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Published in Vadose Zone Journal 3:1082-1092 (2004)
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA

SPECIAL SECTION: HYDROGEOPHYSICS

Ground Penetrating Radar Measurements in a Controlled Vadose Zone

Influence of the Water Content

Olivier Loeffler and Maksim Bano*

Laboratoire Proche Surface, EOST ULP (UMR-7516), 5 rue René Descartes, 67084, Strasbourg Cedex, France
* Corresponding author (Maksim.Bano{at}eost.u-strasbg.fr)

Received 28 January 2004.



   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 DETERMINATION OF THE WATER...
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Ground penetrating radar (GPR) is a nondestructive method, which, as with other geophysical methods, has been successfully used to estimate the water content or hydraulic properties of soils. We performed GPR measurements to calibrate and compare water content estimates with actual water contents in a sand box. A vadose zone was simulated by injecting water in a sand box. We obtained four GPR data sets: for dry sand, for sand with water tables at 72- and 48-cm depths, and for sand after drainage. Using the reflections (or diffractions) from the bottom of the sand box (or objects buried in the sand), mean relative dielectric permittivities were determined at several depths in the sand box. These relative dielectric permittivities were used to calculate "real" mean relative dielectric permittivities of a sand box made up of three layers (dry sand, unsaturated sand, and fully saturated sand), knowing that a layer can be subdivided into more layers depending on the depth of the reflections (or diffractions) recorded. We used three relationships between relative dielectric permittivity and the water content to estimate the mean water content for each layer. From these water contents and the known volume of sand considered, we estimated the amount of water in the sand box for each water table. Subtracting the volume obtained for dry sand from the volume obtained for the different water tables gave estimates of the variations in water quantities in the sand box; these were compared with the quantities injected in the sand box. Despite uncertainties in the determination of the mean relative dielectric permittivities, the calculated variations in water quantities were very similar to those injected in the sand box.

Abbreviations: CMP, common midpoint • CRIM, complex refractive index method • GPR, ground penetrating radar • HBS, Hanai–Bruggeman–Sen


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 DETERMINATION OF THE WATER...
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
GROUND PENETRATING RADAR, a geophysical method based on electromagnetic wave propagation, can provide very detailed and continuous images of the subsurface (Davis and Annan, 1989; Annan, 2002). Since GPR is highly sensitive to the presence of water in the soil, the method has been used successfully in hydrological investigation to locate the water table and to delineate shallow, unconsolidated aquifers (Beres and Haeni, 1991; Van Overmeeren, 1994). Recent case studies have shown that GPR is an efficient method to estimate the water content of the subsurface by using the velocity of radar waves derived from common midpoint (CMP) profiles (Greaves et al., 1996; van Overmeeren et al., 1997; Garambois et al., 2002). The combination of borehole GPR data with conventional borehole tests significantly improves estimates of saturation and has the potential to improve estimates of the permeability and hydraulic conductivity compared with methods using only borehole data (Hubbard et al., 1997; Binley et al., 2002). Stoffregen et al. (2002) used a lysimeter to measure changes in water content for 1 yr and correlated results with GPR measurements. Dannowski and Yaramanci (1999) used reflected radar waves and geoelectrical measurements to determine the water content between the water table and the soil surface.

Ground penetrating radar reflection techniques are able to detect liquid contaminants (Brewster and Annan, 1994; Sneddon et al., 2002) and, if combined with rock physics, can be used to map the water content and salinity of sandy materials (Hagrey and Müller, 2000; Schmalz et al., 2002). Lambot et al. (2004) used an ultra-wide band stepped frequency continuous wave radar (range of 0.8–4 GHz) to estimate by inversion the water content of a homogeneous sand layer subject to different water content levels. Their results showed good accuracy for the relative dielectric permittivity compared with results based on Topp's relationship (Topp et al., 1980). However, none of these studies were performed with a time domain GPR system (pulse radar) in the case of a controlled vadose zone to compare and calibrate the water content obtained from GPR measurements with the actual water content in the soil.

