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).


Introductive review.
Electrical resistivity survey in soil science -The electrical resistivity of a surface is a proxy for the spatial and temporal variability of many other physical properties of the subjacent medium.
Samouëlian (Samouëlian et al., 2005) discusses the basic principles of data interpretation and the main advantages or limits of the analysis. This method allows non-destructive and very sensitive investigation, describing subsurface properties without direct inspection. Various techniques are applied according to the required scales of resolution and heterogeneities of the area. A suitable probe injects generated electric currents into a medium and the resulting potential differences are measured. The information is recovered from the potential difference patterns, which provide the form of medium heterogeneities and their electrical properties (Kearey et al., 2002). The greater the electrical contrast between the subsurface matrix and a heterogeneity, the easier the detection. Other authors (Banton et al. 1997) showed that surface resistivity can be considered as a good indication of the variability of other physical properties. The current pattern distributions depend on the medium heterogeneities and are concentrated in a conductive volume. Some linear distributed arrays use four-electrode cells, which are commonly employed in the laboratory for resistivity calibration (Rhoades et al., 1976) and in the field for vertical electrical sounding (Loke, 2001).
Dielectric permittivity survey in soil science [Middle frequencies (MF), 300kHz<f<3MHz] -Analysis in middle frequencies allows the measurement of dielectric permittivity. Fechant and Tabbagh (Fechant and Tabbagh, 1999) developed an interesting approach. They used a MF band for the characterization of permittivity in the natural media. This approach employs an electrostatic quadrupole probe designed to measure resistivity at several centuries of kHz (Tabbagh, 1994). A quadrupole, working at the frequency 455kHz, measures permittivity for determination of water content (Fechant, 1996). However, this approach requires calibration in laboratory.
Electrical resistivity and dielectric permittivity surveys in soil science [low frequencies (LF), 30kHz<f<300kHz] -Analysis in low frequencies allows simultaneous measurements of both electrical resistivity and dielectric permittivity. Tabbagh and Grard, in their experiments (Grard, 1990, a-b) (Grard and Tabbagh, 1991) (Tabbagh et al., 1993), showed that the resistivity and dielectric constant (complex permittivity) of a surface can be measured by a set of four electrodes.
This novel approach, first introduced by Wenner, improved the existing system which provided only a resistivity assessment. In the new method the four electrodes are manually inserted into the subjacent medium. Permittivity, which is sensitive to the presence of water, can also be determined employing a LF probe (below 300 kHz) and plays an important role in the detection of anomalies in the subsurface.
Vannaroni and Del Vento (Vannaroni et al., 2004) (Del Vento and Vannaroni, 2005) used a dielectric spectroscopy probe to determine the complex permittivity of a surface from measurements of transfer impedance of a four-electrode system electrically coupled to the medium.
They defined transfer impedance as the ratio of the voltage measured across a pair of receiving electrodes to the current transmitted by a second pair of electrodes (Vannaroni et al., 2004). This impedance measurement, performed in AC regime capacitive coupling, strongly depends on the geometry of the electrode array but also on the complex permittivity of the subsurface. The advantages offered by this method are due to the fact that the exciter current can be injected into the surface even in the absence of galvanic contact, and, in AC regime, both conduction and displacement currents of the medium can be measured, obtaining further information on the polarizability. In this case the frequency band is 10kHz-1MHz. The lower limit is effectively imposed by two facts: a) firstly, the Maxwell-Wagner effect which limits probe accuracy (Frolich, 1990): the most important limitation happens because of interface polarization effects that are stronger at low frequencies, say below 1kHz depending of medium conductivity; b) secondly, the need to maintain the amplitude of the current at measurable levels as, given the capacitive coupling between electrodes and soil, the current magnitude is proportional to the frequency. On the other hand, the upper limit is opportunely fixed to allow the analysis of the system in a regime of quasi static approximation and neglect the velocity factor of the cables used for the electrode harness, that in turn degrades the accuracy of the mutual impedance phase measurements. Thus, it is possible to exploit the analysis of the system in the low and middle frequency band where the electrostatic term results considerable. The general electromagnetic (e.m.) calculation provides lower values than the static case; a high resistivity narrows the differences. So, comparing, above 1 MHz the general e.m. calculation must be preferred, while under 500 kHz the static case should be used and between 500 kHz and 1 MHz both methods could be used (Tabbagh et al. 1993).
