Seismic velocity

Methods

Seismic velocity can specified in a number of ways

GeoParams.MaterialParameters.SeismicVelocity.ConstantSeismicVelocity Type

ConstantSeismicVelocity(Vp=8.1 km/s, Vs=4.5km/s)

Set a constant seismic P and S-wave velocity:

$$V_p=cst$$$$V_s=cst$$

where $V_p,V_s$ are the P-wave and S-wave velocities [$km/s$].

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In addition, you can use phase diagram lookup tables to compute seismic velocities as a function of pressure and temperature.

Seismic velocity correction for partial melt

Methods

Both routines use the velocity reduction of Clark & Lesher (2017). melt_correction_Takei models the melt as oblate-spheroidal inclusions of a given aspect ratio (self-consistent model of Dean, 1983, and Phani, 1996); melt_correction uses the equilibrium geometry model for the solid skeleton of Takei (1998), parameterized by contiguity.

GeoParams.MaterialParameters.SeismicVelocity.melt_correction_Takei Function

Vs,Vp = melt_correction_Takei(Kb_L, Kb_S, Ks_S, ρL, ρS, Vp0, Vs0, ϕ, α)

This corrects the Vp/Vs velocities if melt is present. The melt sits in oblate-spheroidal inclusions whose effect on the bulk and shear moduli follows the self-consistent model of Dean (1983) and Phani (1996); the resulting velocity reduction follows Clark & Lesher (2017). The same parameterization is used in Takei (2002) to relate Vp/Vs to pore geometry.

Input arguments:

• Kb_L: bulk modulus liquid

• Kb_S: bulk modulus solid

• Ks_S: shear modulus solid

• ρL: density liquid

• ρS: density solid

• Vp0: P-wave velocity of solid

• Vs0: S-wave velocity of solid

• ϕ: melt content

• α: melt aspect ratio [0.001 - 1)

Output arguments:

• Vs: corrected S-wave velocity

• Vp: corrected P-wave velocity

The reduction is first order in ϕ; corrected velocities that would become negative are clamped to zero. For α = 1 the inclusion model is undefined and Vs0, Vp0 are returned unchanged.

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GeoParams.MaterialParameters.SeismicVelocity.melt_correction Function

    Vp_cor,Vs_cor = melt_correction(  Kb_L, Kb_S, Ks_S, ρL, ρS, Vp0, Vs0, ϕ, α)

Corrects P- and S-wave velocities if the rock is partially molten.

Input:

• Kb_L: adiabatic bulk modulus of melt

• Kb_S: adiabatic bulk modulus of the solid phase

• Ks_S: shear modulus of the solid phase

• ρL : density of the melt

• ρS : density of the solid phase

• Vp0 : initial P-wave velocity of the solid phase

• Vs0 : initial S-wave velocity of the solid phase

• ϕ : melt volume fraction

• α : grain-boundary contiguity of the solid framework, from 0 (grains fully wetted by melt) to 1 (no melt on grain boundaries). At textural equilibrium it decreases with melt fraction, roughly as 1 - A√ϕ with A ≈ 1 - 2.3, i.e. 0.7 - 0.9 for a few percent melt.

Output:

• Vp_cor,Vs_cor : corrected P-wave and S-wave velocities for melt fraction

The velocity reduction is first order in ϕ. It loses accuracy when the product of ϕ and the slopes Λ_K, Λ_G of the framework moduli approaches one, which happens for small contiguities.

The routine uses the reduction formulation of Clark & Lesher, (2017) and is based on the equilibrium geometry model for the solid skeleton of Takei et al., 1998.

References:

• Takei (1998) Constitutive mechanical relations of solid-liquid composites in terms of grain-boundary contiguity, Journal of Geophysical Research: Solid Earth, Vol(103)(B8), 18183–18203

• Clark & Lesher (2017) Elastic properties of silicate melts: Implications for low velocity zones at the lithosphere-asthenosphere boundary. Science Advances, Vol 3 (12), e1701312

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Seismic S-wave velocity correction for (shallow depth) porosity

Methods

The porosity follows the empirical porosity-depth relationship of Chen et al. (2020); the pores are treated as fluid-filled inclusions, as in melt_correction_Takei.

GeoParams.MaterialParameters.SeismicVelocity.porosity_correction Function

    Vs_cor = porosity_correction(Kb_S, Ks_S, ρf, ρS, Vs0, depth, α)

Corrects S-wave velocity at shallow depth for fluid-filled porosity, with the porosity taken from an empirical porosity-depth profile.

