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

Abstract

A technique for calculating traveltime of a seismic wave in three dimensional tilted transversely isotropic (3D TTI) media includes determining a wave vector, defining a unit vector, calculating an angle of the wave vector from an axis and performing a slowness determination. The technique may be practiced as a computer implemented set of instructions, and may be incorporated into measurement equipment.

1 . A method for determining a travel time of a seismic wave in three dimensional transversely isotropic (3D TTI) media, the method comprising: 
 determining a vector for the wave;    defining a unit vector for the wave;    calculating an angle between the wave vector and an axis of symmetry of the media; and,    using the calculated angle, determining a slowness of the wave to determine the travel time of the wave.    
     
     
         2 . The method as in  claim 1 , further comprising at least one of identifying and generating the wave.  
     
     
         3 . The method as in  claim 1 , wherein determining the wave vector comprises using a recursive loop from each previous traveltime determination.  
     
     
         4 . The method as in  claim 1 , 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.    
     
     
         5 . The method as in  claim 1 , wherein calculating the angle α comprises solving the relationship:  
         α=cos −1 [(τ x  cos φ sin θ+τ y  sin φ sin θ+τ z  cos θ)/(τ x   2 +τ y   2 +τ z   2 ) 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;    τ x  represents a traveltime derivative component for an x-axis;    τ y  represents the traveltime derivative component for an y-axis; and,    τ z  represents the traveltime derivative component for an z-axis.    
     
     
         6 . The method as in  claim 1 , wherein determining the slowness S ijk  comprises solving the relationships:  
           S   ijk ( P )=1/[ν p0 (1+ε sin 2    α+D (ε,δ,α,ν p0 ,ν so )) 1/2 ];    S   ijk ( SV )=1/{ν s0 [1+(ν s0 /ν p0 ) 2 ε sin 2  α−(ν s0 /ν p0 ) 2   D (ε,δ,α,ν p0 ,ν s0 )] 1/2 }; and,    S   ijk ( SH )=1/[ν s0 (1+2γ sin 2  α) 1/2 ];  where    ν p0 , ν so  represent vertical velocities for P and SV waves, respectively;    α 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.    
     
     
         7 . The method as in  claim 1 , wherein the media comprises features having at least one of a transverse isotropy (TI) and a tilted symmetric axis isotropy (TTI).  
     
     
         8 . The method as in  claim 1 , wherein determining the travel time comprises determining the travel time for a Kirchhoff pre-stack depth migration.  
     
     
         9 . A computer program product comprising computer readable instructions for determining a travel time of a seismic wave in three dimensional transversely isotropic (3D TTI) media, by: 
 determining a vector for the wave;    defining a unit vector for the wave;    calculating an angle between the wave vector and an axis of symmetry of the media; and,    using the calculated angle, determining a slowness of the wave to determine the travel time of the wave.    
     
     
         10 . The computer program product as in  claim 9 , further comprising at least one of identifying and generating the wave.  
     
     
         11 . The computer program product as in  claim 9 , wherein determining the wave vector comprises using a recursive loop from each previous traveltime determination.  
     
     
         12 . The computer program product as in  claim 9 , 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.    
     
     
         13 . The computer program product as in  claim 9 , wherein calculating the angle α comprises solving the relationship:  
         cos −1 [(τ x  cos φ sin θ+τ y  sin φ sin θ+τ z  cos θ)/(τ x   2 +τ y   2 +τ z   2 ) 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;    τ x  represents a traveltime derivative component for an x-axis;    τ y  represents the traveltime derivative component for an y-axis; and,    τ z  represents the traveltime derivative component for an z-axis.    
     
     
         14 . The computer program product as in  claim 9 , wherein determining the slowness S ijk  comprises solving the relationships:  
           S   ijk ( P )=1/[ν p0 (1+ε sin 2    α+D (ε,δ,α,ν p0 ,ν so )) 1/2 ];    S   ijk ( SV )=1/{ν s0 [1+(ν s0 /ν p0 ) 2 ε sin 2  α−(ν s0 /ν p0 ) 2   D (ε,δ,α,ν p0 ,ν s0 )] 1/2 }; and,    S   ijk ( SH )=1/[ν s0 (1+2γ sin 2  α) 1/2 ];  where    ν p0 , ν so  represent vertical velocities for P and SV waves, respectively;    α 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.    
     
     
         15 . The computer program product as in  claim 9 , wherein the media comprises features having at least one of a transverse isotropy (TI) and a tilted symmetric axis isotropy (TTI).  
     
     
         16 . The computer program product as in  claim 9 , wherein determining the travel time comprises determining the travel time for a Kirchhoff pre-stack depth migration.  
     
     
         17 . A sampling tool comprising: 
 equipment for sampling within a wellbore, the sampling tool further comprising a coupling to an electronics unit, the electronics unit comprising a computer program product comprising computer readable instructions for determining a travel time of a seismic wave in three dimensional transversely isotropic (3D TTI) media, by;    determining a vector for the wave;    defining a unit vector for the wave;    calculating an angle between the wave vector and an axis of symmetry of the media; and,    using the calculated angle, determining a slowness of the wave to determine the travel time of the wave.