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Published in Vadose Zone Journal 3:288-299 (2004)
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA

ORIGINAL RESEARCH PAPERS

Acoustic Tomography Applied to Water Flow in Unsaturated Soils

Andreas Blum*,a, Ivo Flammera, Thomas Friedlib and Peter Germanna

a Institute of Geography of the University of Bern, Hallerstr.12, CH-3012 Bern, Switzerland
b Department of Mathematical Statistics and Actuarial Science, Sidlerstr. 5, CH-3012 Bern, Switzerland
* Corresponding author (andiblumi{at}aol.com).

Received 15 October 2002.



   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES
 
The water content of unsaturated soils influences soil properties like shear strength, elasticity, and the velocity of acoustic waves. They are also suited to investigate the spatial–temporal variations of soil water. In this study we present noninvasive two-dimensional tomography based on acoustic pulse transmission with a spatial resolution of 60 by 60 mm. Ten acoustic transducers were placed in a horizontal plane along the wall of a soil column of an undisturbed loess soil. Acoustic pulses were emitted into the soil and monitored after traveling through the soil column. Two infiltration experiments followed by water redistribution were performed while acoustic signals were monitored at 60-s intervals. The water content was measured with a time domain reflectometer (TDR). Travel times of the acoustic signals were converted to acoustic velocity distributions using a tomography algorithm. The acoustic velocities were transformed to water contents, resulting in 21 time series. Some of the measurements indicated fast (i.e., preferential) flow paths, whose positions varied between the two infiltration experiments.

Abbreviations: TDR, time domain reflectometry


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES
 
WATER FLOW in unsaturated soils occurs by capillary and preferential flow. Capillary flow is slow and uniformly distributed. Preferential flow is much faster, and flow paths are restricted to small areas. The theory of capillary flow is based on the Richards (1931) equation, and ignores preferential flow. Germann and Di Pietro (1996)( 1999), among others, provide theoretical and experimental approaches to preferential flow.

Measurements of soil moisture variations must be fast and simultaneously provide for adequate spatial resolution to distinguish between preferential and capillary flow. For example, soil moisture tracking with TDR equipment or tracer measurements provide adequate temporal or spatial resolution. Tomography, on the other hand, is a well-known noninvasive technique in geophysical, biological, and medical research that produces images of internal structures, and may simultaneously measure spatial and temporal variations in properties.

During the last few decades numerous tomography techniques were developed based on the propagation of various signals like X-rays (Jacobs et al.,1995), magnetic resonance (Amin et al., 1998), electric resistivity (Binley et al., 1997; Zhou et al., 2001), and acoustic waves (Bording et al., 1987; Michelena and Harris, 1991; Dietrich et al., 1995).

Radon's (1917) transformation of line integrals provides the basic mathematics of tomography applied to straight rays. It defines a two-dimensional function f(x,y) through the line projections integrated over all angles. However, if considerable heterogeneity exits in the internal structure, such as in geologic formations or soils, the assumptions of wave propagation along straight lines is no longer valid because of signal diffraction. To remedy this problem computer codes have been developed that account for wave diffraction. Worthington (1984) and Phillips and Fehler (1991) summarized the most common strategies of ray path tomography. They distinguished between matrix inversion (Aki and Richards, 1980), Fourier transform methods (Merserau and Oppenheimer, 1974), convolutional or filtered back-projection methods (Robinson, 1982), and iterative methods (Gilbert, 1972; Dines and Lytle, 1979; Krajewski et al., 1989).

Berkenhagen et al. (1998) and Flammer et al. (2001) successfully applied acoustic waves to soil moisture variations. They found that the velocity of an acoustic wave is related to the elasticity modulus and the total density of the medium. According to Brutsaert (1964), the impact of the elasticity modulus on the acoustic velocity exceeds the soil density influence. Brutsaert (1964) also provided a relationship between acoustic velocities and water content. Here we explore the use of acoustic tomography to measure temporal and spatial soil moisture variations in a column of an undisturbed soil during and shortly after transient infiltration events.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Experiment Set-Up
Figure 1 shows the experimental set-up. Ten transducers were placed in a horizontal plane along the wall of a soil column. The transducers acted as acoustic sources and receivers. A relay circuit separated the source transducer from the receiving transducers and switched between source transducers. A pulse generator, fed by a high voltage power supply, created the electric pulses. The source transducers transferred the electrical to acoustic pulses. The acoustic-to-electrical signals received by the other transducers were amplified and stored on a PC. A sprinkler on top of the soil column simulated rainfall at preset rates and durations, and a pair of TDR wave guides were used to measure soil moisture variations.



