Browsing by Author "Qian, Jianliang"

Now showing 1 - 3 of 3
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    An Adaptive Finite Difference Method for Traveltime and Amplitude
    (1999) Qian, Jianliang; Symes, William W.
    The eikonal equation with point source is difficult to solve with high order accuracy because of the singularity of the solution at the source. All the formally high order schemes turn out to be first order accurate without special treatment of this singularity. Adaptive upwind finite difference methods based on high order ENO (Essentially NonOscillatory) Runge-Kutta difference schemes for the paraxial eikonal equation overcome this difficulty. The method controls error by automatic grid refinement and coarsening based on an a posteriori error estimation. It achieves prescribed accuracy at far lower cost than fixed grid methods. Reliable auxiliary quantities, such as take-off angle and geometrical spreading factor, are by-products.
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    Geometrical optics for quasi-P waves: Theories and numerical methods
    (2000) Qian, Jianliang; Symes, William W.
    The quasi-P wave in anisotropic solids is of practical importance in obtaining maximal imaging resolution in seismic exploration. The geometrical optics term in the asymptotic expansion for the wave characterizes the high frequency part of the quasi-P wave by using two functions: a phase (traveltime) function satisfying an eikonal equation and an amplitude function satisfying a transport equation. I develop theories and numerical methods for constructing the geometrical optics term of quasi-P waves in general anisotropic solids. The traveltime corresponding to the downgoing wave satisfies a paraxial eikonal equation, an evolution equation in depth. This paraxial eikonal equation takes into account the convexity of the quasi-P slowness surface and thus has a built-in reliable indicator of the ray velocity direction. Therefore, high-order finite-difference eikonal solvers are easily constructed by utilizing Weighted Essentially NonOscillating (WENO) schemes. Because the amplitude function is related to second-order derivatives of the traveltime, a third-order accurate eikonal solver for traveltimes is necessary to get a firstorder accurate amplitude. However, the eikonal equation with a point source has an upwind singularity at the source which renders all finite-difference eikonal solvers to be first-order accurate near the source. A new approach combining an adaptive-gridding strategy with WENO schemes can treat this singularity efficiently and can yield highly accurate traveltimes and amplitudes for both isotropic and anisotropic solids. A variety of numerical experiments verify that the new paraxial eikonal solver and adaptive-gridding-WENO approach are accurate and efficient for capturing the anisotropy. Therefore, the two new methods provide tools for constructing the geometrical optics term of the quasi-P wave in general anisotropic solids.
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    Geometrical Optics for Quasi-P Waves: Theories and Numerical Methods
    (2000-04) Qian, Jianliang
    The quasi-P wave in anisotropic solids is of practical importance in obtaining maximal imaging resolution in seismic exploration. The geometrical optics term in the asymptotic expansion for the wave characterizes the high frequency part of the quasi-P wave by using two functions: a phase (traveltime) function satisfying an eikonal equation and an amplitude function satisfying a transport equation. I develop theories and numerical methods for constructing the geometrical optics term of quasi-P waves in general anisotropic solids. The traveltime corresponding to the downgoing wave satisfies a paraxial eikonal equation, an evolution equation in depth. This paraxial eikonal equation takes into account the convexity of the quasi-P slowness surface and thus has a built-in reliable indicator of the ray velocity direction. Therefore, high-order finite-difference eikonal solvers are easily constructed by utilizing Weighted Essentially NonOscillating (WENO) schemes. Because the amplitude function is related to second-order derivatives of the traveltime, a third-order accurate eikonal solver for traveltimes is necessary to get a first-order accurate amplitude. However, the eikonal equation with a point source has an upwind singularity at the source which renders all finite-difference eikonal solvers to be first-order accurate near the source. A new approach combining an adaptive-gridding strategy with WENO schemes can treat this singularity efficiently and can yield highly accurate traveltimes and amplitudes for both isotropic and anisotropic solids. A variety of numerical experiments verify that the new paraxial eikonal solver and adaptive-gridding-WENO approach are accurate and efficient for capturing the anisotropy. Therefore, the two new methods provide tools for constructing the geometrical optics term of the quasi-P wave in general anisotropic solids.