Spatial Time Domain Reflectometry for Monitoring Transient Soil Moisture Profiles Applications of the Soil Moisture Group, Univ. of Karlsruhe R. Becker 1,, S. Schlaeger 1,3, C. Hübner 1,4, A. Scheuermann 1,5, W. Schädel 1,6 1 University of Karlsruhe, Soil Moisture Group, 7618 Karlsruhe, info@smg.uni-karlsruhe.de IMKO, Mikromodultechnik GmbH, Im Stöck, 7675 Ettlingen, r.becker@imko.de 3 SCHLAEGER - mathematical solutions & engineering, Herrenstr. 13, 76133 Karlsruhe, info@stefan-schlaeger.de 4 University of applied sciences, Department of electrical engineering, 68163 Mannheim, c.huebner@fh-mannheim.de 5 University of Karlsruhe, Institute of rock mechanics and soil mechanics, 7618 Karlsruhe, alexander.scheuermann@ibf.uka.de 6 University of Karlsruhe, Institute for water and river basin management, 7618 Karlsruhe, wolfram.schaedel@iwg.uka.de Short Abstract Monitoring of transient soil moisture profiles yields valuable insight into soil hydraulic processes. A recently developed reconstruction algorithm allows to derive water content profiles along extended moisture probes from Time Domain Reflectometry (TDR) signals in time domain. The algorithm is based on inverse modeling of parameter distributions along a numerical transmission-line. The method named Spatial TDR will be explained and practical applications presented, aiming at monitoring spatial and temporal evolution of soil moisture. Time domain reflectometry, water content profile, inverse problem, spatial TDR I. INTRODUCTION The spatial distribution of soil moisture and its evolution in time is a valuable information for many investigations in hydrology, agriculture, and soil science. However the monitoring of a sufficient number of soil moisture profiles can be expensive, laborious, and invasive, especially if the profiles are determined point-wise by a large amount of single probes buried in soil. A recently developed reconstruction algorithm [1,] allows to compute complete soil water content profiles along single moisture probes from time domain reflectometer (TDR) samples in a short time. This method leads to a reduction of probes accompanied by a higher spatial resolution of moisture profiles at the same time. The whole technology of soil moisture profile retrieval has been named Spatial TDR [3]. This method is being developed and applied by the Soil moisture Group (SMG), an interdisciplinary research group at the Univ. of Karlsruhe. In this article we first introduce the basic concept of STDR with emphasis on its algorithmic core and the initial probe calibration by way of a coated 3-rodprobe. Then we assess the theoretical accuracy of the yielded moisture profiles by means of electromagnetic (EM) field simulations of the TDR process. To check the method under field conditions we realized a laboratory experiment in order to compare the reconstructed water content profiles with results from oven drying method in a real soil [3]. Finally two examples of practical STDR applications are demonstrated, a flood warning system [4] and a dike monitoring system [5], two ongoing research projects of the SMG. A. The Inverse Problem II. METHODS A TDR instrument located at x emits a voltage step pulse ( t, x ) V m I via a feeding cable into a waveguide (moisture probe) buried in soil. When the propagating EM wave hits the junction between cable and probe it is generally split due to impedance discontinuity. Part is reflected and traveling back, part is transmitted into the waveguide, interacting with the surrounding soil. When the pulse reaches the probe end it is reflected again. Hence V m ( t, x ) I (input, measured) excites the system under test (SUT) probe/soil which reacts with voltage waves whose superposition V m O ( t, x ) (output, measured) is sampled by the TDR instrument. The elapsed time between first and second main reflection is the pulse travel time forth and back the moisture probe. This travel time can be transformed into average soil moisture by appropriate calibration functions and/or mixing rules. This is generally what common TDR signal evaluation does. But the TDR signal contents more information. The reflectogram, especially the part between first and second main reflection at the probe s beginning and end is a finger print of the dielectric profile along the waveguide, which is mainly ruled by the water content. Unfortunately the moisture distribution cannot be calculated directly from the TDR signal but has to be estimated indirectly. The basic idea of STDR is to transform the sampled output signal V m O ( t, x ) into the soil moisture profile ( x) along the probe by means of inverse modeling. The essence of the approach is to simulate the propagation of the TDR signal along the waveguide in time domain by employing a numerical model (forward problem) based on the telegraph equation.
