6 and 12.2 m (15 and 40 feet) below land surface and is capped with 2 m of native soil. The landfill from which samples were collected was relatively small, with a total area of 2.4 ha and an estimated total volume of refuse of 202 000 m3 (Pima Association of Governments, 1993). Refuse samples were collected from 19 locations throughout one-third of the total area of the landfill. All samples were collected within the upper 6 m of the landfill, beneath the native soil cover.
Samples were collected with a hollow stem auger and stored in tightly sealed 19-L plastic containers. Each container was mixed thoroughly before subsamples were collected and then stored in airtight plastic bags. Nineteen refuse samples were tested. Each sample was packed in the column to a bulk density between 0.799 and 0.984 g cm-3, which corresponds with the range of wet-based density of 0.712 to 0.949 g cm-3 for degraded municipal refuse at landfills with similar depth, water content, and environmental conditions (Landva and Clark, 1990; Pacey, 1982). Note that the low bulk density of refuse is due to the combination of a low solid density, a high porosity, and a moderate volumetric water content (Table 1). Municipal water was added to each refuse sample in 100- to 200-mL increments. The column mass and the phase and amplitude of the transmitted electromagnetic signal were measured for each increment of added water. A series of 10 water content measurements could be measured in approximately 45 min. Ten of the refuse samples were dried with a Yamato DX400 oven (Yamato Scientific, Tokyo, Japan) at 105°C until no further mass change was measured. Nine refuse samples were not oven dried.
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Table 1. Measured properties of refuse based on time domain transmission upward infiltration measurements and the slopes and intercepts of Eq. [12] based on linear fits to paired measurements of the square root of the dielectric permittivity and the volumetric water content.
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To test the ability of the TDT upward infiltration method to measure the dielectric permittivity of a variably saturated porous medium, measurements were made in a column filled with No. 30 sand with incremental additions of municipal water. For each increment of water added, the dielectric permittivity of the sand–water mixture was calculated using Eq. [9]. The volumetric water content of the sand was then determined using Eq. [11], assuming that the sand was initially dry.
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RESULTS
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Measurements were made in variably saturated sand and the results were compared with published correlations of volumetric water content and apparent dielectric permittivity. Based on the excellent agreement of the results with published measurements, the method was then extended to examine the dielectric properties of refuse samples.
Measurements in Variably Saturated Sand
Ferré et al. (1996) showed that the polynomial calibration of Topp et al. (1980) is equivalent to a mixing model based on the square root of the dielectric permittivity (Eq. [12]) with a slope and intercept of 0.1181 and -0.1841, respectively. The slope and intercept of the TDT measurements made in sand (Fig. 4) were in excellent agreement, showing values of 0.1224 and -0.1960, respectively. The differences in inferred water contents derived using these relationships gave rise to a difference in water content of <0.015 cm3 cm-3 for dielectric permittivity values ranging from 4 to 36.
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Fig. 4. Square root of the dielectric permittivity vs. volumetric water content for Industrial Paragon Sand (No. 30) and municipal water during upward infiltration. Each symbol represents measurements made in a separate calibration experiment. The column was repacked with dry sand for each experiment. The regression equation and the goodness of fit (R2) of a linear regression to all of the data are shown. The calibration developed by Topp et al. (1980) is included as a dashed line.
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Measurements in Municipal Refuse
Figure 5 shows the square root of the dielectric permittivity as a function of the volumetric water content for a typical refuse sample, SVNO5. The relationship is highly linear with an R2 > 0.99. Table 1 lists the sample depth, goodness of fit (R2), y-intercept, and the slope of the square root of the dielectric permittivity vs. water content relationships for each calibration experiment. The highly linear relationships suggest that a two-point calibration could be performed using refuse samples as collected in the field and after some period of air drying, greatly simplifying the calibration procedure.
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Fig. 5. The square root of the dielectric permittivity of refuse sample SVNO5 (20–25) vs. the volumetric water content. The regression equation and the goodness of fit (R2) of a linear regression to the data are shown.