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  		Fig. 1. Experimental set-up of acoustic tomography and rainfall simulator.

 
Transducers
A piezo-electrical transducer consisted of four piezo-rings, a front panel, and a seismic mass (Fig. 2) . The seismic mass exerts a force on the piezo-rings during reception of the acoustic signal, thus generating an electrical signal. The piezo-rings were 4 mm thick and had outer and inner diameters of 25 and 10 mm, respectively. To increase the strain of the transducer, the four piezo-rings were shunted according to Waanders (1991). The front panel consisted of a stepped brass cylinder. Its front, having a diameter equal to the piezo-rings, was in contact with the soil (contact area), while its back was used to press the transducer against the soil. A pressure of about 0.1 MPa (1 bar) was applied to improve stability and acoustic contact between soil and transducers. The seismic mass behind the rings consisted of a steel cylinder with a diameter of 37 mm and a length of 77 mm. It was pressed against the piezo-rings by means of a screw mounted on the stepped brass cylinder, and guided through the inside of the piezo-rings and the seismic mass. Various PVC rings and tubes were incorporated to prevent electrical short circuiting.



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  		Fig. 2. Schematic view of a transducer.

 
Mounting of the Transducers
The axes of the 10 transducers pointed to the center of the soil column, with angles of 36° between their axes. The horizontal acoustic measurement plane was 0.27 m below the soil surface. The transducers were fixed to the soil surface with a mounting rack. Figure 3a shows the complete arrangement of the mounting rack, and Fig. 3b and 3c present the details of the guide rails. The mounting rack consisted of a U-shaped aluminum ring with an inner diameter of 0.8 m. The aluminum ring was mounted on four legs while 10 guide rails (Fig. 3b) were attached to the aluminum ring with adjusting screws (Fig. 3a and 3c). A shaft with a groove was guided through each guide rail. A PVC cylinder guided the transducer axially onto the soil column along the shaft. An adjusting screw was used to connect the sleeve with the shaft. Two springs with a spring constant of 1050 N m–1 controlled the pressure of the transducer against the soil which was about 0.1 MPa (1 bar).



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  		Fig. 3. (a) Mounting rack, (b) transducer rack, and (c) guide rail.

 
Electrical Circuit
The electrical circuit consisted of two parts. A pulse generator that produced the electrical-to-acoustic pulse, and a relay circuit that managed the electrical currents of the transducers. The pulse generator produced the acoustic pulses by closing a relay. Figure 4 shows the typical shape of a pulse with a rising limb during 1 µs, a stable voltage of about 150 µs, and a falling signal during 2000 µs. A high-voltage power supply and a capacitor provided and stabilized the electric power with satisfactory reproducibility at U = 780 V. Closing of the relay electrodes produced a line surge of about 1400 V. This surge was not relevant to our measurements because the high frequencies contained in the surge were completely absorbed when the acoustic signal moved through the soil. The relay circuit managed the sources and receivers. For each transducer two relays controlled the connections of the transducer with the pulse generator and the PC, respectively. The source relay was closed after opening the receiver relay, and vice versa, to prevent damage to the PC by high voltage. Amplifiers between the receivers and the PC amplified the received signals by a factor of 100. The amplifiers and the relay circuit required a 12-V power supply.



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  		Fig. 4. Exiting pulse (gray) which drove the source transducer to emit an acoustical signal and a typical acoustic signal received (black) with the transducer.

 
Time Domain Reflectometry
A pair of TDR wave guides were mounted between Transducers 1 and 2, 0.42 m below the surface (Fig. 1 and 5) . The wave guides consisted of a pair of stainless-steel rods, 35 mm apart, each having a diameter of 6 mm and a length of 300 mm. A radio frequency pulse transformer (500 kHz–1 GHz and 50–200 {Omega}) was used to stabilize the signal for the Tektronix 1502B cable-tester (Beaverton, OR). The TDR system was calibrated according to Roth et al. (1990). Measurement frequency was 1/300 Hz during both infiltration experiments.