This simplified model assumes that the relevant properties of the transmission-line can be described by bulk electronic parts like resistors, inductors, and capacitors (Fig. 1). Among the conditions for this electronic circuit model to hold the most important are: wave modes other than the transversalelectromagnetic (TEM) mode may be neglected, and frequency dependence of transmission-line properties may be neglected. The first condition requires a well-behaving waveguide with little distortion on the signal propagation, the second is only met, if the losses in the SUT are not too large. ( t x,g') ( V s ) O, for given C ('x ) and G ('x ). The result of the simulation is compared to the TDR measurement. An optimization algorithm described in [1,] is used to modify the electrical parameters C ('x ) and G ('x ) along the simulated moisture probe until the simulated TDR reflectogram V s O ( t, x ) matches the measurement V m O ( t, x ) sufficiently well. The final parameter distributions resulting from the simulation are the best estimate of the electric properties along the real probe in soil. B. From Capacitance to Dielectric Permittivity To derive the volumetric water content profile θ (x) the dielectric permittivity profile ε (x) of the soil/water/air mixture has to be extracted from the capacitance profile C ('x ) first. For the simple moisture probe it is possible to find a convenient parametric form for C ('ε ) : 1 / C (' ε ) = 1/( ε C 1 )' + 1/ C ' () Figure 1. The simplified moisture probe model consisting of bulk electronic parts. Above: coated 3-rod-probe as an example for a moisture probe (TDR waveguide); below: equivalent circuit of the transmission line. Reference [1] derived the following wave equation from the circuit model for describing the propagation of a voltage pulse V x,t along the buried waveguide: L' t + L' ( x) / x V L' ( x) x x ( x) ( x) +L' ( x) G' ( x) t ( x,t) = Capacitance C ('x ) and effective conductance G ('x ) are influenced by the soil water content distribution ( x) along the waveguide. Inductance L ('x ) is a function of the transmissionline only and piecewise constant for coaxial cable and moisture probe. The spatial derivative of L ('x ) in (1) describes the change of inductance between coaxial cable and probe. Resistance R ' along the waveguide has been neglected. All parameters are given per unit length. Strictly spoken the equivalent circuit of Fig. 1 is not totally correct, because the conductor G ' should be enclosed by two capacitors due to the rod coating. Therefore G ' is not the real ionic conductance of the soil but a kind of correcting parameter in the determination of C '. According to former results we assume that this simplification does not have a large influence on the results. Eqn. (1) is solved numerically with appropriate initial and boundary conditions to simulate a TDR measurement (1) Figure. Capacitance C of a 3-rod-probe as a function of the soil s dielectric permittivity. (a) segment of three parallel rods encompassed by soil; light gray: PVC coating; dark gray: metallic core; (b) equivalent circuit. C 1', C ' : constant capacitance parameters determined by the probe s geometry and material. C. From Dielectric Permittivity to Water Content The second step performs the transition from dielectric permittivity to water content. An empirical relationship between ε and θ often used in TDR applications was found by [7]. We use a more simple but less general empirical formula, which we derived from laboratory experiments with a loamy sand:.31 θ ( ε ) = 3.1 ε 41.1 (%Vol). (3) D. Probe Parameters The rods of the 3-rod-probe under investigation consist of stainless steel cores of 6 mm diameter with a 1 mm thick PVC coating. The rods are 3 mm apart. They are screwed into the probe head which connects them to a 5 Ohms coaxial cable.
