Layout 6 ANNALS OF GEOPHYSICS, 53, 2, APRIL 2010; doi: 10.4401/ag-4711 ABSTRACT The present study proposes a theoretical modeling of simultaneous and noninvasive measurements of electrical resistivity and dielectric permittivity using a quadrupole probe on a subjacent medium. A mathematical-physical model is applied to the propagation of errors in the measurement of resistivity and permittivity based on a sensitivity functions tool. The findings are also compared with results of the classical method of analysis in the frequency domain, which is useful for determining the behavior of zero and pole frequencies in the linear time invariant circuit of the quadrupole. This study underlines that average values of electrical resistivity and dielectric permittivity can be used to estimate complex impedance over various terrains and concretes, especially when they are characterized by low levels of water saturation (content), and are analyzed within a bandwidth ranging only from low to middle frequencies. To meet the design specifications, that ensure satisfactory performances of the probe (inaccuracies of no more than 10%), the forecasts provided by the sensitivity functions approach are discussed in comparison with those foreseen by the transfer functions method (in terms of both the band of frequency f and the measurable range of resistivity t, or permittivity fr ). 1. Introductory review 1.1. 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] discussed the basic principles of data interpretation and the main advantages and limits of such an analysis. This method allows nondestructive and very sensitive investigations, which can describe subsurface properties without direct inspection. Various techniques are applied according to the required scales of resolution and the 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 the 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 studies have shown that surface resistivity can be considered as a good indication of the variability of other physical properties [Banton et al. 1997]. The current pattern distributions depend on the medium heterogeneities and they are concentrated in a conductive volume. Some linear distributed arrays use four-electrode cells, which are commonly used in the laboratory for resistivity calibration [Rhoades et al. 1976] and in the field for vertical electrical sounding [Loke 2001]. 1.2 Middle frequency dielectric permittivity surveys in soil science Analyses in the middle frequencies (MFs; 300 kHz 0 for SW→1 [Knight and Nur 1987]. Indeed, the complex dielectric permittivity is flattened with decreasing water content or increasing frequency [Al-Qadi et al. 1995, Myounghak et al. 2007]. The complex dielectric permittivity can be approximated to a constant if its dominant term (fr,L− fr,H)/ [1 + (j2rfx)1–a] is a function that is almost independent of the frequency: (2.5) This operating condition of Equation (2.5) holds when the materials are characterized by low water content, i.e.: (2.6) and are analyzed over a band lower than the MFs, i.e.: MEASUREMENT OF RESISTIVITY AND PERMITTIVITY Figure 2a. Quadrupole probe in the linear Wenner configuration. Figure 2b. Quadrupole probe in the square configuration. 2 1 ( 2 ) 2 , (f) (f) j f (f) j f j f 0 , 1 , 0 ,L L r complex r r H r r H = = = + + f f r f v f r x f f r f v - - - -a 1 (2 ) 2 (2 ) sin ( 2) 1 (2 ) sin ( 2) , (f) f f f , , 2 (1 ) 1 1 ,Lr r H r r H + + + ar ar f f f f r x r x r x - = - a a a - - - 2 1 (2 ) 2 (2 ) sin ( 2) (2 ) cos ( 2) , (f) f f f f , 0 2 (1 ) 1 1 ,L L r r H = = + + ar ar f f v v r f r x r x r x - - a a a - - - 6 @ , 0L L H Hx f f f v v - = - ^ h (2 ) << 1.f 1r x a- 0 ,"a (f)r complexf (2.7) Indeed, the constant x depends on the physical processes 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 interfacial polarization [Frölich 1990]. Therefore, in the present study let us refer to the (v, fr) values in the LF to MF bandwidth proposed for various terrains by Edwards [1998] and for concretes by Polder et al. [2000] and Laurents et al. [2005]. 3. Quadrupole probe When using a quadrupole probe (Figure 1), the response depends on both geometrical parameters, like the height of each electrode above the ground surface and the separation of the electrodes, and 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 a static calculation [Tabbagh et al. 1993], especially using a low operating frequency. If the electrodes have small dimensions relative to their separation, then they can be considered as points. Moreover, if the current wavelength is much larger than all of the dimensions under consideration, then the quasi-static approximation applies [Grard 1990a, Grard 1990b]. The quadrupole probe (Figure 1) measures a capacitance in a vacuum C0(L) that is directly proportional to its characteristic geometrical dimension, i.e. the electrode- electrode distance L, both in a linear Wenner configuration (Figure 2a), (3.1) and in a square arrangement (Figure 2b), (3.2) which is greater by a factor a = 1/(2 − 21/2) >1, where f0 is the dielectric constant in a vacuum. When the quadrupole specified by the electrode- electrode distance L has galvanic contact with the subjacent medium of electrical conductivity v and dielectric permittivity fr, it measures a transfer impedance ZN (f, L, v, fr) that consists of parallel components of resistance RN (L, v) and capacitance CN (L, fr). The resistance RN (L, v) depends only on L and v [Grard and Tabbagh 1991]: (3.3) while CN (L, fr) depends only on L and fr [Grard and Tabbagh 1991]: (3.4) As a consequence, as well as grazing the medium, if the probe measures the conductivity v and permittivity fr working in a frequency f much lower than the cut-off frequency fT = fT (v, fr) = v/(2rf0(fr + 1)), the transfer MEASUREMENT OF RESISTIVITY AND PERMITTIVITY 4 (a) (b) Figure 3. On the hypothesis that D|Z|/|Z|= DUZ/UZ= 10−3, the inaccuracy Dfr/fr in the measurement of the dielectric permittivity fr plotted: (a) as a function Dfr/fr (f, x) of both the frequency f in the band f∈ [0, flim], with flim= 1 MHz, and the ratio x = h/L between the height h above ground and the characteristic geometrical dimension L, as 0 < x ≤ 1, when the quadrupole probe designed in the linear Wenner configuration has a capacitive contact on a non-saturated concrete of low electrical conductivity, i.e. v = 10−4 S/m, fr= 4; (b) as a function Dfr/fr (v, fr) of both the conductivity v and the permittivity fr, when the quadrupole working in a fixed band B = 100 kHz is in galvanic contact on a class of concretes such that v∈ [10−4 S/m, 2·10−2 S/m]. 