The present 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. This 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 or content (Knight and Nur, 1987) and analyzed within a frequency bandwidth ranging only from LF to MF (Myounghak et al., 2007)(Al-Qadi et al., 1995. In order to meet the design specifications which ensure satisfactory performances of the probe (inaccuracy no more than 10%), the forecasts provided by the theory of error propagation [suggested by (Vannaroni et al., 2004)] applying the sensitivity functions approach, explicitly developed in the paper, are less stringent than those foreseen by the analysis in the frequency domain [suggested by (Grard and Tabbagh, 1991)], deepening here the transfer function method to analyze the zero and pole behaviour (in terms of both a larger band of frequency f and a wider measurable range of resistivity ρ or permittivity ε r ).
The paper is organized as follows. Section 2 discusses the Cole-Cole empiric function: for simplicity of analysis, the dielectric dispersion is assumed very low: this operating condition is satisfied when the electrical spectroscopy is performed only on non-saturated water materials and especially in a suitable band of low and middle frequencies. Section 3 introduces the quadrupole probe. Sec. 4 provides a theoretical modelling which applies to both the sensitivity functions approach (Sec 4.a.) and the transfer function method (4.b.). In Sec 5, the configurations of the quadrupole are defined and discussed. Conclusions are drawn in Sec. 6. Finally, an outline of the somewhat lengthy calculations is presented in the Appendices A and B.

Discussing Cole-Cole empiric function.
Even if, according to Debye polarization mechanisms (Debye, 1929) or Cole-Cole diagrams (Auty and Cole, 1952), the complex permittivity of various materials in the frequency band from VLF to VHF exhibits several intensive relaxation effects and a non-trivial dependence on the water saturation (Chelidze and Gueguen, 1999) , anyway 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 content (Knight and Nur, 1987) and analyzed within a frequency bandwidth ranging only from LF to MF (Myounghak et al., 2007)(Al-Qadi et al., 1995.
Many functions have been proposed to fit the data of dielectrics. Among them, there are those obtained by attempts to model the physical processes or those of simple empirical functions, which are used to parameterize the data without the knowledge of the involved mechanisms. A widely used empirical function has been proposed by brothers Cole and it is based on the theory of Debye relaxation, the first one to have treated this phenomenon.
The Cole-Cole empiric function defines the first order dielectric response of materials in frequency domain, consisting of real and imaginary parts: where ε 0 is the dielectric constant in vacuum.
The electrical conductivity σ(f) and dielectric permittivity ε r (f) exhibit limit values at low and high frequencies, σ L , ε r,L and σ H , ε r,H , which are linked by the relaxation time τ, so, as it can be noticed, permittivity and conductivity can not vary independently of each other (Frolich, 1990).
At the characteristic frequency of relaxation, f c =1/(2π·τ), the permittivity ε r assumes an intermediate value between the values of high and low frequency, ε r,L and ε r,H . Alternately, the relaxation frequency f c could be considered as that frequency at which the conductivity σ assumes the middle value between the two limit values, σ L and σ H .
In reality, the relationship (2.1) is a generalization of Debye equation, having the purpose to take into account, through the introduction of another parameter α (inclusive between 0 e 1), the enlargement of dispersion region due to the complexity of structure and the composition of materials. Note that, for α = 0, eq. (2.1) can be exactly reduced to Debye equation. It is to be underlined that the parameter α is a increasing function of the water saturation S W , such that α(S W =0)→0, reaching a limit value α L >0 for S W →1 (Knight and Nur, 1987 (2 ) 1 f α π τ − << . (2.7) In fact, the constant τ depends on the physical process under consideration and it has an order of magnitude that varies from a few picoseconds for the orientation of electrons and small dipolar molecules, up to a few seconds for the effects of counter-ions or for the interfacial polarization (Frolich, 1990

Quadrupole probe.