In this study, using GPR measurements performed on a controlled vadose zone (in our case a sand box), we attempted to estimate the water contents from relative dielectric permittivities ({kappa} = {epsilon}/{epsilon}0, where {epsilon}0 is the dielectric permittivity of free space and {epsilon} the dielectric permittivity of the medium) at several depths. We considered that our sand box is composed of a dry sand layer, an unsaturated layer, and a fully saturated layer. The relative dielectric permittivities were calculated from average velocities derived from reflection hyperbolae on CMP profiles and/or diffraction hyperbolae due to different objects buried inside the sand. In general, the GPR velocity decreases rapidly with depth, which is primarily a result of an increasing water content with depth. To estimate the volumetric water content, we combined the GPR measurements with three relationships for the relative dielectric permittivity of the sample: a semi-empirical (two- and three-phase) complex refractive index method (CRIM) equation, the (two- and three-phase) Hanai–Bruggeman–Sen (HBS) mixing formula based on self-similar theory (Hanai, 1968; Sen et al., 1981), and the experimental Topp equation (Topp et al., 1980). We next compared results obtained using these relationships with the actual amount of water injected into the sand box. Thus, we were able to estimate the validity of the results obtained with the three relations. We show that good information can be obtained for the vertical distribution of the water content.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
DETERMINATION OF THE WATER...
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
The Sand Box Experiment and GPR Measurements
To characterize the influence of soil moisture on electromagnetic waves reflected by buried objects, we performed an experiment with a sand box. The resin box had a diameter of 2 m and was 0.98 m high (Fig. 1a) . A tap installed on the side of the box, 5 cm from its bottom, allows filling and draining of water. The water was injected by imposing a low hydraulic gradient (Fig. 1a).



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  		Fig. 1. (a) General view of the sand box with the water filling system. The resin box has a diameter of 2 m and a height of 0.98 m. (b) Plane view of the sand box with the adopted measurement grid and different objects. The depth of each object is also shown.

 
The box was filled with fine calibrated sand having diameters between 0.3 and 0.5 mm. Several objects were buried in the sand: a water-filled PVC pipe (WPVC in Fig. 1b), a steel pipe (Steel), an air-filled PVC pipe (APVC), three steel balls (P1, P2, P3), and a clay cake (A). The characteristics of the different objects (size and depth) are summarized in Table 1.


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  		Table 1. Size and depth for each object buried in the sand.

 
The GPR measurements were performed with the PulseEkko1000 system (Sensors & Software, Mississauga, ON, Canada). After comparing the results obtained with 900- and 1200-MHz shielded antennas, we chose to use the 1200-MHz monostatic antenna. This antenna had the best resolution and enough penetration to reach the bottom of the sand box (at least when dry). The antenna was moved manually, and the points of measurement were drawn on a plastic sheet to improve the accuracy of the data. We performed 71 parallel monostatic profiles (Pxx in Fig. 1b) separated by 2 cm, while the distance between measurements was also 2 cm.

We started GPR acquisition by performing measurements on the initial state of the sand, considered at this moment to be dry. In this way, we obtained the first three-dimensional GPR data set, which is shown in Fig. 2 . The in-line profiles are parallel to the x axis. We note that the time of the first breaks (first arrivals) had drifted slightly along the cross-line direction (y axis). This is due to the time drift of the system. The diffraction events visible in the figure are due to the different objects buried in the dry sand.



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  		Fig. 2. Three-dimensional GPR data set acquired for the dry sand box. The profiles are parallel to the x axis. The diffraction events visible in this figure are due to different objects buried in the sand. The reflection from the bottom of the sand box is marked by B.

 
We next injected water (340 L) up to a level of 26 cm from the bottom of the box (i.e., the water table was at a depth of 72 cm) and acquired a second three-dimensional data set. By injecting 240 L more water we subsequently raised the water table to 48 cm above the bottom of the sand box and acquired another data set. The GPR measurements were always performed in hydrostatic equilibrium (we waited about 2 wk between the injections and the measurements). Upon completion of these experiments, we allowed the water to drain for several days and performed a final set of measurements. In this way we obtained four three-dimensional data sets involving 71 parallel profiles each. For each data set, we also performed three perpendicular profiles (TA, TP, and T0 in Fig. 1b) and two CMPs using 900-MHz bistatic shielded antenna (positions shown as P1 and P2 in Fig. 1b).

During the injection and drainage operations, the sand compacted twice. After the second water injection to raise the water table, we measured a settling of about 2 cm. Additional compaction occurred during drainage. The total change in height of the sand during the experiment was approximately 3 cm.

