Estimating core-mantle boundary temperature from seismic shear velocity and attenuation

The temperature at Earth’s core-mantle boundary (CMB) is a key parameter to understand the dynamics of our planet’s interior. However, it remains poorly known, with current estimate ranging from about 3000 K to 4500 K and more. Here, we introduce a new approach based on joint measurements of seismic shear-wave velocity, V S, and quality factor, Q S, in the lowermost mantle. Lateral changes in both V S and Q S above the CMB provide constraints on lateral temperature anomalies with respect to a reference temperature, T ref, defined as the average temperature in the layer immediately above the CMB. The request that, at a given location, temperature anomalies inferred independently from V S and Q S should be equal gives a constraint on T ref. Correcting T ref for radial adiabatic and super-adiabatic increases in temperature gives an estimate of the CMB temperature, T CMB. This approach further relies on the fact that V S-anomalies are affected by the distribution of post-perovskite (pPv) phase. As a result, the inferred T ref is linked to the temperature T pPv at which the transition from bridgmanite to pPv occurs close to the CMB. A preliminary application to V S and Q S measured beneath Central America and the Northern Pacific suggest that for T pPv = 3500 K, T CMB lies in the range 3,470–3880 K with a 95% likelihood. Additional measurements in various regions, together with a better knowledge of T pPv, are however needed to determine a precise value of T CMB with our method.


Introduction
The temperature at the Earth's core-mantle boundary (CMB), T CMB , is a key property for a better understanding of the dynamics of our planet's mantle. Combined with thermal boundary layer (TBL) models, it can further be used to estimate the heat flux at the CMB, which is controlling (at least partially) core dynamics, geodynamo, and our planet's thermal evolution (see Frost et al., 2022 for a multidisciplinary study on these questions). CMB temperature remains however poorly constrained, with current estimates ranging from 2,500-3000 K to 4,000-4500 K, depending on the method used to estimate it. At other major boundaries, temperature may be deduced from phase diagrams of appropriate materials. For instance, the temperature at the boundary between the upper and lower mantle, around a depth of 660 km, may be deduced from the phase transformation from ringwoodite to bridgmanite and ferro-periclase. Similarly, the temperature at the limit between the inner and outer cores (ICB), at a depth of 5,150 km, can be estimated from the liquid to solid transition of iron alloys. By contrast, because the CMB is a material boundary between silicate rocks and molten iron alloys, T CMB cannot be deduced directly from a specific phase diagram.
Instead, estimating T CMB requires the combination of different observations and modelling, including seismic data, properties of core and mantle materials, and core and mantle dynamics. First estimates from seismic observations consisted in building adiabatic mantle geotherms that fit the mantle average seismic structure (for instance PREM, Dziewonski and Anderson, 1981) given a mantle average composition and thermo-elastic properties of mantle minerals (e.g., Brown and Shankland, 1981;Anderson, 1982;Shankland and Brown, 1985;Jackson, 1998;Deschamps and Trampert, 2004). These studies lead to CMB temperatures in the range 2,500-3200 K, to which a non-adiabatic contribution related to the presence of a TBL at the bottom of the mantle should be added. Other constraints may be obtained from the presence in the lowermost mantle of postperovskite (pPv), a high-pressure phase of bridgmanite (Oganov and Ono, 2004;Tsuchiya et al., 2004). Depending on temperature, pPv may transform back to bridgmanite a few kilometers or tens of kilometers above the CMB, forming a pPv lens (Hernlund et al., 2005). Such lenses imply a double crossing between the mantle geotherm and the post-perovskite phase boundary. Combined with an analytical modeling of the lower mantle TBL, and given pPv phase boundary properties, the depths of the lenses upper and lower sides provide estimates of T CMB and CMB heat flux. Possible pPv lenses reported beneath Central America (van der Hilst et al., 2007) and the central Pacific ocean (Lay et al., 2006) lead to T CMB around 3950 K and 4100 K, respectively. However, Buffet (2007) pointed out that the flow beneath the pPv lens and the release of latent heat by the transition from pPv to bridgmanite at the lower side of this lens would modify the thermal structure in this region, leading to higher temperature gradient and CMB heat flux. Still on the mantle side, maximum possible values of T CMB may be obtained from the solidus of mantle rocks at CMB pressures. Mineral physics experiments lead to maximum values from 3570 K (Nomura et al., 2014) to 4200 K (Fiquet et al., 2010;Andrault et al., 2011). On core side, estimates of T CMB rely on core thermodynamic properties. More specifically the melting temperature of iron alloys at ICB pressures measured from mineral physics experiments (e.g., Brown and McQueen, 1986;Boehler, 1993;Anzellini et al., 2013) is extrapolated to CMB pressures assuming that the outer core is adiabatic. A difficulty is that the exact melting temperature depends on the outer core content in light elements (S, O, Si and C), which is poorly known. Experimental results have been obtained for various iron alloys, including Fe-FeO (Morard et al., 2017), Fe-Fe 3 S (Kamada et al., 2012), Fe-FeSi (Fischer et al., 2013), and Fe-Fe 3 C (Fischer, 2016;Morard et al., 2017). Following these results, ICB temperature may range from 5,150 to 6200 K (Fischer, 2016), depending on the core composition, leading to T CMB in the range 3,850-4600 K for a purely adiabatic outer core. Integration along an adiabatic profile further requires knowledge of core material properties, including its density, bulk modulus and Grüneisen parameter, whose values may, again, depend on the core exact composition. For three different Fe-O-Si alloys, Davies et al. (2015) found T CMB between 4,290 and 3910 K.
Here, we propose a new approach to the determination of T CMB . This approach is based on the analysis of observed lateral variations in seismic shear-wave velocity, V S , and attenuation, measured with the quality factor Q S , in the lowermost mantle. Shear-wave velocity is sensitive to the presence of postperovskite, with shear waves travelling faster in pPv than in bridgmanite (see Cobden et al., 2015 for a compilation). Because the stability field of pPv strongly depends on temperature (see, again, Cobden et al., 2015 for a compilation of Clapeyron slope values), the presence of this phase and its impact on V S provides a constraint on the local and horizontally averaged temperature in the lowermost mantle. Another constraint on local and horizontally averaged temperatures may be obtained from seismic attenuation, which is a thermally activated process (Minster and Anderson, 1981;Anderson and Given, 1982), implying that its amplitude depends on temperature. In the reminder of this paper, we detail this method, and we perform a preliminary application using models of V S and Q S obtained beneath Central America (Borgeaud and Deschamps, 2021) and the Northern Pacific (Deschamps et al., 2019).

