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henrybrent
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Homework Statement
A magnetic data set is believed to be dominated by a strong periodic tidal signal of known tidal period[tex]\Omega[/tex] The field strength [tex]F(t)[/tex] is assumed to follow the relation:
[tex]F=a+b\cos\Omega t + c\sin\Omega t[/tex]
If the data were evenly spaced in time, then Fourier analysis would enable simple determination of the three parameters {a, b, c}. For non-uniform data, one technique to obtain the parameters is to calculate a generalized matrix inverse.
a) Define the model vector m for this problem.
b) Assume we have three measurements [tex]{F_1, F_2, F_3}[/tex] at times [tex]{t_1, t_2,t_3}[/tex]. Write down the data vector [tex]\gamma[/tex] and matrix A you would derive for these three measurements.
c) Hence, calculate the normal equations Matrix [tex]A^T A[/tex] and right-hand side vector [tex]A^T \gamma[/tex].
d) By generalizing your argument to N data, write down the normal equations matrix.
f) Imagine you now have many evenly spaced data over one full period of the oscillation. Explain why the off leading-diagonal terms of the matrix are now 0. What are the diagonal terms?
g) when the data are evenly spaced, explain why the estimates of the parameters {a,b,c} are independent.
h) What physical properties of the tidal signal could be derived from the values for b and c?
(20 marks)
Homework Equations
Given a vector of model parameters m, a data vector [tex]\gamma[/tex] and a matrix A to connect the two vectors, such that [tex]\gamma = Am[/tex]
a solution for the model parameters can be obtained by solving (inverting) the equation [tex](A^T A)m = A^T \gamma[/tex]
The Attempt at a Solution
[/B]
Starting with a), I'm trying to define my model vector.
[tex] m = 1/(A^T A) * A^T \gamma [/tex] ??
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