Presentation of GPR Results
Figure 3 shows the central profile (P36) for each data set (dry, saturated, more saturated, and drained sand), which is perpendicular to the pipes. The data were processed with in-house interactive GPR software developed by Girard (2002) for use on a PC. The quality of the data was improved by removing the low-frequency component (known as the DC component) with a running average filter in time, and by applying a linear gain to the GPR data. The same parameters were used for all data to keep the relative amplitudes of the signals the same for all data sets.



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  		Fig. 3. Data acquired on the sand box (Profile P36), with different saturation levels: (a) with dry sand, (b) with a water table at the 72-cm depth, (c) with a water table at the 48-cm depth, and (d) after draining.

 
The three diffraction hyperbolae noticeable in Fig. 3 represent reflections from three buried pipes (from left to right: WPVC, Steel, and APVC pipes; see also Fig. 1b). An increase in saturation of the sand leads not only to an increase in attenuation, but also in the travel time. For example, the reflections from the pipes in Fig. 3c (saturated sand) are not as strong as those from the pipes in Fig. 3a (dry sand). Additionally, the travel time of the diffractions presented in Fig. 3c is larger than the travel time of the diffractions in Fig. 3a. This is due to the velocity of the radar waves decreasing with increasing water saturation. The images presented in Fig. 3b (water table at 72-cm depth) and Fig. 3d (drained sand) are very similar. Figure 3b and 3c, for water tables at the 72- and 48-cm depths, respectively, do not show any clear reflections from the top of the saturated zone. This might be because of the existence of a capillary fringe above the water table. The capillary fringe is the zone in which water rises by capillarity from the water table toward the surface. The degree of saturation of the capillary fringe decreases gradually upwards, and, as will be shown later, this zone extends to the surface when the water table is at the 48-cm depth. Furthermore, in Fig. 3c just below the first arrivals (capillary fringe) some reflections can be seen that are not present in the other images. It seems that an increase in water saturation leads to an increase in reflectivity (or heterogeneity) inside the capillary fringe (see Fig. 3c at about 5 ns). By contrast, no reflections are present inside the saturated zones (just above the bottom marked by the dashed line) in Fig. 3b and 3c.

The bottom of the sand box is well imaged (dashed line in Fig. 3). The reflections are clear for dry sand and more attenuated and recorded later when the water table is higher. The objects clearly disturb the travel ways of the waves—a broken distorted line is present instead of a continuous flat reflection and there is more reflection when there is no object above. For the profiles of areas without objects, we recorded the bottom as a continuous flat reflection (as will be shown in Fig. 5c and 5d). We also noticed two dipping reflections on each side of the profiles. These were caused by diffracted energy from the bottom corners of the sand box.



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  		Fig. 5. Two common midpoints (CMP) obtained with a 900-MHz antenna (a) for dry sand and (b) for the water table at 48 cm depth, and the corresponding monostatic Profile P16 performed with a 1200-MHz antenna (c) for dry sand and (d) for the water table at the 48-cm depth. B indicates the reflection from the bottom of the sand box. The CMP is positioned in the middle of the profile.

 
In Fig. 4 we present three traces (taken from Fig. 3a, dry sand) recorded over each pipe. The upper trace was obtained for the water-filled PVC pipe, the middle trace is for the steel pipe, and the bottom trace for the air-filled pipe. In this case no processing was applied to the data. The direct arrivals are very similar for the three traces. The signals marked by arrows are reflections from the pipes. The strongest reflection came from the steel pipe, which is the most conductive object. The polarity of this reflection is the same as the polarity of the reflection from the water-filled pipe. The polarity for the air-filled pipe is opposed to the others, and not so strong. This is consistent with the equation of the reflection coefficient for normal incidence of the electromagnetic waves to the boundary of two dielectric media (neglecting the magnetic permeability contrast) given by (Straton, 1941):

[1]
where {kappa}1 is the relative dielectric permittivity of the first medium (in our case sand, {kappa}s = 4.6 when dry), and {kappa}2 is the relative dielectric permittivity of the second medium (here water {kappa}w = 81 or air {kappa}a = 1). The reflection coefficient (R) obtained from Eq. [1] is negative when {kappa}2 = {kappa}w and positive when {kappa}2 = {kappa}a. Its absolute value is larger in the case of dry sand–water contact than in the case of dry sand–air.



