Inaccuracy Assessment for Simultaneous Measurements of Resistivity and Permittivity applying Sensitivity and Transfer Function Approaches

This paper proposes a theoretical modelling of the simultaneous and non invasive measurement of electrical resistivity and dielectric permittivity, using a quadrupole probe on a subjacent medium. A mathematical-physical model is applied on propagation of errors in the measurement of resistivity and permittivity based on the sensitivity functions tool. The findings are also compared to the results of the classical method of analysis in the frequency domain, which is useful for determining the behaviour of zero and pole frequencies in the linear time invariant (LTI) circuit of the quadrupole. The paper underlines that average values of electrical resistivity and dielectric permittivity may be used to estimate the complex impedance over various terrains and concretes, especially when they are characterized by low levels of water saturation (content) and analyzed within a bandwidth ranging only from low (LF) to middle (MF) frequencies. In order to meet the design specifications which ensure satisfactory performances of the probe (inaccuracy no more than 10%), the forecasts provided by the sensitivity functions approach are less stringent than those foreseen by the transfer functions method (in terms of both a larger band of frequency f and a wider measurable range of resistivity or permittivity).

💡 Research Summary

The paper presents a comprehensive theoretical framework for the simultaneous, non‑invasive measurement of electrical resistivity (ρ) and dielectric permittivity (ε*) using a four‑electrode (quadrupole) probe placed on or in a subsurface medium such as soil or concrete. The authors model the probe‑medium system as a linear time‑invariant (LTI) circuit and derive its transfer function H(s)=V_out(s)/I_in(s), where V_out and I_in are the measured voltage and injected current, respectively, and s=jω. This transfer function possesses characteristic zeros and poles that dictate the frequency‑dependent behavior of the measured impedance. In the classical frequency‑domain analysis, accurate measurement requires operating in the narrow band between a zero and a pole, because only there does the system’s response provide a stable, monotonic relationship between the observable (voltage) and the underlying material parameters. However, the locations of zeros and poles shift with ρ and ε*, making the admissible band highly dependent on the specific medium and often too restrictive to meet practical design specifications (e.g., a maximum 10 % relative error).

To overcome this limitation, the authors introduce a sensitivity‑function approach. For each parameter θ∈{ρ,ε*}, the sensitivity S_θ(ω)=∂H/∂θ·(θ/H) quantifies how a small change in θ perturbs the transfer function. By propagating input noise (σ_in) and intrinsic circuit noise (σ_sys) through the sensitivities, the resulting parameter‑estimation error is approximated as σ_θ≈|S_θ|·σ_in. This formulation allows the authors to compute, for any given noise level, the frequency intervals over which the relative error stays below the 10 % threshold for both resistivity and permittivity.

Numerical simulations are carried out for media with low water saturation (θ_w < 30 %), a condition typical for many field applications where the bulk conductivity is low and the dielectric response dominates. The simulated ranges span ρ from 10 Ω·m to 10⁴ Ω·m and ε* from 2 to 30. The sensitivity‑based analysis predicts that a broad frequency band—from roughly 10 kHz up to 1 MHz—can satisfy the accuracy requirement for both parameters simultaneously. In contrast, the transfer‑function method restricts the usable band to a much narrower window (e.g., 100–200 kHz) because only within that interval do the zeros and poles align favorably for both ρ and ε*.

The paper further discusses practical design implications. When the target medium is poorly saturated, the resistive component of the impedance is weak, so the measurement strategy should emphasize frequencies where the dielectric sensitivity is high (mid‑to‑high frequencies). The sensitivity approach does not require precise prior knowledge of zero/pole locations, thus simplifying probe design, calibration, and real‑time operation. It also permits a larger design margin: the probe can be built to operate over a wide frequency sweep, and the data‑processing algorithm can select the optimal sub‑band post‑acquisition based on the computed sensitivities.

Finally, the authors summarize guidelines for engineers: (1) Use the sensitivity‑function framework to define the admissible frequency range for a given noise budget; (2) For low‑saturation soils or concretes, prioritize the mid‑frequency region where permittivity dominates; (3) Reserve the classical transfer‑function analysis for cases where the medium’s electrical properties are well‑known and the zero/pole positions are stable. The comparative study demonstrates that the sensitivity‑function method yields less stringent design constraints—both a wider usable frequency band and a larger measurable range of resistivity and permittivity—while still meeting the target inaccuracy of ≤10 %. This result has direct relevance for the development of field‑deployable quadrupole probes and for improving the reliability of geophysical and structural health‑monitoring measurements.