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  		Fig. 5. Configuration of the 25 grids and 31 ray paths.

 
Sprinkler
A sprinkler was mounted on top of the soil column (Fig. 1). Water was pumped through a hose from a water reservoir to the distribution cylinder. From there the water was guided through hoses (1-mm diam.) to 40 outlets in a rotating disk. The distance between the outlet and soil surface was 150 mm. A motor rotated the disk with a frequency of 0.048 Hz. The outlets were arranged such that each drop covered the same surface area of the soil column. The velocity of the outer circular path of the outlets was 6 mm s–1.

Soil and Column Preparation
The soil was a Typic Hapludalf (Soil Survey Staff, 1975), derived from loess with particle diameters <1 mm. According to Germann (1976) the soil contains 15 to 21% clay, 60 to 65% silt, and 19 to 24% sand. The bulk density, {rho}b, and the porosity, {epsilon}, varied between 1050 and 1500 kg m–3 and between 0.6 and 0.43 m3 m–3, respectively, as determined with cylindrical samples of 1 L and 0.1 m high. At the depth of the acoustic measurement plane {rho}b was 1400 kg m–3, and {epsilon} was 0.47 m3 m–3. The soil column was collected at the site of Germann's (1976) experiments between Möhlin and Wallbach (in Kanton Aargau, near the northern border of Switzerland). The column was carved out with a diameter of 0.3 m and a length of 0.62 m. The column was encased in fiberglass and polyester resin in the pit, and subsequently brought to the laboratory. A sheet metal cylinder (0.2 m tall, 0.3-diam.) was pushed about 0.1 m over the top of the column to prevent water from running off during infiltration. The column was placed on a metal grid. The volumetric water content before the first infiltration experiment was 0.315 m3 m–3.

Acoustic Travel Time Measurements
Holes larger than the contact areas of the transducers were cut into the fiberglass mantle to prevent acoustic signals from traveling along the mantle. The contact areas were smoothed with a wet soil paste where the transducers were pressed against the soil. The transducers were removed after the soil paste had dried. Final contact was achieved by gluing the contact area of the transducer onto the dried soil paste with an epoxy resin. As compared with direct contact or gel-contacting glue markedly improved the travel time measurements in that the amplitude is rising up.

One full data set consisted of the following. One transducer served as emitter while seven other transducers received the acoustic signals. The two transducers next to the source were not used as receivers because of their short distance to the emitter. The acoustic signal received by each transducer was digitized at a sampling rate of 1.2 x 106 s–1. The procedure was repeated for the remaining nine source transducers. Because each transducer acted as a source as well as a receiver, two sets of ray paths with corresponding acoustic signals were identical within one complete data set. Therefore, one complete data set consisted of only 35 travel distances. The acoustic signals were afflicted by both systematic and random noise. To eliminate the systematic noise, a calibration was conducted by measuring signals without soil contact. This signal was subsequently subtracted from the corresponding soil measurements. The random noise was largely eliminated by time averaging. The time interval for measurement of one complete data set was 12 s, that is, short enough to track the velocity of a wetting front.

According to Cowan et al. (1998) and Page et al. (1997), the acoustic travel time is interpreted as the first arrival of the acoustic signal. It is defined as the crossing point of the zero line with a line defined by two points of the rising limb of the signal at 0.35 and 0.85 of the peak amplitude (Fig. 4). The acoustic signals of the four travel distances S0-R2, S1-R3, S1-R8, and S5-R7 were discarded due to their extreme attenuation (where "S" and "R" denote source and receiver, respectively, with the numbers of the position of the transducers corresponding with those shown in Fig. 5). Thus, a total of 31 acoustic travel times were combined to yield one tomography data set. Note, that acoustic travel times are presented in seconds, whereas times related to water content variations are given in hours and minutes.

