We simulate two levels of water table (at 72 and 48 cm depth) by injecting water in a sand box that also contains several buried objects. GPR profiles acquired with a 1200 MHz antenna at the top of the sand box do not show any clear reflections from the water table. This is because of the existence of a ‘transition zone’ in which the velocity is a continuously decreasing function of depth. The reflection coefficient in this case decreases with increasing frequency and even vanishes for a cut-off frequency f0 which itself increases with decreasing transition zone thickness. By modeling in the frequency domain, we explain the absence of the high-frequency GPR reflections from the top of the saturated zone. When the wavelength is small (high frequency) compared to the thickness of the transition layer, the reflection coefficient is negligible and hence no reflections from the water table will be observed.
 Ground penetrating radar (GPR), a geophysical method based on electromagnetic (EM) wave propagation, can provide very detailed and continuous images of the subsurface [e.g., Davis and Annan, 1989; Greaves et al., 1996; Van Overmeeren, 1998]. Since GPR waves are highly sensitive to the presence of water in the subsurface, the method has been successfully used in hydrological investigations to locate the water table and to delineate shallow, unconsolidated aquifers [Beres and Haeni, 1991; Tsoflias et al., 2001]. Good reviews of GPR applications to hydrogeological studies are given by Huisman et al.  and Annan . Examples showing GPR reflections from the water table are given by Van Overmeeren . Endres et al.  and Bevan et al. , have used GPR profiling to image the temporal and spatial response of an unconfined aquifer during a pumping test. In all these studies the maximum frequency used to image the water table was 200 MHz. Loeffler and Bano  have observed that high-frequency (900 and 1200 MHz) GPR profiles acquired on a sand box do not show any clear reflections from the top of the saturated zone. This is because of the existence of a capillary fringe and a transition zone located above the water table.
 The capillary fringe is the nearly-saturated zone right above the water table in which water rises by capillarity from the water table toward the surface [de Marsily, 1986]. Daniels et al. , using a 500 MHz antenna, have shown GPR reflections from the top of the capillary fringe. Above this zone the saturation decreases gradually upward implying an increase of the velocity of EM waves. This is the “transition zone” which thickness depends on the pore-size distribution and pore shapes of the medium. The finer the grain sizes, the more important the thickness of the transition zone [Bear, 1972]. Bevan et al. [2003, 2005], during a pumping and recovery test, have shown that this zone migrates vertically without being altered in thickness or shape.
 The propagation of EM waves in inhomogeneous dielectric media has widely been treated by several authors [e.g., Hashish, 2003; Bass and Resnick, 2004]. In order to design thin-layer filters for microwaves, Bilotti et al. [1999, 2002] have investigated the propagation of EM waves for the case of a non-homogeneous dielectric layer between two homogenous half-spaces.
 We used a 1D wavelet modeling method in the frequency domain [Bano and Girard, 2001; Bano, 2004] and considered different frequencies from 100 to 1000 MHz. For each frequency the simulation is performed by using two models: i) with a transition zone such that the velocity of EM waves decreases linearly with depth and ii) without a transition zone. Afterward we compare the results obtained for each model and discuss the reflection coefficient of the transition zone. Examples of real and synthetic GPR data will be shown to illustrate the methodology presented here. Our analytical mathematical expression for the transition zone, combined with EM modeling gives hydrogeophysicists the means to simply analyze any transition zone and antenna combination. More complex formulas [Van Genuchten, 1980; Annan et al., 1991] can be combined with the frequency modeling method (or any Finite-Difference Time-Domain scheme) but they will not change the conclusions drawn here.
2. Real Data Examples
Figure 1 is a 100 MHz GPR profile located near an exposed deposit of a gravel-pit trench in which the water table was at 5 m depth (Rhine valley in eastern France). The mode of the acquisition was in common offset configuration with 1 m between antennas and a trace spacing of 0.25 m. The strong reflection, at about 100 ns, is from the top of the saturated zone, while the diffraction event (with its apex at about 35 ns and 13 m horizontally) is due to a pipe perpendicular to the profile direction.