When using a quadrupole probe [ fig. 1] the response depends on geometric parameters, like the height of each electrode above the ground surface and the separation of the electrodes, and on physical parameters including frequency, electrical conductivity and dielectric permittivity. When a medium is assumed to be linear and its response linearly dependent on the electrical charges of the two exciting electrodes, the simplest approach is static calculation (Tabbagh et al., 1993), especially using a low operating frequency. If the electrodes have small dimensions relative to their separations, then they can be considered as points. Moreover, if the current wavelength is much larger than all the dimensions under consideration, then quasi-static approximation applies (Grard, 1990, a-b).
The quadrupole probe [ fig. 1] measures a capacitance in vacuum C 0 (L), which is directly proportional to its characteristic geometrical dimension, i.e. the electrode-electrode distance L, both in a linear (Wenner) configuration [ fig. 2.a], and in a square arrangement [ fig. 2.b], which is greater by a factor α=1/(2-2 1/2 )>1, where ε 0 is the dielectric constant in vacuum.
When the quadrupole, specified by the electrode-electrode distance L, has a galvanic contact with the subjacent medium, of electrical conductivity σ and dielectric permittivity ε r , it measures a transfer impedance Z N (f,L,σ,ε r ), which consists of parallel components of resistance R N (L,σ) and capacitance C N (L,ε r ). The resistance R N (L,σ) depends only on L and σ (Grard and Tabbagh, 1991) while C N (L,ε r ) depends only on L and ε r (Grard and Tabbagh, 1991) As a consequence, if the probe, besides grazing the medium, measures the conductivity σ and permittivity ε r working in a frequency f much lower than the cut-off frequency f T =f T (σ,ε r )= σ/(2πε 0 (ε r +1)), the transfer impedance Z N (f,L,σ,ε r ) is characterized by the phase Ф N (f,σ,ε r ) and modulus |Z| N (L,σ). The phase Ф N (f,σ,ε r ) depends linearly on f with a maximum value of π/4 and is directly proportional to the ratio (ε r +1)/σ; while |Z| N (L,σ) does not depends on f, and is inversely proportional to both L and σ. In fact, if Z N (f,L,σ,ε r ) consists of the parallel components of R N (L,σ) (3.3) and C N (L,ε r ) (3.4), then it is fully characterized by the high frequency pole f T =f T (σ,ε r ), which cancels its denominator: the transfer impedance acts as a low-middle frequency band-pass filter with cut-off f T =f T (σ,ε r ), in other words the frequency equalizing Joule and displacement current. In the operating conditions defined in sec 2, average values of σ may be used over the band ranging from LF to MF, therefore |Z| N (L,σ) is not function of frequency below f T .
Instead, when the quadrupole probe [ fig. 1] has a capacitive contact with the subjacent medium and the geometry of the probe is characterized by the ratio x between the height above ground h and the Actually, Grard and Tabbagh preferred to introduce the complementary δ(x) of the geometrical So, if the quadrupole works in the pulse frequency ω=2πf, which can be normalized with respect to the cut-off ω T =2πf T (Grard and Tabbagh, 1991), then the probe measures a transfer impedance Z(Ω,x,σ,ε r ) which consists of the resistance R(Ω,x,σ,ε r ) and capacitance C(Ω,x,σ,ε r ) parallel components (Grard and Tabbagh, 1991), (3.11) Inverting eqs. (3.10) and (3.11), σ and ε r can be expressed as functions of R and C, i.e. (3.13) In our opinion, once fixed the pair's (f, x) degrees of freedom, it is not suitable to choose (R,C) as independent variables and then (σ, ε r ) as dependent variables [eqs. (3.12)-(3.13)]. Instead, it is more appropriate to consider (σ, ε r ) as quantities of physical interest and consequently eqs. (3.10)-(3.11) as starting point for the theoretical development. In fact, even if the physics does not forbid to choose (R,C) as independent variables, running the way (R,C) → (σ, ε r ), anyway the procedures of design should choose (σ, ε r ) as independent variables, running a preferential way (σ, ε r ) → (R,C).