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  		Fig. 4. Three traces (of Fig. 3a) showing reflections from three different pipes: (a) a water-filled PVC pipe, (b) a steel pipe, and (c) an air-filled PVC pipe. The black arrows indicate the strongest reflected phase.

 
Figure 5 shows two CMPs obtained with a 900-MHz antenna located at the same point P1 of Profile P16 (see position in Fig. 1b). The first CMP is for dry sand (Fig. 5a) and the second one for the most saturated case (Fig. 5b). The strong reflection marked by B is the signal coming from the bottom of the sand box. The signal is more attenuated and recorded later (about 26 ns) for a saturated layer than for dry sand (around 17 ns). We observed also a few reflections between direct arrivals and the bottom of the sand box, which are due to the buried objects located out of plane of the profile direction. We also show in Fig. 5 the monostatic profiles P16 for each case (performed with the 1200-MHz antenna). Results in Fig. 5c are for dry sand, and those in Fig. 5d are for a water table at the 48-cm depth.

Determination of the Average Relative Dielectric Permittivity
Relative dielectric permittivities were calculated from average velocities derived from the direct ground wave and the reflection hyperbolae in the CMP profiles on the one hand, and from the diffraction hyperbolae (due to different objects buried inside the sand) observed on monostatic profiles on the other hand. By using the techniques based on the curvature of reflections and diffractions hyperbolae (shown by the CMPs and monostatic profiles) we determined the average velocities at several depths in the sand box. For example, the diffractions from the steel ball P3 (at the 38-cm depth) and from the three pipes (APVC, WPVC, and Steel at 50 cm) allowed us to estimate the average velocities from the surface to 38- and 50-cm depths, respectively. Only in the case of dry sand did we observe diffractions from the deepest steel balls (P1 and P2 at the 68-cm depth), which helped us to determine the average velocity from the surface to the 68-cm depth. The reflection hyperbolae (observed in the CMPs) from the bottom of the box were next used to determine the average velocity applicable to the entire sand box. Finally, the dips in the direct ground wave (observed in the CMPs) could also be used to determine a velocity on the surface of the sand. All of these methods gave similar velocities at the same depth, with a mean precision on the velocities between ± 0.01 and ± 0.02 m ns–1.

In Fig. 6 we present the same GPR data as in Fig. 5 superimposed with modeled reflections from the bottom of the sand box (black curves). The best fit between the modeled reflections and observed hyperbolae was found for velocities of 0.116 m ns–1 (dry sand, Fig. 6a) and 0.075 m ns–1 (highest saturation level, Fig. 6b). It is worth noting here that the precision of the estimated velocities is much higher for the case of a low-velocity medium (high curvature of hyperbola, Fig. 6b) than for a high-velocity medium (low curvature of hyperbola, Fig. 6a). The dashed dipping lines present the modeled direct ground waves using velocities of 0.14 m ns–1 (Fig. 6a) and 0.12 m ns–1 (Fig. 6b), respectively. The velocities estimated by using the direct ground waves in fact represent velocities from the surface of the sand. The average velocities (from the surface to the buried objects/or to the bottom of the sand box) estimated from the GPR data varied between 0.14 m ns–1 for dry sand and 0.075 m ns–1 when the water table is at the 48-cm depth (i.e., with the highest quantity of water inside the sand box: 580 L).



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  		Fig. 6. Modeling of reflections of the common midpoints (CMP) from Fig. 5 (a) for dry sand and (b) with a water table at the 48-cm depth. The black dashed line is for direct arrivals from the surface, and the continuous black lines are calculated reflections from the bottom of the sand box.

 
Average relative dielectric permittivities (from the surface to a certain depth), {kappa}, of the sand at several depths were calculated by using the relation {kappa}1/2 = c/V, where c is the velocity of the electromagnetic waves in free space (c = 0.3 m ns–1) and V is the velocity of the radar waves in the sand, estimated from the GPR data. The results for each data set are summarized in Table 2.


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  		Table 2. Average relative dielectric permittivities at several depths in the sand for each data set obtained from the velocities of GPR waves. The estimates are made from the surface to the given depth.

 
Relative dielectric permittivities after drainage decreased and then increased with depth (see Table 2). This is due to nonlinear variations of the water content in the sand box. A schematic diagram of the saturation profile versus depth is shown on Fig. 7 .