Experimental Procedure
Acoustic and TDR signals were measured for a total time period of 2894 min. During this time two runs were performed, each consisting of an infiltration and redistribution period. The infiltration rate was 30 mm h–1 (8.3 x 10–6 m s–1) and each infiltration lasted 30 min. Runs 1 and 2 lasted 1135 min (309–1444 min); 1450 min (1444–2894 min), respectively. The time intervals between measurements of a complete data set were 1 min during infiltration, 2 min during the first 30 min of redistribution, and 1 h otherwise. A total of 151 data sets were collected.

Tomography
A tomographic algorithm transformed the 31 travel times of one tomographic data set (i.e., a snapshot) to 25 acoustic grid unit velocities. The 151 snapshots resulted in time series of the acoustic grid unit velocities, which were converted to volumetric water contents.

We assumed that the acoustic velocity varied in space, possibly due soil heterogeneity, resulting in curved ray paths because of wave diffraction. We applied a modified version of the SIRT code. This code, developed by GeoTom, LLC (Apple Valley, MN), provides algorithms for both straight and curved rays. According to Krajewski et al. (1989), the straight ray assumption is no longer valid if the acoustic velocity as a result of heterogeneities is larger than 1.5 times the uniform acoustic velocity. We assumed that the acoustic velocity was randomly distributed. The curved ray procedure was used whenever the maximum acoustic velocity of a single tomogram exceeded 1.5 times the minimum acoustic velocity, as determined from the first tomographic data set (t = 0 min) using both the straight ray and curved ray algorithm. The acoustic velocity ranged between 280 and 490 m s–1 and 310 and 460 m s–1 for the straight ray and curved ray computations, respectively. Both cases required application of the curved ray procedure, which was subsequently used for all computations.

SIRT requires an initial acoustic velocity model to start the iteration procedure. We assume a uniform acoustic velocity distribution for that purpose. In the first iteration step travel times were initiated for each ray path on the basis of previous experience. The residuals between initial and measured travel times were used for fitting the acoustic velocities along each ray path. The resulting acoustic velocity distribution was used as the initial model for the second iteration, for which the acoustic velocities were fitted again to the residuals. The iteration procedure was repeated nine times.

The 31 travel times of the first six snapshots were averaged, assuming stagnant water. SIRT computed the acoustic grid unit velocities for the averaged snapshots based on the initially uniform distribution. The resulting velocity distribution was taken as the initial guess for all 151 snapshots. The computation started with one straight ray followed by eight curved ray iterations. Acoustic velocities were computed for the 25 grid units (G1–G25). The four corner grid units (G1, G5, G21, and G25) were discarded from further analysis because their larger parts lay outside the cross section of the soil column (Fig. 5). A grid unit size of 60 by 60 mm was selected, so that at least two rays passed through each grid unit.


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Time Domain Reflectometry Measurements
Table 1 lists several variables related to TDR and acoustic measurements. Figure 6 shows soil moisture variations {theta}(t) at the 0.42-m depth due to the two 30-min infiltration events of 15 mm each. Soil moisture increased significantly at {Delta}tr = 24 and 25 min after the start of infiltration. The two wetting fronts had moved with average velocities of 17.5 and 16.8 mm min–1, respectively, between the surface and the TDR depth (Table 2). This is typical for preferential flow (e.g., Germann et al., 2002). The water content increases of {Delta}{theta}m = 0.0075 and 0.0055 m3 m–3 are modest. Steady-state distributions in both runs were reached after {Delta}ts = 300 and 400 min, respectively. We interpret the soil moisture variations as being caused by preferential flow within a soil subjected to simultaneous imbibition of water from preferential flow paths into the unsaturated soil matrix.


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  		Table 1. List and definitions of variables.

 


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  		Fig. 6. {theta}TDR(t) measured with TDR probe during Runs 1 and 2.

 

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  		Table 2. Summary of time domain reflectometry (TDR) parameters.

 
Interpretation of the Acoustic Measurement
Acoustic Velocity Variations
The two contour plots in Fig. 7 depict for both runs the differences between the acoustic velocities immediately before and 20 min after the cessation of sprinkling. The contour plots demonstrate that the paths of the wetting front of Run 1 are different from those of Run 2. The acoustic velocity distributions at the beginning of the infiltrations at t0 = 309 min and t0 = 1445 min were set to zero, so that the contour plots present the velocity differences, {Delta}vm.



