Homework 1

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# Theoretical part

You can either write your answers on paper or edit them in the file hw1/paper.tex. Please show all the mathematical derivations that you perform.

1. In class, we used a mysterious parameter to represent a variable continuously increasing along a ray. There are other variables that can play a similar role.
1. Transform the isotropic ray tracing system
 (1) (2) (3)

into an equivalent system that uses instead of , where is constrained by equation (4):
 (4) (5) (6)

What are the physical units of ?
2. Suppose you are given - the traveltime from the source to all points in the domain of interest. Your task is to find for all . Derive a first-order partial differential equation that connects and .

2. The so-called parabolic'' or eikonal equation (Bamberger et al., 1988; Claerbout, 1985; Tappert, 1977) has the form
 (7)

where is a point in space, is the traveltime, and is slowness.
1. Derive the ray tracing system for equation (7)
 (8) (9) (10) (11)

where represents and represents .

2. Assuming constant slowness , solve the ray tracing system for a point source at the origin .

3. Using the ray tracing solution, find the shape of the wavefronts defined by equation (7 in the case of a constant slowness.

4. The isotropic eikonal equation
 (12)

describes wavefronts of the wave equation
 (13)

with omitted possible first- and zero-order terms. What wave equation corresponds to equation (7)?
 (14)

 Homework 1

Next: Computational part Up: Homework 1 Previous: Prerequisites

2013-09-17