Methods
At a given depth, lateral variations in temperature trigger changes in seismic shear-wave velocity, V S , and seismic attenuation, measured with the quality factor Q S . Deviations of Q S and, if post-perovskite is present, V S from their reference (horizontally averaged) values depend on a reference temperature, T ref , that can be defined as the horizontally averaged temperature at that depth. Local Q S and shear-wave velocity anomalies, dlnV S , may then be used to estimate T ref .
More specifically, the request that, at a given location, the temperature deviations derived from dlnV S and Q S should be equal provide a constraint on T ref . Our method is sketched in Figure 1 and summarizes as follow. At each selected location where V S and Q S measurements are available, and for a prescribed a priori range of T ref , we first calculate probability density functions (pdfs) of the temperature anomalies dT VS and dT Q predicted by the deviations of V S and Q S deviations from Flow chart of the method used to evaluate the temperature at the CMB, T CMB , from observed shear velocity (V S ) and seismic quality factor (Q S ) measurements at different locations i. For each location, we first calculate a probability density function (pdf) of the average temperature in the layer above the CMB, T ref , by comparing the pdfs of temperature anomalies predicted by the deviations of V S and Q S from PREM. We then combine the local pdfs of T ref to calculate a total pdf of T ref . Finally, because T ref is sampling a layer whose thickness is fixed by the resolution of seismic models, we make a correction for adiabatic and super-adiabatic temperature increase throughout this layer.

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frontiersin.org 03 their PREM values. We then calculate a pdf of T ref at this location from the overlap between the pdfs of dT VS and dT Q . Next, we derive a total (i.e., constrained by all selected measurements) pdf of T ref by combining the pdfs of T ref at each selected location. Finally, because T ref is sampling a layer whose thickness is fixed by the resolution of seismic models, we apply a correction for adiabatic and super-adiabatic temperature increase throughout this layer. We now detail these different steps.