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  		Fig. 7. Contour plots which show the acoustic velocity variation 20 min after the first (left panel) and second infiltration (right panel).

 
Conversion to Water Contents
Bourbié et al. (1987) approximated the acoustic velocity in an unsaturated porous media with

[1]
where M is the elasticity modulus, and {rho}t the total density defined as

[2]
where {rho}w = 1000 kg m–3 is the density of water. The decrease in the acoustic velocities due to the increase in the total density, {Delta}{rho}t ~= 6 kg m–3, amounts only to a few m s–1. This cannot explain the variations in the acoustic velocities in the range of 0 to –45 m s–1 (see {Delta}vm in Table 3).


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  		Table 3. Summary of the parameters of the acoustic measurements.

 
According to Brutsaert (1964), the acoustic velocity of the first compressional wave is

[3]
where Z describes the impact of the liquid, gas, and bulk modulus on the acoustic velocity, and a compensates for inaccuracies due to the approximations in the derivation of Brutsaert's (1964) approach. Flammer et al. (2001) assumed for the same soil Z = 1 and a = 1, values that we used also for this study. The effective pressure pe is composed of the total pressure pt, the gas pressure pg, and the capillary pressure pc({theta}) weighted with the degree of saturation S = {theta}/{epsilon}, or

[4]

The total pressure is derived from the overburden soil. It amounts to pt = 4100 Pa, assuming an average bulk density and water content values of 1230 kg m–3 and 0.316 m3 m–3, respectively. The bulk density of the overburden soil increases with depth from 1050 to 1400 kg m–3. Brutsaert (1964) increased the gas pressure pg experimentally, whereas in our case we assumed pg = 0. The parameter b accounts for the elastic property of the solid particles. We further assumed a uniform porosity, {epsilon}, of 0.47 m3 m–3. Assuming the relationship pc = c1exp(c{theta}) for the retention curve (Germann, 1976). Flammer et al. (2001) previously determined values of c1 = –2.5 x 109 and c = –33.2 for our soil. Here we assume c1 = –2.5 x 109 and c as a matching parameter for the fitting procedure. Thus, Eq. [3] transforms to

[5]
The v({theta}) relation was obtained by fitting the parameters b and c, as discussed below.

Calibration of v({theta})
Application of Brutsaert's (1964) modified approach (Eq. [5]) requires the parameters b and c. The calibration procedure is in need of pairs of acoustic velocities and corresponding water contents. However, the two parameters are not available at the same spatial resolution. We therefore first resort to their averages and add their spatial variations later.

The average water content was assumed equal to the water content measured with the TDR device. Figure 6 shows three static levels of the water content at (i) 0.315, (ii) 0.323, and (iii) 0.328 m3 m–3. The absolute values of the water contents may deviate considerably from the values reported here; however, the precision of the water content differences is high. For instance, using the same TDR-device, Germann et al. (2002) assessed the water content differences to be <0.002 m3 m–3. The corresponding average acoustic velocities were obtained by averaging 10 snapshots, each containing the 21 acoustic velocities of the grid units, during (i) 0 to 313 min, (ii) 883 to 1443 min, and (iii) 2294 to 2894 min. The time intervals include the three static water contents. Figure 8 shows the three data pairs. The parameters b = 9.81 x 10–10 Pa–1 and c = –11.28 follow from calibration of Eq. [5] against the three averaged data pairs (i), (ii), and (iii). Flammer et al. (2001) found that a value of 8.1 x 10–10 Pa–1 for b agreed well with our result, particularly in view of the considerable spatial variation within 1 m2 as reported by Blum (2002).



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  		Fig. 8. Calibration of the water content. The maximum and minimum acoustic grid velocities (G14 and G23) as well as the acoustic grid velocities of G2, G15, G17, and G24 are represented.

