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