Rotating solid Euler predicted free nutation of the

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) + i. Y(t). Compute the location differences, Z(t), and then the finite FT d. ZT( ) = t=0 T-1 exp {-i t}[Z(t+1)-Z(t)] Periodogram IZZT( ) = (2 T)-1|d. ZT( )|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} = w 2 h=0 f( ) = w 2 and otherwise = 0

d. F( )/d = f( ) if differentiable d. F( ) = 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) d. P(x) = g(x) dx (x) density function = d. H(x)/dx

Approximant X N(0, 2 ) (x/ )/ with small E{g(X)} g(0) cov{d. Z( 1), d. Z( 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 Mean“correction” “sample spectral density”

Non parametric spectral estimation. L = 2 m+1