Input:

• Kb_S: adiabatic bulk modulus of the solid phase

• Ks_S: shear modulus of the solid phase

• ρf : density of the pore fluid

• ρS : density of the solid phase

• Vs0 : initial S-wave velocity of the solid phase

• depth: in kilometers

• α : aspect ratio of the pores, modelled as oblate spheroids (0 < α < 1)

Output:

• Vs_cor : S-wave velocity corrected for fluid-filled porosity, clamped to be non-negative

The pores are treated like melt inclusions in melt_correction_Takei: the shear-modulus reduction of the self-consistent oblate-spheroid model (Dean, 1983; Phani, 1996) enters the velocity reduction of Clark & Lesher (2017). For α = 1 the inclusion model is undefined and Vs0 is returned unchanged.

References:

• Chen et al. (2020) Empirical porosity-depth model for continental crust, Hydrogeology Journal

• Clark & Lesher (2017) Elastic properties of silicate melts: Implications for low velocity zones at the lithosphere-asthenosphere boundary. Science Advances, Vol 3 (12), e1701312

• Dean (1983) Elastic moduli of porous sintered materials as modeled by a variable-aspect-ratio self-consistent oblate-spheroidal-inclusion theory, Journal of the American Ceramic Society, 66(12), 847–854

• Phani (1996) Porosity-dependence of ultrasonic velocity in sintered materials - a model based on the self-consistent spheroidal inclusion theory, Journal of Materials Science, 31, 262–266

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Correcting the velocities of a phase diagram

GeoParams.MaterialParameters.SeismicVelocity.correct_wavevelocities_phasediagrams Function

PD_corrected = correct_wavevelocities_phasediagrams(PD::PhaseDiagram_LookupTable;
                            apply_porosity_correction=true, ρf=1000.0, α_porosity=0.1,
                            apply_melt_correction=true, α_melt=nothing, melt_correction_takei=true,
                            apply_anelasticity_correction=false, water=0,
                            combine=:sequential)

This applies various corrections to the seismic velocities specified in the phase diagram lookup table PD, and returns a new lookup table in which the fields Vp, Vs and VpVs hold the corrected values. The original Vp,Vs is stored in Vp_uncorrected,Vs_uncorrected; PD itself is not modified.

The following corrections can be applied (together with potential options)

• apply_anelasticity_correction: applies an anelasticity correction (anelastic_correction) to the S-wave velocity of the solid, with the optional parameter water: 0 = dry; 1 = dampened; 2 = water saturated

• apply_porosity_correction: applies a correction for fluid-filled pores (porosity_correction) to the S-wave velocity. Optional parameters are ρf (fluid density, [kg/m3]) and α_porosity (pore aspect ratio)

• apply_melt_correction: applies a correction to the P- and S-wave velocities for the melt fraction of the diagram. melt_correction_takei=true uses melt_correction_Takei, where α_melt is the aspect ratio of the melt inclusions (default 0.1); false uses melt_correction, where α_melt is the contiguity of the solid framework (default 0.84, textural equilibrium at about 1% melt)

combine selects how the porosity and melt corrections act together:

• :sequential: the melt correction is applied to the porosity-corrected velocities

• :weighted: both corrections start from the velocities of the solid, and the results are averaged with the porosity and the melt fraction as weights

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Seismic velocity correction for anelasticity

Methods

The routine uses the reduction formulation of karato (1993), using the quality factor formulation from Behn et al. (2009)

Computational routines

To compute a correction of S-wave velocity for anelasticity, use this:

GeoParams.MaterialParameters.SeismicVelocity.anelastic_correction Function

    Vs_anel = anelastic_correction(water::Integer, Vs0, P, T)

This routine computes a correction of S-wave velocity for anelasticity

Input:

• water: water flag, 0 = dry; 1 = dampened; 2 = water saturated

• Vs0 : S-wave velocitiy of the solid phase (with or without melt correction)

• Pref : pressure given in Pa

• Tref : temperature given in °K

Output:

• Vs_anel : corrected S-wave velocity for anelasticity

The routine uses the reduction formulation of Karato (1993), using the quality factor formulation from Behn et al. (2009)

References:

• Karato, S. I. (1993). Importance of anelasticity in the interpretation of seismic tomography. Geophysical research letters, 20(15), 1623-1626.

• Behn, M. D., Hirth, G., & Elsenbeck II, J. R. (2009). Implications of grain size evolution on the seismic structure of the oceanic upper mantle. Earth and Planetary Science Letters, 282(1-4), 178-189.

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