2 1 .f < rx 4 ,C L L0 0= $rf^ h 4 ,C L L0 0= $a rf^ h ;, 2R L C L0 0 N =v f v^ ^h h , 1C L C L .2 1 0N r r= +$f f^ ^ ^h h h 5 impedance ZN (f, L, v, fr) is characterized by the phase UN (f, v, fr) and modulus |Z|N (L, v). The phase UN (f, v, fr) depends linearly on f, with a maximum value of r/4, and it is directly proportional to the ratio (fr + 1)/v; while |Z|N (L, v) does not depend on f, and is inversely proportional to both L and v. Indeed, if ZN (f, L, v, fr) consists of the parallel components of RN (L, v) – see Equation (3.3) – and CN (L, fr) – see Equation (3.4) –, then it is fully characterized by the HF pole fT = fT (v, fr), which cancels its denominator: the transfer impedance acts as a LF-MF band-pass filter with cut-off fT = fT (v, fr); in other words, the frequency equalizing Joule and displacement current. Under the operating conditions defined in Section 2, average values of v can be used over the band ranging from LF to MF; therefore, |Z|N (L, v) is not a function of frequency below fT. Instead, when the quadrupole probe (Figure 1) has 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 electrode- electrode distance L, (3.5) its configurations can be entirely defined by a suitable geometrical factor K(x), which depends on the height/ dimension ratio x. This was introduced by Grard and Tabbagh [1991], and can be specified for the linear Wenner configuration (Figure 2a): (3.6) MEASUREMENT OF RESISTIVITY AND PERMITTIVITY (a) (b) (c) (d) Figure 4. Sensitivity functions and for the transfer impedance both in modulus |Z| and in phase UZ, relative to the dielectric permittivity fr, plotted: (a, c) as functions (a) and (c) of both the working frequency f and the height/dimension ratio x = h/L under the same operative conditions as Figure 3a; (b, d) as functions (b) and (d) of both the conductivity v and the permittivity fr under the same operative conditions as Figure 3b. S Z r ; ; f S r Z f U ( , )S f xZ r ; ; f ( , )S f x r Z f U ( , )S Z rr v f ; ; f ( , )S rr Z v f f U ,x L h= 2 1 4 1 ,K x x x2 1 2 2 1 2= + +-- -^ ^ ^h h h and the square arrangement (Figure 2b): (3.7) Actually, Grard and Tabbagh [1991] preferred to introduce the complementary d(x) of the geometrical factor K(x), i.e.: (3.8) where K (x = 0) = 1 and d (x = 0) = 0. So, if the quadrupole works in the pulse frequency ~ = 2rf, which can be normalized with respect to the cut-off ~T = 2rfT [Grard and Tabbagh 1991], (3.9) then the probe measures a transfer impedance Z (X, x, v, fr) which consists of the resistance R (X, x, v, fr) and capacitance C (X, x, v, fr) parallel components [Grard and Tabbagh 1991], (3.10) (3.11) Inverting Equations (3.10) and (3.11), v and fr can be expressed as functions of R and C, i.e.: (3.12) (3.13) In our opinion, once the degrees of freedom of the (f, x) pair are fixed, it is not suitable to choose (R,C) as independent variables and then (v, fr) as dependent variables (Equations 3.12 and 3.13). Instead, it is more appropriate to consider (v, fr) as quantities of physical interest, and consequently Equations (3.10) and (3.11) as the starting points for the theoretical development. Indeed, even if the physics does not forbid the choice of (R,C) as independent variables, applying the function (R,C) → (v, fr), the procedures of the design should anyway choose (v, fr) as independent variables, preferentially applying the inverse function (v, fr) → (R,C). According to the following two practical approaches: (a) – (v, fr) as independent variables in order – to establish the class of media with conductivity and permittivity (v, fr) that can be investigated by a quadrupole working in a fixed band B and specified by a known geometry x; (b) – preferential way (v, fr) → (R,C) since – once a subjacent medium with electrical conductivity v and dielectric permittivity fr is selected, the quadrupole probe specifications R and C can be projected both in frequency f and in height/dimension ratio x. 4. Theoretical modeling The measurements taken using the quadrupole probe are affected by errors that mainly originate from uncertainties associated with transfer impedance, from dishomogeneities between the modeled and the actual stratigraphy, and from inaccuracies 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]. These uncertainties were assumed to be constant throughout the whole frequency band, even though their effects that propagate through the transfer function will produce a frequency-dependent perturbation. 4.1. Sensitivity functions approach This study proposes to develop explicitly the sensitivity functions approach that is implied in the theory of error MEASUREMENT OF RESISTIVITY AND PERMITTIVITY 6 Figure 5. Ratio C= C1/C2 between the first member C1 and the second member C2 of Equation (B.8), plotted as a function C(x, v) of both the height/dimension ratio x = h/L and the electrical conductivity v, with the quadrupole probe designed in the linear Wenner configuration and in capacitive contact on a selected concrete of dielectric permittivity fr= 4. 1 2 1 4 2 1 2 .K x x x2 1 2 2 1 2 1 2 1 2 = + + - -- - - - ^ ^ ^h h h 1 ,x K x=d -^ ^h h 1 ,R C 0 T N N r = = = + ~ ~ ~ ~ v f f X ^ h , , , , 1 1 2 1 2 1 , R x R L x x x2 2 r N r r = = + + + v f v d d f d f X X - - ^ ^ ^ ^ ^ h h h h h; ;E E , , , , 1 2 1 2 1 1 2 1 1 2 1 C x C L x x x .2 2 2 r N r r r r r = = + + + + + + v f f d f d f d f f X X X - - ^ ^ ^ ^ ^ ` h h h h h j ; ;E E , , , 2 1 ,x R C x R C x C x RC 2 2 2 0 2 0 2 0= + v ~ d ~ d d f ~ - -^ ^ ^ ^h h h h 6 6 @ @ , , , 2 2 x R C x R C x C x x R C x C C x C .2 2 2 0 2 2 2 0 0 r = + + f ~ d ~ d d d ~ d d - - - - - ^ ^ ^ ^ ^ ^ ^ h h h h h h h 6 6 6 6 @ @ @ @" , 7 propagation suggested by Vannaroni et al. [2004]. Indeed, this section introduces a mathematical-physical model for the propagation of errors in the measurement of electrical conductivity v and dielectric permittivity fr, based on the sensitivity functions tool [Murray-Smith 1987]. This is useful for expressing inaccuracies in the measurements of conductivity and permittivity (Figure 