5.1. Electrical Conductivity of Mantle Minerals
 For electrical conductivity the equations of state for dry mantle minerals are reviewed in detail in Jones et al. [2009a], and, for small polaron conduction, are given as
based on the work of Hirsch et al.  for Ol, Xu and Shankland  for Opx, Xu et al.  for Cpx, and Romano et al.  for Gt (but with a correction applied, see Jones et al. [2009a]), where mineral XFe is the iron number (related to the magnesium number Mg# by XFe= 1 − Mg#/100; can be different for each mineral), k is Boltzmann's constant, and T is temperature in Kelvin. Note that the conductivities of the pyroxenes are not iron-dependent – this is because either the measurements have not been made or the laboratories are inconsistent in their results (see discussion inJones et al. [2009a]), so we have little choice but to assume iron dependency is negligible. Also, spinel is ignored as the evidence from mantle xenoliths is that its occurrence is well below 1% at the depth of interest in this paper (100 km).
 Fullea et al.  revised the above equations taking into account newer laboratory studies and also the measurements of the effects of water in the minerals. The revised equations are of the general form
The first term in equation (10) describes small polaron conduction, and includes a dependency on mineral iron number XFe. The second term describes the contribution of Mg vacancies at high temperatures (and lacks XFedependency). The third term describes proton conduction in the water-present case, as described in the general case byequation (1).
 At the conditions of interest in this study, the temperatures are sufficiently low that conduction from Mg vacancies is negligible (it only becomes important at temperatures beyond 1200°C).
5.2. Variation of Olivine Conductivity With Temperature
Table 4. Parameters for Conductivity Equations of Fullea et al. a
| ||Small Polaron Conduction||Mg Vacancies||Proton Conduction|
|Karato Model||Yoshino/Poe Model|
|log σ0,sp||a||b||c||d||e||f||ΔV||log σ0i||ΔHi||ΔHwet (eV)||log σ0||r||ΔHwet (eV)||log σ0||α|
|Ol||2.4–3.0||1.642||0.246||−4.85||3.259||–||–||0.68||4.73||2.31||0.87 ± 0.05||3.0 ± 0.4||0.62 ± 0.15||1.19||2.35||1.1|
|Gt||(2.67–3.39) + log10(1–0.044P)||2.6||−15.33||80.4||−194.6||202.6||−75||20.5||40.96||20.05||00.725||20.29||00.63||10.19||20.35||10.1|
Figure 4. Variation of conductivity with temperature for olivine at a pressure of 3.25 GPa (∼100 km depth) with an Mg# of 92.5 and a water content of 100 ppm. Small polaron conductivity from the Fullea et al.  (green solid line) and Jones et al. [2009a] (blue solid line) models plus the SEO3 model of Constable  for both QFM (black solid line) and IW (red solid line) oxygen fugacity buffering. Proton conductivity from the Karato (dashed black line), Yoshino (dashed red line) and Poe (dashed blue line) models.
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 Also shown in Figure 4 are the conductivity estimates for proton conduction for 100 ppm water from the three models, Karato (Figure 4, dashed black line), Yoshino (Figure 4, dashed red line) and Poe (Figure 4, dashed blue line). Comparing Figure 1 with Figure 4, proton conduction dominates over small polaron conduction for even small amount of water; <45 ppm for the Poe AV model and far less for the other two models. Thus, the differences between the polaron models and the errors introduced by adopting a particular polaron conductivity model, and also ignoring iron effects on pyroxene conductivity, are irrelevant at the water contents of interest in this study given the dominance of proton conduction.
 Note that for the water contents of interest, the proton conduction term has a relatively small gradient with temperature, regardless of model chosen, of approx. 0.37–0.42 log units for a 100°C change. Thus, a 50°C error in our present-day temperature estimates for either Jagersfontein (740°C) or Gibeon (850°C) will result in an error of 0.2 log units in our estimate of conductivity. This is the same order as the experimental errors of our resistivity estimates (0.20 log units for Jagersfontein and 0.09 log units for Gibeon, see below).
 The complete conductivity calculations in temperature-water content (T-Cw) space for the three proton conduction models are shown in Figure 5. For all of them, at low water contents temperature is virtually immaterial. This changes at high water contents, where there is a strong gradient with both temperature and water content for the Poe model, but for the Karato and Yoshino models conductivity becomes insensitive to actual water content, as discussed above.