According to the two following practical approaches: a) [(σ, ε r ) as independent variables in order] to establish the class of media with conductivity and permittivity (σ, ε r ) which are investigable by a quadrupole working in a fixed band B and specified by a known geometry x; b) [preferential way (σ, ε r ) → (R,C) since] once a subjacent medium with electrical conductivity σ and dielectric permittivity ε r is selected, one can project the quadrupole probe specifications R and C both in frequency f and in height/dimension ratio x.

Theoretical modelling.
The measurements taken using the quadrupole probe are affected by errors mainly originating from uncertainties associated with transfer impedance, from dishomogeneities between the modelled and actual stratigraphy, and from inaccuracy of the electrode array deployment above the surface (Vannaroni et al., 2004). Errors in impedance result mainly from uncertainties in the electronic systems that perform the amplitude and phase measurements of the voltages and currents (Del Vento and Vannaroni, 2005). The above uncertainties were assumed constant throughout the whole frequency band even though their effects, propagating through the transfer function, will produce a frequency dependent perturbation.

Sensitivity Functions Approach .
This paper proposes to develop explicitly the sensitivity function approach which is implied in the theory of error propagation suggested by (Vannaroni et al., 2004). In fact, the section introduces a mathematical-physical model for the propagation of errors in the measurement of electrical conductivity σ and dielectric permittivity ε r , based on the sensitivity functions tool (Murray-Smith, 1987). This is useful for expressing inaccuracies in the measurement of So, the inaccuracies Δσ/σ, in the measurement of the electrical conductivity σ, and Δε r /ε r , in the dielectric permittivity ε r , can be expressed as a linear combination of the inaccuracies Δ|Z|/|Z| and ΔΦ Z /Φ Z in the measurement of the transfer impedance, respectively in modulus |Z| and in phase Φ Z , are the pairs of sensitivity functions for the transfer impedance, both in |Z| and Φ Z , relative to the conductivity σ and permittivity ε r , whose expressions are reported in Appendix A. The conditions σ=const and ε r =const in eqs. (4.1) and (4.2) underline not so much that constant values of electrical conductivity and dielectric permittivity are used to estimate the complex impedance over various terrains and concretes under the operating conditions defined in Sec. 2, as that the quantities σ and ε r are not independent of each other, since the electrical displacement shows a phase-shift with respect to the electrical field (Frolich, 1990); so, for need to distinguish the inaccuracies in measurements of conductivity and permittivity, the inaccuracy Δσ/σ can only be calculated assuming there is not uncertainty for ε r (Δε r =0 ⇔ ε r =const) and vice versa. can be applied. Necessarily, the inaccuracy Δσ/σ, in the measurement of the electrical conductivity σ, is calculated assuming ε r =const, and then the inaccuracy Δε r /ε r , for the dielectric permittivity ε r , assuming σ=const. As a consequence, the mathematical calculations should be done recalling the fact that eqs. (3.10)-(3.11) have been considered as starting point for the theoretical development.
The inaccuracies Δσ/σ for the conductivity σ and Δε r /ε r for the permittivity ε r can be more directly expressed as functions of (f, x, σ, ε r ) by calculating the sensitivity functions ( The interesting physical results obtained using this sensitivity functions approach are discussed below. If the quadrupole probe has a galvanic contact with the subjacent medium, i.e. h=0, then the inaccuracies Δσ/σ in the measurement of the electrical conductivity σ and Δε r /ε r in the dielectric permittivity ε r are minimized in the frequency band B of the quadrupole, for all its geometric configurations and media; and, even if h≠0, the design of the probe must still be optimized with respect to the minimum value of the inaccuracy Δε r /ε r in ε r , which is always higher than the corresponding minimum value of the inaccuracy Δσ/σ in the band B of the probe, for all its configurations and media (Tabbagh et al., 1993) (Vannaroni et al., 2004).
