Traveltime calculation in three dimensional transversely isotropic (3d tti) media by the fast marching method

A computer program product comprising computer readable instructions stored on machine readable media, the instructions for determining slowness of a wave in three dimensional transversely isotropic (3D TTI) media, by:

• determining a vector for the wave;
• calculating an angle between the wave vector and an axis of symmetry of the media;
• and,
• using the calculated angle to determine a slowness of the wave.

A tool adapted for use within a wellbore comprising:

• a transducer and a processor in communication with the transducer;
• and
• a computer program product for execution by the processor, the product comprising computer readable instructions for determining a slowness of a wave in media comprising features having at least one of a transverse isotropy (TI) and a tilted symmetric axis isotropy (TTI), by;
• at least one of identifying and generating the wave;
• determining a vector for the wave, wherein determining the vector comprises using a recursive loop from a previous slowness determination;
• wherein a unit vector for a symmetry axis is defined as:
• (cos φ sin θ, sin φ sin θ, cos θ);
• where φ represents the azimuth of the symmetry axis measured from the x direction;
• and, θ represents the dip angle of the symmetry axis measured from the z direction;
• calculating an angle α between the wave vector and an axis of symmetry of the media, wherein calculating the angle α comprises solving the relationship:
• cos−1 [(τx cos φ sin θ+τy sin φ sin θ+τz cos θ)/(τx2+τy2+τz2)1/2];
• where φ represents the azimuth of the symmetry axis measured from the x direction;
• θ represents the dip angle of the symmetry axis measured from the z direction;
• τx represents a traveltime derivative component for an x-axis;
• τy represents the traveltime derivative component for an y-axis;
• and, τz represents the traveltime derivative component for an z-axis;
• and,
• using the calculated angle to determine a slowness of the wave;
• wherein determining the slowness Sijk comprises solving the relationships:
• Sijk(P)=1/[vp0(1+ε sin2 α+D(ε, δ, α, vp0, vso))1/2];
• Sijk(SV)=1/{vs0[1+(vs0/vp0)2 ε sin2 α−(vs0/vp0)2D(ε, δ, α, vp0, vs0)]1/2};
• and, Sijk(SH)=1/[vs0(1+2 γ sin2 α)1/2];
• where vp0, vso represent vertical velocities for P and SV waves, respectively;
• a represents an angle between the wave vector and an axis of symmetry of the media;
• and, ε, δ, γ and D comprises relationships of components of stress and strain for the media.