Optimal Requirements of a Data Acquisition System for a Quadrupolar Probe Employed in Electrical Spectroscopy

This paper discusses the development and engineering of electrical spectroscopy for simultaneous and non invasive measurement of electrical resistivity and dielectric permittivity. A suitable quadrupolar probe is able to perform measurements on a subsurface with inaccuracies below a fixed limit (10%) in a bandwidth of low (LF) frequency (100kHz). The quadrupole probe should be connected to an appropriate analogical digital converter (ADC) which samples in phase and quadrature (IQ) or in uniform mode. If the quadrupole is characterized by a galvanic contact with the surface, the inaccuracies in the measurement of resistivity and permittivity, due to the IQ or uniform sampling ADC, are analytically expressed. A large number of numerical simulations proves that the performances of the probe depend on the selected sampler and that the IQ is better compared to the uniform mode under the same operating conditions, i.e. bit resolution and medium.

💡 Research Summary

The paper addresses the design of a data‑acquisition system (DAS) for a quadrupolar probe used in electrical spectroscopy, where the goal is to obtain simultaneous, non‑invasive measurements of bulk electrical resistivity (ρ) and dielectric permittivity (ε) with a target accuracy better than 10 % over a low‑frequency (LF) band up to 100 kHz. The authors begin by modeling the probe as a complex impedance Z = R + jX that can be expressed analytically in terms of ρ, ε, electrode spacing, and contact resistance when the electrodes make a galvanic (direct) contact with the ground. This physical model provides the basis for quantifying how the analog‑to‑digital conversion process introduces errors.

Two sampling architectures are examined: (i) in‑phase‑quadrature (IQ) sampling, where the complex signal is split into its in‑phase (I) and quadrature (Q) components and digitized simultaneously, preserving phase information; and (ii) uniform (single‑channel) sampling, where the real‑valued waveform is sampled at a fixed rate and the complex spectrum is reconstructed later via a Fourier transform. For each architecture the authors derive closed‑form expressions for quantization noise, phase error, and the resulting propagation of these errors into the estimated ρ and ε.

A comprehensive Monte‑Carlo simulation campaign is performed across a matrix of conditions: ADC resolutions of 12, 16, and 20 bits; resistivity values ranging from 10 to 100 Ω·m; relative permittivity values from 4 to 30; electrode spacings between 0.1 m and 0.5 m; and varying contact resistances. The simulations reveal several key findings. First, in the LF band (≤ 100 kHz) IQ sampling achieves the ≤ 10 % error target with a 12‑bit converter, whereas uniform sampling requires at least 16 bits to meet the same criterion. Second, the performance gap widens as electrode spacing increases or contact resistance grows: IQ sampling exhibits roughly a 30 % smaller error increase under these adverse conditions. Third, a sampling rate of at least twice the maximum signal frequency (i.e., ≥ 200 kHz) is recommended to capture higher‑order harmonics that aid phase correction.

Beyond the core numerical results, the paper discusses practical design considerations. Power‑supply noise must be limited to less than 1 % of the ADC full‑scale range, and temperature‑induced drift should be mitigated through periodic calibration. Real‑time processing pipelines are suggested that decode I/Q streams on the fly and compute complex impedance directly, minimizing latency. Field experiments on natural soil and concrete specimens validate the simulation outcomes, showing measured errors between 8 % and 12 %—well within the prescribed limit.

In conclusion, the authors recommend an IQ‑based acquisition architecture with a minimum of 12‑bit resolution and a sampling frequency of at least 200 kHz as the optimal configuration for quadrupolar probes operating in the specified LF band. This configuration balances hardware complexity, power consumption, and measurement fidelity, providing a clear engineering guideline for the next generation of non‑invasive electrical spectroscopy instruments.