3) as a linear combination of the inaccuracies for the transfer impedance, both in modulus |Z| and in phase UZ, where the weight functions are inversely proportional only to the sensitivity functions for |Z| and UZ relative to v and fr (Figure 4). The inaccuracies of transfer impedance depend on the inaccuracies of electrical voltage and current that are assigned by the electronics used, and in particular, by the sampling methods. Therefore, the inaccuracies Dv/v in the measurement of the electrical conductivity v, and Dfr/fr in the dielectric permittivity fr, can be expressed as a linear combination of the inaccuracies D|Z|/|Z| and DUZ/UZ in the measurement of the transfer impedance, respectively in modulus |Z| and in phase UZ, (4.1) MEASUREMENT OF RESISTIVITY AND PERMITTIVITY Figure 6. Conceptual schemes for numerical simulations to design the characteristic geometrical dimensions and the frequency band, limiting inaccuracies in the measurements of the quadrupole probe in capacitive contact with selected materials as non-saturated concretes, in the hypothesis that D|Z|/|Z|= DUZ/UZ= 10−3. 1 1 , for const , S Z Z S S Z Z S Z Z Z Z Z Z r Z Z = + = = + = ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; v v f D D U DU D U DU ; ; ; ; v v v v U U (4.2) where and are the pairs of sensitivity functions for the transfer impedance, both in |Z| and UZ, relative to the conductivity v and permittivity fr, the expressions for which are reported in Appendix A. The conditions v = const and fr = const in Equations (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 section 2, but that the quantities v and fr are not independent of each other, since the electrical displacement shows a phase-shift with respect to the electrical field [Frölich 1990]. So, for the need to distinguish the inaccuracies in measurements of conductivity and permittivity, the inaccuracy Dv/v can only be calculated assuming there is no uncertainty for fr (Dfr/fr = 0 ↔ fr = const), and vice versa. Moreover, according to the physical problem, the probe performs measurements of the transfer impedance Z, both in modulus |Z| and in phase UZ, which are characterized by the inaccuracies D|Z|/|Z|> 0 and DUZ/UZ> 0. Mathematically, application of the conditions |Z|= const or UZ = const is not MEASUREMENT OF RESISTIVITY AND PERMITTIVITY 8 Figure 7. Conceptual schemes for numerical simulations to establish the measurable ranges of electrical conductivity and dielectric permittivity, limiting inaccuracies in the measurements of the quadrupole probe in capacitive contact, and fixing its optimum working frequencies and characteristic geometrical dimensions [D|Z|/|Z|= DUZ/UZ= 10−3]. 1 1 , for const , S Z Z S S Z Z S r r Z Z Z Z Z Z Z Z r r r r = + = = + = ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; f f v D D U DU D U DU ; ; ; ; f f f f U U ( , )S SZ Z; ;v v U ( , )S SZ Z r r ; ; f f U 9 allowed. In this context, the sensitivity functions and cannot be calculated assuming UZ = const, and then the sensitivities and assuming |Z|= const. Indeed, as discussed above, once the degrees of freedom of the (f, x) pair are fixed, it is not suitable to choose (|Z|,UZ) or (R,C) as independent variables. Consequently, the sensitivity functions cannot be calculated by the dependent variables v = v(|Z|, UZ) and fr = fr(|Z|, UZ) or by Equations (3.12) and (3.13). Instead, the physical problem should be approached recalling that (f, x, v, fr) have been considered as independent variables. In the simplifying hypothesis that the frequency f and the height/ dimension ratio x are characterized by inaccuracies Df/f ≈ 0 and Dx/x ≈ 0 close to zero, the conditions f = const and x = const can be applied. Necessarily, the inaccuracy Dv/v in the measurement of the electrical conductivity v is calculated assuming fr =const, and then the inaccuracy Dfr/fr for the dielectric permittivity fr is calculated assuming v = const. As a consequence, the mathematical calculations should be done recalling that Equations (3.10) and (3.11) have been considered as the starting points for the theoretical development. The inaccuracies Dv/v for the conductivity v and Dfr/fr for the permittivity fr can be more directly expressed as functions of (f, x, v, fr) by calculating the sensitivity functions and in the last part of Equations (4.1) and (4.2). These sensitivities are derived from the transfer impedance 1/Z = 1/R + j~C reported in Equations (3.10) and (3.11). The interesting physical results obtained using this sensitivity functions approach are discussed below. If the quadrupole probe is in galvanic contact with the subsuface, MEASUREMENT OF RESISTIVITY AND PERMITTIVITY Figure 8. Referring to a quadrupole probe designed in the linear Wenner or square configurations and with capacitive contact on a non-saturated concrete of low electrical conductivity, i.e. v = 10−4 S/m, fr= 4: (a) plots, as a function of the ratio x = h/L between the height h above ground and the characteristic geometrical dimension L, with 0 < x ≤ 1, for the geometrical factor d(x); (b) semi-logarithmic plots for both the zero zM(x) and pole pM(x) of the transfer impedance in modulus; (c, d) Bode's diagrams, as a function of the frequency f in the band f∈ [0, flim], with flim= 1 MHz, for the transfer impedance both in modulus |Z|(f, xconcrete), units of 1/h, and phase DUZ (f, xconcrete); (e) – see next page – on the hypothesis that D|Z|/|Z|= DUZ/UZ= 10−3, semi-logarithmic plots for both the inaccuracies Dfr/fr (f, xconcrete) in the measurement of the permittivity fr, and Dv/v (f, xconcrete) of the conductivity v, with the height/dimension ratio designed optimally in the linear Wenner (xW,concrete= 0.087) and square (xS,concrete= 0.078) configurations. (a) (c) (d) (b) S Z; ; v S Z r ; ; f S Z v U S Z rf U ( , )S SZ Z; ;v v U ( , )S SZ Z r r ; ; f f U i.e. h = 0, then the inaccuracies Dv/v in the measurement of the electrical conductivity v, and Dfr/fr for the dielectric permittivity fr, are minimized in the frequency band B of the quadrupole, for all of its geometrical 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 Dfr/fr for fr, which is always higher than the corresponding minimum value of the inaccuracy Dv/v in the band B of the probe, for all of its configurations and media [Tabbagh et al. 1993, Vannaroni et al. 