Constraint from shear-wave velocity anomalies
At a given depth, shear-wave velocity anomalies with respect to a reference velocity, dlnV S , may be expressed as a function of changes in temperature, composition, and phase with respect to the average (or reference) values of these parameters. Because phase changes depend on the pressure and temperature, the contributions of a phase change to dlnV S implicitly depend on both the local and reference temperatures, T and T ref . In the lowermost mantle bridgmanite, the most abundant minearal of the lower mantle, may transform its high-pressure phase, postperovskite (pPv; Oganov and Ono, 2004;Tsuchiya et al., 2004). Mineral physics data indicate that in regions where pPv is present shear waves travel faster than in regular (bridgmanite dominated) mantle. Available measurements are relatively dispersed (see Cobden et al., 2015 for a compilation), but show that, on average, shear velocity increases by 2-3% as bridgmanite transforms to pPv. Due to its large Clapeyron slope, in the range 8-13 MPa/K (Cobden et al., 2015), the depth at which this phase transition occurs strongly depends on temperature and may thus sharply vary in space. In addition, pPv may transform back to bridgmanite a few kilometers or tens of kilometers above the CMB (Hernlund et al., 2005). As a result, the thickness of the pPv lens may strongly vary from one place to another. Radial and lateral parameterizations of seismic models imply that pPv might not be present everywhere in the region sampled by the seismic data. It is therefore meaningful to define a local fraction of pPv, X pPv . Changes in the local fraction of pPv, for instance due to variations in the thickness of pPv lens, may then contribute to lateral changes in seismic velocities. One may further define a local anomaly in pPv fraction anomaly, dX pPv , from the difference between the local and reference (i.e., horizontally averaged) fractions of pPv, which depend on T and T ref .
Assuming that only temperature changes and related changes in the stability field of pPv are present, the temperature anomaly deduced from observed dlnV S is where S T and S pPv are sensitivities of shear-wave velocity to temperature and pPv, respectively, defined as the logarithmic partial derivatives of shear-velocity with respect to temperature and pPv. Sensitivities may be deduced from mineral physics data and equation of state modelling. Taking into account dispersion and error bars in mineral physics data provides both means and uncertainties in these sensitivities. For calculations (Section 3), mean and uncertainties in S T are taken from Deschamps et al. (2012) and mean and uncertainties in S pPv are deduced from the compilation of Cobden et al. (2015) ( Table 2). With these values, a 500 K temperature increase induces a reduction in shear velocity anomaly between 1.2 and 1.5%, and the transition from bridgmanite to pPv triggers a shear velocity increase between 0.1 and 4.6%. Again, the resolution of seismic models implies that pPv may not be present throughout the region sampled by seismic data, implying that the contribution of the pPv phase change to the observed seismic velocity anomalies is a fraction of the velocity change measured by mineral physics data. Transformation of bridgmanite to pPv occurs over a narrow range of temperature, or thermal width, centered on temperature T pPv . Here, we describe the local fraction of pPv (between 0 and 1) at temperature T with where T pPv is, again, the temperature of the transition to pPv, and δT pPv is a typical temperature anomaly modeling the thermal width of the phase transition. Compilation of experimental and ab initio data (Cobden et al., 2015) suggests that at the bottom of the mantle T pPv may range from 3,000 to 4500 K. Following Eq. 2, and taking T pPv = 3500 K and δT pPv = 20 K, X pPv goes to one for temperatures lower than 3450 K, and to zero for temperatures larger than 3550 K. Because the distribution of pPv depends on the distribution of temperature, we defined the pPv reference (or horizontally averaged) fraction, X pPv,ref , according to the lowermost mantle distribution in temperature anomaly, dT = T -T ref , obtained by Mosca et al. (2012). Noting that this distribution is nearly bimodal (as two peaks can clearly be distinguished; Figure 2A), we first modeled these anomalies with the normalized sum f of two Gaussian distributions. For a given T ref , we then modulate the fraction of pPv associated with a temperature anomaly dT with the function f (dT), and sum these modulated values over a range of temperature ΔT following (3) Figure 2B shows X pPv,ref as a function of T ref for ΔT = 3000 K and T pPv = 3500 K. For lower (higher) values of T pPv , X pPv,ref is similar to the curve plotted in Figure 2B, but shifted towards lower (higher) T ref .
Noting that dX pPv = X pPv -X pPv,ref and replacing X pPv by its expression in Eq. 2, Eq. 1 provides an expression for dT VS as a function of dlnV S and T ref , where X pPv,ref is given by Eq. 3. Note that the temperature anomaly in Eq. 3 is a summation (dummy) variable specific to this equation,  Deschamps et al. (2012), and sensitivity to pPv is from the compilation of Cobden et al. (2015). The modelled distribution is given by the normalized sum of two Gaussian distributions (thin orange curves), based on the fact that the observed distribution is nearly bimodal. The amplitude, mean and standard deviations of these Gaussian are A 1 = 2.987×10 -2 , dT 1 = 60 K, σ 1 = 110 K, and A 2 = 1.173×10 -2 , dT 2 = -225 K, σ 2 = 60 (K) (B) Reference fraction of post-perovskite, X pPv,ref , calculated by Eq. 3 with T pPv = 3500 K and δT pPv = 20 (K) Frontiers in Earth Science frontiersin.org and is therefore different from the dT VS that we aim to determine. Equation 4 can then easily be solved for dT VS using classical zerosearch methods.

