A Comparison of Second-Order Derivative Based Models for Time Domain Reflectometry Waveform Analysis

Adaptive waveform interpretation with Gaussian filtering (AWIGF) and second-order bounded mean oscillation Z(2)(u,t,r) are time domain reflectometry (TDR) analysis models. The AWIGF was originally designed for relatively long-probe (>150 mm) TDR waveforms, while Z(2)(u,t,r) was originally designed for relatively short-probe (<50 mm) TDR waveforms. The performances of AWIGF and Z(2)(u,t,r) on both long and short TDR probes have not been evaluated. The main objective of this study was to evaluate theoretically and experimentally the AWIGF and Z(2)(u,t,r) performances on long and short TDR probes. The evaluations are performed via mathematical analysis, and measurements of long probe and short probe waveforms in CaCl2 solutions with various electrical conductivities (EC), adding Gaussian noise, and testing the stability of Z(2)(u,t,r) and AWIGF. A corner-preserving filter (CPF) is proposed to improve the stability of AWIGF on short-probe TDR waveforms. The CPF preserves the second order derivatives of the waveforms, and emphasizes the reflection positions (t(2)) compared to the original Gaussian filter. Both theoretical and experimental tests illustrate the consistency of Z(2)(u,t,r) and AWIGF. The standard deviations of relative permittivity (er) are <5% for all noise levels. In conclusion, Z(2)(u,t,r) and AWIGF can provide stable analysis for long and short probe TDR waveforms. The AWIGF with CPF is capable of stably analyzing challenging short probe TDR waveforms. The original AWIGF exhibits the lowest standard deviation of er at a given EC, whereas AWIGF with CPF filter exhibits the lowest bias of er across solutions varying in EC.