Parameterizations

There exists a whole zoo of different parameterizations to describe the acoustic or elastic properties of a medium.

Salvus offers a lot of flexibility and supports various parameterizations. Because conversion can be a tedious and error-prone task, we list several parameter options and how the translate into each other below.

Acoustic parameters

Internally, Salvus uses an abstract formulation of the acoustic wave equation, that allows to treat different parameterizations within the same implementation:

m0t2ϕ(m1ϕ)=f.{` m_0 \\partial_t^2 \\phi - \\nabla \\cdot \\big(m_1 \\nabla \\phi \\big) = f.`}

Different choices of m0{`m_0`} and m1{`m_1`} lead to a different physical meaning of ϕ{`\\phi`} and f{`f`}.

parametersymbolunitSalvus key
m0{`m_0`}m0{`m_0`}variableM0
m1{`m_1`}m1{`m_1`}variableM1
densityρ{`\\rho`}kgm3{`kg \\,m^{-3}`}RHO
velocity (sound speed)vp{`v_p`} or c{`c`}ms1{`m \\,s^{-1}`}VP
impedanceI{`\\mathcal{I}`}kgm2s1{`kg \\,m^{-2}\\,s^{-1}`}coming soon
compressibilityβ{`\\beta`}ms2kg1{`m\\,s^2\\,kg^{-1}`}coming soon

Below, we list conversion formulas for a displacement potential ϕ{`\\phi`}.

m0={`m_0 =`}m1={`m_1 =`}ρ={`\\rho =`}vp={`v_p =`}I={`\\mathcal{I} =`}β={`\\beta =`}
(m0m1){`\\left(\\begin{array}{cc}m_0\\\\m_1\\end{array}\\right)`}m11{`m_1^{-1}`}m1m0{`\\sqrt{\\frac{m_1}{m_0}}`}1m0m1{`\\frac{1}{\\sqrt{ m_0 \\, m_1 }}`}m0{`m_0`}
(ρvp){`\\left(\\begin{array}{cc}\\rho\\\\v_p\\end{array}\\right)`}1ρvp2{`\\frac{1}{\\rho\\, v_p^2}`}1ρ{`\\frac{1}{\\rho}`}ρvp{`\\rho\\, v_p`}1ρvp2{`\\frac{1}{\\rho \\, v_p^2}`}
(Ivp){`\\left(\\begin{array}{cc}\\mathcal{I}\\\\v_p\\end{array}\\right)`}1Ivp{`\\frac{1}{\\mathcal{I}\\, v_p}`}vpI{`\\frac{v_p}{\\mathcal{I}}`}Ivp{`\\frac{\\mathcal{I}}{v_p}`}1Ivp{`\\frac{1}{\\mathcal{I}\\, v_p}`}
(Iρ){`\\left(\\begin{array}{cc}\\mathcal{I}\\\\\\rho\\end{array}\\right)`}ρI2{`\\frac{\\rho}{\\mathcal{I}^2}`}1ρ{`\\frac{1}{\\rho}`}Iρ{`\\frac{\\mathcal{I}}{\\rho}`}ρI2{`\\frac{\\rho}{\\mathcal{I}^2}`}
(βρ){`\\left(\\begin{array}{cc}\\beta\\\\\\rho\\end{array}\\right)`}β{`\\beta`}1ρ{`\\frac{1}{\\rho}`}1ρβ{`\\frac{1}{\\sqrt{\\rho\\,\\beta}}`}ρβ{`\\sqrt{\\frac{\\rho}{\\beta}}`}

Elastic parameters

Salvus can handle isotropic and anisotropic elastic media and supports different parameterizations.

Isotropic material

In isotropic media, the elastic properties reduce to three parameters, which can either be expressed using the Lamé coefficients or by the velocities of compressional (P) and shear (S) waves.

parametersymbolunitSalvus key
first Lamé coefficientλ{`\\lambda`}Pa{`Pa`}LAMBDA
second Lamé coefficient, shear modulusμ{`\\mu`}Pa{`Pa`}MU
densityρ{`\\rho`}kgm3{`kg \\,m^{-3}`}RHO
P-wave velocityvp{`v_p`}ms1{`m \\,s^{-1}`}VP
S-wave velocityvs{`v_s`}ms1{`m \\,s^{-1}`}VS

Here, the following conversion formula apply.

λ={`\\lambda =`}μ={`\\mu =`}vp={`v_p =`}vs={`v_s =`}
(λμρ){`\\left(\\begin{array}{c}\\lambda\\\\\\mu\\\\\\rho\\end{array}\\right)`}λ+2μρ{`\\sqrt{\\frac{\\lambda + 2\\mu}{\\rho}}`}μρ{`\\sqrt{\\frac{\\mu}{\\rho}}`}
(ρvpvs){`\\left(\\begin{array}{c}\\rho\\\\v_p\\\\v_s\\end{array}\\right)`}ρ(vp2vs2){`\\rho\\,(v_p^2-v_s^2)`}ρvs2{`\\rho\\,v_s^2`}

Anistropic material

Due to the symmtetry relations

Cijkl=Cklij=Cjikl{`\\mathbf{C}_{ijkl} = \\mathbf{C}_{klij} = \\mathbf{C}_{jikl}`}

the fourth-order elastic tensor C{`\\mathbf{C}`} reduces to at most 21 (in 3-D) or 6 (in 2-D) independent parameters. Using Voigt notation, these are fully specified by

parametersymbolunitSalvus key
densityρ{`\\rho`}kgm3{`kg \\,m^{-3}`}RHO
Elastic tensorcij{`c_{ij}`}Cij

with 1 {`\\leq`} i {`\\leq`} j {`\\leq`} 6 in 3-D, and with 1 {`\\leq`} i {`\\leq`} j {`\\leq`} 3 in 2-D, respectively.

Tilted transversely isotropic material is an important special case of anisotropic media with are only two additional parameters compared to isotropic material that distinguish the wave speeds of horizontally and vertically traveling waves.

parametersymbolunitSalvus key
first Lamé coefficientλ{`\\lambda`}Pa{`Pa`}LAMBDA
second Lamé coefficient, shear modulusμ{`\\mu`}Pa{`Pa`}MU
densityρ{`\\rho`}kgm3{`kg \\,m^{-3}`}RHO
horizontal P-wave velocityvph{`v_{ph}`}ms1{`m \\,s^{-1}`}VPH
vertical P-wave velocityvpv{`v_{pv}`}ms1{`m \\,s^{-1}`}VPV
horizontal S-wave velocityvsh{`v_{sh}`}ms1{`m \\,s^{-1}`}VSH
vertical S-wave velocityvsv{`v_{sv}`}ms1{`m \\,s^{-1}`}VSV
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