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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
|
(1) |
In this homework, you will consider another special case, that of a
constant gradient of velocity
|
(2) |
It is convenient to change the parameterization of the ray tracing
system, with parameter defined by equation (3) below:
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
|
(6) |
and express velocity along the ray as a function of
, , and :
|
(7) |
- Let
. Using
the chain rule, find the expression for
|
(8) |
and solve it to show it that
|
(9) |
and
|
(10) |
- 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
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)
|
(15) |
Your result will be the analytical two-point
traveltime
|
|
|
(16) |
- In class, we discussed the hyperbolic traveltime approximation for normal moveout
|
(17) |
More accurate approximations, involving additional parameters, are possible.
- 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
- Let . Show that the function can be approximated to the same accuracy by
|
(22) |
Find , , and .
Next: Computational part 1
Up: Homework 2
Previous: Homework 2
2019-09-26