A short note on the pressure-depth conversion for geophysical interpretation

Authors


Corresponding author: F. Cammarano, Department of Geosciences and Natural Resource Management, University of Copenhagen, Øster Voldgade 10, Copenhagen, Denmark. (fc@geo.ku.dk)

Abstract

[1] Databases of material properties based on mineral physics are rapidly becoming an essential tool for interpreting geophysical observations. The conversion of physical properties from pressure to depth is usually based on preliminary reference Earth model. We quantify the error that is introduced with this assumption. The corrected pressure profile is obtained by updating the starting one with the inferred density distribution and iterating until convergence (which is attained in few steps). We show two examples. The first is related to the interpretation of average seismic Earth models. The second refers to the interpretation of a 2-D thermal structure of the oceanic lithosphere as predicted by a half-space cooling model. The effects are overall small, nevertheless important for interpretation since they are systematic, they have a feedback with thermal interpretation and they can shift the location of predicted mineralogical phase transitions that are associated with abrupt density and seismic velocity jumps. For example, variations up to 0.6 GPa at 2500 km depth are obtained by changing the potential mantle temperature of 100 K.

1 Introduction

[2] The knowledge of material properties at high pressures (P) and temperatures (T) is fundamental for understanding the evolution and the current physical conditions inside the Earth and other planetary bodies. Steady advances in experimental techniques and ab initio computations [Manghnani and Syono, 2013; Wentzcovitch and Stixrude, 2010] are contributing to augment the amount of information we have on material properties at high pressure and temperature. The different pieces of information are modeled together in a thermodynamical self-consistent manner to obtain the stable mineralogical phases and the properties of rock assemblages at P-T typical of the Earth's deep interior [Connolly, 2005; Holland and Powell, 1998, 2011; Matas et al., 2007; Stixrude and Lithgow-Bertelloni, 2005, 2011]. The material databases can be incorporated in geodynamic modeling tools [e.g., Afonso et al., 2013; Baumann et al., 2010; Bunge et al., 2007; Nakagawa et al., 2009; Tackley et al., 2005] and are increasingly used to interpret geophysical observations [Afonso et al., 2013; Cammarano et al., 2003, 2009; Forte et al., 2010; Khan et al., 2009; Steinberger and Calderwood, 2006, and many others].

[3] A straightforward implementation of the P-T and composition databases in numerical simulations can be done since it is possible to have direct control on pressure, both in space than in time (for example, this is done in the mantle convection code STAGYY) [Nakagawa et al., 2009]. However, all geophysical models involve depth. Therefore, a pressure-to-depth conversion is required to use the material properties databases.

[4] This is usually done by using a scaling relation given by a 1-D pressure profile of our planet. In most of the cases, using the preliminary reference Earth model (PREM) pressure profile [Dziewonski and Anderson, 1981] is sufficiently accurate, but it must be recognized that we are thus introducing an unnecessary bias in our interpretation. It is desirable, and certainly more rigorous, to use directly the experimentally determined P-dependent density to compute the correct pressure profile. The goal of this paper is to quantify accurately, for the first time, what are the effects of assuming the PREM pressure profile on geophysical interpretation. Note that we do not aim to refine the physical interpretation of existing seismic models or data.

[5] The (static) pressure profile is simply given by:

display math(1)

that can be discretized in small steps in which the density can be assumed to be constant. Both the density, ρ, than the gravity acceleration, g, are a function of depth (z). We choose not to invert for the g profile, which requires tedious iteration because of the feedback with the density profile. Instead, we approximate the gravity profile by using the PREM profile. Note that g does not vary much throughout the Earth's mantle, slightly increasing when approaching the dense core. Not inverting for g does not introduce any observable variations in our results. We also use the P(z) profile of PREM as starting one. The density distribution associated to the PREM pressure profile for a given thermal and compositional structure is then inferred from the material-property database and the pressure profile updated. We continue iteratively this procedure up to convergence that takes place very rapidly, usually in few steps, for a tolerance given by a normalized root-mean-squared-error of 10−6. Note that the steps in depth should be very small and should be chosen according to the rate of density variation. Care should be taken to the drastic jumps in density that are associated with mineralogical phase transitions.

[6] In what follows, we present two examples. A first one referring to the 1-D interpretation of seismic models and a second to the interpretation of a 2-D thermal structure of oceanic lithosphere, as predicted by a half-space cooling model.