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  		Fig. 7. Schematic representation of the water content profile after drainage. Sw is the degree of saturation of the soil. At depth b (bottom of the sand box), the sand is considered to be fully saturated (Sw = 1).

 

   DETERMINATION OF THE WATER SATURATION OF THE SAND
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 DETERMINATION OF THE WATER...
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Determination of "Real" Relative Dielectric Permittivity by Layers
The values of the relative dielectric permittivity estimated previously are average values from the surface to a given depth. We can divide our sand box into three layers: a dry layer with constant permittivity (near the surface), an unsaturated layer (capillary fringe, above the water table) where the permittivity varies with water content (the permittivity increases from the base of the dry layer to the water table), and finally a fully saturated sand layer where the permittivity is again constant (above the bottom of the sand box). The thicknesses of the saturated layers are known: 26 and 48 cm (i.e., for water tables at 72 and 48 cm, respectively).

The thickness h of the capillary rise of water in a tube is given by the relation:

[2]
where {gamma} is surface tension (for water, {gamma} = 0.072 N m–1), {theta} is the angle of contact between the meniscus and the wall of the tube (for air and water, {theta} = 0°), r is the radius of the meniscus, g is the gravity constant (9.81 m s–2), and {rho} the density of water (1000 kg m–3). In our case we do not know the radius r of the meniscus. Packwood (1983) gave r = d{phi}/2, which relates the radius of the meniscus r to the porosity {phi} and to the mean grain diameter d. Taking d = 4 x 10–4 m and {phi} = 0.42 in Packwood's relation, we obtain r = 8.4 x 10–5 m, and by using Eq. [2] we find h = 17.5 cm. Mavis and Tsui (1939) proposed the following equation to determine the maximum capillary rise h (m) from the porosity {phi} and the mean grain diameter d (m):

[3]

In our case, {phi} = 0.42 and d = 4 x 10–4 m, and hence h = 17.3 cm, which is consistent with the result found by combining Eq. [2] with Packwood's relation. For our sand box, the mean velocity at the 50-cm depth when the water table is at 72 cm is lower than the velocity for the dry case. This is due to the influence of the capillary fringe, which rose from the 72-cm depth to at least the 50-cm depth. The mean velocity from the surface to the 38-cm depth was not affected. Hence, the capillary fringe can have a height from 22 to 34 cm. For the most saturated case, the velocity is lower than for the dry case, regardless of the depth considered. The capillary fringe affects the layer from the 48-cm depth to the surface. Consequently, we fixed the thickness of the capillary fringe at 29 cm, which is larger than the one given by Todd (1980) for sandy soils (around 24.6 cm) and close to the value of 30 cm used by Gloaguen et al. (2001). The thicknesses of the dry layers varied with saturation: 68 cm with no saturation (no water injected), 43 cm for the second case, and 19 cm for the most saturated case.

To estimate the "real" relative dielectric permittivity for each layer, we used the following formula (Chan and Knight, 1999):

[4]
where {kappa}m is the average relative dielectric permittivity (estimated previously), and {kappa}l and {theta}l are the relative dielectric permittivity and the volumetric fraction, respectively, of the lth layer. The volumetric fraction of the lth layer is the ratio of the volume from the lth layer to the total volume of the sand box. Chan and Knight (2001) demonstrated the validity of Eq. [4] for a ratio of the wavelength ({lambda}) to the thickness of the layer (h) of <3. For a medium with velocities (V) varying from 0.14 m ns–1 (dry sand) to 0.075 m ns–1 (saturated sand), and a dominant frequency (fd) of 1100 MHz (determined from the spectrum of the traces at the highest saturation levels), the dominant wavelength ({lambda}d = V/fd) varied from 12.7 to 6.8 cm. Therefore, the ratio wavelength/thickness is in our case always <3, which justifies the use of Eq. [4]. The real permittivities are summarized in Table 3.


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  		Table 3. Comparison of average relative dielectric permittivities of the sand box and relative dielectric permittivities calculated using Eq. [4].

