Rotating solid Euler predicted free nutation of the rotating Earth in 1755 Discovered by Chandler in 1891 Data from International Latitude Observatories setup in 1899

Monthly data, t = 1 month. Work with complex-values, Z(t) = X(t) + iY(t). Compute the location differences, Z(t), and then the finite FT dZT() = t=0T-1 exp {-it}[Z(t+1)-Z(t)] Periodogram IZZT() = (2T)-1|dZT()|2

variance

Appendix C. Spectral Domain Theory

4.3 Spectral distribution function Cp. rv’s

f is non-negative, symmetric(, periodic) White noise. (h) = cov{x t+h, xt} = w2 h=0 and otherwise = 0 f() = w2

dF()/d = f() if differentiable dF() = f()d

Dirac delta function, () a generalized function simplifies many t.s. manipulations r.v. X Prob{X = 0} = 1 P(x) = Prob{X  x} = 1 if x  0 = 0 if x < 0 = H(x) Heavyside E{g(X)} = g(0) =  g(x) dP(x) =  g(x) (x) dx (x)  density function = dH(x)/dx

Approximant X  N(0,2 ) (x/)/ with  small E{g(X)}  g(0) cov{dZ(1),dZ(2)} = (1 – 2) f(1) d 1 d 2 Means 0 cov{X,Y} = E{X conjg(Y)} var{X} = E{|X|2}

Example. Bay of Fundy

flattened

Periodogram “sample spectral density” Mean“correction”

Non parametric spectral estimation. L = 2m+1