tecnicas de optimización para trazar rayos y resolver problemas inversos

A 3-D Ray Tracing and Inverse Problem ApproachUniversidad Simón Bolívar
Un nuevo enfoque para trazar rayos en 3D y otros problemas inversos en geofísica
Debora Cores Carrera
Ellipsoidal velocity
Adavantages of the solver
Universidad Simón Bolívar
along the ray
Tomography Inverse problem
Solving the eigenvalue problem:
Ellipsoidal Velocity
Approximating the eigenvalues of the Christoffel equation and using the Byun Transformation, Contreras et al. in 1997 obtained an ellipsoidal group velocity:
is the group velocity in the layer delimited by interfaces i-1 and i. is the i-th component of the normal move out velocity in the symmetry plane [X,Z] with wave propagation mode j=P,SV or SH .
where,
A More general ellipsoidal velocity
The distance segment between two consecutive points at interfaces i-1 and i,
where,
More recently
The Projected Spectral Gradient (PSG) Method (Raydan et al. (2000))
Considered a low cost and storage technique as any of the extensions of conjugate gradient methods (Polak-Ribiere, Hestenes-Stiefel) for a nonlinear optimization problem.
Local Storage requirements
Fast Local Convergence
Do not require to solve a linear system of equation per iteration
Where: P is the projection on and
Given , and
If , stop
The projection over is simple and has low computational cost
The objective function does not decrease monotonicaly because of step lenght and the non monotone line search (step 4). Implying less function evaluations to converge from any initial point (Global convergence).
The step size is not the classical choice for the steepest descent method. It speeds up the convergence of the PSG method.
The PSG method is related to the Quasi Newton methods. It can be view as a two point method.
The PSG method is competitive and many times out performs the extensions of CG methods (CONMIN and PR+)
The method converge to the global minimun if we have an stratified and dipped model with constant velocity between layers
Numerical Results for Ray Tracing
5 layer synthetic model where P-S converted waves velocities are considered
157 recievers and 3 sources randomly genereted at the surface.
The average CPU time for 1 shot is 3 s (from different initial rays).
Convergence to the global minimum is obtained.
5 layer synthetic model where P-S converted wave velocities are considered
157 recievers and 5 sources randomly generated at the surface.
Lateral heterogeneous model :
We can not guarantee convergence to the global minumum.
The average CPU time for the first shot was 50 s (from different initial rays).
4 layer synthetic lateral heterogeneous model of complex stratigraphy
We consider a 5 layer ellipsoidal anisotropic medium,where the velocities are
given by the formula:
Where and denote the polar and azimuthal rotation angles in the
layer i, and j=P,SV,SH, i=1,2,...,2n+1
If the medium is an stratified or dipped model, the approach converges to a
global minimum
5 layer synthetic ellipsoidal anisotropic medium
157 receivers at the surface and 1 source in the origen.
for i=2,...,n+1
grid size (500x500) to observe
the reduction in the gradient
and the residual during that
period of time
Real velocities
Initial velocities
The initial velocities have an error of 50% from the real velocities
Final velocities (PSG)
Final velocities (PR+)
The quality of the solution by the 2 methods are almost the same
Numerical Results for the tomography inversion
SIRT has low computational cost per iteration but requires too many iterations and therefore consumes more CPU time.
PSG, PR+ and CONMIN reach quickly a good precision (10e-03) when compared to SIRT and Gauss Newton methods.
Gauss Newton is fast, in CPU time, for very small size of the grid.
The PSG and PR+ methods outperform CONMIN for very large problems.
The PSG method is always slightly faster , in CPU time, than PR+.
Anisotropic tomography inversion
Stopping criteriun:
4 −4 −3 −2 −1 0 1 2 3 4
0
1
2
3
4
5
6
−4 −3 −2 −1 0 1 2 3 4 −4
−3
−2
−1
0
1
2
3
0 0.5 1 1.5 2 2.5 3 3.5 4 0
0.5
1
1.5
2
2.5
3
3.5
i
2
1.5
1.5
1.7
1.69
1.9
1.86
1.49
1.69
1.9
3
2
1.91
2.3
2.22
2.5
2.37
1.81
2.06
2.21
4
3
2.86
2.8
2.85
3.3
3.21
3.01
2.81
3.55
5
2.7
2.83
2.9
2.85
3.1
3.19
2.73
2.88
2.91
6
1.8
1.89
2
2.07
2.3
2.42
2.01
2.27
2.59
7
1.3
1.29
1.6
1.61
1.8
1.84
1.29
1.6
1.81
i
2
5
5.84
20
21.49
8.46
20.1
3
7
7.36
15
15.46
4.93
13.46
4
3
6
25
20.31
2
25.13
5
3
5.78
25
19.14
6
20.43
6
7
7.47
15
14.29
4.47
12
7
5
5.01
20
18.94
7.64
19.86
Numerical results on the anisotropic tomography inversion
This is a highly nonlinear problem that has many solutions, so regularization of the problem and priori information is required.
The SPG method gets a good precision for estimating the velocities using small number of rays.
The problem for obtaining a better estimate of the polar angle vector is not the optimization scheme used, it depends on the seismic data acquisition.
Increasing the number of rays, the error in the velocity vector and in the azimuthal angle vector can be reduced, but the CPU time increase.
None of the mesh distribution used here give enough information for obtaining a good estimate of the polar angle vector. May be the travel time information is not appropiate for estimating fracture orientation.
Minimize
Where,
is the pressure wavefield at the velocity matrix c (computed by a finite difference scheme (Luo and Schuster 1991) ), and the matrix is a covariance operator.
Full wave inversion
Conclusions
The PSG method is a simple, global and fast method for large scale problems (Example: inversion and ray tracing).
The PSG method reachs quickly to a good precision (For example 10e-02 or 10e-03).
The PSG method only requires firts order information.
The PSG method does not require exhastive line search which implies less function evaluations per iteration.
We also used the SPG method for Full waveform inversion, obtaining very good results.
j
l
5
Minimize

tecnicas de optimización para trazar rayos y resolver problemas inversos

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