 
Relative Dielectric Permittivity of Moist Soils
The volumetric water content {theta}w is the ratio of the volume of water (Vw) present in a sample to the total volume (Vtotal) of the sample ({theta}w = Vw/Vtotal). The water content is related to the degree of water saturation Sw and the porosity {phi} of the medium by {theta}w = {phi}Sw. Saturation varied between 0 to 1, and hence the water content from 0 to {phi}. The relative dielectric permittivity for moist soils increases with increasing water content and is generally within the range 6 to 30. This is because the relative dielectric permittivity of water ({kappa}w = 81) is much larger than that of dry soils (3–5). Many mixing formulas for the bulk permittivity of moist soils have been reported in the literature. Topp et al. (1980) determined the following empirical relationship between the relative dielectric permittivity of the sample and the water content {theta}w:

[5]

They proposed a reciprocal formula to estimate the water content from the relative dielectric permittivity {kappa}:

[6]

Another relationship, analogous to the time average relation of Wyllie (used to predict the acoustic velocity in porous media), is the CRIM. For a three-phase unsaturated medium (water, air, and sand), the relative dielectric permittivity {kappa} of the medium is related to water saturation Sw, porosity {phi}, and the relative dielectric permittivities of water, air, and the solid phase ({kappa}w, {kappa}a, and {kappa}g). Mavko et al. (1998) gave the following equation for {kappa}:

[7]

Even if {kappa}w and {kappa}g are known, it is impossible in practice to derive both the sample porosity {phi} and saturation Sw from the relative dielectric permittivity of the sample. Thus, knowledge of either {phi} or Sw is necessary to estimate the water content ({theta}w) from the relative dielectric permittivity of the sample using Eq. [7]. In the case of a fully saturated medium (Sw = 1 or {phi} = {theta}w), the CRIM Eq. [7] becomes

[8]

However, in the case of the CRIM Eq. [7] and (8), we can specify only the volume fractions and the constituent relative dielectric permittivities without accounting for the geometry of internal structures of rocks and microscopic fluid distributions. Endres and Knight (1992) showed that these features have significant effects on the dielectric properties of unsaturated rocks. To partially resolve this problem, the HBS mixing formula, based on self-similar theory (Hanai, 1968; Sen et al., 1981), may be used to estimate the porosity of water-saturated rock. For a two-phase mixture of grains (sand) and water, the porosity is given by

[9]
where the cementation index m is a function of the grain shape and varies between 1.5 for unconsolidated sands with spherical grain shapes and 2 for sandstones with oblate grain shapes. A value of 2.56 for m was found to be the best fit between the measured data and HBS (Eq. [9]) for a sand–clay mixture saturated with water (Carcione et al., 2003). Samstag and Morgan (1991) used the HBS relation twice for unsaturated media (see also Greaves et al., 1996). They first determined the pore relative dielectric permittivity {kappa}p obtained by mixing air and water and then calculated the relative dielectric permittivity of rock {kappa} by mixing {kappa}p with the relative dielectric permittivity of mineral {kappa}g. Thus, we have the following two relationships:

[10]
and

[11]

The cementation factor m1 in Eq. [10] is related to the microscopic shape of the air phase and may vary with saturation (Endres and Knight, 1992), while m2 is related to the shape of the mineral grains.


   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 DETERMINATION OF THE WATER...
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
To calculate the water content, we use Eq. [6] through [11] with the relative dielectric permittivities given in Table 3. For the HBS equation, we first determined the relative dielectric permittivity of the pores {kappa}p from Eq. [11]. The degree of water saturation was next calculated using Eq. [10]. The following parameters were used:

In Table 4 we compare water contents (%) obtained using Eq. [6] through [11]. Equations [8] and [9] are used only for the saturated layers, while Eq. [7], [10], and [11] are used for unsaturated media. It is worth noting here that Eq. [6] does not depend on the degree of saturation of the medium. This actually is an advantage of Eq. [6], which can be used irrespective of the degree of the saturation of the medium (i.e., fully or partially saturated).


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  		Table 4. Water contents estimated with Eq. [6] through [11].{dagger}

 
Very similar results were obtained with the different equations. The Topp and CRIM (Eq. [6], [7], and [8]) gave nearly the same results, while the HBS formulae gave lower values for the water content. For the lowest water saturation level (water table at the 72-cm depth), we found that the water content was larger than the porosity of the saturated layer. For example, with CRIM (Eq. [8]), we found {theta}w = 0.449, while the porosity {phi} was 0.42. The Topp equation produced the best estimates for the water content in this case. Results for the second saturation case were better, with the water content being very close to the porosity (in this case, {phi} = 0.39, after settling of the sand), although still overestimated with the CRIM Eq. [8]. Results obtained with the HBS Eq. [9] for the saturated layers were very similar to those obtained with the Topp equation. The Topp and CRIM equations (Eq. [6] and [7], respectively) produced very similar results for the unsaturated or dry layers, whereas the HBS Eq. [10] and [11] underestimated the water contents for these cases.

