/
background_userinputs.txt
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/
background_userinputs.txt
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.. _ref-methods-user-input:
User input
----------
Mineralogical composition
^^^^^^^^^^^^^^^^^^^^^^^^^
A number of pre-defined minerals are included in the mineral library and users can create their own.
The library includes wrapper functions to include a transition from the high-spin mineral to the low-spin mineral :cite:`Lin2013` or to combine minerals for a given iron number.
*Standard minerals* -- The 'standard' mineral format includes a list of parameters given in the above table.
Each mineral includes a suggested EoS with which the mineral parameters are derived.
For some minerals the parameters for the thermal corrections are not yet measured or calculated, and therefore the corrections can not be applied.
An occasional mineral will not have a measured or calculated shear moduli, and therefore can only be used to compute densities and bulk sound velocities.
The mineral library is subdivided by citation.
BurnMan includes the option to produce a \LaTeX\; table of the mineral parameters used.
BurnMan can be easily setup to incorporate uncertainties for these parameters.
*Minerals with a spin transition* -- A standard mineral for the high spin and low spin must be defined separately.
These minerals are "wrapped," so as to switch from the high spin to high spin mineral at a give pressure.
While not realistic, for the sake of simplicity, the spin transitions are considered to be sharp at a given pressure.
*Minerals depending on Fe partitioning* -- The wrapper function can partition iron, for example between ferropericlase, fp, and perovskite, pv.
It requires the input of the iron mol fraction with regards to Mg, :math:`X_\mathrm{fp}` and :math:`X_\mathrm{pv}`, which then defines the chemistry of an Mg-Fe solid solution according to (:math:`\mathrm{Mg}_{1-X_{\mathrm{Fe}}^{\mathrm{fp}}},\mathrm{Fe}_{X_{\mathrm{Fe}}^{\mathrm{fp}}})\mathrm{O}` or :math:`(\mathrm{Mg}_{1-X_{\mathrm{Fe}}^{\mathrm{pv}}},\mathrm{Fe}_{X_{\mathrm{Fe}}^{\mathrm{pv}}})\mathrm{SiO_3}`.
The iron mol fractions can be set to be constant or varying with P and T as needed.
Alternatively one can calculate the iron mol fraction from the distribution coefficient :math:`K_D` defined as
.. math::
K_{D} = \frac{X_{\mathrm{Fe}}^{\mathrm{pv}}/X_{\mathrm{Mg}}^{\mathrm{pv}}}{X_{\mathrm{Fe}}^{\mathrm{fp}}/X_{\mathrm{Mg}}^{\mathrm{fp}}}.
:label: KD
We adopt the formalism of :cite:`Nakajima2012` choosing a reference distribution coefficient :math:`K_{D0}` and standard state volume change (:math:`\Delta \upsilon^{0}`) for the Fe-Mg exchange between perovskite and ferropericlase
.. math::
K_{D}={K_D}_0 \:\exp\left(\frac{(P_0-P)\Delta \upsilon^{0}}{RT}\right),
:label: KD2
where :math:`R` is the gas constant and :math:`P_0` the reference pressure.
As a default, we adopt the average :math:`\Delta \upsilon^{0}` of :cite:`Nakajima2012` of :math:`2\cdot10^{-7}` :math:`m^3 mol^{-1}` and suggest using their :math:`{K_D}_0` value of :math:`0.5`.
The multiphase mixture of these minerals can be built by the user in three ways:
1. Molar fractions of an arbitrary number of pre-defined minerals, for example mixing standard minerals mg\_perovskite (:math:`\mathrm{MgSiO_3}`), fe\_perovskite
(:math:`\mathrm{FeSiO_3}`), periclase (:math:`\mathrm{MgO}`) and wüstite (:math:`\mathrm{FeO}`).
2. A two-phase mixture with constant or (:math:`P,T`) varying Fe partitioning using the minerals that include Fe-dependency,
for example mixing :math:`\mathrm{(Mg,Fe)SiO_3}` and :math:`\mathrm{(Mg,Fe)O}` with a pre-defined distribution coefficient.
3. Weight percents (wt\%) of (Mg, Si, Fe) and distribution coefficient (includes (P,T)-dependent Fe partitioning).
This calculation assumes that each element is completely oxidized into its corresponding oxide mineral
(:math:`\mathrm{MgO}`, :math:`\mathrm{FeO}`, :math:`\mathrm{SiO_2}`) and then combined to form iron-bearing perovskite and ferropericlase taking into account some Fe partition coefficient.