 
We assume that the spatial variation of the water content was negligible before Run 1 since the installation and the testing of the instruments took several months. Therefore, we assume that the parameters c and c1 of the retention curve are spatially invariant and that the spatial variation of the grid unit velocities before Run 1 was exclusively due to spatial variations in parameter b. Thus, each grid unit was assigned a b parameter by inverting Eq. [5] and assuming that the average water content (i), {theta} = 0.315 m3 m–3, applies to all grid units. Figure 8 shows the resulting v({theta}) relationship for grid units G14 (upper limit) and G24 (lower limit) with bmin = bG14 = 5.27 x 10–10 Pa–1 and bmax = bG24 = 1.69 x 10–10 Pa–1. The intermediate parameter values are compiled in Table 4.


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  		Table 4. Summary of the b parameters of the grid units (x 10–10 Pa–1).

 
The water content measured with the TDR device increased by 0.013 m3 m–3 from before Run 1 to after Run 2. The averaged water content which follows from the acoustics data at the beginning (t = 0 s) and end (t = 2894 s) of the measurements were 0.316 and 0.328 m3 m–3 (Table 3). The difference, {Delta} = 0.012 m3 m–3, is in good agreement with the TDR measurement. In contrast, application of v({theta}) shows already a water content increase of 0.075 m3 m–3 in grid unit G3 during Run 1 (Table 3).

Error Estimation
In this paragraph we assess numerically the errors in v({theta}) due to spatial variations in b, c, {theta}, and {epsilon}. The water content ranges between 0.305 and 0.394 m3 m–3 when considering all grid units during the measurement period. Thus, our measurements are located in the area between the four corners of the vertical lines at {theta} = 0.305 and 0.394 m3 m–3 with the maximum and minimum v({theta}) relations (Fig. 8). We numerically determine the partial derivatives {partial}v/{partial}b, {partial}v/{partial}c, {partial}v/{partial}{theta}, and {partial}v/{partial}{epsilon} at these corners.

Since we had no information about the impact of b and c variations, we varied b and c with factors of 0.9 and 1.1. The results are compiled in Table 5. The variations in b and c result in variations in the acoustic velocity of about ±17 and ±28 m s–1. They do not exceed the maximum acoustic velocity variation of grid unit G3. However, varying b by about 10% considerably affects the absolute values of v({theta}). We now consider the upper derivative, {Delta}v/{Delta}b = |(v1.1b – vb)/(1.1b – b)|, and the lower derivative, {Delta}v/{Delta}b = |(v0.9b – vb)/(0.9b – b)|, at {theta} = 0.305 and 0.394 m3 m–3. Here, [1.1b; b] and [0.9b; b] are the intervals of the derivations. The derivatives do not differ greatly. The ratios of the slopes never exceed 1.16 for all derivatives (Table 6). This ratio is rather small and confirms the approximate linear behavior of the v({theta}) relations. We also compared the upper and lower derivatives at the corners (Table 6). The ratios were 1.14, which are also relatively small. Similar results hold for the partial derivatives {partial}v/{partial}c. We conclude that varying b and c have only minor impact on the v({theta}) expression.


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  		Table 5. Error estimation.

 

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  		Table 6. Results of the numerical consideration.{dagger}

 
The variations in b and c above were estimated. However, Blum (2002) reported standard deviations of the porosity and water content of 0.014 and 0.007 m3 m–3 for the same type of soil. The standard deviation of the water content is thought to be fairly constant. Thus, we define 0.97 and 1.03 to be somewhat arbitrarily as the intervals for the derivatives {Delta}v/{Delta}{theta} and {Delta}v/{Delta}{epsilon} (Table 5). The ratios of the derivatives, {Delta}v/{Delta}{theta} at {theta} = 0.305 and 0.394 m3 m–3 are always below 1.09 (Table 6). The ratios of the lower and upper {theta} derivatives are practically zero at the corners. Similar ratios are obtained for the {epsilon} derivatives. Thus, spatial variations in {theta} and {epsilon} also seem to have relatively minor impacts on the v({theta}) expressions.

The above procedure suggests that the absolute v({theta}) relation are subjected to some uncertainty. However, the relative proportions among the 21 v({theta}) relations are only weakly affected by errors in b, c, {theta}, and {epsilon}.