 We ran two experiments to simulate a water table (water level at 72 and 48 cm depth respectively) by injecting water from the bottom of a sand box that contains three buried pipes at 48 cm depth (a water-filled PVC pipe, a steel pipe and an air-filled PVC pipe). The height and the diameter of the resin box were 0.98 m and 2 m, respectively. Details of this sand box experiment were presented by Loeffler and Bano . The box was filled with fine calibrated sand having diameters between 0.3 and 0.5 mm. GPR profiles (with 1200 MHz monostatic antenna) are performed in order to estimate the depth of the water level. The diffraction events visible in Figure 2 are due to different objects buried in the sand. The presence of the reflections from the bottom (dashed lines) and diffractions events (from buried pipes) show the absence of the attenuation in the wet sand. The GPR data of Figure 2, for the water tables at the 72 and 48 cm depth, respectively, do not show any clear reflections from the top of the saturated zone. In consequence, for lossless media, this might be because of the existence of a capillary fringe and a transition zone both above the water table.
3. Capillary Rise and “Transition Zone”
 Above the level of the water table there is first a nearly-saturated zone, the capillary fringe [Bear, 1972; de Marsily, 1982]. The thickness hc of the capillary rise of water in a tube is given by the relation:
Where γ is surface tension (for water, γ = 0.072 N/m), θ is the angle of contact between the meniscus and the wall of tube (for air and water, θ = 0°), r is the radius of the meniscus, g is the gravity constant (9.81 m/s2) and ρ the density of water (1000 kg/m3). In our case, we do not know the radius r of the meniscus. Packwood  gives r = dϕ/2 linking the radius of the meniscus r to the porosity ϕ and to the mean grain diameter d. Taking d = 4 × 10−4 m and ϕ = 0.4 in Packwood's relation, we obtain r = 8.0 × 10−5 m and by using equation (1) we find h = 18.3 cm.
 Above the capillary rise, the saturation decreases gradually upward until it reaches the residual saturation, this is the ‘transition zone’. A schematic presentation of the retention curve (saturation profile) in a soil is shown in Figure 3. When the profile is in hydrostatic equilibrium with the saturated zone, the transition zone can be expressed analytically by the characteristic water retention function of the soil. Most authors use the model of Van Genuchten  to describe the saturation profile of the transition zone. The water table and capillary fringe, together, form the saturated zone which is susceptible to be imaged by GPR reflections; hence the top of the capillary fringe is indeed the top of the saturated zone which is often different from the hydraulic water table defined in an observation well. Endres et al.  and Bevan et al. , during a pumping test, have observed GPR reflections which are delayed relative to the water table directly measured from piezometers. They have referred to this GPR event as the transition zone reflection, while Bentley and Trenhlom  defined it as capillary fringe reflection. Throughout this paper we refer to saturated zone (top of the capillary fringe) simply as water table.
 As for the capillary fringe, the thickness (h) of the transition zone depends on the pore-size distribution and pore shapes of the medium. In hydrostatic equilibrium, it is more important for fine-textured soil than for coarse-textured soil. For a given sandy soil the thickness and the shape of the transition zone are not affected by the depth of the water table. Endres et al.  and Bevan et al. [2003, 2005], in their pumping test, have observed a thickness of 40 cm for the transition zone. For fine calibrated sand having diameters between 0.3 mm and 0.5 mm this zone extends to the surface when the water table is at 48 cm depth [Loeffler and Bano, 2004] which, for a capillary rise of 18 cm, gives a thickness h = 30 cm, a value we will keep for the following analyses.