.. _ref-methods-geothermal:
Geotherm
^^^^^^^^
Unlike the pressure, the temperature of the lower mantle is relatively unconstrained.
As elsewhere, BurnMan provides a number of built-in geotherms, as well as the ability to use user-defined temperature-depth relationships.
A geotherm in BurnMan is an object that returns temperature as a function of pressure.
Alternatively, the user could ignore the geothermal and compute elastic velocities for a range of temperatures at any give pressure.
Currently, we include geotherms published by :cite:`Brown1981` and :cite:`anderson1982earth`.
Alternatively one can use an adiabatic gradient defined by the thermoelastic properties of a given mineralogical model.
For a homogeneous material, the adiabatic temperature profile is given by integrating the ordinary differential equation (ODE)
.. math::
\left(\frac{\text{d}T}{\text{d}P}\right)_S = \frac{\gamma T}{K_S}.
:label: geoth
This equation can be extended to multiphase composite using the first law of thermodynamics to arrive at
.. math::
\left(\frac{\text{d}T}{\text{d}P}\right)_S = \frac{ T \displaystyle\sum_{i} \frac{ n_i C_{Pi} \gamma_i }{K_{Si}}}{ \displaystyle\sum_{i} n_i C_{Pi} },
:label: geoth2
where the subscripts correspond to the :math:`i` th phase, :math:`C_P` is the heat capacity at constant pressure of a phase, and the other symbols are as defined above.
Integrating this ODE requires a choice in anchor temperature (:math:`T_0`) at the top of the lower mantle (or including this as a parameter in an inversion).
As the adiabatic geotherm is dependent on the thermoelastic parameters at high pressures and temperatures, it is dependent on the equation of state used.
.. _ref-methods-seis:
Seismic Models
^^^^^^^^^^^^^^^^^^^^^^^^^
BurnMan allows for direct visual and quantitative comparison with seismic velocity models.
Various ways of plotting can be found in the examples.
Quantitative misfits between two profiles include an L2-norm and a chi-squared misfit, but user defined norms can be implemented.
A seismic model in BurnMan is
an object that provides pressure, density, and seismic velocities (:math:`V_P, V_\Phi, V_S`) as a function of depth.
To compare to seismically constrained profiles, BurnMan provides the 1D seismic velocity model PREM :cite:`dziewonski1981`.
One can choose to evaluate :math:`V_P, V_\Phi, V_S, \rho, K_S` and/or :math:`G`.
The user can input their own seismic profile, an example of which is included using AK135 :cite:`kennett1995`.
Besides standardized 1D radial profiles, one can also compare to regionalized average profiles for the lower mantle.
This option accommodates the observation that the lowermost mantle can be clustered into two regions, a 'slow' region, which represents the so-called Large Low Shear Velocity Provinces, and 'fast' region, the continuous surrounding region where slabs might subduct :cite:`Lekic2012`.
This clustering as well as the averaging of the 1D model occurs over five tomographic S wave velocity models (SAW24B16: :cite:`megnin2000`; HMSL-S: :cite:`houser2008`; S362ANI: :cite:`kustowski2008`; GyPSuM: :cite:`Simmons2010`; S40RTS: :cite:`Ritsema2011`).
The strongest deviations from PREM occur in the lowermost 1000 km.
Using the 'fast' and 'slow' S wave velocity profiles is therefore most important when interpreting the lowermost mantle. Suggestion of compositional variation between these regions comes from seismology :cite:`to2005,He2012` as well as geochemistry :cite:`Deschamps2012,jackson2010`.
Based on thermo-chemical convection models, :cite:`Styles2011` also show that averaging profiles in thermal boundary layers may cause problems for seismic interpretation.
We additionally apply cluster analysis to and provide models for P wave velocity based on two tomographic models (MIT-P08: :cite:`Li2008`; GyPSuM: :cite:`Simmons2012`).
The clustering results correlate well with the fast and slow regions for S wave velocities; this could well be due to the fact that the initial model for the P wave velocity models is scaled from S wave tomographic velocity models.
Additionally, the variations in P wave velocities are a lot smaller than for S waves.
For this reason using these adapted models is most important when comparing the S wave velocities.
While interpreting lateral variations of seismic velocity in terms of composition and temperature is a major goal :cite:`Trampert2004,Mosca2012`, to determine the bulk composition the current challenge appears to be concurrently fitting absolute P and S wave velocities and incorporate the significant uncertainties in mineral physical parameters).