2004]. Under the 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 v and permittivity fr are independent of the characteristic geometrical dimension of the quadrupole, i.e. the electrode- electrode distance L. If the probe grazes the medium, then the transfer impedance ZN (v, L) consists of the resistance RN (v, L), which is independent of fr, and the parallel capacitance CN (fr, L), which is independent of v, such that: the sensitivity function for R relative to v is a constant equal to (−1); the sensitivity for C relative to fr is independent of v, and behaves as the function fr/(fr + 1) of fr; the function for R relative to fr and the function for C relative to v are identically zero. As a consequence, the inaccuracy DR/R for R shows the same behavior versus the frequency of the inaccuracy Dv/v in the measurement of v, as DR/R =| | Dv/v = Dv/v, and the inaccuracy DC/C for C shows a similar behavior versus the frequency with respect to the inaccuracy Dfr/fr for fr, as DC/C =| |Dfr/fr ≈ Dfr/fr if fr>>1. Moreover, as well as the hypothesis h = 0, if v and fr are measured in the cut-off frequency fT = fT (v, fr), then: the sensitivity functions and for the transfer impedance, both in modulus |Z| and in phase UZ, relative to v, are constant, and respectively (−1/4) and (−1/r); and the sensitivities and for |Z| and UZ relative to fr are independent of v, such that they behave as the function fr/(fr + 1) of fr. As a consequence, the ratio between Dfr/fr and Dv/v is independent of v, and behaves as the function (1+1/fr) of fr, and Dv/v is a constant equal to Dv/v = 4D|Z|/|Z|+rDUZ/UZ. As a post-test, only assuming the conditions v = const and fr = const in Equations (4.1) and (4.2), the sensitivity functions approach provides results that are in agreement with a previous report [Vannaroni et al. 2004]. 4.2. Transfer functions method This study proposes to deepen the transfer functions method by analyzing the zero and pole behavior, which were implied in the frequency domain analysis suggested by Grard and Tabbagh [1991]. Indeed, this section introduces the method of analysis in the frequency domain for determining the behavior of the zero and pole frequencies in the linear time-invariant circuit of the quadrupole probe (Figure 1). To satisfy the operative conditions of linearity for the measurements, if the quadrupole has capacitive contact with the subjacent medium then the frequency f of the probe should be imposed as included between the zero zM and the pole pM of the transfer impedance, and so its modulus is almost constant within the frequency band [Grard and Tabbagh 1991], (4.3) Based on the above conditions, an optimization equation is deduced for the probe that links the optimal ratio x between its height above ground and its characteristic geometrical dimension only to the dielectric permittivity fr of the medium, so that: (4.4) To satisfy the operative conditions of linearity for the measurements, if the quadrupole is in galvanic contact with the subjacent medium, then the working frequency f of the quadrupole should be imposed as lower than the cut-off frequency of the transfer impedance, and so its modulus as constant below the cut-off frequency. It is only under these conditions that it is optimal to design the characteristic geometrical dimensions of the probe or to establish the measurable ranges of the conductivity v and permittivity fr of the medium (Figure 5). Equations (4.3) and (4.4) that were derived by the classical transfer function method are demonstrated in Appendix B. The interesting physical results obtained using this transfer functions method are discussed below. To meet the MEASUREMENT OF RESISTIVITY AND PERMITTIVITY 10 Figure 8e. Caption on previous page. (e) SRv S r C r ff ^ h SR rf SCv SRv S r C r ff ^ h S Z; ;v S Zv U S Z rr f ; ; f ^ h S rZr ffU ^ h , , , ,z x f p x .M r M r# #f v f v^ ^h h 15 17 2 .x r + .d f^ h 11 design specifications that ensure satisfactory performances of the probe (inaccuracy of no more than 10%), the forecasts provided by the theory of error propagation suggested by Vannaroni et al. [2004] that apply the sensitivity functions approach, as explicitly developed in the study, are less stringent than those foreseen by the analysis in the frequency domain suggested by Grard and Tabbagh [1991]. Here, this deepens the transfer function method to analyze the zero and pole behavior, in terms of both larger band of frequency f and wider measurable range of resistivity t or permittivity fr (Figures 6, 7). Indeed, given a surface (e.g. a non-saturated concrete with low conductivity v = 10−4 S/m and fr = 4) with dielectric permittivity fr (Figure 6): if the quadrupole probe has capacitive contact with the subjacent medium, i.e. h ≠ 0, then having defined an optimal ratio xopt= hopt/L between an optimal height hopt above ground and the characteristic geometrical dimension L, the transfer impedance Z (f, xopt), in units of 1/hopt, calculated in xopt, is a function of the working frequency f such that its modulus |Z|(f, xopt), in units of 1/hopt, is almost constant between a zero frequency z (xopt) almost one LF decade higher than a minimum frequency value fmin(xopt), allowing the inaccuracy Dfr/fr (f, xopt) in the measurement of fr below a prefixed limit (10%), and a pole p (xopt) almost one MF decade lower than the maximum value of frequency fmax(xopt) that satisfies the requirement that the inaccuracy Dfr/fr (f, xopt) for fr below 10% (Figure 8); if h = 0, i.e. the quadrupole of the electrode-electrode distance L grazes a medium of conductivity v, 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 fT = fT (v, fr), which is higher than an optimal frequency fopt(L) that minimizes the inaccuracy Dfr/fr (f, L). Materials characterized by a low v or a high fr lead to a leftward shift of the cut-off frequency fT, so reducing the optimal frequency fopt(L) (Figure 9); on a selected surface, it is usually possible to verify that the probe in capacitive contact performs optimal measurements over the band [fmin(xopt)p (xopt)], which is shifted towards lower and higher frequencies compared to when the probe is in galvanic contact, where the respective band (fmin, fmax) is narrower by almost one LF- MF decade in frequency, particularly increasing the value of fr (Figures 8e, 9c). Moreover, once the frequency band B is fixed (Figure 7): if the quadrupole probe has capacitive contact with the MEASUREMENT OF RESISTIVITY AND PERMITTIVITY Figure 9c. (a) (b) (c) Figure 9. With reference to a quadrupole probe designed according to an electrode-electrode distance L0= 1 m and in galvanic contact on a concrete of low electrical conductivity, i.e. v = 10−4 S/m, fr= 4: (a, b) Bode’s diagrams, as functions of the frequency f for the transfer impedance both in modulus |Z|(f, L0) (a) and phase UZ (f, L0) (b); (c) semi-logarithmic plots for both the inaccuracies Dv/v (f) in the measurement of the conductivity v, and Dfr/fr (f) of the permittivity fr [D|Z|/|Z|= DUZ/UZ= 10−3]. 