Constraint from seismic attenuation
The presence of small defects in the crystalline structure of mantle rocks results in the dissipation of a small fraction of the energy carried by seismic waves, leading, in turn, to seismic attenuation as these waves travel through the mantle. Attenuation depends on the frequency of seismic waves and is high only within a range of frequencies, or absorption band (Anderson and Given, 1982). In addition, it is a thermally activated process with a relaxation time that is well described by an Arrhenius law (Anderson and Given, 1982). Practically, seismic attenuation is measured with the quality factor Q. The higher the attenuation, the lower Q. Assuming that it follows a power-law with exponent α of the frequency ω and of the relaxation time (Minster and Anderson, 1981), the quality factor may be written where Q 0 is a constant, R = 8.32 Jmol −1 K −1 the ideal gas constant, T the temperature, and H = E + pV the activation enthalpy, with E and V being the activation energy and volume, respectively, and p the pressure. We model the anomaly in quality factor with respect to a reference value Q ref following the approach of Deschamps et al. (2019). Eq. 5 may be used model the quality factor at any temperature T, including at a reference temperature T ref , which defines the reference quality factor Q ref . Noting dT Q the temperature anomaly (T -T ref ) at a location where Q S is measured, we define the anomaly in shear quality factor with the ratio between Q S and Q ref , Inversion of Eq. 6 gives dT Q from Q S following, For calculations, we set Q ref to its PREM value in the lower mantle, which is equal to 312 (Dziewonski and Anderson, 1981) and is consistent with models built from a probabilistic approach (Resovsky et al., 2005). A difficulty is that the values of α and H are poorly constrained. At lowermost mantle depths α was found to be equal to 0.1 for periods in the range 300-800 s, and around 0.3 for a period of 200 s (Lekić et al., 2009). In their anelastic model Dannberg et al. (2017) used α = 0.274 to fit the shear-wave velocity of PREM (Dziewonski and Anderson, 1981) and the quality factor of QL6 (Durek and Ekström, 1996). Dannberg et al. (2017) further used activation energy and volume activation equal to 286 kJ/mol and (1.2 ± 0.1)×10 -6 m 3 /mol, respectively, leading to activation enthalpy at the bottom of the mantle around 440 kJ/mol. Possible range of these two parameters are however larger and may lead to values of H in the range 250-900 kJ/mol (for discussion on E and V values, see Matas and Bukowinski, 2007 and supplementary material of Deschamps et al., 2019). Interestingly, Eq. 7 indicates that only the product αH matters for calculations of dT Q . Here, we explored values of this product in the range 20-400 kJ/mol, covering conservative ranges of α and H, 0.1-0.4 and 200-1,000 kJ/mol, respectively.
Attenuation may be affected by the presence of volatiles, most particularly water. The amount of water in the deep mantle may however be very limited, less than about 30 ppm weight in bridgmanite (Panero et al., 2015). For such low concentrations, rocks may be considered as dry, and volatiles would have no or very limited effects. This hypothesis is further supported by recent mineral physics experiments on olivine showing that for water contents relevant to the Earth's interior, attenuation and seismic velocities are not sensitive to the water content (Cline et al., 2018). Finally, attenuation of mantle minerals may also be sensitive to grain size. Jackson et al. (2002) quantified this effect for olivine, but to date the grain size sensitivities of lower mantle minerals remain unconstrained. Using olivine data and an extended Burgers model (Faul and Jackson, 2015), Lau and Faul (2019) showed that grain-size may affect attenuation at lower mantle pressures and periods ranging from seismic to tidal timescales. For periods around 10 s, their results further indicate that grain-size dependence is reduced as pressure increases (see their Figure 5), suggesting no or limited grain-size dependence close to the CMB. In addition, because grain-size dependence is controlled by a grain-size exponent, and since we quantify anomalies in quality factor as the ratio between local and reference quality factors (Eq. 6), grain-size effects should not affect dT Q provided that the grain-size does not vary substantially at a given depth.

Probability density functions of the reference temperature T ref
The request that, at a given location, the dT VS and dT Q deduced from Eqs. 4 and Eq. 7 are equal provides, in principle, an estimate of T ref . However, depending on the observed dlnV S and Q S and on the assumed values of the model parameters (mainly α, H, and T pPv ), dT VS = dT Q may have more than one solution or no solution at all. To solve this issue, we follow an approach based on distributions of dT VS and dT Q as a function of T ref , which further allow to take into account uncertainties on observed data. For a set of dlnV S and Q S measured at different locations, we then Frontiers in Earth Science frontiersin.org obtain a probability density function (pdf) for an a priori range of T ref . The different steps leading to such pdfs are detailed below. We first estimate distributions in dT VS and dT Q at a given location i and reference temperature T ref . For this, we randomly vary observed dlnV S and Q S around their average values following Gaussian distributions with prescribed standard deviations. For each sample we then calculate the corresponding dT VS and dT Q , and bin these temperature anomalies in normalized frequency histograms, leading to individual pdfs, P i VS and P i Q . Figure 3 plots example of such pdfs built from a set of 1 million dlnV S and Q S samples for Central America location S2 (Table 1) and for different values of T ref . Standard deviations for dlnV S and Q S are fixed to the estimated uncertainties in these data, which are here equal to 0.1% and 20, respectively, and the values of parameters in Eqs. 4 and Eq. 7 are listed in Table 2. Interestingly, dT Q pdfs are Gaussian with a good approximation. If the pPv anomaly (dX pPv ) is null, dlnV S is only due to temperature changes, and dT VS pdfs are also nearly Gaussian. By contrast, distributions in dT VS strongly deviate from Gaussian distributions if the local pPv fraction is different from its horizontally average (dX pPv is different from zero).
We then define the likelihood that T ref is equal to a specific value by the overlap between the integrals of the distributions obtained for dT VS and dT Q (see Figure 3), which, for location i, may be expressed as where dT 0 and dT 1 are the assumed lower and upper bounds in temperature anomaly, ΔT (dT 1 − dT 0 ), and C i n a normalisation constant obtained by summing P i Tref over the entire explored ranges of T ref and, if applying, other parameters (for instance the product αH). Eq. 8 can be used to estimate likelihoods at all locations where joint measurements of dlnV S and Q S are available. The products of these local likelihoods for each value of T ref finally provide an estimate of the likelihood for each value of T ref , where N is the number of locations for which measurements are available.

