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Measurement and Modeling of the TDR Signal Propagation through Layered Dielectric Media

^a George E. Brown Jr., Salinity Lab., 450 W. Big Springs Rd., Riverside, CA 92507
^b The Institute of Soil, Water and Environmental Sciences, (ARO) The Volcani Center, Bet Dagan, Israel

View larger version (62K): [in a new window]   Fig. 1. Schematic diagram showing two-layer and multiple-layer dielectric configurations with reference to the TDR electrodes.

View larger version (22K): [in a new window]   Fig. 2. The relative permittivity () or electrical conductivity (EC) measured using a TDR probe. Zero degrees represents the probe on the water surface, half in, half out. A negative angle indicates the probe in air and a positive angle the probe in water.

View larger version (87K): [in a new window]   Fig. 3. Photograph of the multi-layer probe with 5-cm diam. Acrylic disks spaced 0.38 cm apart.

View larger version (27K): [in a new window]   Fig. 4. Permittivity measured for two layers using a 20-cm probe sequentially inserted into water with the upper portion in air.

View larger version (21K): [in a new window]   Fig. 5. Measurements of three layers, white paraffin, water, and air. The length fraction amounts of white paraffin are indicated and the lines are the predicted permittivity () according to Eq. [2].

View larger version (38K): [in a new window]   Fig. 6. Modelled and measured waveforms, (A) Modelled waveforms for homogeneous media, (B) Modelled waveforms for layered media, (C) Measured waveforms for layered media corresponding to those modelled in 6B.

View larger version (26K): [in a new window]   Fig. 7. Apparent permittivity increasing as the layers become proportionally thinner, refractive index and arithmetic averaging acting as bounds.

View larger version (28K): [in a new window]   Fig. 8. Normalized wave velocity (Eq. [14]) as a function of layer thickness with bounds provided by the refractive index and arithmetic-averaging regime. Modeled results reflect 2,4,6...98,100 layers, with layers becoming proportionally thinner. Measurements of Chan and Knight (2001) for interchanging layers of wet and dry sand included.

View larger version (28K): [in a new window]   Fig. 9. Signal delay (left axis) and the root of the apparent permittivity (right axis) versus frequency. The horizontal lines represent the expected delay for refractive (lower line) and arithmetic averaging (upper line) for a volume with 50% Acrylic and 50% water. Because the probe length was 30.24 cm the numerical values in left and right axes are nearly identical.