Under quasi static approximation, only if the quadrupole probe is in galvanic contact with the subjacent medium, i.e. h=0, and considering that the sensitivities functions are defined as normalized functions, then our mathematical-physical model predicts that the sensitivities of the transfer impedance relative to the conductivity σ and permittivity ε r are independent of the characteristic geometrical dimension of the quadrupole, i.e. electrode-electrode distance L.
If the probe grazes the medium, then the transfer impedance Z N (σ,L) consists of the resistance R N (σ,L), which is independent of ε r , and parallel capacitance C N (ε r ,L), which is independent of σ, such that: the sensitivity function R S σ for R relative to σ is a constant equal to (-1); the sensitivity ( ) r C r S ε ε for C relative to ε r is independent of σ, behaving as the function ε r /(ε r +1) of ε r ; the r R S ε function for R relative to ε r and the C S σ function for C relative to σ are identically null. As a consequence, the inaccuracy ΔR/R for R shows the same behaviour versus frequency of the inaccuracy Δσ/σ in the measurement of σ, as ΔR/R=| R S σ |Δσ/σ=Δσ/σ, and the inaccuracy ΔC/C for C shows a similar behaviour versus frequency with respect to the inaccuracy Δε r /ε r for ε r , as for |Z| and Φ Z relative to ε r are independent of σ, such that they behave as the function ε r /(ε r +1) of ε r . As a consequence, the ratio between Δε r /ε r and Δσ/σ is independent of σ, behaving as the function (1+1/ε r ) of ε r , and Δσ/σ is a constant equal to Δσ/σ=4Δ|Z|/|Z|+πΔΦ Z /Φ Z . As post-test, only assuming the conditions σ=const and ε r =const in eqs.

Transfer Function method.
This paper proposes to deepen the transfer function method, by analyzing the zero and pole behaviour, which is implied in the frequency domain analysis suggested by (Grard and Tabbagh, 1991). In fact, the section introduces the method of analysis in the frequency domain for determining the behaviour of the zero and pole frequencies in the LTI circuit of the quadrupole probe [ fig. 1]. In order to satisfy the operative conditions of linearity for the measurements, if the quadrupole has a capacitive contact with the subjacent medium, then one should impose the frequency f of the probe to be included between the zero z M and the pole p M of the transfer impedance, so its modulus to be almost constant within the frequency band (Grard and Tabbagh, 1991), Based on the above conditions, an optimization equation is deduced for the probe, which links the optimal ratio x between its height above ground and its characteristic geometrical dimension only to the dielectric permittivity ε r of the medium, so that The interesting physical results obtained using this transfer function method are discussed below. In order to meet the design specifications which ensure satisfactory performances of the probe (inaccuracy no more than 10%), the forecasts provided by the theory of error propagation [suggested by (Vannaroni et al., 2004)] applying the sensitivity functions approach, explicitly developed in the paper, are less stringent than those foreseen by the analysis in the frequency domain [suggested by (Grard and Tabbagh, 1991)], deepening here the transfer function method to analyze the zero and pole behaviour (in terms of both a larger band of frequency f and a wider measurable range of resistivity ρ or permittivity ε r ) [figs. 6,7].