Corrections for radial averaging
Strictly speaking our method provides an estimate of the average temperature within the mantle lowermost few tens of

FIGURE 3
Probability density functions of temperature anomalies deduced from dlnV S (dT VS , orange distributions) and Q S (dT Q , blue distributions) measured in corridor S2 beneath Central America (see Borgeaud and Deschamps 2021 for the exact location). Mean and standard deviation in dlnV S and Q S are 1.0% and 0.1% and 470 and 20, respectively (Table 1), and different reference temperature T ref are considered in each plot: 2600 K (A); 300 K (B); 3700 K (C); and 4100 K (D). Other modeling parameters are set to their preferred values listed in Table 2. The overlap between the pdfs for dT VS and dT Q at a specific value of T ref is taken as a measure of the probability p that T ref is equal to this specific value. The transition temperature to post-perovskite is set to T pPv = 3500 K. Note that the frequency (y-axis) is plotted in logarithmic scale.

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frontiersin.org kilometers, the exact thickness of this layer depending on the radial resolution of the observed V S and Q S . It can however give access to T CMB provided that a small correction ΔT z is applied to account for the temperature increase within this layer. This increase includes an adiabatic contribution due to the pressure increase, and a super-adiabatic temperature increase due to the presence at the bottom of the mantle of a thermal boundary layer associated with mantle convection. Practically, and ignoring the effects of spherical geometry, the horizontally averaged temperature obtained by the method described in the previous sections may be written T ref where T top is the temperature at the top of the sampled layer. Noting that T CMB T top + ΔT z , one gets T CMB T ref + ΔT z /2. As an illustration, for models of V S and Q S with a radial resolution of 50 km, and assuming adiabatic and super-adiabatic temperature gradient of 0.3 K/km and 2.5 K/km (typical of the gradient obtained in numerical simulations of mantle convection), respectively, ΔT z is about 140 K and a correction of 70 K should be applied.

