1. Geology 5600/6600 Signal Analysis 16 Sep 2015 © A.R. Lowry 2015 Last time: A process is ergodic if time averages equal ensemble averages. Properties of weakly WSS ergodic processes: and hence, statistical properties can be derived by averages (We don’t need the pdf a priori to evaluate expectation!) The convolution of two functions f and g is defined as: so we can also write: The convolution theorem : convolution in the time domain is equivalent to multiplication in the freq domain.

2. Geology 6600/7600 Signal Analysis 16 Sep 2015 © A.R. Lowry 2015 Last time (Continued): The Auto-Power Spectrum of a random variable is the Fourier transform of the autocorrelation function, given by the Wiener-Khinchin relation : which we can also write as:

3. Grokking the Fourier Transform: Power spectra and the Fourier Transform to the frequency domain are fundamental to signal analysis, so you should spend a little time familiarizing yourself with them. For the following functions, I’d like you to first evaluate the integral by hand, & then calculate and plot the Fourier transform using Matlab. (Send me by class-time Monday Sep 28). 1) Autocorrelation of a zero-mean, WSS white noise process (use   2 = 3 ) 2) A constant (use a = 3 ) 3) A cosine function (use amplitude 1 ;  ) 4) A sine function (as above) 5) A box function ( 0 on [– ,–  /2] & [  /2,  ] ; 1 on [–  /2,  /2] )

4. Use the continuous Fourier transform to do calculations by hand; use the DFT for the matlab exercises Matlab functions you’ll need to learn: fft fftshift Be sure to turn in matlab scripts so if you did something wrong I can figure out what happened!

5. Important: Defns of forward/inverse Fourier transform Useful: Euler’s eqn, function product relations Some key things to recognize: Integral of a sinusoid on [– ,  ] (hence [–∞,∞] ) is always zero unless the sinusoid is cos(0). Ergo, integral of a product of sin ’s/ cos ’s is nonzero only when . In the case of FT, the Fourier transform Acos(   t) and Asin(   t) is thus nonzero only at  0. By definition this means that sinusoids are a class of orthogonal functions. (Their inner product is zero unless f = g !) This makes them a very important class of functions…

6. As a shorthand for the forward and inverse Fourier transform, we will use e.g.: Some properties of the Fourier transform: Recalling Euler’s relation, e –i  t = cos(  t) – isin(  t), the FT of an even function will always be even (and real), and the FT of an odd function will always be odd and imaginary. Hence, because the autocorrelation function R xx is real and even, the autopower spectrum S xx will always be real and even as well! Note however this also implies that the power spectrum does not contain any phase information about the signal…

7. Some practical applications of cross-correlation in geophysics: If we assume ergodicity of WSS random processes that are sampled at N regular intervals in time (latter is commonly the case for geophysical data), the cross-correlation R xy [l] at lag l can be estimated as: Note that for zero-mean signals, this simplifies to:

8. Example: The change in reflection travel-time with offset is given by the Normal MoveOut (NMO) equation: NMO correction cross-correlates signals shifted using a range of assumed velocities to identify the “stacking velocity” that maximizes the cross-correlation.

9. Example: Pleistocene Lake Bonneville shorelines exhibit erosional and depositional features that are easily recognized in plots of elevation slope and curvature…. Cross-correlation of slope & curvature allows accurate estimates of  -height!

10. Generally, because both the autocorrelation function and the autopower spectrum are real and even, we can write: The average power of is simply We can derive this by recalling that and plugging zero in for  in the Wiener-Khinchin relation: Hence the average power is the mean square of x. ~

11. The Cross-Power Spectrum relating two random processes x and y is given by: Recall that R xy (  ) = R yx (–  ) : If we substitute and let  – , However so consequently ~~