2 Effects on Interpretation of Average 1-D Seismic Models

[7] Average 1-D seismic models are extremely important since most of the seismic observations (around 90%) can be explained by average structure and because they are used as reference models for 3-D tomography models. Their physical interpretation in terms of thermal and/or compositional structure has been performed, providing evidence for a deviation from a chemically homogenous and adiabatic structure [Cammarano et al., 2005; Cobden et al., 2009; Matas et al., 2007; Khan et al., 2008].

[8] In this paper, we simply use a reference thermal structure corresponding to a 60 Myr old oceanic geotherm for the top lithospheric part and with an adiabatic thermal gradient below (with a surface potential temperature of 1300°C). We assume a classic pyrolite composition [Xu et al., 2008] for the mantle. The mineral-physics database is modeled with a six oxides (NCFMAS) equation-of-state by Stixrude and Lithgow-Bertelloni [2011] as implemented in Perple-X [Connolly, 2005]. Thermal effects on density, and thus on pressure, are also tested by using the same reference thermal structure but decreasing and increasing the temperature of 100°C.

[9] We do not model density in the top 80 km and in the bottom 150 km of the mantle, thus eliminating the structural and compositional complexity at the boundary layers that reflects into large lateral variations in terms of seismic structure and anisotropy. In fact, we remove almost all crustal effects and part of the lithosphere which are compositionally different compared to the normal mantle, and we also remove the D” that is also probably compositional different [e.g., Cobden et al., 2009].

[10] The cumulative effect of the pressure profile highlights the systematic differences between the predicted density structure and the PREM one. Throughout the upper mantle, in particular down to 400 km, our predicted density exceeds the PREM one. Consequently, we have the largest divergence with PREM, which amount to ∼0.35 GPa, at the base of the upper mantle (Figure 1, left). In the lower mantle, density is only slightly but steadily lower compared to PREM, and therefore, the discrepancy with the PREM model reduces (Figure 1, left). For the reference thermal structure (black line), the discrepancy is fortunately very small. Nevertheless, the newly computed density structure has (a) a systematic effect, being the pressure always higher than PREM at any depth and (b) shift upward all the mineralogical phase transitions associated with seismic discontinuities. In Figure 1(right), we highlight the structure of the upper-to-lower mantle transition around 660 km depth (PREM discontinuity is placed at 670 km) which is characterized by multiple phase transitions, the ringwoodite to perovskite and ferropericlase transition being the most important. Note the systematic effect and the shift of ∼10 km of the transition.

Figure 1.

(left) Absolute pressure variations compared to PREM obtained with an adiabatic thermal structure of 1300°C and pyrolite (in black). Effects of varying mantle potential of ±100°C are shown in red. (right) Zoom on the density structure computed with the PREM pressure profile and after correction with consistent pressures for the reference T structure. PREM densities are shown as reference.

[11] Systematic effects and shifts of mineralogical transitions also characterize the seismic structure and play a role on seismic interpretation. Global seismic observations are extremely sensitive to variation of the depth-integrated average seismic velocities (<VP(z)> or <VS(z)>) [e.g., Cammarano et al., 2009]. For example, the produced difference in <VS(z)> between the old and new pressure profiles for the 1300°C adiabatic T structure, causes a variation of 1 s for a teleseismic S traveltime at 30° of epicentral distance (the turning point of seismic ray at this distance is around 770 km depth) and half a second for P (traveltimes have been computed with the code TauP, by Crotwell et al. [1999]). The shift of the phase transitions toward surface also introduces an unwanted bias when studies of mantle seismic discontinuities wants to be conducted.

[12] If thermal effects are taken into consideration, the pressure profiles that are consistently computed with the new density distribution have, not surprisingly, relatively large variations. A variation up to almost 0.6 GPa in pressure at 2500 km depth is obtained when varying the mantle potential temperature of ±100 K (Figure 1, left).

[13] Finally, redefining the density distribution with its appropriate pressure profile, increases the total mass of around 0.16% in the reference case.

3 Effects on Interpretation of Thermal Structure of Oceanic Lithosphere

[14] The thermal structure of the oceanic lithosphere is a remarkable feature of plate tectonics. Hot material erupted at the mid-ocean ridges moves away from the ridge and becomes colder (and denser) with time. A solution can be obtained by solving the 1-D heat-conduction equation,

display math(2)

being T the temperature, t time, and z depth and which excludes radiogenic internal heat sources and lateral exchange of heat. The relevant physical properties, namely the thermal conductivity κ, density ρ, and heat capacity CP, are all dependent on depth and temperature.

