Euler predicted free nutation of the rotating Earth in 1755

Discovered by Chandler in 1891

Data from International Latitude Observatories setup in 1899

Rotating solid

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=0

T-1 exp {-it}[Z(t+1)-Z(t)]

Periodogram

IZZT() = (2T)-1|dZ

T()|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 and otherwise = 0 f() = w

2

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