## CS 5390/4390, Exam #2

Date: Fall 2005
Name (please type legibly, ideally in block letters): ______________________________________________________________________

Please place your name on all extra sheets. One page of notes allowed (with notes on both sides).

1) Let us assume that the initial value of slowness in the cell is 0.9. There are two paths going through this cell:

• the first path has length 10, and on this path, the actual traveltime is 1 unit smaller than the predicted one;
• the second path has length 5, and on this path, the actual traveltime is 1 unit longer than the predicted one.
If we use Hole's algorithms, what will be the value of the slowness in this cell on the next iteration? Explain your answer step by step.

2-4) Use the Least Squares Methods (LSM) to solve the following over-determined system of equations: x1 ~ 1.0; x2 ~ 1.1; x1 - x2 ~ 0.1. Explain where the formulas for LSM come from. Describe the above system in matrix form, and show that the matrix-based LSM leads to exactly the same solution.

5) Use the regularization technique to find the values x1 and x2 from the under-determined system of equations x1 - x2 ~ 0.1.

6-8) We are interested in the difference y = x1 - x2 between the slownesses x1 and x2 at two different locations. The measured value of x1 is 3.5, the measured value of x2 is 2.5. The standard deviation of the first measurement is 0.3, the standard deviation of the second measurement is 0.4. Compute the resulting standard deviation of y by using the following two methods:

• sensitivity analysis method, and
• method using partial derivatives.
What are the main drawbacks of these methods? Describe the Monte-Carlo method and explain how it can avoid these drawbacks.

9-11) Let us now assume that the measured values of x1 and x2 are the same as in Problem 6)-8), but that instead of the probabilistic uncertainty, we only know the upper bounds on the measurement errors:

• for measuring x1, the upper bound is 0.3;
• for measuring x2, the upper bound is 0.4.
Compute the interval of possible values of y = x1 - x2 by using the following three methods:
• sensitivity analysis method,
• method using partial derivatives, and
• commonsense analysis of the largest and the smallest values of x1 - x2.
What are the main drawbacks of these methods? Briefly explain how the interval Monte-Carlo method can avoid these drawbacks.

12) Briefly describe what is the main topic of your project and what you have done so far as part of the project.