According to (1) and () it is necessary to determine the three parameters C 1 ', C ', and L ' for the rod probe. This can be done empirically by measuring TDR pulse propagation velocities v = ε ) for two different media with dielectric i v( i permittivities ε 1, and ε, respectively. The pulse propagation velocity along the coated probe rods is: v( ε ) = 1/ L' C (' ε) (4) The pulse velocity is determined empirically by measuring the time span between the first two main reflections in the TDR reflectogram. Combining (4) for the two materials one yields: 1 = ( ε ε1)/( ε ε1( v1 v ) L )' and = ( ε ε )/(( ε v ε v ) )' (5) 1 1 1 L The rod impedance Z can be used to get L ': Z ( ε ) = L /' C (' ε ) (6) The impedance mismatch between coaxial cable and probe rods leads to a partial reflection of the incident excitation pulse. The amplitude of incident and reflected signal are denoted by A I and A R, respectively. Then the reflection coefficient yields: r ε ) = A / A = ( Z( ε) Z )/( Z( ε ) ) (7) ( I R + Z which can be determined experimentally from TDR measurements. The combination of the last equations yields: L' = (1 + r( ε )) /(1 r( ε )) Z / v( ε ) (8) Eqns. (5) and (8) are sufficient to determine the parameters for the coated 3-rod-probe from TDR reflectograms. E. Empirical Relationship between Capacitance and Effective Conductance The wave equation (1) needs two parameter distributions C ('x) and G ('x ). These parameter distributions could be found simultaneously by inverse modeling, if two independent TDR measurements were available for the same moisture probe, which is possible only with special probes (double sided). In case of single sided probes it is reasonable to assume a relationship between C ('x ) and G ('x ), since both parameters are linked by soil moisture: higher water content leads to higher dielectric permittivity and higher conductivity. The following relationship is proposed: With this relationship a given capacitance profile can be transformed into an effective conductance profile. Both parameter distributions are inserted into (1). F. Electrodynamic Simulation of the TDR Measuring Process with Microwave Studio To test the Spatial TDR method together with the 3-rodprobe several TDR reflectograms are simulated with Microwave Studio (MWS), an EM simulation tool based on the full wave solution of Maxwell s equations. In the numerical model the 3-rod-probe is embedded in a three layered material, whose dielectric permittivity and ionic conductance can be modified ideally. A voltage step pulse of 1 Volt amplitude and 1 GHz bandwidth is fed into the probe. The simulated TDR reflectograms are used for three purposes: 1. determination of the probe parameters according to (5) and (8),. determination of the empirical C -G -relationship (9), and 3. generation of test reflectograms to assess the quality of the Spatial TDR algorithm. III. RESULTS To assess the quality of the algorithm which determines the water content profile from a TDR reflectogram by inverse parameter estimation, the MWS is fed with three soil layers of different moisture. Tab. I shows the applied soil parameters. TABLE I. MATERIAL PARAMETERS USED IN MICROWAVE STUDIO Table Moisture state Head θ (%vol) ε (-) σ ms/m dry.5.9 moist 8 4.9 14 wet 13 6.8 3 Dielectric permittivity and ionic conductivity of a soil for different volumetric water contents. The parameters were derived from loamy sand in laboratory experiments. For the sake of simplicity the moisture states are named dry, wet, and moist, respectively. Each simulated reflectogram together with the excitation pulse was fed into the Spatial TDR algorithm and a reconstruction process was conducted to retrieve the soil moisture profiles, which should match the predefined as close as possible. G' (1 exp( ( ) / Cd ), if, G (' C )' = (9), if. Figure 3. Moisture profile, sequence wet/moist/dry, predefined in the MWS model, and reconstructed by means of the Spatial TDR algorithm.
The predefined and reconstructed soil moisture profile for the sequence wet/moist/dry is shown in Fig. 3. Two cases were realized: one with and the other without consideration of ionic conductivity σ (lossy and lossless case). Fig. 4 displays the corresponding TDR reflectograms simulated with MWS and reconstructed by the Spatial TDR algorithm. IV. APPLICATIONS A. Measuring of water content for flood warning The high variability of the rainfall-runoff process can be partly explained by the catchments state which is linked to soil moisture (cf. [8]). Not only the mean moisture content is crucial but also its distribution (cf. [6]). At the Goldersbach catchment near Tübingen 46 twin rod probes were installed in order to measure the extension of a saturation zone. A brook divides the measurement site, which is dominated by podzolic soils. The measurements of water content were reconstructed spatially along the 6 cm long twin rod probes in a distribution of mm and interpolated into a quasi three-dimensional soil moisture space. Fig. 6 shows the soil moisture distributions as a cut in the depth of 15 cm for dry and wet conditions. As a result of the measurements most of the soil moisture dynamic was found on the top 3 cm of the soil horizon. Figure 4. TDR reflectograms simulated by MWS and the corresponding signal approximations resulting from the reconstruction algorithm. Material sequence wet/moist/dry. Energy losses due to ionic conductance lead to a strong falling trend of the TDR signals. To test Spatial TDR in laboratory a box with three chambers of.m length each was