Homework 1 |

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.

- 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.
- Transform the isotropic ray tracing system

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

What are the physical units of ? - 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 .

- Transform the isotropic ray tracing system
- The so-called ``parabolic'' or eikonal
equation (Bamberger et al., 1988; Claerbout, 1985; Tappert, 1977) has the form

where is a point in space, is the traveltime, and is slowness.- Derive the ray tracing system for equation (7)

where represents and represents . - Assuming constant slowness
, solve
the ray tracing system for a point source at the origin
.
- Using the ray tracing solution, find the shape of the
wavefronts defined by equation (7 in the case of a
constant slowness.
- The isotropic eikonal equation

describes wavefronts of the wave equation

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

- Derive the ray tracing system for equation (7)

Homework 1 |

2013-09-17