Calculation of Seismic Velocities with Perple_X: Data Preparation and Computational Algorithm

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Contents

Introduction
Thermodynamic data file preparation
A comment on missing data
Computation of Seismic Velocities
 
Figures:
Figure 1: Standard thermodynamic data file 
Figure 2: Data as modified for velocity calculations

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Introduction

This document describes how the standard thermodynamic data files read by Perple_X must be modified for seismic velocity calculations. Four data files available at this site that have been prepared for such calculations:

seismo.dat – the Holland & Powell (1998) thermodynamic data base, with shear modulii as summarized in Connolly & Kerrick (2002).

hp02ver.dat – the Holland & Powell (2002) thermodynamic database, with shear modulii as summarized in Connolly (2005).

sfo05ver.dat - Stixrude & Lithgow-Bertelloni [2005a, JGR 110, art. no.-2965] thermodynamic data base augmented by endmember data for phases stable in the mid- to lower-mantle as described in Khan et al. [2006, JGR]. The data structure of this file is distinct from the standard G(P,T) data bases implemented in Perple_X and is not documented here. Refer to Stixrude & Lithgow-Bertelloni [2005] and the header of the data file for more information. See example #23 for applications of this data base.

stx07ver.dat - Stixrude & Lithgow-Bertelloni [2007, EPSL] thermodynamic data base augmented with endmember data for post-perovskite. The data structure of this file is distinct from the standard G(P,T) data bases implemented in Perple_X and is not documented here. Refer to Stixrude & Lithgow-Bertelloni [2005b,2007] and the header of the data file for more information. Use of this data base requires solution model choices as detailed in the header section of the file.

stx08ver.dat – NCFMAS mantle model, refer to the Perple_X update file for more information.

The file seismo.dat has been superceded by hp02ver.dat and is provided only because it is used in the example detailed in the second part of this document. Neither of these files is complete, i.e., shear modulii are not provided for all the phases in the data base. The hp02ver.dat database can be used for calculations to depths approaching the 440 km transition, but cannot be used for greater depths because it lacks data for the relevant high pressure phases. The advantage of the hp02ver.dat data base is that it provides a complete chemical model and can also be used to compute melting phase relations. The sfo05ver.dat base is for a simpler chemical model (CaO-FeO-MgO-Al2O3-SiO2) and therefore may not be appropriate for some near surface problems, but it includes data for the major phases likely to be present at any depth within the earth’s mantle.

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Modification of the thermodynamic data file

The Perple_X thermodynamic data file format is described in detail in the program documentation (vdoc.ps), for present purposes it is only necessary to know that the file consists of a header section followed by entries for the thermodynamic entities (e.g., fluid and mineral species) described by the data base (Figure 1).

Figure 1. An abridged example of a Perple_X thermodynamic data file (hp98ver.dat). The header section contains various data (the first 13 lines here), which may be followed by comments that describe the data base and its entries, the header section is terminated by a left justified line beginning with the word “end” (marked here in red). The header section is followed by entries (shown in blue) that define the thermodynamic properties of the mineral and fluid species included in the data base. The first line of each entry is formatted and begins with the name of the species (usually abbreviated). The remaining numeric data is unformatted. Comments may be placed between entries, as between the entry for fo (forsterite) and fa (fayalite).

To create a thermodynamic data base for seismic velocity calculations it is necessary to modify the entries for all entities of interest to include the parameters necessary to compute the shear modulus as a function of pressure (p) and temperature (T) (the units of pressure and temperature are defined by the data base header section, typically these are bars and Kelvin). Optionally, the user may also provide parameters for the adiabatic bulk modulus (K_s) and density (r). Currently Perple_X expects the following functional forms

(1)

(2)

(3)

where p_r and T_r are the reference pressure and temperature for the data base as defined in the header (typically 1 bar, 298.15 K); r₀, m₀ and K₀ are the density, shear modulus and bulk modulus at the reference conditions and the pressure and temperature derivatives are constants. If the user provides empirical data for K_s and r₀ then Perple_X computes seismic velocities using phase compositions, and proportions determined by free energy minimization in combination with the shear modulii and density computed from equations 1 and 2. However, if data is not provided for either or both K_s and r, then the requisite parameters are computed by differentiation of the free energy function as

(4)

(5)

where G is the molar free energy and N is the molar formula weight.

The first line of each modified entry must be altered to indicate the nature of the modifications. This line is formatted, such that the first 8 characters are reserved for the entry name (left justified), the name is followed by four integer numbers, each of which occupies two columns, and the remaining part of the line may be used to for comments, e.g., for forsterite the first line reads

fo 1 0 0 0 H= -2171830.

If the entry is to be modified to include only parameters for the shear modulus, the second integer (JEMOD) after the name must be changed to 1, i.e.

fo 1 1 0 0 H= -2171830.

Alternatively, if the entry is to include parameters for both shear and bulk modulus the JEMOD must be changed to 2, and if the entry is to include parameters for the computation of density JEMOD must be 3. The parameters are then appended to the end of the entry in the order

the last 4-9 values are only read if JEMOD > 1, there are no formatting constraints for this data. Note that the amount of numeric data following the initial line may vary depending on the other integers specified on the first line (for a complete discussion of these parameters refer to the program documentation).

WARNING 1: If JEMOD = 0 for a mineral or fluid species Perple_X will not compute seismic velocities for any phase or phase assemblage involving the species.

WARNING 2: be careful to modify the entries consistently, Perple_X expects JEMOD*3 additional parameters at the end of each entry. Any inconsistency will cause the programs to crash with a data file format error.

Figure 2. Thermodynamic data file entries modified from Figure 1 to permit seismic velocity calculations (modifications in red; the header section is not modified). A more complete version of this data base (seismo.dat) is described elsewhere (Connolly & Kerrick, 2002). Here JEMOD has been set to 1 for all phases except “q” and “ftat”, for which it is zero. JEMOD = 1 instructs Perple_X to read only the parameters for the shear modulus (Eq 1) and to compute the density and bulk modulus from the thermodynamic data used for phase equilibria calculations. Because JEMOD = 0 for “q” (quartz) and “ftat” (a talc endmember species), with this file Perple_X would not compute seismic velocities for any assemblage involving quartz or talc (if the talc contains ftat species).

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Computation of Seismic Velocities

As customary in seismic velocity computations, in Perple_X it is assumed that all phases are elastically isotropic. Currently no corrections for anelastic behavior are made, though such corrections could be implemented by modification of WERAMI. Compressional- and shear-wave velocities (v_p and v_s) through an elastically isotropic fluid or homogeneous single crystal are given by

(6)

(7)

As a consequence of the pseudocompound approximation employed in pre-07 versions of Perple_X a homogeneous solution phase may be represented by more than one pseudocompound. When this is the case the parameters used in equations 6 & 7 are computed as

(8)

where is the value of the property in the phase of interest, is the molar proportion of the i^th pseudocompound representing the phase, and is the value of property in that pseudocompound. Once the seismic velocities in each phase of an assemblage have been computed from equations 6 & 7, the elastic moduli of the aggregate are computed as

(9)

where is the moduli of interest, x_j is the volume fraction of the j^th phase, is the corresponding modulus, and is a weighting factor taken to be 0.5 for the Voigt-Reuss-Hill averaging scheme.

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