Figure 5. Electrical conductivity from proton conduction with temperature and water content using equation (1) with the parameters listed in Table 1 for (a) the Karato [Wang et al., 2006], (b) the Yoshino [Yoshino et al., 2009], (c) the averaged Poe [Poe et al., 2010; Fullea et al., 2011], and (d) this paper's model parameters. Also shown are the expected temperatures and water contents for Jagersfontein (Jag, 740°C, 80 ppm, yellow square) and Gibeon (Gib, 750°C, 80 ppm, purple square), and the resistivities at 100 km depth below Jagersfontein (−3.41 ± 0.2 log units, solid yellow line with one sigma errors shown as dashed yellow lines) and Gibeon (−2.78 ± 0.09 log units, solid purple line with one sigma errors shown as dashed purple lines).
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5.3. Whole Rock Conductivity Calculation
 The parameters used for equations (10) and (11) for the four minerals are listed in Table 4. In implementing the generalized form for proton conduction, σp, in equation (11) for the different conduction models derived by the three laboratories, the following formulations were used (with parameters specified in Table 3): (1) For the Karato model, a constant value of ΔHwet (i.e., α assumed to be zero) and f(Cw) = σ0Cwr. (2) For the Yoshino and Poe models, ΔHwet = ΔH0 − αCw(1/3) and f(Cw) = σ0Cw (i.e., r assumed to be one).
 Following the Jones et al. [2009a] approach, we calculate whole rock conductivity for the four samples and two averages using the (P,T) conditions and modal compositions from Table 3, the formulae in equations (10) and (11), and aggregate the individual minerals using second-order, multiphase Hashin-Shtrikman extremal bounds [Berryman, 1995], taking into account possible interfacial effects [Salje, 2007]. Simple mixing laws, such as arithmetic or geometric relations, Archie's Law, and Voigt and Reuss estimates, give first-order variational bounds, whereas Hashin-Shtrikman represents a member of the class of second-order variational bounds. There exist third-order variational bounds [Beran, 1965; Beran and Molyneux, 1966] that have been applied in seismology [X F Xu, 2011] but not yet in electromagnetism. An additional term was added to the Hashin-Shtrikman equations ofBerryman  by Salje to explain observations that were found to lie outside the Hashin-Shtrikman bounds. The term considers interfacial effects that are shown to scale withf(1-f), where f is the volume proportion of the individual phase. For intermediate cases, where both volume and interfacial effects are important, the volume proportion f in the averaging scheme is replaced by f(1 − S) + Sf2, where S ranges from 0 (no interfacial effects) to 1 (only interfacial effects).
 Extremal bounds are more important for electrical conductivity than for bulk moduli, as the influence of the highest conductivity phase, even if only a minor component (1%), can be dramatic if the minor phase is sufficiently well interconnected. However, olivine (Ol) and pyroxene (Opx, Cpx) minerals when dry have conductivities that are within an order of magnitude of each other at any given (P,T) conditions, and, for the mineral assemblages considered herein, Ol and Opx make up the bulk of the whole rock, over 95% in most cases, so the extremal bounds are not wide. We derive estimates using the geometric means of the upper and lower Hashin-Shtrikman bounds with and without interfacial effects; the individual values are so close to each other that the geometric and arithmetic means are virtually the same.
 Our average petrologically based estimates for log10(conductivity) at 100 km depth beneath Jagersfontein and Gibeon, under water-absent, dry conditions, i.e., small polaron conduction only, are −5.30 and −4.39 log units respectively (Table 5).
Table 5. Hashin-Shrikman Estimates of Log10(Conductivity [S/m]) at 100 km Depth Beneath Jagersfontein (FRB Samples) and Gibeon (KGG Samples), Assuming Dry Conditions (No Water)a
|Sample||T (°C)||HS− (S = 0)||HS+ (S = 0)||HS− (S = 1)||HS+ (S = 1)||HS GAV|
 Given the observations of water in the lithosphere, we must derive the expected conductivities under “slightly damp,” conditions. None of the data in Peslier et al. [2010b]are from xenoliths sourced as shallow as 100 km nor as cold as 740°C, so we have to extrapolate from their results at higher temperatures (900°C) beneath Kimberley that show evidence for around 80 ppm water in the olivine. Given the lack of any depth-dependence shown for the upper lithosphere inPeslier et al. [2010b], we assume a value of 80 ± 10 ppm later in the text, but calculate at this point conductivity for water content from 0–200 ppm in olivine.