4. Modeling Radar Reflections From the Water Table
 To model the reflections from a water table we consider a lossless medium composed of a dry sand layer (upper layer with residual saturation of 0.1; homogeneous layer), an unsaturated layer (transition zone of 30 cm thick; inhomogeneous layer) of velocity changing linearly with depth and a fully-saturated layer (capillary fringe and water table with saturation of 1; homogeneous layer).
 The velocities (V1 > V3) of the first (three-phase: sand, water and air) and third (two- phase: sand and water) layers are considered to be constant and calculated from the dielectric constants of the layers which are assumed to follow the CRIM (Complex Refractive Index Method) relationship. The porosity of the sand was ϕ = 40%, the values of the dielectric constants for the sand (κs) and water were taken as 4.5 and 81, which gives κ1 = 3.97 and κ3 = 23.74. Hence, we have for V1 and V3 the values of 0.1505 and 0.0615 m/ns, respectively.
 The propagation of electromagnetic waves for a non-homogeneous dielectric layer between two homogenous half spaces has been investigated by several authors, for example, Bilotti et al. [1999, 2002]. In order to design thin-layer filters for microwaves, they proposed an approximated explicit expression for the reflection coefficient of this model. We consider a model for the transition zone, in which the velocity decreases linearly (with depth) from V1 to V3 (see Figure 4). For that special case we choose here to consider the expression of the reflection coefficient R (at vertical incidence) given by Wolf  and Officer :
With m = 2, and a = the velocity gradient.
 On the other hand we consider another three-layer model of constant velocities V1, Vaverage = (V1 + V3)/2 and V3, respectively, shown in Figure 4b. The middle layer, in this figure, has the same thickness h = 30 cm as the transition zone. The results of the Fourier modeling, using two different frequencies 100 and 1000 MHz respectively, are shown in Figure 5. For the 100 MHz source the water table (top of capillary fringe) was located at 8 m depth, while for the 1000 MHz source it was at 0.8 m depth. Owing to this choice, we keep the same intrinsic attenuation (the product of multiplication between frequency and travel time is constant), consequently the differences observed on both reflected wavelets (frequencies) are due only to the reflection coefficient of the transition zone. This justifies our choice and the use of different time scales in Figure 5. In our analysis we recognize that two factors such as intrinsic and geometric attenuation [Bano, 2004] are not being accounted for. Both are wave propagation phenomena and their effect, the same on both models (with and without transition zone), will not affect our observations. We remark that for a 100 MHz source (2nd order Ricker wavelet) the strength of the reflections is nearly the same for both models, while for a 1000 MHz source the reflections for the case of the transition zone are very small (mostly invisible) compared to the reflections of three-layer model of constant velocities (Figure 5d). This is because the reflection coefficient of the transition zone is a function of the wavelength (frequency) of the incident waves. When the wavelength is very long compared to the thickness of the transition layer, the presence of the latter is clearly of no importance (see also Figure 6).
5. Discussion on Reflection Coefficient R(ω) of the Transition Zone
Figure 6a shows the modulus of the reflection coefficient R(ω) as a function of frequency. In the limit as ω ⇒ 0 or h ⇒ 0, R ⇒ (V3 − V1)/(V3 + V1). As frequency increases from zero, the modulus of the reflection coefficient decreases and we have found that it becomes zero at
Where Vaverage = (V3 + V1)/2 is the average velocity of the transition zone. The modulus of R(ω) then goes through a series of zeros for frequencies fn = nf0 (n = 1, 2, …) with decreasing amplitude between the zeros. It looks like the modulus of a ‘sinc’ function whose zeros are for the same frequencies (fn), which is given by:
The modulus of the sinc function is shown in Figure 6b.