In fact, given a surface (for example, a non-saturated concrete with low conductivity σ=10 -4 S/m and ε r =4) with dielectric permittivity ε r [ fig. 6]: • if the quadrupole probe has a capacitive contact with the subjacent medium, i.e. h≠0, then, having defined an optimal ratio x opt =h opt /L between an optimal height h opt above ground and the characteristic geometrical dimension L, the transfer impedance Z(f,x opt ) in units of 1/h opt , calculated in x opt , is a function of the working frequency f such that its modulus |Z|(f,x opt ), in units of 1/h opt , is almost constant between a zero frequency z(x opt ), almost one decade higher than a minimum frequency value f min (x opt ) allowing the inaccuracy Δε r /ε r (f,x opt ) in the measurement of ε r below a prefixed limit (10%), and a pole p(x opt ), almost one decade lower than the maximum value of frequency f max (x opt ) satisfying the requirement that the inaccuracy Δε r /ε r (f,x opt ) for ε r is below 10% [ fig. 6][fig. 8]; • if h=0, i.e. the quadrupole of electrode-electrode distance L grazes a medium of conductivity σ, then the transfer impedance Z(f,L), calculated in L, is a function of the working frequency f such that its modulus |Z|(f,L) is constant down to the cut-off frequency f T =f T (σ,ε r ), which is higher than an optimal frequency f opt (L) minimizing the inaccuracy Δε r /ε r (f,L). Materials characterized by a low σ or a high ε r lead to the effect of leftward shifting of the cut-off frequency f T , so reducing the optimal frequency f opt (L) [ fig. 9]; • usually, on a selected surface, it is possible to verify that the probe in capacitive contact Moreover, once the frequency band B is fixed [ fig. 7]: • if the quadrupole probe has a capacitive contact with the subjacent medium, then the ratio x=h/L, between the height h above ground and the characteristic geometrical dimension L, ranges from the lower limit x low , corresponding to water (ε r =81).
In a preliminary analysis, based on the transfer functions approach, it follows that the quadrupole, designed with the height/dimension ratio x=h/L, optimally measures dielectric permittivity ε r,opt ; the modulus |Z|(x,σ,ε r,opt ), in units of 1/h, of its transfer impedance, calculated in x and ε r,opt , function of the electrical conductivity σ, is characterized by a zero z(σ,ε r,opt ) and a pole p(σ,ε r,opt ) frequency, which respectively fall near the lower and upper limit of B when σ is measured within a range of lower limit low σ ′ and upper limit up σ ′ .
In a deeper analysis, based on the sensitivity function method, it is possible to verify, still designing the quadrupole with the ratio x=h/L for an optimal measurement of ε r,opt , the measurable range of σ; the inaccuracy Δε r /ε r (x,σ,ε r,opt ) in the measurement of ε r,opt , a function of σ, is below a prefixed limit (10%)  • If h=0, i.e. the probe of electrode-electrode distance L grazes a medium of conductivity σ and permittivity ε r , then the transfer impedance Z(L,σ,ε r ), calculated in L, is a function of σ and ε r such that its cut-off frequency f T =f T (σ,ε r ), a function of both σ and ε r , ranges from f T,min =100kHz to f T,max =1MHz for the materials belonging to an (σ,ε r )-domain, almost superimposable with the corresponding one within which the inaccuracy Δε r /ε r (L,σ,ε r ,) for ε r is below about 10% [ fig. 11].
• Usually, having fixed the frequency band, the probe in capacitive contact performs optimal measurements over surfaces of lower conductivities compared to the case when the probe is in galvanic contact, as the respective conductivities are higher even of almost one magnitude order [tabs. 1, 3].

Quadrupole configurations.
The transfer impedance of a quadrupolar array can be evaluated for any arbitrary configuration. As a general rule it is assumed that subsurface electrical sounding becomes scarcely effective at depths greater than the horizontal distance between the electrodes (Grard and Tabbagh, 1991) (Vannaroni et al., 2004). This paper considers two kinds of probes, i.e. square and linear (Wenner) configurations. The square configuration is an array of two horizontal parallel dipoles with the four electrodes positioned at the corners of a square (Grard and Tabbagh, 1991). Instead, the Wenner arrangement consists of four terminals equally spaced from one another along a straight horizontal line (Vannaroni et al., 2004).