 The conductivity calculations for wet conditions for olivine for the Karato, Yoshino and Poe (average) models that include the effects of proton conduction using equations (10) and (11) for temperatures of 740°C and 850°C, i.e., our Jagersfontein and Gibeon estimated ambient temperatures at 100 km, are shown as the dashed lines in Figure 6.
Figure 6. (a) Variation of conductivity at Jagersfontein parameters at 100 km depth of T = 740°C, P = 3.2 GPa, for olivine water concentration varying from 0 – 200 ppm for the Karato (black lines), Yoshino (red lines) and Poe (blue lines) models (equation (10)) for olivine alone (Mg# = 93.2) (dashed lines) and for the Jagersfontein composition assemblage (Mg# = 93.2, except for Gt where Mg# = 75.0) (solid lines). Note that the abscissa annotates olivine water concentration, and that for the composition assemblage calculations, water concentrations for Opx, Cpx and Gt are taken as 2 × Ol, 3 × Ol and zero respectively, as discussed in the text. The observation datum (green point) is for a water content in Ol of 80 ± 20 ppm (from xenoliths) and a log(conductivity) of −3.41 ± 0.41 (from mapping of MT observations, with 2σ error bounds). The dashed green line represent the revised Karato water model for olivine, with log(σ0) = 2.70 (±0.26), r = 0.70 (±0.12), and ΔHwet = 0.91 (±0.04). The solid green line represents the revised water model for olivine that best fits equation (1) with 1og10(σ0) = 3.05 (±0.16), r = 0.86 (±0.08), ΔHwet = 0.91 (±0.04), and α = 0.09 (±0.07), valid for water content in damp lithosphere of around 60–100 ppm. The dotted lines are ±1σ. (b) Variation of conductivity at Gibeon parameters at 100 km depth of T = 850°C, P = 3.335 GPa, for water varying from 0 – 200 ppm. The observation datum (green point) is for a water content in Ol of 80 ± 20 ppm (assumed to be the same as for Jagersfontein in the absence of xenolith water content measurements at Gibeon) and a log(conductivity) of −2.78 ± 0.18 (from mapping of MT observations, with 2σ error bounds). Curves as for Figure 6a. The dotted lines are ±1σ.
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 Olivine is not the only mantle mineral, even though it comprises >70% (Table 3) of the rock mass, so consideration must be given to the contribution to proton conduction from the other mantle minerals, pyroxenes and garnet, of which Opx dominates. Water in pyroxenes is higher than in olivine, with averages of 200 ppm in Opx and 300 ppm in Cpx, and virtually zero in garnet, in the Kaapvaal peridotite xenolith samples reported in Peslier [2010, supplementary material]. A recent abstract lists a narrow range for both Opx and Cpx of 119–251 ppm and 149–398 ppm respectively [Peslier et al., 2010a], bracketing the average given above, with the intriguing suggestion of an increase in water content in pyroxenes from Liqhobong (Lesotho), through Finsch Mine, Jagersfontein, to Kimberley xenoliths. In contrast, far lower values are reported for xenoliths from the Letseng kimberlite in Lesotho [Sundvall and Stalder, 2011], from 20 ppm. As discussed above, we assume a water partitioning of 1:2:3 between the Ol, Opx and Cpx.
 Computing the resistivities for the composition given in Table 3 for water contents of x ppm in Ol, 2x ppm in Opx, 3x ppm in Cpx, and 0 ppm in Gt (values observed are 0–15 ppm at 4 Gpa in Peslier , but given the very low volumes of these minerals we set the water contents to zero), where x varies from 0 to 200 and the increase in water in pyroxenes is assumed to be linearly related to the increase in water in olivine, yields the solid lines shown in Figure 6a for the conductivity beneath Jagersfontein with varying water content. Holding the water contents of the pyroxenes fixed at the averages of 200 ppm and 300 ppm for Opx and Cpx respectively and varying Ol only from 0 to 200 ppm introduces very little change in our conductivity estimates over assuming a 1:2:3 water partitioning, especially at the water contents of interest, around 80 ppm.
 The water content of minerals from xenoliths beneath Gibeon is unknown, but it is reasonable to assume that it will be of the same order, if not somewhat less. Undertaking the computations for Gibeon parameters (Table 3) yields the solid lines in Figure 6b for the Karato, Yoshino and Poe models.