##### Department of Mathematics,

University of California San Diego

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### Math 278 - Numerical Analysis

## Jianliang Qian

#### UCLA

## A Local Level Set Eulerian Method for Paraxial Geometrical Optics

##### Abstract:

Geometrical optics and its ingredients, eikonals and amplitudes, have wide applications, such as optimal control, robotic navigation and computer vision. We propose a local level set method for constructing the geometrical optics term in the paraxial formulation for the high frequency asymptotics of 2-D acoustic wave equations. The geometrical optics term consists of two multivalued functions: a traveltime function satisfying the eikonal equation locally and an amplitude function solving a transport equation locally. The multivalued traveltimes are obtained by solving a level set equation and a traveltime equation with a forcing term. The multivalued amplitudes are computed by a new Eulerian formula based on the gradients of traveltimes and takeoff angles. As a byproduct the method is also able to capture the caustic locations. The proposed Eulerian method is second-order accurate and has complexity of $O(N^2 Log N)$. Several examples including the well known Marmousi synthetic model illustrate the accuracy and efficiency of the Eulerian method. We will also discuss the extension of the method to anisotropic elastic wave equations and other possible future directions.

Host:

### January 27, 2004

### 10:00 AM

### AP&M 7321

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