Verify that the matrix is Hermitian. It has been conjectured that the earthquake has an origin time t , latitude x , and longitude y . The theoretical travel time may be looked up in a travel time table where T is the travel time and is the great circle angle. Find equations to solve for dx , dy , and dt which minimize the sum-squared time residuals. Show that the matrix of the normal equation which fits the data to spherical harmonics may be written as a sum of a column times its transpose, as in the preceding problem.
Ocean tides fit sinusoidal functions of known frequencies quite accurately.
Associated with the tide is an earth tilt. The observed complex time series may be fitted to an analytical form . Find a set of equations which may be solved for the A j which gives the best fit of the formula to the data. Show that some elements of the normal equation matrix are sums which may be summed analytically. The general solution to Laplace's equation in cylindrical coordinates for a potential field P which vanishes at is given by Find the potential field surrounding a square object at the origin which is at unit potential.