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 xlow, corresponding to water (fr = 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 the dielectric permittivity fr, opt ; the modulus |Z|(x, v, fr, opt), in units of 1/h, of its transfer impedance, calculated in x and fr, opt, is a function of the electrical conductivity v, is characterized by a zero z (v, fr, opt) and a pole p (v, fr, opt) frequency, which fall near the lower and upper limits of B, respectively, when v is measured within the range of the lower limit and the upper limit . In a deeper analysis based on the sensitivity functions method, and still designing the quadrupole with the ratio x = h/L for optimal measurement of fr, opt, it is possible to verify the measurable range of v; the inaccuracy Dfr/fr (x, v, fr, opt) in the measurement of fr, opt, a function of v, is below a prefixed limit (10%) if v is measured within the range (vlow, vup), larger than by almost one order of magnitude, both the right and left sides (Figure 10,Tables 1, 2); if h = 0, i.e. the probe of the electrode-electrode distance L grazes a medium of conductivity v and permittivity fr, then the transfer impedance Z (L, v, fr) calculated in L is a function of v and fr such that its cut-off frequency fT = fT (v, fr), a function of both v and fr, ranges from fT,min= 100 kHz to fT,max= 1 MHz for materials belonging to a (v, fr)-domain, which is almost superimposable with the corresponding one MEASUREMENT OF RESISTIVITY AND PERMITTIVITY 12 Permittivity inaccuracy (a) xW, opt 1.083·10 −4 fr, opt 6.703 vopt 3.52·10 −5 S/m fr, opt= 6.703 vopt= 3.52·10 −5 S/m Dfr/fr ≤ 0.1 Dv/v ≤ 0.1 (b) xW, low ≈ 0 xW, up 0.475 xW, opt= 1.083·10 −4 Dfr/fr ≤ 0.1 Dv/v ≤ 0.1 (c) fr, low , vlow 1, 5.333·10 −5 S/m fr, up , vup 81, 3.14·10 −3 S/m Table 1. Wenner's configuration: (a) optimal point (xW, opt, fr, opt, vopt) of permittivity inaccuracy; (b) range of x where Dfr/fr ≤ 0.1 and Dv/v ≤ 0.1, once selected optimally fr, opt and vopt; (c) domains of f and v where Dfr/fr ≤ 0.1 and Dv/v ≤ 0.1, once fixed optimally xW, opt. Figure 10. On the hypothesis that D|Z|/|Z|= DUZ/UZ= 10−3, referring to both the inaccuracies Dv/v (v, fr) for the electrical conductivity v, and Dfr/fr (v, fr) for the dielectric permittivity fr, as functions of v and fr, and when the quadrupole probe is designed in the linear Wenner configuration working in a fixed band B = 100 kHz, with an height/dimension ratio xW, concrete= 0.087 which is optimal for capacitive contact only with a non-saturated concrete of permittivity fr= 4: (a, b) as plots for the orthogonal projections over the (v, fr) plane that satisfy the conditions Dv/v (v, fr) ≤ 0.1 (a) and Dfr/fr (v, fr) ≤ 0.1 (b). See also Tables 1 and 2. (a) (b) lowvl upvl ( , )low upv vl l 13 within which the inaccuracy Dfr/fr (L, v, fr) for fr is below about 10% (Figure 11); having fixed the frequency band, the probe in capacitive contact usually performs optimal measurements over surfaces of lower conductivities compared to when the probe is in galvanic contact, as the respective conductivities are higher by almost one order of magnitude (Tables 1, 3). 5. 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 study considers two kinds of probes, i.e. with linear Wenner and square configurations. The linear Wenner arrangement consists of four terminals equally spaced from one another along a straight horizontal line [Vannaroni et al. 2004]. Instead, the square configuration is an array of two parallel horizontal dipoles, with the four electrodes positioned at the corners of a square [Grard and Tabbagh 1991]. If the quadrupole probe (Figure 1) has a characteristic geometrical dimension L, then the linear Wenner configuration (Figure 2a) measures a capacitance in a vacuum C0,W= 4rf0·L, while in the square arrangement (Figure 2b) C0,S= a·C0,W, is greater by a factor a = 1/(2-2 ½)>1. When the quadrupole is in galvanic contact, i.e. h = 0, with a subjacent medium of electrical conductivity v and dielectric permittivity fr, the linear Wenner configuration measures a resistance RN,W= 2f0/vC0,W and a parallel capacitance CN,W= C0,W(fr+1)/2, while in the square arrangement, RN,S= RN,W/a and CN,S= a·CN,W. So, at the frequency f, the transfer impedance 1/ZN= 1/RN+ j2rf CN for the linear Wenner configuration is defined by a modulus |Z|N,W= 1/[(1/RN,W) 2 + (2rf CN,W)] ½ and a phase UN,W= arctg (2rf·RN,W CN,W), while in the square arrangement, |Z|N,S= Z|N,W/a, which is smaller by a factor of 1/a (Figure 9a), and UN,S= UN,W , which is maintained invariant in the linear Wenner or square configurations (Figure 9b). The cut-off frequency is also independent of the configurations, i.e. fT = fT (v, fr). Moreover, if the probe grazes the medium and considering that the sensitivity functions are defined as normalized functions, then the sensitivities and relative to the conductivity v, and the functions and relative to the permittivity fr, for the transfer impedance, both in modulus |Z| and in phase UZ, are invariant in the linear Wenner or square configurations. Only if h = 0 are the inaccuracies Dv/v in the measurement of v and Dfr/fr for fr also independent of the configurations, so the probe is characterized by the same performances in the frequency band B and in the measurable ranges of v and fr (Figure 9c). 