A preliminary estimate
We applied the method detailed in section 2 to the measurements of V S and Q S listed in Table 1. Our goal is not to provide a conclusive value for the CMB temperature, as more measurements of V S and Q S together with a transition temperature to pPv more precise than current estimates may be needed for this, but rather to test our method. The measurements of V S and Q S we used are the lowermost layer of 1D radial profiles obtained beneath the Northern Pacific (Deschamps et al., 2019) and beneath Central America (Borgeaud and Deschamps, 2021) by full-waveform inversions of seismic data. In both cases, the inversion method is similar to that used in Konishi et al. (2017), and the radial resolution of the 1D models is 50 km, i.e. the V S -anomalies and Q S in Table 1 sample a 50 km thick layer above the CMB. Note that the method used to recover Central America profiles further includes travel-time corrections for the 3D mantle structure beneath Central America, spectral amplitude misfit to better constrain Q S , and corrections for focusing effects in the lowermost mantle (Borgeaud and Deschamps, 2021). We converted V S to relative anomalies dlnV S with respect to PREM (Dziewonski and Anderson, 1981) shear-wave velocity, which, in the lowermost 50 km, is equal to 7.26 km/s. We then built dT VS and dT Q distributions for values of T ref in the range 2,500-4500 K by randomly generating 1 million dlnV S and Q S samples at each T ref . Samples distributions follow Gaussian distributions centered on the observed dlnV S and Q S and with standard deviations fixed to 0.1% for dlnV S and 20 for Q S , on the basis of observed error bars. Values of the modelling parameters used in Eqs 4, 7 are listed in Table 2. In particular, we explored values of the frequency exponent, α, and activation enthalpy, H, of attenuation in ranges leading to values of the product αH between 20 and 400 kJ/mol. Because temperature of the transition to pPv, T pPv , is uncertain, and to quantify its influence on T ref , we performed calculations for several values of this parameter in the range 3,000-4250 K.
Left column in Figure 4 shows the median in dT VS and dT Q (defined as the 0.5 pdf quartile, meaning that there is 50% likelihood that T ref lie on each side of this value) as a function of T ref and at different locations. Modeling parameters are fixed to T pPv = 3500 K, α = 0.274, and H = 440 kJ/mol. The coloured areas cover 68.3% of the pdfs around their median values, which, for Gaussian distributions, correspond to one standard deviation. Right column plots the corresponding likelihood normalized with the maximum likelihood, P max , for each case. The evolution of dT VS as a function of T ref is controlled by the anomaly in the fraction of pPv, dX pPv . If T ref is too low or too large, pPv is either fully covering the CMB (X pPv,ref = 1) or nowhere stable (X pPv,ref = 0). In both cases dX pPv = 0 and dlnV S is only affected by the temperature anomaly, implying that dT VS does not depend on T ref . The lower (T low ) and upper (T up ) bounds of temperature for a non-zero dX pPv are depending on both the assumed lower mantle temperature distribution and T pPv . Taking the temperature distribution from Mosca et al. (2012) and T pPv = 3500 K, these bounds are around T low = 3000 K and T up = 4000 K ( Figure 2B). For intermediate temperatures, an excess (deficit) in pPv triggers an increase (decrease) in seismic velocity, such that part of the observed dlnV S is due to the pPv anomaly. As a result, for locations colder than the reference temperature, dT VS is lower (in absolute value) than its purely thermal value if dX pPv > 0, and larger if dX pPv < 0. For location hotter than T ref , the opposite trends occur. Note that discontinuity in dT VS may occur (for instance, corridor N1 in Central America) as the dX pPv falls to zero for T ref ≤ T low or T ref ≥ T up . For each location, overlaps between dT VS and dT Q distributions at a given value of T ref provide an estimate of the likelihood for this specific value of T ref , with larger overlaps leading to higher likelihoods (Section 2.3). For locations with V S and Q S close to PREM (for instance, Northern Pacific and corridor N4 in Central America), overlap between dT VS and dT Q occur for a wide range of T ref .
Implying that these locations bring few constraints to T ref . The distributions in dT VS further depend on the assumed value of T pPv , which, again, is fixed to 3500 K in Figure 4. For higher (lower) T pPv , these distributions keep the same shape but shifts to larger (smaller) T ref . In other words, and as one would expect, higher T pPv favors higher T ref . Similarly, the distributions in dT Q depends on the product αH, with temperature anomalies getting smaller (blue curves in Figure 4 move towards dT Q = 0) as αH increases. The sign of dT Q , on another hand, is controlled by the ratio between the observed and reference quality factors, Q S /Q ref . Figure 5A plots the total likelihood (Eq. 9), defined as the product of individual local likelihoods, obtained for T pPv =3500 K as a function of both T ref and the product αH. For clarity, we also show in Figure 6 likelihoods for selected Frontiers in Earth Science frontiersin.org values of αH. Interestingly, the most likely range of T ref depends relatively little on αH. For αH ≤ 100 kJ/mol, likelihood is very low (at least one order of magnitude lower than the maximum likelihood) whatever the value of T ref , suggesting that such values of αH can be ruled out. Above this value, the most likely T ref slightly decreases with increasing αH, from~3800 K at αH = 100 kJ/mol, to~3600 K at αH = 400 kJ/mol. Likelihood is largest for αH around 140 kJ/mol, but remains high throughout the range 100-400 kJ/mol (Figure 6; note the logarithmic scale). High likelihood may still be found for αH ≥ 400 kJ/mol (not explored in this study), but this would imply values of α and H in excess of 0.4 and 1,000 kJ/mol, respectively, which appear unlikely (Section 2.2). We did another calculation using the temperature distribution of Trampert et al. (2004) to calculate X pPv,ref (and  Table 2. Frontiers in Earth Science frontiersin.org therefore dX pPv ; Section 2.1), but did not find substantial differences in the total likelihood ( Figure 5B). By contrast, and as one would expect, T pPv has a strong impact on the estimated likelihood, the most likely value of T ref increasing with T pPv (see plots c and d in Figure 5, obtained for T pPv equal to 3,000 and 4000 K, respectively). Note also that lower values of T pPv allow lower values of αH. Following our approach, a precise inference of T ref therefore requires an accurate knowledge of T pPv .
To estimate the most likely range of T ref we summed up obtained likelihoods over the explored range of αH, and calculated cumulative likelihoods (Figure 7; note the logarithmic scale in plot a). For T pPv = 3500 K, the summed likelihood has two peaks, around T ref = 3600 K and T ref = 3750 K. The cumulative likelihood indicates that T ref lies within the range 3,400-3810 K with a 95% likelihood, and that its median value, defined as the 0.5 quartile (meaning that there is 50% likelihood that T ref lies on each side of this value), is around 3650 K. Again, for larger (lower) values of T pPv , these range and median value both shift to higher (lower) values. Figure 8, plotting the median value of T ref as a function of T pPv , shows that the median T ref increases nearly linearly with T pPv . The grey band in Figure 8 covers values of T ref ranging from quartiles 0.025 to 0.975, meaning, again, that there is 95% likelihood that T ref lies within this range. As discussed in section 2.4, T ref is an estimate of the reference temperature averaged out in the mantle lowermost 50 km. Adding the estimated adiabatic and super-adiabatic temperature jumps to the median T ref , the CMB temperature may be given by A least square fit of our calculations for T pPv in the range 3,000-4250 K leads to a 0 = -125 K and a = 1.077. For instance, taking T pPv = 3500 K and ΔT z = 140 K (Section 2.4) leads to a CMB temperature of 3715 K and a 95% likelihood range of 3,470-3880 K.