 The frequency f0 is sort of a cut-off frequency and it increases with decreasing h. In our case, for h = 30 cm, its value is f0 = 166 MHz. For frequencies lower than f0 the transition zone will be of no importance and the saturated zone (water table) will be well imaged, this is what we see in Figure 5a, while for frequencies larger than f0 the reflection coefficient is negligible and no (or only very small) reflections will come from the water table, as shown in Figure 5b. Therefore, in order to image the water table, the dominant frequency (central frequency) fd of the antennas must verify the following equation:
where λd the dominant wavelength of the antenna estimated for the average velocity (Vaverage) of the transition zone. Our choice of equation (2) corresponds to a permittivity profile of the form:
This is a square hyperbolic permittivity profile with c = 0.3 m/ns, that is, the velocity of EM waves of the free space. We have combined equation (6) with the reflection coefficient (which has a sinc function form) given by Bilotti et al.  and found that the cut-off frequency in that case has the following form:
which gives exactly the same value for the cut-off frequency (166 MHz) as in equation (3). On the other hand we have calculated (for κ1 = 3.97, κ3 = 23.74 and h = 0.3 m) the cut-off frequencies for the permittivity profiles used by Bilotti et al. [1999, Table 2] and found frequencies of 138 and 145 MHz respectively for κ(z) = b(z + z0)m (m = 1, 2). In that case, the larger the integer m, the higher the cut-off frequency, while for κ(z) = b/(z + z0)n (n = 1, 3) we have values of 176 and 162 MHz respectively, thus, the larger the integer n, the smaller the cut-off frequency. Here, b and z0 are two real constants. From this analysis we can see that the cut-off frequency depends on the shape of the transition zone, but that it does not vary greatly for different permittivity profiles. One can easily verify that the maximum value (f0max = 176 MHz) of the cut-off frequency has exactly the same value as the upper limit of criterion in equation (5): f0max = (Vaverage)/2h. Consequently, the dominant frequency of the antennae (fd) should be chosen to be less than the minimum value (f0min) of the cut-off frequency. Since the cut-off frequency given by equation (3) or (7) (f0 = 166 MHz) and the minimum value f0min = 138 MHz are quite close (17%), criterion (5) can be written:
with f0 the cut-off frequency given by equation (3) or (7). This is the λd/2 criterion shown in equation (5), except for the sign “less” which has now been replaced by “much less”. This criterion is independent on the shape of the transition zone and is different from the vertical resolution criterion which is λd/4 for a homogeneous layer.
Annan et al.  have assumed a “cosine-type” water content variation and used a numerical modeling approach with the transition zone approximated by thin layers. They have shown [Annan et al., 1991, Figure 6] that the reflection amplitude decreases with increasing the thickness of the transition zone and the water table is most detectable with GPR if the ratio of the thickness of the transition zone to the pulse length is less than 0.3. Their conclusions are in good agreement with our criterion in equation (5).
 We have observed, on a sand-box experiment, that GPR frequencies as high as 1200 MHz do not show any clear reflections from the water table. Assuming, on hydrological grounds, that there is a ‘transition zone’ in which the velocity decreases linearly with depth, we show the importance of this zone to the reflection coefficient and, using wavelet modeling, we explain the absence of high-frequency GPR reflections from the water table. The strength of the GPR reflections coming from the transition zone decreases with increasing frequency and/or thickness of the transition zone. The work presented here shows that when the dominant wavelength (λd) of the GPR antenna is very long compared to the thickness (λd/2 ≫ h) of the transition layer, its presence is clearly of no importance and the water table will be well imaged. However, for small wavelengths (λd/2 < h), the transition layer reflection coefficient is negligible and no/small reflections will be seen from the water table. The analytical approach proposed here combined with EM modeling gives a means to simply analyze any transition zone and antenna combination. Furthermore, we have shown that the λd/2 criterion is always valid whatever the shape of the transition zone (permittivity profile) to be used in the forward modeling. It has been pointed out also that the classical vertical resolution λd/4 criterion for a homogeneous layer is replaced by the λd/2 criterion for an inhomogeneous layer.
 I would like to thank three anonymous reviewers for their constructive comments and helpful suggestions that were much appreciated. The first reviewer provided me with some useful references of EM wave's propagation in thin dielectric layers.