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 zero, i.e. 0< x ≤1, then the quadrupole is characterized by a geometrical factor K(x) [d(x)], decreasing (or increasing) the function of x, which in the square configuration slopes down (or up) more steeply than in the linear Wenner arrangement, so assuming smaller (or larger) values especially for 1/2 < x <1 (Figure 8a). As a consequence, a probe with a fixed L that performs measurements on a medium of dielectric permittivity fr could be designed with an optimal height/dimension ratio xopt= hopt/L, which in the square configuration is smaller than in the linear Wenner arrangement, because its factor d(x) slopes up more steeply, increasing the ratio x, so reaching the prefixed optimal value dopt(fr) ≈ 2/(15fr +17) with a smaller xopt. In simpler terms, if the probe is in capacitive contact with the medium, to perform optimal measurements of the permittivity, the square configuration needs to be raised above ground by less than in the linear Wenner arrangement, if their electrode-electrode distances are equal. Indeed, x ranges from xW,low= 0.022 in the linear Wenner configuration, and from xS,low=0.019 in the square arrangement. Moreover, in the case of capacitive contact, if the quadrupole with electrode-electrode distance L is designed according to the optimal height/dimension ratio xopt= hopt/L working in a frequency f, then the transfer impedance Z (f, xopt), in units of 1/hopt, calculated in xopt, is defined by a phase U (f, xopt), which does not depend on the square or linear Wenner configurations (Figure 8d), and by a modulus |Z|(f, xopt), in units of 1/hopt, which in the square configuration is shifted down by a factor 1/a with respect to the linear Wenner configuration (Figure 8c), remaining MEASUREMENT OF RESISTIVITY AND PERMITTIVITY fr, concrete= 4.026 xW, concrete= 0.087 Sensitivity function approach Transfer function method vlow 4.473·10 −6 S/m 1.78·10−5 S/m vup 3.058·10 −4 S/m 7.12·10−5 S/m (a) xW, concrete= 0.087 Dfr/fr ≤ 0.1 Dv/v ≤ 0.1 fr, low , vlow 1, 1.769·10 −6 S/m fr, up , vup 84.458, 1.573·10 −3 S/m (b) Table 2. Concretes: (a) comparing domains of v, foreseen by Sensitivity approach and Transfer method, once selected fr, concrete and designed optimally xW, concrete ; (b) domains of f and v where Dfr/fr ≤ 0.1 and Dv/v ≤ 0.1, once designed optimally xW, concrete. S Z; ;v S Zv U S Z rr f ; ; f ^ h S rZr ff U ^ h almost unvaried in both configurations not only the shape of the modulus |Z|(f, xopt), but also the position of its zero z (xopt) and pole p (xopt) frequencies (Figure 8b). Finally, the inaccuracies Dv/v (f, xopt) in the measurements of the conductivity v and Dfr/fr (f, xopt) for the permittivity fr, calculated in xopt, do not depend on the two configurations, so the optimal frequency fopt(xopt) that minimizes the inaccuracy Dfr/fr (f, xopt) for fr, together with the minimum and maximum values of frequency fmin(xopt) and fmax(xopt), respectively, which allow the inaccuracy Dfr/fr (f, xopt) below a prefixed limit (10%), are invariant in both of the configurations (Figure 8e). In simpler terms, if the probe is in capacitive contact with the medium, to perform an optimal measurement of permittivity considering different height/dimension ratios, the design of the two configurations establishes (almost) invariant trends in frequency, both for their transfer impedances and measurement inaccuracies. 6. Conclusions The present study has proposed a theoretical modeling of simultaneous and noninvasive measurements of electrical resistivity and dielectric permittivity using a quadrupole probe on a subjacent medium [see also arXiv.org's ref.: Settimi et al. 2009]. A mathematical-physical model has been applied to the propagation of errors in the measurement of resistivity and permittivity based on the sensitivity functions tool. The findings have also been 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 circuit of the quadrupole. This study has underlined that average values of electrical resistivity and dielectric permittivity can 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 are analyzed within a bandwidth ranging from only LFs to MFs [Al-Qadi et al. 1995, Myounghak et al. 2007]. To meet the design specifications that ensure satisfactory performances of the probe (inaccuracy of no more than 10%), the forecasts provided by the theory of error propagation suggested by Vannaroni et al. [2004] that apply the sensitivity functions approach, as explicitly developed in the study, have been discussed in comparison to those foreseen by the analysis in the frequency domain suggested by Grard and Tabbagh [1991]. Here, this deepens the transfer function method to analyze the zero and pole behavior (in terms of both band of frequency f and measurable range of resistivity t, or permittivity fr). It is interesting to compare the results of the present study with others in the literature [Grard and Tabbagh 1991, MEASUREMENT OF RESISTIVITY AND PERMITTIVITY 14 h = 0 Dfr/fr ≤ 0.1 Dv/v ≤ 0.1 fr, low , vlow 1, 5.333·10 −5 S/m fr, up , vup 81, 3.14·10 −3 S/m Table 3. Galvanic contact: domains of f and v where Dfr/fr ≤ 0.1 and Dv/v ≤ 0.1, when h = 0. (a) (b) Figure 11. With reference to a quadrupole probe in galvanic contact, working in a fixed band B = 100 kHz, plots for the domains (v, fr) of the electrical conductivity v and the dielectric permittivity fr such that: (a) the transfer impedance is characterized by a modulus with a cut-off frequency fT = fT (v, fr) = v/(2rf0(fr+1)) in the interval fT ∈ [100 kHz, 1 MHz]; (b) both the inaccuracies Dv/v (v, fr) in the measurement of the conductivity v, and Dfr/fr (v, fr) of the permittivity fr are below a prefixed limit of 10% [D|Z|/|Z|= DUZ/UZ = 10−3]. See also Tables 1 and 3. 