On numerical solving the complex eikonal equation using ray tracing methods

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On numerical solving the complex eikonal equation using ray tracing methods

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On numerical solving the complex eikonal equation using ray tracing methods

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On numerical solving the complex eikonal equation using ray tracing methods

Václav Vavryčuk

Institute of Geophysics, Czech Acad. Sci., Prague

Wave propagation in dissipative media

Description of dissipative media

Tensor of viscoelastic parameters

– elastic parameters

– viscous parameters

Quality matrix (Q-matrix)

all quantities

and

equations

are complex

Christoffel tensor and the eikonal

equation

Ray tracing in isotropic elastic media

Eikonal equation

non-linear partial differential equation

G (x, p) is the eigenvalue of the Christoffel tensor

c, x, p, τare real

Ray tracing equations

system of linear ordinary differential equations

Ray tracing in isotropic viscoelastic media

Eikonal equation

non-linear partial differential equation

G (x, p) is the eigenvalue of the Christoffel tensor

c, x, p, τ are complex

Ray tracing equations

system of linear partial differential equations

Solution of complex ray tracing equations are complex rays !

Troubles of complex ray theory

Rays are curves in complex space

3D real space 6D complex space

Physical meaning of rays is more involved

Rays are not parallel to energy flux

Model of the medium must be defined in complex space

For simple models: analytic continuation

It is not clear how to apply the complex ray theory to inhomogeneous media with interfaces

Analytic continuation is not possible

Real ray tracing in dissipative media

Approximate ray tracing equations

Approximate approaches:

1. Perturbation approach of elastic ray theory

(Gajewski & Pšenčík, 1992)

- rays are calculated in an elastic reference medium
- dissipation effects are included by perturbations
- weak attenuation

2. Real ray approach

(Vavryčuk, 2008, 2010, 2012)

- complex rays are approximated by real rays
- no perturbations are used
- includes strong attenuation

Approximate ray tracing equations

Ray tracing equations

same equations as for elastic medium

Perturbation approach

Real ray approach

Numerical examples

Model of the medium

Isotropic medium with a constant

gradient of c-2:

real part of c-2

imaginary part of c-2

imaginary part of c

real part of c

Complex travel time: exact solution

real part

of τ

imaginary part

of τ

Complex rays: exact solution

propagation rays

(curves of the gradient of the real travel time)

attenuation rays

(curves of the gradient of the imaginary travel time)

Errors of approximate travel time

Perturbation approach

errors of real travel time

errors of imaginary travel time

(%)

(%)

Real ray approach

errors of real travel time

errors of imaginary travel time

(%)

(%)

Errors in geometry of approximate rays

Perturbation approach

propagation rays

attenuation rays

Real ray approach

propagation rays

attenuation rays

exact

approximate

Conclusions

- The real ray approach deals with real rays and works in isotropic as well as anisotropic dissipative media
- The real ray approach is approximate but highly accurate
- The real ray approach is applicable to a broad family of models including 3-D heterogeneous structures with interfaces
- Computational costs of the real ray approach are the same as for perturbationsapproach