Homework 2 |

You can either write your answers on paper or edit them in the file
`hw2/paper.tex`

. Please show all the mathematical
derivations that you perform.

- In class, we derived analytical solutions for one-point and
two-point ray tracing problems for the special case of a constant
gradient of slowness squared

In this homework, you will consider another special case, that of a constant gradient of velocity

Recall the ray tracing system from Homework 1

and consider the one-point ray tracing problem with the initial conditions and .- Show that the solution of
equation (3) for the constant gradient of velocity is

and express velocity along the ray as a function of , , and :

- Let
. Using
the chain rule, find the expression for

(8)

and

- One way to seek the solution for the one-point ray tracing
problem is to look for scalars and in the representation

Under what condition does the linear system of equations

have a unique solution for and ? Solve the system to find and and obtain an analytical expression for the ray trajectory . - Express the squared distance between the ray end points

in terms of , , and . - In the two-point problem, the unknown parameters are and . Express from your equation (7) and substitute it into your equation (14). Solve for .
- Finally, use and
expressed in terms of
,
,
, and and
substitute them into the one-point traveltime solution obtained by
integrating equation (5)
^{}

Your result will be the analytical two-point traveltime

- Show that the solution of
equation (3) for the constant gradient of velocity is
- In class, we discussed the hyperbolic traveltime approximation for normal moveout

More accurate approximations, involving additional parameters, are possible.- Consider the following three-parameter approximation

where is the so-called ``heterogeneity'' parameter.Evaluate parameter in terms of the velocity and the reflector depth .

by expanding equation (18) in a Taylor series around the zero offset and comparing it with the corresponding Taylor series of the exact traveltime. The exact traveltime is given by the parametric equations

- Let . Show that can be approximated to the same accuracy by

Find , , and .

- Consider the following three-parameter approximation

Homework 2 |

2013-09-19