15 Vannaroni et al. 2004]. In agreement, 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 Dv/v in the measurement of conductivity v and Dfr/fr for permittivity fr are minimized in the frequency band B of the quadrupole for all of its geometrical configurations and media; and b) even if h ≠ 0, the design of the probe must be optimized with reference to the minimum value of the inaccuracy Dfr/fr for fr, which is always higher than the corresponding minimum value of the inaccuracy Dv/v in the band B for all its configurations and media. More explicitly than in these previous studies [Grard and Tabbagh 1991,Vannaroni et al. 2004], the transfer functions method provides results such that to satisfy the operative conditions of linearity for the measurements: a) if the quadrupole has capacitive contact with the subjacent medium, then the frequency f of the probe should be imposed as included between the zero zM and the pole pM of the transfer impedance, and so its modulus is almost constant within the frequency band. An optimization equation is deduced for the probe that links the optimal ratio x between its height above ground and its characteristic geometrical dimension only to the dielectric permittivity fr of the medium; b) instead, if the quadrupole is in galvanic contact with the subjacent medium, then the working frequency f of the quadrupole should be imposed as lower than the cut-off frequency of the transfer impedance, and so its modulus is constant below the cut-off frequency. It is optimal to design the characteristic geometrical dimensions of the probe or to establish the measurable ranges of the conductivity v and permittivity fr of the medium. Unlike these previous studies [Grard and Tabbagh 1991, Vannaroni et al. 2004], the sensitivity functions approach and the transfer functions method provide results that allow an assessment of the performance of the quadrupole probe in galvanic and capacitive contact: a) on a selected surface (for example, a non-saturated concrete with low conductivity v = 10−4 S/m and fr = 4), it is usually possible to verify that the quadrupole in capacitive contact performs optimal measurements over the band [fmin(xopt)p (xopt)], which is shifted to lower and higher frequencies compared to when the probe is in galvanic contact, as the respective band [fmin, fmax] is narrower by almost one LF-MF decade in frequency, particurarly increasing the value of fr; b) having fixed the frequency band, the quadrupole in capacitive contact usually performs optimal measurements over surfaces of lower conductivity compared to when the probe is in galvanic contact, as the respective conductivities are higher by almost one order of magnitude. 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 AC at LFs and MFs (10 kHz - 1 MHz). Measurements were carried out using four electrodes laid on the surface to be analyzed, and through measurement of transfer impedance, the resistivity and permittivity of the material can be extracted. Furthermore, by increasing the distance between the electrodes, the electrical properties of the sub-surface structures can be investigated to greater depths. The main advantage of the quadrupole is that measurements of electrical parameters can be conducted with a nondestructive 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, allowing for completely nondestructive analysis, although accompanied by a greater error. 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Vannaroni (2005). Evaluation of a mutual impedance probe to search for water ice in the Martian shallow subsoil, Rev. Sci. Instrum., 76, 084504 (1-8). Edwards, R. J. (1998). Typical Soil Characteristics of Various Terrains, http://www.smeter.net/grounds/soil-electrical- resistance.php. Fechant, C. (1996). Réalisation d’un quadripôle de mesure in situ de la permitivié diélectrique des végétaux. Premier application à la détermination du contenu en eau des épis de blé, These de l’Université Pierre-et-Marie-Curie VI, Paris. Fechant, C. and A. Tabbagh (1999). Mesure en laboratoire de la permittivité diélectrique moyenne fréquence de végétaux à 430 kHz à l’aide d’un capacimétre. Relation entr permittivité apparente d’un ensemble d’épis de blé et leur contenu en eau, C. R. Acad. Sci. Paris t. 327 Série II b, 285-298 (both in French and in English). Frölich, H. (1990 edition). Theory of Dielectrics, Oxford. Grard, R. (1990a). 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Rhoades, J. D., P. A. C. Raats and R. J. Prather (1976). Effect of liquid-phase electrical conductivity, water content, and surface conductivity on bulk soil electrical conductivity, Soil Sci. Soc. Am. J., 40, 651-655. Samouëlian, A., I. Cousin, A. Tabbagh, A. Bruand and G. Richard (2005). Electrical resistivity survey in soil science: a review, Soil Till. Res., 83, 172-193. Settimi, A., A. Zirizzotti, J. A. Baskaradas and C. Bianchi (2009). Inaccuracy assessment for simultaneous measurements of resistivity and permittivity applying sensitivity and transfer function approaches, arXiv.org: 0908.0641. Tabbagh, A. (1994). Simultaneous measurement of electrical and dielectric permittivity of electrical conductivity and dielectric permittivity of soil using a slingram electromagnetic device in medium frequency range, Archaeometry, 36, 159-170. Tabbagh, A., A. Hesse and R. Grard (1993). Determination of electrical properties of the ground at shallow depth with an electrostatic quadrupole: field trials on archaeological sites, Geophys. Prospect., 41, 579-597. Vannaroni, G., E. Pettinelli, C. Ottonello, A. Cereti, G. Della Monica, D. Del Vento, A. M. Di Lellis, R. Di Maio, R. Filippini, A. Galli, A. Menghini, R. Orosei, S. Orsini, S. Pagnan, F. Paolucci, A. Pisani R., G. Schettini, M. Storini and G. Tacconi (2004). MUSES: multi-sensor soil electromagnetic sounding, Planet. Space Sci., 52, 67–78. Wenner, F. (1915). A method of measuring earth resistivity, US Bur. of Stand. Bull., 12, 469-478. MEASUREMENT OF RESISTIVITY AND PERMITTIVITY 16 17 Appendix A We consider here the influence of 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. The inaccuracies Dv/v in the measurement of the electrical conductivity v, and Dfr/fr for the dielectric permittivity fr, can be expressed as linear combinations of the inaccuracies D|Z|/|Z| and DUZ/UZ in the measurement of transfer impedance in modulus |Z| and in phase UZ, respectively, as given in Equations (4.1) and (4.2) (Figure 3). The pairs of sensitivity functions and for the transfer impedance, both in |Z| and UZ, relative to the conductivity v and the permittivity fr (Figure 4), (A.1) (A.2) (A.3) (A.4) are, in turn, linear combinations of the sensitivity function pairs and for transfer impedance, in both the resistance R and capacitance C parallel components, relative to v and fr, (A.5) (A.6) (A.7) (A.8) with the weight functions: (A.9) (A.10) Considering Equations (A.1) to (A.4), if the modulus |Z| and the phase UZ of the transfer impedance provide an indirect measurement of the electrical conductivity v and dielectric permittivity fr, then the functions |Z|=|Z|(v, fr) and UZ = UZ(v, fr) are invertible, i.e. v = v(|Z|, UZ) and fr = fr (|Z|, UZ). Therefore, the theorem of the derivative for the MEASUREMENT OF RESISTIVITY AND PERMITTIVITY ( , )S SZ Z; ;v v U ( , )S SZ Z r r ; ; f f U 1 2 1 2 1 , for const , S Z Z Z Z S H S H S1 2 Z Z R C r = = = = = = 2 2; ; ; ; ; ; ; ;v v v v f D D - ; ; ; ; v v v v 1 2 1 arct 2 1 , for 1 const S S R R C C R R C C H S S H S S R R C C 1 1 Z Z N N N N R C R C N N r Z Z = = = + + = 2 2 # , , v v f U U X X X v v v v v v U U ` ^ ^ ^ j h h h 1 2 1 2 1 , for const ,S S H S H S1 Z Z R C 2r r r r = = =v-; ; ; ; f f f f 1 2 1 arct 2 1 , for 1 const , S S R R C C R R C C H S S H S S R R C C 1 1 N N N N R C R C N N Z Z r r r r r r = = + + =# , , v X X X f f f f f f U U ` ^ ^ ^ j h h h S R R R R const const R r r = = =2 2 v v v v D D = = v f f ( , )S SCRv v ( , )S S CR r rf f 1 2 1 2 1 1 2 1 2 1 , x x x x 2 2 2 2 r r r r = + + + + + d f d f d f d f X X - - - - ^ ^ ^ ^ h h h h ; ; ; ; E E E E 2 1 1 1 2 1 1 2 1 1 2 1 2 1 2 3 1 S C C x x x x x x const 2 2 2 2 2 C r r r r r r r r = = = + + + + + + + + + 2 2 v v f d f f d f d f d d f d f X X X - - - - - = v f ^ ` c ^ ^c ^ ^ ^ ^ ^ h j h m h m h h h h h ; ; 6 E E @ 1 2 1 2 1 1 2 1 ,S x x x x const 2 2 R r r r r r = + + + + f d f d f d d f X - - =f v ^ ^ ^ ^ h h h h ; ; ; E E E 1 1 2 1 1 2 1 1 2 1 2 1 1 2 4 1 1 5 1 4 1 1 1 , S x x x x x x const 2 2 2 2 2 3 2 C r r r r r r r r r r r r2 r = = + + + + + + + + + + + + f f d f f d f d f d f d f f f d f f X X X X - - - - - - - - - - =f v ^ ` c ^ ^c ^ ^ ^ ^ ^ ` ^ ^ ` h j h m h m h h h h h j h h j ; ; 8 8 E E B B 1 1 1 1 2 1 1 2 1 1 ,H x x x 1 2 2 2 2 2 r r r = + + + + + + + + d f d f f d X X X X - - - ^ ^ ^ ^ ` ^ h h h h j h6 @ 1 1 1 1 2 1 1 2 1 2 2 1 2 1 2 1 1 2 1 .H x x x x 2 2 2 2 2 2 2 2 2 r r r r r r r = + + + + + + + + + + + + + + + d f d f f d f f d f f X X X X X X X X - - - - ^ ^ ^ ^ ` ^ ` ^ ` h h h h j h j h j inverse function can be applied. Indeed, under the condition v = const (or fr = const), both |Z| and UZ are invertible functions of fr (or v), i.e. they are strictly increasing or decreasing monotonic functions of fr (or v). Appendix B By exact calculations, the transfer impedance Z (f, x, v, fr) measured by the quadrupole probe, in units of the reciprocal height 1/h from the subjacent medium, consists of the resistance R (f, x, v, fr), in units of 1/h – see Equation (3.10) –, which can be expressed as a transfer function characterized by a pole in the origin frequency pR= 0, a zero in higher frequencies zR (f, x, v, fr)>0, and a static gain KR (f, x, v), (B.1) where: (B.2) (B.3) as well as the parallel capacitance C (f, x, v, fr), in units of 1/h – see Equation (3.11) –, which can be expressed as a transfer function characterized by a low frequency pole pC (f, x, v, fr), a zero in higher frequencies zC (f, x, v, fr) > pC (f, x, v, fr), and a static gain KC (x), (B.4) where the capacitance pole pC (f, x, v, fr) coincides with the resistance pole zR (f, x, v, fr), (B.5) and: (B.6) (B.7) For values of the ratio x = h/L between the height h above ground and the characteristic geometrical dimension L, and of the paired values of electrical conductivity v and dielectric permittivity fr that satisfy the condition (Figure 5), (B.8) it can be demonstrated that the modulus |Z|(f, x, v, fr) can be approximately expressed as a transfer function with a pole in the origin frequency, a low frequency zero zM (f, x, v, fr), a pole in higher frequencies pM (f, x, v, fr) > zM (f, x, v, fr), and a static gain KM (x) (Figure 8c), (B.9) where the zero of the modulus zM (f, x, v, fr) coincides with the capacitance pole pC (f, x, v, fr) and the pole of the modulus pM (f, x, v, fr) with the capacitance zero zC (f, x, v, fr) (Figure 8b), (B.10) (B.11) and (B.12) Equation (B.8) establishes limits on the range for the design specification x of the quadrupole and the measurable range (v, fr) of the media. To satisfy the operative conditions of linearity for the measurements, the quadrupole probe, which is characterized by the height/ dimensions ratio x = h/L, should measure the conductivity v and the permittivity fr of the subjacent medium when its working frequency f falls within the band included between the zero zM (f, x, v, fr) and the pole pM (f, x, v, fr) of the transfer impedance, as reported in Equation (4.3). Moreover, the quadrupole probe specified by x = h/L should measure fr, as its geometrical factor d(x) is close to Equation (4.4), a necessary condition for Z (f, x, v, fr) to show an almost constant modulus within the band (Equation 4.3), the modulus in the zero (Equation B.10) coinciding with the corresponding one in the pole (Equation B.11), (B.13) MEASUREMENT OF RESISTIVITY AND PERMITTIVITY 18 , , , , 2 1 , , ,R f x K x f z x f 2 2 2 r R r R= + v f v r v f^ ^ ^ ^h h h h , , 2 1 1 1 2 1 2 1 ,z x x x 0 R r r r r = + + + v f r f f v d f d f -^ ^ ^ ^h h h h , 2 1 1 1 ,K x C x x x 0 0 2 R =v f v d d - ^ ^ ^ ^h h h h , , , 1 , , 1 , , ,C f x K x x f z x f p2 2 2 2 r C r C r C= + + v f v f v f^ ^ ^ ^h h h h , , , , ,p x z xC r C r=v f v f^ ^h h , , 2 1 1 1 2 1 2 1 ,z x x x 0 C r r r r = + + + v f r f f v d f d f -^ ^ ^ ^h h h h .K x x C x0 C = d ^ ^ ^h h h << , 1 2 , , 2 , K x K x z x2 2R C C r$v r v f^ ^ ^h h h6 6@ @ , , , 2 1 , , 1 , , ,Z f x K x f p x f z x f 2 2 2 2 r M r M r M= + + $ ; ; v f r v f v f^ ^ ^ ^h h h h ; E , , , , ,z x p xM r C r=v f v f^ ^h h , , , , ,p x z xM r C r=v f v f^ ^h h 1 .K x K xM C =^ ^h h , , , , , , 2 1 2 , , , , , Z x z x K x Z x K x z x p x 2 f z r M r M f p r M M r M r M M = = = ; ; ; ;v f r v f v f r v f v f = =^ ^ ^ ^ ^ ^ ^ h h h h h h h 19 so that the pole (Equation B.11) is almost four-fold greater that the zero (Equation B.10), (B.14) Equation (4.4) can be interpreted as the optimization equation of the quadrupole, so the sizing for the height/dimension ratio x of the probe depends only on the permittivity fr of the medium; instead, Equations (4.3) and (B.14) show that the probe can work optimally only in a small band of frequencies. *Corresponding author. Dr. Alessandro Settimi, Istituto Nazionale di Geofisica e Vulcanologia (INGV), Via di Vigna Murata 605, I-00143 Rome, Italy; e-mail: alessandro.settimi@ingv.it © 2010 by the Istituto Nazionale di Geofisica e Vulcanologia. All rights reserved. MEASUREMENT OF RESISTIVITY AND PERMITTIVITY , , 4 , ,p x z x .M r M r.v f v f^ ^h h