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New approach to point-to-curve raytracing
Tijmen Jan Moser
Fugro-Jason, The Netherlands
29 September 2004
Charles University, Prague, Czech Republic
Biography
1987 - masters degree Geophysics, State University of Utrecht, The Netherlands
(Nolet)
1992 - PhD, "Shortest path method for seismic ray tracing in complicated media"
State University of Utrecht, The Netherlands
(Helbig, Nolet)
1992 - Postdoc Amoco, Tulsa, OK
(Gray, Treitel)
1992-4 - Institut Français du Pétrole and Institut de Physique du Globe, Paris
(Lailly, Madariaga, Tarantola)
1994-5 - Institute of Solid Earth Physics and Norsk Hydro
(Hanyga, Pajchel)
1996-7 - Alexander von Humboldt fellow - Karlsruhe
(Hubral, Karrenbach, Shapiro)
1997-2000 - Geophysical Institute of Israel/Norsk Hydro
(Landa, Keydar, Gelchinsky, Pajchel)
2001-present - Fugro-Jason
Two-point raytracing
xR
xS
3D Ray field
x(γ 1 , γ 2 , τ ) = x R
γ 1 , γ 2 initial angles, τ independent monotonic parameter along the ray
nonlinear system of 3 equations with 3 unknowns
Point-to-curve raytracing (Hanyga and Pajchel, GP, 1995)
Profile plane f 2 (x, y, z, ) = 0
Target plane f 1 (x, y, z, ) = 0
f = 0
Profile  1
 f2 = 0
3D Ray field up to target plane
f 1 (x(γ 1 , γ 2 , τ )) = 0
(solve for τ 0 = τ )
Distance from profile plane
f 2 (x(γ 1 , γ 2 , τ 0 ))
→
F(γ 1 , γ 2 ) = f 2 (x(γ 1 , γ 2 , τ 0 )) = 0
nonlinear system of 1 equation with 2 unknowns
implicit plane curve
Horizontal shot profiling
t
x
surface topography
shot point
Vertical seismic profiling (VSP)
shot point
t
borehole
z
Exploding reflector modeling
t
x
surface
exploding reflector
Solution of F(γ 1 , γ 2 ) = 0
• implicit curve tracing
d γ 1   Fγ 1 
=
dq γ 2   Fγ 2 
q parameter along curve.
Predictor-corrector, because first integral is known.
Predictor-corrector
Bifurcation
Isolated branches
• Contouring of F(γ 1 , γ 2 )
• Algebraic rasterization
(An Accurate Algorithm for Rasterizing Algebraic Curves and Surfaces, by G. Taubin.
IEEE Computer Graphics & Applications, March 1994.)
• Lipschitz condition
|F(γ ) − F(γ ′)| ≤ L||γ − γ ′||
where γ = (γ 1 , γ 2 )
Recursive triangulation of γ plane, rejecting triangles for which Lipschitz condition
predicts there is no solution.
x^4 + x^2 y^2 - x ( x^2 - y^2 ) + 0.0001 = 0
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