When the quadrupole is in galvanic contact, i.e. h=0, with a subjacent medium of electrical conductivity σ and dielectric permittivity ε r , the Wenner configuration measures a resistance R N,W =2ε 0 /σC 0,W and a parallel capacitance C N,W =C 0,W ·(ε r +1)/2, while in the square arrangement Instead, when the quadrupole is in capacitive contact with the subjacent medium, and so the ratio x=h/L between its height h above ground and its electrode-electrode distance L is not null, i.e. 0<x≤1, then the quadrupole is characterized by a geometrical factor K(x) [δ(x)], decreasing (increasing) function of x, which, in the square configuration, slopes down (up) more swiftly than the Wenner arrangement, so assuming smaller (larger) values especially for 1/2<x<1 [ fig. 8.a]. As a consequence, a probe with a fixed L, which performs measurements on a medium of dielectric permittivity ε r , could be designed with an optimal height/dimension ratio x opt =h opt /L which, in the square configuration, is smaller than the Wenner arrangement, because its factor δ(x) slopes up more swiftly increasing the ratio x, so reaching the prefixed optimal value δ opt (ε r )≈2/(15ε r +17) in correspondence with a smaller x opt . In simpler terms, if the probe is in capacitive contact with the medium, in order to perform optimal measurement of the permittivity, then the square configuration needs to be raised above ground less than the Wenner arrangement, their electrode-electrode distance being equal. In fact, x ranges from x W,low =0.022 in the linear configuration and from  -Qadi et al., 1995). In order to meet the design specifications which ensure satisfactory performances of the probe (inaccuracy no more than 10%), the forecasts provided by the theory of error propagation [suggested by (Vannaroni et al., 2004) applying the sensitivity functions approach, explicitly developed in the paper, are less stringent tham those foreseen by the analysis in the frequency domain [suggested by (Grard and Tabbagh, 1991)], deepening here the transfer function method to analyze the zero and pole behaviour (in terms of both a larger band of frequency f and a wider measurable range of resistivity ρ or permittivity ε r ).
It is interesting to compare the results of the present paper with those published in scientific literature (Grard and Tabbagh, 1991) (Vannaroni et al., 2004). In accordance, the sensitivity functions approach, provides the following results: a) if the quadrupole probe is in galvanic contact with the subsurface, i.e. h=0, then the inaccuracies Δσ/σ in the measurement of conductivity σ and Δε r /ε r for permittivity ε r are minimized in the frequency band B of the quadrupole, for all its geometric configurations and media; b) and, even if h≠0, the design of the probe must be optimized with reference to the minimum value of the inaccuracy Δε r /ε r for ε r , which is always higher than the corresponding minimum value of the inaccuracy Δσ/σ in the band B, for all its configurations and media.
More explicitly than in referred papers, the transfer functions method provides results for which, in order to satisfy the operative conditions of linearity for the measurements: a) if the quadrupole has a capacitive contact with the subjacent medium, then one should impose the frequency f of the probe to be included between the zero z M and the pole p M of the transfer impedance, so its modulus to be almost constant within the frequency band, so an optimization equation is deduced for the probe, which links the optimal ratio x between its height above ground and its characteristic geometrical dimension only to the dielectric permittivity ε r of the medium; b) instead, if the quadrupole is in galvanic contact with the subjacent medium, then one should impose the working frequency f of the quadrupole to be lower than the cut-off frequency of the transfer impedance, so its modulus to be constant below the cut-off frequency, so it is optimal to design the characteristic geometrical dimensions of the probe or establish the measurable ranges of the conductivity σ and permittivity ε r of the medium.
Unlike referred papers, the sensitivity functions approach and the transfer functions method provide results which permit an assessment of the performance of the quadrupole probe in galvanic and capacitive contact: a) usually, having selected the surface (for example, a non-saturated concrete with low conductivity σ=10 -4 S/m and ε r =4), it is possible to verify that the quadrupole in capacitive contact performs optimal measurements over the band [f min (x opt )<z(x opt ), f max (x opt )>p(x opt )], which is shifted to lower and higher frequencies compared to the case when the probe is in galvanic contact, being the corresponding band [f min , f max ] narrower of almost one decade in frequency, especially increasing the value of ε r ; b) usually, having fixed the frequency band, the quadrupole in capacitive contact provides optimal measurements over surfaces of lower conductivity compared to when the probe is in galvanic contact, being the respective conductivities higher even of almost one magnitude order.