prepared in accordance to the MWS numerical model and filled with soil of predefined moisture (s. Tab. I). A 3-rod-probe of.6m length (s. Fig. 1) was installed such that it crossed all chambers. TDR measurements were performed with a Tektronix metallic cable tester 15B. With each material sequence four soil samples of known volume were taken from each chamber. Their volumetric water content was determined by oven drying. Fig. 5 shows the result for the material sequence dry/moist/wet. Figure 6. Horizontal cut of the three dimensional soil moisture space in the depth of 15 cm, for dry conditions (left) and wet conditions (right) as volumetric soil mositrue Figure 5. Spatial TDR application to a real soil moisture profile. Material sequence dry/moist/wet. Reconstruction results compared to volumetric water content of soil samples determined by oven drying. Differences up to 3%vol are due to imperfect calibration of the real 3-rod-probe. The overall accuracy of Spatial TDR with coated rod probe is sufficient for many applications in soil science. A lysimeter experiment with 1 m³ loamy sand showed that the method is capable of tracking transient soil moisture profiles under irrigation with high spatial and temporal resolution [3]. B. Measuring of water content on a full-scale Dike model In order to investigate transient hydraulic processes in river dikes during a flood, a full-scale dike model at the Federal Waterways and Research Institute was equipped vertically with 1 sensor cables between.7 m and 3. m in length. Both sides of the sensor cables were connected to the TDR device. The model was built up homogeneously with uniform sand (grain size. to mm) and it is based on a waterproof sealing of plastic. As a result of this construction, the water infiltrating into the dike will flow to a drain at the toe of the land side slope and directed to a measuring device (cf. Fig. 7). In the course of the investigation mentioned above, flood simulation tests and sprinkling tests were carried out on the dike model (cf. [4] and [5]). Fig. 8 shows the steady state of seepage condition during a flood simulation test in December. The measurements of water content were reconstructed spatially along the sensor cable (cf. [1,]) and interpolated as distribution of saturation in the considered cross section (assumed porosity of 37 %). The dots represent the position of the sensor cables together with the saturation at theses points. On the water side the water level in the basin is shown, and the
3 cm are possible. The profiles of saturation can be determined with an average uncertainty of + 4 % compared to other independent field measurements. REFERENCES Figure 7. Full-scale dike model at the Federal Waterways and Research Institute (BAW) in Karlsruhe during a flood simulation test in December (steady state of seepage condition). position of the phreatic line within the body of the dike is given, as estimated from the pore water pressure measurements in the base of the dike. It can be seen that the measurements of the water content correspond very well with the position of the phreatic line. During the flood simulation tests on the dike model it was discovered that the percolation area below the phreatic line does not become fully saturated. Up to 15 % of the pore space remained filled with air. This observation was verified by independent measurements during the steady state condition. With this system it is possible to record the distribution of water content in a cross section within 5 minutes. This way, measurements of the water content with a spatial resolution of [1] S. Schlaeger, Inversion von TDR-Messungen zur Rekonstruktion räumlich verteilter bodenphysikalischer Parameter, Ph.D. thesis, Institute of Soil Mechanics and Rock Mechanics, Univ. of Karlsruhe,. [] S. Schlaeger, A fast TDR-inversion technique for the reconstruction of spatial soil moisture content, Hydrology and Earth System Sciences, vol. 9, pp.481-49, 5. [3] R. Becker, Spatial Time Domain Reflectometry for Monitoring Transient Moisture Profiles, Ph.D. thesis, Institute for Water Resources, Hydraulic and Rural Engineering, Univ. of Karlsruhe, 4. [4] A. Scheuermann, S. Schlaeger, C. Huebner, A. Brandelik, and J. Brauns, Monitoring of the spatial soil water distribution on a full-scale dike model, Proceeding of the Fourth International Conference on Electromagnetic Wave Interaction with Water and Moist Substances, edited by K. Kupfer, 343-35, Weimar, 1. [5] A. Scheuermann, Instationäre Durchfeuchtung quasi-homogener Erddeiche, Ph.D. thesis, Institute of Soil Mechanics and Rock Mechanics, Univ. of Karlsruhe, 5. [6] W. Schädel, Schritte zur Verbesserung der Hochwasserwarnung mittels Online-Bodenfeuchtemessungen, Ph.D. thesis, Institute of Water and River Basin Management, Univ. of Karlsruhe, 6. [7] G.C. Topp, J.L. Davis, and A.P. Annan, Electromagnetic determination of soil water content: Measurement in coaxial transmission lines, Water Resources Research, vol. 16(3), pp. 574-58, 198 [8] G. Peschke, Hydrological processes of storm runoff generation. PIK Report, 48, 75-87, Potsdam, 1998. Figure 8. Interpolated distribution of saturation (from measured volumetric water content with an assumed porosity of 37 %) in the dike model received from the cable sensor measurements during a flood simulation test in December. The points represent the positions of the cable sensors, and the numbers next to the points show the corresponding values. The phreatic line was derived from the piezometer measurements in the base of the model.