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From Eq. [12], the slope of the square root of the dielectric permittivity vs. water content relationship can be used directly to determine the change in water content at a given location on the basis of two dielectric permittivity measurements made at two different times. All of the refuse samples were used to examine the slope of the square root of the dielectric permittivity vs. water content relationship. The slope was found to vary significantly, from a maximum of 0.1256 to a minimum of 0.0872. Considering, for example, a dielectric permittivity change from 5 to 25, these slopes would indicate a volumetric water content change ranging from 0.241 to 0.347 cm3 cm-3. These results demonstrate that, even for the relatively small landfill sampled in this study, there can be significant variation in the slope of the relationship between the square root of the dielectric permittivity and the volumetric water content among refuse samples. Therefore, it is inappropriate to use a single refuse sample for calibration, even for determining water content changes. The densities and slopes presented on Table 1 show poor correlation (R2 = 0.02). This differs from the findings of Li and Zeiss (2001), who suggested that the sample density or porosity could be used to correct variations in the dielectric permittivity vs. water content calibrations in refuse.
Rather than using a single sample for calibration, we recommend that many samples collected throughout the landfill be used to construct a generic calibration for the landfill. Such a calibration, formed using all nine dried refuse samples, gave a global calibration with a slope and intercept of 0.0998 and 0.0273, respectively, with an R2 of 0.9302 (Fig. 6). This slope and intercept differ considerably from those found by Topp et al. (1980) in soils. That is, the "universal" relationship found to describe the dependence of the dielectric permittivity of a wide variety of soils on their water content cannot be used to infer the volumetric water content of refuse based on dielectric permittivity measurements; medium-specific calibration is required. Using this calibration, the root mean square volumetric water content measurement error for all of the dried sample measurement points is 0.039 cm3 cm-3. While this shows less accuracy than is generally reported for the TDR method in soils (e.g., Topp et al., 1980), it may provide sufficient accuracy to meet some monitoring objectives. It is recommended that a similar analysis be performed on multiple refuse samples to establish the method accuracy for a specific site before applying dielectric permittivity methods to monitor the water content in municipal landfills.
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Fig. 6. The square root of the dielectric permittivity of all of the dried refuse samples vs. the volumetric water content. The regression equation and the goodness of fit (R2) of a linear regression to all of the data are shown. Each data series is labeled with the sample identification and depth of the sample, in parentheses. The calibration developed by Topp et al. (1980) is included as a dashed line.
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SUMMARY AND CONCLUSIONS
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An upward infiltration time domain transmission method is shown to be a rapid and effective means of determining the dependence of the dielectric permittivity of municipal refuse on its volumetric water content. A two-parameter, linear mixing model was found to describe the relationships measured in 19 refuse samples. The slope and intercept of this relationship differed significantly from that found for soils, demonstrating that site-specific calibration is necessary to use electromagnetic time domain methods to infer the water content of municipal refuse. In addition, there was considerable variability in both the slope and intercept of all calibration relationships determined for the refuse samples, even though the samples were collected within a small municipal landfill. As a result of this variability, large errors in estimation of water content change may result from the use of the slope of a calibration determined on a single sample. In contrast, a robust calibration relationship was found using measurements made on nine oven-dried samples. The root mean square volumetric water content measurement error that resulted from the use of this calibration was <0.04 cm3 cm-3. While this error is greater than the error associated with TDR measurements in soils, it may be sufficient to meet some landfill monitoring objectives. On the basis of our results, we recommend that multiple samples be used to construct a general calibration relationship for a site before a dielectric permittivity method, such as TDR, GPR, or TDT, is used for monitoring water content at a municipal landfill. If the expected error is acceptable for the monitoring needs, then the general calibration can be used to improve water content measurement accuracy. The highly linear relationship between the square root of the dielectric permittivity and the volumetric water content measured in this study suggests that a two-point calibration could be performed using refuse samples as collected in the field and after some period of air drying, greatly simplifying the calibration procedure.
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ACKNOWLEDGMENTS
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This material is based upon work supported by the National Science Foundation under Grant no. 0097171.
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REFERENCES
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ABSTRACT
INTRODUCTION