   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
CONCLUSIONS
 REFERENCES
 
To group the grid units according to their hydraulic behavior, we first established the threshold, ci, for a significant water content increase. The standard deviations of the 21 water contents of the grid units within 10 snap shots before arrival of the wetting front (i.e., 0 < t < 313 min) ranged from 0.0013 to 0.0045 m3 m–3; thus, ci = 2 x 0.045 m3 m–3. We defined three classes: Class 1 {Delta}{theta}m > 0 and |{Delta}{theta}m| > ci, Class 2 {Delta}{theta}m < 0 and |{Delta}{theta}m| > ci, and Class 3 |{Delta}{theta}m| ≥ 0 and |{Delta}{theta}m | < ci. dFigure 9a through 12d provide some examples. The considerable noise in the original time series of {theta}(t) was smoothed using Friedman's Super Smoother, which is part of the S-Plus 2000 program language (Friedman, 1984). Table 3 lists the grid units assigned to either Class 1 or Class 2. Class 3 comprises those grid units without significant variations in the water content, and are not listed. The modest decreases in soil water after the cessation of infiltration indicates water redistribution that lasted longer than the acoustic measurements of the corresponding infiltration experiments. We therefore used the acoustic water contents at 1444 min (Run 1) and at 2894 min (Run 2) as a reference for {Delta}{theta}d.



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  		Fig. 9. {theta}(t) of G2: (a, b) Run 1 and (c, d) Run 2. Both runs are examples of Class 1.

 


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  		Fig. 12. {theta}(t) of G24: (a, b) Run 1, which is an example of Class 3, and (c, d) Run 2, which is an example of Class 1.

 


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  		Fig. 10. {theta}(t) of G15: (a, b) Run 1, which is an example of Class 2, and (c, d) Run 2, which is an example of Class 1.

 


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  		Fig. 11. {theta}(t) of G17: (a, b) Run 1 and (c, d) Run 2. Both runs are examples of Class 3.

 
Figure 8 demonstrates that v({theta}), Eq. [5], is essentially linear within the range of the measurements. Moreover, ci is considerably larger than the differences between the plateaus of the {theta}TDR(t) relation (Fig. 6), most likely due to the larger volume covered by the TDR wave guides in comparison with the size of the grid units.

Run 1
The water contents increased during Run 1 in the grid units G2, G3, G4, G7, G8, G9, and G10 (Table 3 and Fig. 5). The travel times of the wetting fronts were in the range of 7 ≤ {Delta}tr ≤ 8 min for G2, G3, G7, and G8, amounting to an average wetting front velocity vw = 38.6 mm min–1. The water contents increased in the range of 0.04 ≤ {Delta}{theta}m ≤ 0.075 m3 m–3 in G2, G3, and G8. The high velocity of the wetting front and the locally pronounced increase in the water contents indicate the presence of preferential flow. The increase of {theta} in G7 is considerable lower. We believe that G7 is only slightly involved in the preferential flow process. In contrast, first wetting occurred 14, 16, and 29 min after the beginning of sprinkling in the neighboring grid units G4, G9, and G10. The delays and the smaller increase in the water content ({Delta}{theta}m = 0.012 m3 m–3) indicate flow from the paths of preferential flow to the grid units G4 and G9 and from there to G10. The water content of all grid units of Class 1 redistribute after infiltration (Table 3). Values of {Delta}{theta}d were in the range of 0.003 to 0.048 m3 m–3. It is worth noting that the grid units of Class 1 indicated decreasing water contents and redistribution, whereas the TDR device showed a relatively constant soil moisture content.

The grid units that surround the preferential flow path (G6, G11, G12, G13, G14, and G15) show somewhat decreasing water contents. The grid units G6, G11, G12, and G13 are located in Class 3 (Fig. 11a and 12a), whereas the grid units G14 and G15 belong to Class 2 (Fig. 10a) whose water contents decreased significantly. Flammer et al. (2001) and Brutsaert and Luthin (1964) did not report increasing acoustic velocities with increasing water contents. Berkenhagen et al. (1998), in contrast, observed acoustic velocities to increase with increasing water content. However, this occurred only in previously oven-dried and pulverized soil material with lose grain to grain contacts. Capillarity presumably exerted some stress on the particles, and pulled them together as the water becomes more evenly distributed. After that the acoustic velocity decreased with increasing water content. We conclude from this that the grid units in Class 2 are probably artifacts since tomographic procedures tend to amplify the differences among the acoustic velocities when the ratio of the number of acoustic paths to the number of grid units (31 to 21 in our case) approaches 1 (Bregman et al., 1989). The acoustic velocities decreased during redistribution in the grid units G6, G11, G12, G13, G14, and G15, suggesting lateral capillary flow. The variations in grid units G16, G17, G18, G19, G20, G22, G23, and G24 (Fig. 11a, 11b, 12a, 12b) are very small, although weakly increasing and decreasing temporal trends are discernible for grid units G16 and G24.