[15] For our purposes, it is sufficient to use a reference thermal structure based on the analytical solution of the equation for the well-known half-space cooling model [Turcotte and Schubert, 1982], i.e.,

display math(3)

where κ, ρ, and CP are assumed constant and equal, respectively, to 3.7 Wm−1°C−1, 3300 kgm−3, and 1170 Jkg−1°C−1. erfc is the complementary error function and T0and T1 are, respectively, the temperature at z=0 and at inline image. T1 is given by a mantle potential temperature of 1363°C and T0fixed to 0°C in our thermal structure.

[16] As shown in previous studies [McKenzie et al., 2005], having variable ρCP affects only slightly the thermal structure. According to our P-T dependency, we computed a variation up to 15°C, at maximum. For doing this, we model equation (2) numerically with an implicit finite-difference scheme, discretizing the heat equation forward in time (steps of 0.1 Myr) and with a central scheme in depth (1 km steps) (results are presented in a separate publication: [F. Cammarano and M. Guerri, paper in preparation, 2013]).

[17] In our mineral-physics modeling, we assume a 7 km thick crust, a constant density of 2900 kg/m3and we use a depleted (dry) harzburgitic composition [Xu et al., 2008] down to 40 km (depth consistent with Ringwood [1982]) and the same pyrolitic mantle below. We do not introduce any other change in chemical compositional of the mantle, but note that the implementation is straightforward. The stable mineralogical phases and their composition in the depleted harzburgitic layer are computed at mid-ocean-ridge conditions by using the same equation-of-state of the previous example. The mineral phases formed at the oceanic spreading centers are then kept frozen in the cooling lithosphere. Mineralogy at thermodynamic equilibrium is instead computed for the normal mantle, pyrolitic layer. More detailed melting models have been developed for an accurate chemical evolution [e.g., Asimow et al., 2004; Afonso et al., 2008; Langmuir et al., 1992; Olugboji et al., 2013]. Reduction in density due to the presence of partial melt below mid-oceanic-ridge (however, small, ∼1%) [Spiegelman and Kelemen, 2003] has been also considered [e.g, Afonso et al., 2008]. These secondary effects are here not considered.

[18] We start again by computing the 2-D density structure using the PREM pressure profile for the depth-pressure conversion. Then, we correct the pressure profile at each time step, paying attention to use very small depth steps (1 km) to capture the complexities due to the phase transitions. The change in pressure profile is negligible when a finer grid is used. The hydrostatic pressure change up to ∼3.6% at a depth of 70 km compared to the lightest (because hotter) column at time zero (Figure 2, top). Absolute P variations (in GPa) compared to PREM pressure profile are shown in Figure 2(bottom).

Figure 2.

Effects on pressure due to density structure of oceanic lithosphere. (top) Absolute pressure variations compared to PREM. (bottom) Percent variation of pressure compared to pressure values at time zero.

[19] The density structure predicted by the mineral-physics model (Figure 3) is complicated by the pressure effects and, more importantly, by mineralogical phase transitions and/or compositional heterogeneity. Modeling the density consistently with the static pressure has small effects on the structure when no phase changes occurs, although can modify the location of the abrupt density changes (Figure 3). The small effects are on the order of ∼0.15% in density. For instance, increasing the pressure of 0.15 GPa at ∼66.5 km (PREM pressure is 2 GPa) gives an increase in density of around 0.15% at 1000 K when a thermodynamically equilibrated pyrolite is assumed. To note that the exact phase composition (e.g., in terms of iron partitioning) has a trade-off with pressure. As an example, a variation in Mg number of olivine from 0.91 to 0.90 causes an increase in density around 0.35% at 2 GPa and 1000 K. In order to obtain the same increase in density at a fixed Mg# = 0.91, an increase of almost 0.45 GPa would be required.

Figure 3.

(top) Thermal structure of oceanic lithosphere as predicted by a half-space cooling model. (middle and bottom) The density structure predicted by using PREM pressures and after corrections with consistent pressures, respectively. Composition in both cases is harzburgitic (depleted) down to 40 km and pyrolite below. Seismic structures (VP and VS) are shown in “supporting information Figure S1.”

