On the effective measurement frequency of time domain reflectometry in dispersive and nonconductive dielectric materials

Time domain reflectometry (TDR) is one of the most commonly used techniques for water content determination in the subsurface. The measurement results in a single bulk permittivity value that corresponds to a particular, but unknown, "effective" frequency (f_eff). Estimating f_eff using TDR is important, as it allows comparisons with other techniques, such as impedance or capacitance probes, or microwave remote sensing devices. Soils, especially those with high clay and organic matter content, show appreciable dielectric dispersion, i.e., the real permittivity changes as a function of frequency. Consequently, comparison of results obtained with different sensor types must account for measurement frequency in assessing sensor accuracy and performance. In this article we use a transmission line model to examine the impact of dielectric dispersion on the TDR signal, considering lossless materials (negligible electrical conductivity). Permittivity is inferred from the standard tangent line fitting procedure (K_aTAN) and by a method of using the apex of the derivative of the TDR waveform (K_aDER). The permittivity determined using the tangent line method is considered to correspond to a velocity associated with a maximum passable frequency; whereas we consider the permittivity determined from the derivative method to correspond with the frequency associated with the signal group velocity. The effective frequency was determined from the 10-90% risetime of the reflected signal. On the basis of this definition, f_eff was found to correspond with the permittivity determined from K_aDER and not from K_aTAN in dispersive dielectrics. The modeling is corroborated by measurements in bentonite, ethanol and 1-propanol/water mixtures, which demonstrate the same result. Interestingly, for most nonconductive TDR measurements, frequencies are expected to lie in a range from 0.7 to 1 GHz, while in dispersive media, f _eff is expected to fall below 0.6 GHz.