Discussion and concluding remarks
In this study, we built a method to infer CMB temperature (T CMB ) from measurements of seismic shear-wave velocity (V S ) and quality factor (Q S ) in the lowermost mantle. Both these two observables bring constraints on the local temperature anomaly Total likelihood (P, color scale), given by Eq. 9, as a function of the reference temperature (x-axis) and of the product of the frequency exponent α and activation enthalpy H (y-axis). Three values of the transition temperature to post-perovskite (pPv), T pPv , are considered, 3500 K (A,B), 3000 K (C), and 4000 K (D). The temperature distribution used to model the pPv average fraction is either from Mosca et al. (2012) [plots (A,C,D)] or Trampert et al. (2004) (B). Other modeling parameters are listed in Table 2. In all cases, likelihoods are normalized with their maximum value P max over the range of explored T ref and αH, and plotted with logarithmic scale.

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frontiersin.org with respect to a horizontally averaged temperature T ref , and the combination of these constraints provides estimates of T ref at that depth. A correction for radial averaging related to the radial resolution of V S and Q S then gives T CMB . To account for uncertainties in the modelling parameters of V S and Q S , our method calculates probability density functions (pdfs) of T ref (and thus T CMB ), rather than mean, single values. Because it is based on the fact that seismic velocity is affected by lateral changes in the depth of the phase transition from bridgmanite to post-perovskite (pPv), and therefore by lateral changes in the amount of pPv at a given location, the pdfs of T ref deduced from our approach depend on the transition temperature from bridgmanite to pPv close to the CMB, T pPv . Applying our method to measurements of V S and Q S beneath Central America (Borgeaud and Deschamps, 2021) and the Northern Pacific (Deschamps et al., 2019), we found that for T pPv = 3500 K the CMB temperature should be in the range 3,470-3880 K with a 95% likelihood. Because in our approach the value of T CMB depends on the value of T pPv right above the CMB, which remains poorly known, comparison between our results and available estimates is not straightforward. It is however interesting to note that the range of T CMB we obtained for T pPv = 3500 K (grey shaded area in Figure 9), is coherent with the upper range of T CMB estimated from radial seismic velocity profiles to which a 500 K amplitude thermal boundary layer (TBL) is added, with the maximum possible T CMB deduced from the solidus of pyrolite, and with the lower bound of the range of T CMB estimated from the melting temperature T ICB of iron alloys at the inner core boundary. Figure 9, further indicate possible ranges of T pPv (according to Eq. 10) for T CMB obtained by different methods. For instance, the range of T CMB estimated Total likelihoods are normalized with the absolute maximum, P max , and plotted with a logarithmic scale. The temperature of the transition to pPv is T pPv = 3500 K. For other details of the calculations, see text and Table 2.
Frontiers in Earth Science frontiersin.org from the pyrolite solidus and from T ICB implies T pPv from 3,350 to 3950 K, and 3,600-4300 K, respectively. The values of T pPv compatible with the CMB temperatures inferred from seismic profiles are overall lower, but depend on the assumed thermal amplitude of the TBL added to the adiabatic temperatures deduced from these profiles. For ΔT TBL = 500 K, T pPv is in the range 2,800-3500 K, i.e., and as one would expect, on the lower range of experimental measurements.
Certainly the biggest unknown in our modelling is T pPv , which at the CMB may range from 3,000 to 4500 K (Cobden et al., 2015). A complication is that the exact transition temperature depends on the composition of the aggregate. For instance, higher iron content within bridgmanite increases T pPv , while aluminium has the opposite effect. In addition, laboratory experiments are made at fixed pressures, and extrapolation to the CMB requires knowledge of the Clapeyron slope, Γ pPv , which may range between 8 and 13 MPa/K. For pyrolytic compositions and a temperature of 2500 K, transition to pPv is usually observed in the pressure range 115-130 GPa (Ohta et al., 2008;Catalli et al., 2009;Sinmyo et al., 2011), which, assuming Γ pPv = 10 MPa/K leads to T pPv in the range 3,000-4500 K close to the CMB. Still for pyrolytic composition, the recent experiments of Kuwayama et al. (2021) favor a low Clapeyron, 6.5 ± 2.2 MPa/K, and a T pPv in excess of 4000 K. If our approach is correct, too large value of T pPv may however be difficult to reconcile with experimental solidus of pyrolite (Fiquet et al., 2010;Andrault et al., 2011;Nomura et al., 2014), as it would lead to CMB temperatures in excess of this solidus. Alternatively, T CMB may be lower than predicted by our approach (Figure 8), in which case pPv may be present all around the CMB. This, however, is difficult to reconcile with the values of V S and Q S observed beneath Central America, which cannot be explained by Frontiers in Earth Science frontiersin.org temperature changes only and instead require changes in the depth of the pPv lens lower boundary (Borgeaud and Deschamps, 2021). Other modelling parameters, most particularly the frequency exponent and activation enthalpy of the quality factor, α and H, are still poorly known (Section 2.2). Because it depends on the product αH, and not on individual values of α and H, our approach can accommodate part of these uncertainties. In addition, the recovered T ref depends only slightly on this product ( Figure 5). Nevertheless, more precise values of α and H would refine the possible range of T ref for a given T pPv .
Our approach implicitly assumes that seismic velocity is not affected by the presence of compositional changes. Unless the compositional effects and their contributions to V S -anomalies are well identified and quantified, thus allowing to correct V S , our method may not be applied to measurements obtained within regions that are chemically different from the average (pyrolytic) mantle. This is likely the case of large low shear-wave velocity provinces (LLSVPs) observed in the lowermost mantle beneath Africa and the Pacific (e.g., Garnero et al., 2016), and which are thought to be regions simultaneously hotter than average mantle and chemically differentiated, possibly enriched in iron by a few percent (e.g., Trampert et al., 2004;Deschamps et al., 2012;Mosca et al., 2012). Mineral physics data indicate that shear velocity decreases with increasing iron content. Following the seismic sensitivities to iron from Deschamps et al. (2012), a 3% enrichment in iron decrease shear velocity by 0.8-1.1%. If not accounted for, an excess in iron oxide would then result in overestimated temperature anomalies. Correction for this effect would shift local temperatures, and thus, temperature anomalies, to higher values (red curves in Figure 4 would shift upwards), changing in turn the estimated T ref . Such corrections however require a precise knowledge of the iron excess, which, to date, is not available. Estimates from seismic normal modes Mosca et al., 2012) suggest an enrichment around 2 to 4 wt%, but these estimates poor of lateral and vertical resolutions. Another potential source of chemical heterogeneities is mid-ocean ridge basalt (MORB) that may be entrained with slabs down to the CMB. This might be the case of the region explored by Borgeaud and Deschamps (2021), which was associated with the subduction of the Farallon slab to the CMB (e.g., Hung et al., 2005;FIGURE 8 Reference temperature in the lowermost 50 km of the mantle, T ref , estimated from attenuation and shear velocity measured beneath Central America (Borgeaud and Deschamps, 2021) and the Northern Pacific (Deschamps et al., 2019) as a function of the transition temperature to post-perovskite, T pPv . The crosses show T ref median values, meaning that there is 50% likelihood that T ref is on each side of this value. The dashed red curve shows the linear fit to these median values, and the gray area covers values of T ref ranging from quartiles 0.025 to 0.975, i.e., the likelihood that T ref is within this range is 95%. The CMB temperature may be estimated by adding a small depth correction to T ref (Section 2.4 and Eq. 10).

