1 / 43

Euler predicted free nutation of the rotating Earth in 1755 Discovered by Chandler in 1891

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).

flann
Download Presentation

Euler predicted free nutation of the rotating Earth in 1755 Discovered by Chandler in 1891

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 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

  2. 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-1exp {-it}[Z(t+1)-Z(t)]  = 2s/T , s = 0, 1, 2, …, T-1 Periodogram IZZT() = (2T)-1|dZT()|2

  3. variance

  4. Appendix C. Spectral Domain Theory

  5. 4.3 Spectral distribution function Cp. rv’s

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

  7. dF()/d = f() if differentiable dF() = f()d Cramer representation/Spectral representation

  8. Dirac delta function, () 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

  9. 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}

  10. Example. Bay of Fundy

  11. flattened

  12. Periodogram “sample spectral density” Mean“correction”

  13. Non parametric spectral estimation. L = 2m+1

  14. Fire video Comb5 start about 13:00

  15. Weighted average. Expected value  ( K( /B) /B) f(-) d

  16. Kernel(“modified daniel”, c(3,3))

  17. Bivariate series. Two-sided case as well AKA

  18. Bivariate example. Gas furnace

  19. Linear filters Impulse response: {aj} Transfer function. amplitude, phase A() = |A()| exp{ ()}

  20. Cramer representations Xt = exp {i t}dZx () Yt = exp {i t} dZy() =  at-uexp{i u} dZx() =  A() exp {i t} dZx() dZy() = A() dZx() Cov{ dZx(), dZx() ] =  ( – } fxx () d d fyy() = |A()|2fxx() Interpretation of power spectrum

  21. ARMA process f yy () = |A()|2fxx ( ) z = exp{ -I  )

  22. Xt = exp {i t}dZx () d() =

More Related