Obtaining the Spatial Distribution of Water Content along a TDR Probe Using the SCEM-UA Bayesian Inverse Modeling Scheme

along TDR probes, corresponding with a scale of cubic centimeters (10 6 m3) to cubic decimeters (10 3 m3). Time domain reflectometry (TDR) has become one of the standard Two different approaches to obtaining soil water conmethods for the measurement of the temporal and spatial distribution of water saturation in soils. Current waveform analysis methodology tent estimates from measured TDR waveforms have gives a measurement of the average water content along the length found widespread use. The first estimates the dielectric of the TDR probe. Close inspection of TDR waveforms shows that permittivity of the soil (and thus the water content) heterogeneity in water content along the probe can be seen in the TDR by detecting characteristic reflections in the waveform waveform. We present a comprehensive approach to TDR waveform (Topp et al., 1980; Heimovaara and Bouten, 1990; Hook analysis that gives a quantitative estimate of the dielectric permittivity et al., 1992; Robinson et al., 2003). In particular, the profile along the length of the probe and, therefore, the distribution travel time is used to estimate the dielectric permittivity, of water content. The approach is based on the combination of a while the amplitude changes in the waveform are used multisection scatter function model for the TDR measurement system to assess the bulk electrical conductivity. This method with the shuffled complex evolution Metropolis algorithm (SCEMis relatively simple to implement and can be used for UA). This combined approach allows for the estimation of the 40 parameters in the transmission line model using a series of simple all known TDR probes (Heimovaara and Bouten, 1990; calibration measurements. The proof of concept is given with measureBaker and Allmaras, 1990; Jones et al., 2002; Robinson ments in a layered system consisting of air and water. Finally, TDR et al., 2003). However, in the presence of layered soils waveforms from layered soil samples were analyzed to estimate the and porous media with a relatively high electrical condistribution of the water content along the length of the probe. Results ductivity, the method exhibits difficulties with the corshow that the proposed method provides much more reproducible rect interpretation of the waveforms, often leading to results than obtained with the traditional travel time method. Because erroneous water content estimates (i.e., Schaap et al., the proposed method can be fully automated, it increases the applica2003). Coating the wave-guides with an insulating matebility of the TDR method, especially in applications where detailed rial can solve some of the problems related to high (real-time) data are required on heterogeneous infiltration. electrical conductivity (Ferré et al., 1996). Nevertheless, the intrinsic uncertainty of this method requires a substantial commitment of human resources to find and K of the spatial and temporal variability correct outliers in the results due to erroneous (autoof water saturation in soils is important to obtain mated) analyses. This is a problem for fully automated improved estimates of water flow (and its dissolved commeasurement and analysis. The second, perhaps more ponents) through the vadose zone. Because of its high sophisticated method is based on the modeling of the accuracy and potential for automated measurement, waveform using the physical properties of the TDR system TDR has become one of the standard methods to mea(Giese and Tiemann, 1975; Heimovaara, 1994; Heimosure the spatial and temporal variability of water convaara et al., 1996; Friel and Or, 1999; Feng et al., 1999; tents in laboratory soil cores and experimental field plots. Schlaeger et al., 2001; Lin, 2003a, 2003b). In this apThe method is based on the measurement of a reflected proach, the dielectric permittivity and bulk electrical voltage-wave from a probe installed in the soil. Detailed conductivity are model parameters estimated by fitting analysis of this reflected waveform enables the estimaagainst measured waveforms. Although the method retion of the bulk-electrical conductivity and dielectric quires significant computer resources, the number of permittivity of the soil, which in porous media primarily outliers due to erroneous analyses is substantially redepends on the water content. Traditionally, waveforms duced (Huisman et al., 2002). are analyzed such that an average water content estiWe follow the latter approach, using deterministic mate of the soil along the TDR probe is obtained. In this modeling of the TDR system in the frequency domain. work we show that measured TDR waveforms contain This inverse modeling method originated in the 1970s sufficient information to estimate water content profiles (Giese and Tiemann, 1975; Clarkson et al., 1977) and has been improved in the 1990s (Heimovaara, 1994; T.J. Heimovaara, Royal Haskoning, P.O. Box 8520, 3009 AM RotterHeimovaara et al., 1996). Recently, major steps forward dam, The Netherlands; S.A. Huisman, J.A. Vrugt, and W. Bouten, have been made so that it is possible to explicitly account Institute for Biodiversity and Ecosystem Dynamics (IBED), University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, for the multiple sections in the measurement setup with The Netherlands; J.A. Huisman, now with Institute for Landscape the so-called multi-section scatter function (MSSF) (Feng Ecology and Resource Management, Justus Liebig University, Heinet al., 1999; Lin, 2003a, 2003b). rich-Buff-Ring 26-32, D-35392 Giessen, Germany. Received 13 May While this inverse modeling approach results in a 2004. Special Section: Hydrogeophysics. *Corresponding author ([email protected]). Abbreviations: MSSF, multi-section scatter function; SCEM-UA, Published in Vadose Zone Journal 3:1128–1145 (2004). © Soil Science Society of America shuffled complex evolution Metropolis algorithm; SR, scale-reduction [score]; TDR, time domain reflectometry. 677 S. Segoe Rd., Madison, WI 53711 USA