FIGURE 9
Comparison between our results and CMB temperatures, T CMB (bottom scale), estimated from radial seismic profiles, post-perovskite (pPv) lenses, pyrolite solidus at the CMB, and measurements of iron alloys melting temperature at the inner core boundary. The top scale shows the pPv transition temperature at the CMB, T pPv , calculated by inverting Eq. 10. The gray area covers the range of T CMB (95% likelihood) we obtained for T pPv = 3500 K (the range of uncertainty does not apply to the top scale).

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frontiersin.org Borgeaud et al., 2017). However, MORBs represent only a thin layer on top of the slab. In addition, if post-perovskite is present in the lowermost mantle, the sensitivity of V S to MORB may be very small (Deschamps et al., 2012). Overall, the contribution of recycled MORB pieces may be limited and much smaller than that of temperature and post-perovskite changes. Finally, reactions between core molten iron alloys and mantle silicate rocks, if they happen, may also impact seismic velocity anomalies. High pressure experiments indicate that such reactions could be a source of iron alloys (FeO and FeSi; Knittle and Jeanloz, 1991) and iron-aluminum alloy (Dubrovinsky et al., 2001). Core-mantle chemical reactions may then result in local excess in iron oxides, with consequences on the interpretation of shear velocity anomalies similar to those for iron-enriched LLSVPs. To date, however, there is no seismic evidences for the presence of such regions, and no quantitative constraints on the possible excess of iron at these locations. Finally, the relationship between T ref and T pPv we obtained (Figure 8) was built from observations made beneath Central America (Borgeaud and Deschamps, 2021) plus one observation made beneath the Nothern Pacific (Deschamps et al., 2019). Additional measurements obtained in different regions would be needed to confirm this trend, and to avoid potential bias related to the area explored by Borgeaud and Deschamps (2021). Ideally, this would include regions with V S and Q S away from PREM values, which do not bring strong constraints, and LLSVPs, for which part of the seismic velocity anomalies may originate from compositional differentiation.
Despite the difficulties discussed in this section and the fact that it relies on an accurate knowledge of T pPv , the approach we developed in this study offers an alternative way to estimate the CMB temperature. In addition, it may be easily adapted or modified for other purposes, for instance mapping postperovskite or chemical fields in the deep mantle.

Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.