Homework 4 |

- Consider the elastic wave equation

in the case of an isotropic elasticity

Using the geometric representation

and assuming the P-wave polarization in the direction of the gradient of , derive the elastic P-wave amplitude transport equation and show its similarity to the corresponding equation for the case of acoustic variable-density wave propagation. - Consider a 2-D common-midpoint gather , which
contains a geometric event
with a
constant amplitude along a hyperbolic shape

The gather gets transformed by the slant-stack (Radon transform) operator

where is a waveform-correcting half-order derivative operator.Using the theory of geometric integration, show that will contain a geometric event . Find and .

- Using the hyperbolic traveltime approximation

(6)

(7) Suppose that you switch to the more accurate shifted-hyperbola approximation

(8) - How would you need to modify the eikonal equation?
- How would you need to modify the following expressions for the escape time and location for use in the angle-domain Kirchhoff time migration?

(9) (10)

Homework 4 |

2013-10-29