On this basis, some constraints were established to design a quadrupole probe for conducting measurements of electrical resistivity and dielectric permittivity in a regime of alternating current at low and middle frequencies (10kHz-1MHz). Measurement is carried out using four electrodes laid on the surface to be analyzed and, through a measurement of transfer impedance, there is the possibility of extracting the resistivity and permittivity of the material. Furthermore, increasing the distance between the electrodes, it is possible to investigate the electrical properties of the subsurface structures to greater depth. The main advantage of the quadrupole is being able to conduct measurements of electrical parameters with a non destructive technique, thereby enabling characterization of precious and unique materials. Also, in appropriate arrangements, measurements could be carried out with electrodes slightly raised above the surface, enabling completely nondestructive analysis, although with a greater error. The probe is able to perform measurements on materials with high resistivity and permittivity in an immediate way, without subsequent stages of post-analysis of data.

Appendix A.
There follows a discussion of the influence of the inaccuracies in transfer impedance in modulus and phase on the measurement of electrical conductivity and dielectric permittivity. The mathematical tool best suited to this purpose applies the so-called sensitivity functions (Murray-Smith, 1987), which formalize the intuitive concept of sensitivity as the ratio between the percentage error of certain physical quantities (due to the variation of some parameters) and the percentage error of the same parameters.
are, in turn, linear combinations of the sensitivity function pairs ( R S σ , C S σ ) and ( r R S ε , r C S ε ) for transfer impedance, in both the resistance R and capacitance C parallel components, relative to σ and ε r , r r 2 2  Therefore, the theorem of the derivative for the inverse function can be applied. In fact, under the condition σ=const (or ε r =const), both |Z| and Φ Z are invertible functions of ε r (or σ), i.e. strictly increasing or decreasing monotonic functions of ε r (or σ).
besides the parallel capacitance C(f,x,σ,ε r ), in units of 1/h [see eq. (3.11)], which can be expressed as a transfer function characterized by a low frequency pole, p C (f,x,σ,ε r ), a zero in higher frequencies z C (f,x,σ,ε r )>p C (f,x,σ,ε r ), and a static gain K C (x), where the capacitance pole p C (f,x,σ,ε r ) coincides with the resistance pole z R (f,x,σ,ε r ),           zed by a modulus w m Fig. 10. In the hypothesis that Δ|Z|/|Z|=ΔΦ Z /Φ Z =10 -3 , referring to both the inaccuracies Δσ/σ(σ,ε r ), for the electrical conductivity σ, and Δε r /ε r (σ,ε r ), for the dielectric permittivity ε r , as functions of σ and ε r , and when the quadrupole probe is designed in the Wenner linear configuration, working in a fixed band B=100kHz, with an height/dimension ratio x W,concrete =0.087, which is optimal for a capacitive contact only with a concrete of permittivity ε r =4: plots for the orthogonal projections o Fig. 11. With reference to a quadrupole probe, in galvanic contact, working in a fixed band B=100kHz, plots for the domains (σ,ε r ) of the electrical conductivity σ and the dielectric permittivity ε r such that: the transfer impedance is characteri ith a cut-off frequency f T =f T (σ,ε r )=σ/(2πε 0 (ε r +1)) ranging in the interval T f [100kHz,1MHz] ∈ (a); both the inaccuracies Δσ/σ(σ,ε r ), in the measure of the conductivity σ, and Δε r /ε r (σ,ε r ), of the permittivity ε r , result below a prefixed limit of 10% [Δ|Z|/|Z|=ΔΦ Z /Φ Z =10 -3 ] (b) [Tabs. 1, 3].