6. Seismic Anisotropy A.Stovas, NTNU 2005

Definition (1) Seismic anisotropy is the dependence of seismic velocity upon angle This definition yields both P- and S-waves

Definition (2) Saying ”velocity” we mean ray veocity and wavefront velocity group velocity and phase velocity interval velocity and average velocity NMO velocity and RMS velocity

Definition (3) We have to distinguish between anisotropy and heterogeneity Heterogeneity is the dependence of physical properties upon position Heterogeneity on the small scale can appear as seismic anisotropy on the large scale

Notations h i – layer thickness v i – layer velocity t 0 – vertical traveltime v P0 – vertical velocity v NMO – normal moveout velocity ,  – anisotropy parameters S 2 – heterogeneity coefficient

Simple example of anisotropy (two isotropic layers model) isotropic VTI h i – layer thickness v i – layer velocity t 0 – vertical traveltime v P0 – vertical velocity v NMO – normal moveout velocity ,  – anisotropy parameters S 2 – heterogeneity coefficient

Elasticity tensor Equation of motion Stress-strain relation (Hook’s law) stress strain The elasticity tensor

Symmetry We convert stiffness tensor C ijmn to the stiffness matrix C  The best case: Isotropic symmetry 2 different elements The worst case: Triclinic symmetry 21 different elements Lame parameters: and 

Seismic anisotropy symmetries Orthorombicic symmetry 9 different elements (shales, thin-bed sequences with vertical crack-sets) Trasverse isotropy symmetry 5 different elements (shales, thin-bed sequences

VTI and HTI anisotropy VTI HTI symmetry axis symmetry plane

The phase velocities (velocities of plane waves) Cij – stiffness coefficients v i – phase velocity  – phase angle

Parametrization (Thomsen, 1984) Vertical velocities Anisotropy parameters

Interpretation of anisotropy parameters Isotropy reduction Horizontal propagation

Weak anisotropy approximation

Weak anisotropy for laminated siltstone

Mesaverde shale/sandstone

Nonellipticity

Wave propagation in homogeneous anisotropic medium Wavefront normal Wavefront tangent k – wavenumber V group – group velocity  – angular frequency v phase – phase velocity p – horizontal slowness  – group and phase angles

The anisotropic moveout The hyperbolic moveout The Taylor series coefficient The moveout velocity (x – offset ot source-receiver separation)

The anisotropic qP-traveltime in p-domain The horizontal slowness The offset The traveltime S  – deviation of the slowness squared between VTI and isotropic cases a j – coefficients for expansion in order of slowness

The anisotropic traveltime parameters The P-wave The S-wave The vertical Vp-Vs ratio

The moveout velocity The critical slowness

The velocity moments v 0 =  0

The heterogeneity coefficients

The Taylor series The normalized offset squared

The traveltime approximations Shifted hyperbola Continued fraction

The continued fraction approximations Tsvankin-Thomsen Ursin-Stovas Correct

The heterogeneity coefficient S 2 Alkhalifah and Tsvankin: Ursin and Stovas: S 2 (Ursin and Stovas) reduces to S 2 (Alkhalifah and Tsvankin) if    is large (acoustic approximation)

The traveltime approximations (single VTI layer) Bold – two terms Taylor Empty circles – shifted hyperbola Filled circles – Tsvankin-Thomsen Empty stars – Stovas-Ursin

The traveltime approximations (stack of VTI layers)

Layering against anisotropy

V NMO for dipping reflector Tsvankin, 1995  is the angle for dipping reflector

VTI DMO operator Stovas, 2002 Operator shape depends on the sign of 

If we ignore anisotropy in post- stack time migration  Dipping reflectors are mispositioned laterally. Mislocation is a function of: - magnitude of the average  for  overburden - dip of the reflector - thickness of anisotropic overburden  Diffractions are not completely collapsed, leaving diffraction tails, etc. Alkhalifah and Larner, 1994

Determining  Use V P-NMO from well-log The residual moveout gives 

Dix-type equations (1) Ursin and Stovas, 2004

Dix-type equations (2) (error in parameters due to error in S 2

Wave propagation in VTI medium U z and U r are transformed verical and horizontal displacement components; S z and S r are transformed vertical and horizontal stress components Stovas and Ursin, 2003

Up- and down-wave decomposition q  and q  are verical slownesses for P- and S-wave With linear transformation

The transformation matrix with the symmetries

Scattering matrices Symmetry relations

The vertical slowness The vertical slownesses squared are the eigenvalues of the matrix and are found by solving the characteristic equation

The R/T coefficients where superscripts (1) and (2) denote the upper and lower medium, respectively with

The weak-contrast R/T coefficients q is the vertical slowness

The weak-contrast R/T coefficients (Rueger, 1996)  is shear wave bulk modulus:

Parametrization Stiffness coefficients Velocities Impedances Mixed

Effect of anisotropy

Different parametrizations

Contribution from the contrasts

Second-order R/T

Effect of second-order R/T

Visco-elastic parameters Linear solid model (Carcione, 1997)The real coefficients The modified comples Zener moduli The relaxation times

Complex stiffness coefficients versus frequency Clay shale (real part is to the top, imaginary part is to the bottom)

The effect of viscoelasticity (1)

The effect of viscoelasticity (2)

Transmission fot the stack of the layers

Conclusion In practice the weak-anisotropy approximation is very useful