[20] Commonly, only thermal contraction based on constant or linear laws of thermal expansion are considered [e.g., McKenzie et al., 2005]. Including depth and T-dependent physical properties according to a self-consistent mineral-physics model provides the opportunity to better constrain the thermochemical evolution of the oceanic lithosphere. For what concern the message of this paper, we choose the pragmatic decision to use the same equation-of-state [Stixrude and Lithgow-Bertelloni, 2011] also for this example and a simple 2-layer compositional structure. However, it is opportune to stress that the lack of minor oxides, such as chrome, that is known to play an important role on the plagiocase stability [Borghini et al., 2010] plus overall uncertainties in the phase equilibria should be considered as well for a correct interpretation [Afonso et al., 2008; Holland and Powell, 2011]. The effects on subsidence of oceanic floor and on gravity and assessment of the seismic signature (also including anelasticity effects) with more complex compositional models will be presented elsewhere.

4 Discussion and Conclusions

[21] In order to improve the interpretation of geophysical observations and in particular of seismic data, an appropriate use of available information from mineral physics is required. Self-consistent thermodynamic models are, on this regard, a precious tool, since they combine the delicate phase equilibria existing between mineralogical phases and their elastic properties and density. While in numerical modeling is possible to keep track of the static pressure and thus use directly the database, the conversion from pressure to depth is ineluctable for the interpretation of seismic and other geophysical observations.

[22] For several applications, using the PREM pressure profile to convert experimentally determined physical properties from pressure to depth is a reasonable, albeit unnecessary assumption. Indeed, it is straightforward to correct the pressure profile by using the P-dependent densities through an iterative scheme. Considering a pressure-to-depth conversion that is consistent with the density distribution ensures a rigorous use of available thermodynamical databases in geophysical interpretations. This is particularly important when the databases want to be used for the prediction and testing of impedance (density × seismic velocity) jumps related to mineralogical transitions. Additionally, we have shown that varying thermal structure has, not surprisingly, a feedback on the pressure-to-depth conversion that needs to be considered.

[23] We presented the effects on the interpretation of average 1-D seismic models and of oceanic lithosphere. In both cases, we observe systematic effects that modify the location of mineralogical phase transitions and the overall density structure.

[24] In the first case, we show that:

  1. [25] The trade-off with thermal structure affects density and, in turn, also the associated pressure profile. Variations of ± 100°C modify the pressures of the lower mantle on the order of 0.6 GPa.

  2. [26] The pressure-to-depth conversion correction has fortunately smaller effects if the choice of the thermal structure is appropriate. In any case, even small inconsistencies in the P-to-depth conversion produce systematic effects that alter the integrated seismic velocities with depth, the depth of mineralogical phase transitions and even the total mass of the Earth's mantle. For the considered homogeneous (pyrolite)–adiabatic mantle, we obtain a shift upward of ∼ 10 km of the ringwoodite to perovskite and ferropericlase transition, responsible for the so-called 660 km discontinuity.

[27] The second example evidences the effects due to lateral variations in thermal structure on the pressure-to-depth conversion. The precise location of changes in density and seismic properties due to phase transitions again present corrections that cannot be discarded for geophysical interpretation. Variations in density structure are fortunately very small. An increase in pressure of 0.15 GPa at ∼66.5 km (PREM pressure is 2 GPa at this depth) gives an increase in density for thermodynamically equilibrated pyrolite of around 0.15% at 1000 K. We remind that the largest lateral variations in pressure, obviously, are near the Earth's surface, where the geobaric gradient (together with temperature) governs metamorphic processes. Hence, the corrected pressure should be considered in crustal studies as well.

[28] Interpretation of geophysical observations in terms of temperature and composition remains challenging due to the still very large uncertainties in the inferred properties from geophysical observations (e.g., seismic velocities and density) and in the mineral-physics relationship to convert them to temperature and composition (e.g., high P-T elastic and anelastic properties). Most of the efforts in geophysical and mineral-physics communities are devoted to reduce, as much as possible, those uncertainties. In this sense, the combined interpretation of different geophysical observations based on insight from mineral physics is useful to assess (and eventually reduce) the overall uncertainties and to account for the feedback between different physical parameters. Applying a rigorous method in our interpretative effort is essential for this purpose. Not introducing artifacts that can be avoided is a small advance in terms of interpretation, but one that can be done right now.

[29] The effect of using an appropriate pressure-to-depth conversion in geophysical interpretation is undoubtedly small; however, it does not represent an uncertainty rather an unnecessary assumption and therefore should be corrected.

Acknowledgments

[30] This work is supported by Danish Research Council, Sapere Aude grant 11-105974.

[31] The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.

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