Values obtained for the dry sand ({theta}w = 0.07) for the first and second data sets were consistent with values found by Hagrey et al. (1999), who measured water contents of a sand box at several depths using TDR, and compared those with GPR velocities. For a velocity of 0.143 m ns–1 for dry sand (i.e., {kappa} = 4.4), they found a water content of 7%.

Using the water contents as calculated above, and knowing the volumetric fraction for the layers considered in each case, we can now calculate the volume of water present in the sand box for each saturation case. Since the water contents varied according to each invoked calibration, the volume of water present in the sand box will also depend on the equation being used. Results are shown in Table 5. Notice that the HBS equation gave the lowest estimates in each case.


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  		Table 5. Water quantities present in the sand box (for different saturation cases) as obtained using the Topp, complex refractive index method (CRIM), and Hanai–Bruggeman–Sen (HBS) equations.

 
By subtracting the volume for the case of dry sand from the volumes when the water tables were at the 72- and 48-cm depths, we found estimates of the amounts of water injected into the sandbox (see Table 6). For the water table at the 72-cm depth, we found 284 L (Topp), 305 L (HBS), and 314 L (CRIM) of water, instead of the 340 L injected. For the second saturation case (water table at the 48-cm depth), we obtained 484 L (Topp), 506 L (HBS), or 520 L (CRIM), instead of the 580 L injected. Hence, using GPR in each case we underestimated the amounts of water in the sand box.


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  		Table 6. Estimates of the amounts of water injected in the sand box (for different saturation cases) as obtained using the Topp, complex refractive index method (CRIM), and Hanai–Bruggeman–Sen (HBS) equations. V1 is the amount of water for the data set with the water table at the 72-cm depth minus that of the dry sand case; V2 is the amount of water for the data set with the water table at the 48-cm depth minus the amount of water for the dry sand case.

 
A major source of uncertainty stems from estimates of the velocity derived from the curvature of the reflection and/or diffraction hyperbolae. A low velocity (high curvature of the hyperbola; see Fig. 6b) is much easier to estimate reliably than a high velocity (low curvature of hyperbola; see Fig. 6a). Consequently, the error in estimating the velocity for the GPR data of the dry sand was larger than for the other two cases (water tables 26 and 48 cm above the bottom). Because of this error, the velocities for the case of dry sand were underestimated more severely than the velocities for the other cases. Water contents of the dry sand hence were more overestimated than the water contents of the other cases. This explains the lower amount of water found in each case after subtraction of the water content of the dry sand. However, the water quantities found for different saturation states (by using Eq. [6] through [11]) were still relatively close to the amounts of water injected, with the best result obtained with CRIM relation (Table 6).


   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 DETERMINATION OF THE WATER...
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
The work presented here shows that GPR is an effective method to assess and monitor vadose zone water contents. By repeating the same GPR measurements over a controlled vadose zone (sand box experiment), one can compare and calibrate the water content obtained from GPR measurements with the actual water content present in the soil. In this study we attempted to relate the GPR velocities (obtained from reflections and diffractions hyperbolae) with theoretical and laboratory relationships (the CRIM, HBS, and Topp equations) between the water content and the relative dielectric permittivity of the sample.

Despite uncertainties in estimates of the velocity derived from the curvature of the reflection and/or diffraction hyperbolae in the case of the dry sand, the final results for the amounts of water found were very close to the amount of water injected. The CRIM calibration curve produced the best result. A next logical step would be to extend the proposed method to radar data for contaminated media and to combine this with suitable laboratory measurements.


   ACKNOWLEDGMENTS
 
This research was financed by the European Community under the HYGEIA (HYbrid Geophysical technology for the Evaluation of Insidious contaminated Areas) project. The PulseEKKO 1000 GPR system was funded by Université Louis Pasteur and INSU/CNRS program. We thank two anonymous reviewers as well as Associate Editor Ty Ferre and Editor Rien van Genuchten for their constructive comments and helpful suggestions that were much appreciated. This paper constitutes EOST contribution 2004.07-UMR7516.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 DETERMINATION OF THE WATER...
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 




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