Homework 2

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# Theoretical part

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.

1. 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
 (1)

In this homework, you will consider another special case, that of a constant gradient of velocity
 (2)

Recall the ray tracing system from Homework 1
 (3) (4) (5)

and consider the one-point ray tracing problem with the initial conditions and .
1. Show that the solution of equation (3) for the constant gradient of velocity is
 (6)

and express velocity along the ray as a function of , , and :
 (7)

2. Let . Using the chain rule, find the expression for
 (8)

and solve it to show it that
 (9)

and
 (10)

3. One way to seek the solution for the one-point ray tracing problem is to look for scalars and in the representation
 (11)

Under what condition does the linear system of equations
 (12) (13)

have a unique solution for and ? Solve the system to find and and obtain an analytical expression for the ray trajectory .
4. Express the squared distance between the ray end points
 (14)

in terms of , , and .
5. In the two-point problem, the unknown parameters are and . Express from your equation (7) and substitute it into your equation (14). Solve for .
6. Finally, use and expressed in terms of , , , and and substitute them into the one-point traveltime solution obtained by integrating equation (5)
 (15)

Your result will be the analytical two-point traveltime
 (16)

2. In class, we discussed the hyperbolic traveltime approximation for normal moveout
 (17)

More accurate approximations, involving additional parameters, are possible.
1. Consider the following three-parameter approximation
 (18)

where is the so-called heterogeneity'' parameter.

Evaluate parameter in terms of the velocity and the reflector depth .

 (19)

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
 (20) (21)

2. Let . Show that can be approximated to the same accuracy by
 (22)

Find , , and .

 Homework 2

Next: Computational part 1 Up: Homework 2 Previous: Homework 2

2013-09-19