Run 2
Water in Run 2 followed paths that were distinctly different from the paths of run 1 as clearly demonstrated by the comparison in Fig. 7, and between Fig. 12a and 12c. A distinct path of preferential flow occurred in grid units G13, G18, G19, and G24. The small but still significant increase in the water content in grid unit G13 of only {Delta}{theta}m = 0.01 m3 m–3 indicates its indirect involvement in the preferential flow process, similar to G7 during Run 1. The wetting fronts needed {Delta}tr = 11, 13, 17, and 33 min to arrive at grid units G19, G14, G20, G15, respectively. The wetting front moved fastest in G13, having a velocity vw of 30.0 mm min–1. We believe that water flowed from G19 and G14 into G20 and G15. Redistribution and drainage reduce soil moisture beyond the period of measurements, and no steady states were achieved, as illustrated by Fig. 9d and 12d. A minor path of preferential flow appears in grid unit G2 (Fig. 9c and 9d). Here we also observed the arrival of the draining front at td = 1510 min (i.e., at {Delta}td = 35 min after the cessation of sprinkling).

Only grid unit G22 belongs to Class 2 with a significantly decreasing water content. In grid units G16 and G23 the water contents decreased whereas they increased in G6, G7, and G12. However, the variations were insignificant and the grid units belong to Class 3.

Run 1 vs. Run 2
The fastest arrivals to the acoustic plane of the wetting front were {Delta}tr = 7 and 9 min for Runs 1 and 2, respectively. This corresponds to front velocities of vw = 38.6 and 30.0 mm min–1. Average soil moisture content increased by 0.011 and 0.01 m3 m–3 during Run 1 and Run 2, respectively, and decreased during redistribution by 0.006 and 0.004 m3 m–3. The averages were taken at the beginning of the infiltrations at t = 309 and 1444 min, 40 min after the cessation of sprinkling at t = 379 and 1514 min, as well as at t = 2894 min. The water content increased to 0.028 and 0.015 m3 m–3 if only grid units in Classes 1 and 2 were considered.


   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSIONS
REFERENCES
 
Two prescribed infiltration runs were performed on a column containing an undisturbed loess soil. Water content increases reduced the acoustic velocities within a cross section in agreement with Flammer et al. (2001), Berkenhagen et al. (1998), and Brutsaert (1964). An acoustic tomography procedure applied to grid units of 60 by 60 mm revealed preferential flow in both runs but along different paths. Grid unit G2 during the second run showed an increase in the water content followed by a decrease shortly after the cessation of sprinkling, whereas the remaining 20 grid units indicated infiltration followed by typical water redistribution.

Variations in the acoustic velocity ranged from 310 to 460 m s–1. They were converted into variations in the water content according to Brutsaert (1964). The distribution of the acoustic soil property, b (Pa–1), was derived before the first infiltration experiment when soil moisture presumably was spatially equilibrated.

This study shows that acoustic tomography is a potentially powerful method for studying flow processes. The method allows for simultaneous spatial and temporal measurements of soil moisture variations without interfering with the flow process, at resolutions of better than 1 min–1.


   ACKNOWLEDGMENTS
 
The Swiss National Science Foundation supported the project under grant numbers 21-50699.97 and 21-56950.99. We thank Jürg Schenk (electronics), Martin Ötliker and his crew (mechanics), Roger Grandjean and Andreas Leiser (coding) for continuing support. This research was initiated by Dr. Abdallah Alaoui's investigations.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES
 




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D. A. Robinson, C. S. Campbell, J. W. Hopmans, B. K. Hornbuckle, S. B. Jones, R. Knight, F. Ogden, J. Selker, and O. Wendroth
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