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## Extension to cross-spectra

The same procedure as the one described above can be applied to cross-spectra. Let us first rewrite (1) as:

 f(at)=s+f'(at)(at-at+1) (9)

This is clearly nothing but a first order Taylor expansion around (s,at+1). We can write a similar relation for a two-dimensional function as:

 f(at,bt)=s+f(a)(at,bt)(at-at+1)+f(b)(at,bt)(bt-bt+1) (10)

where f(a) and f(b) denote the partial derivative with respect to a and b.

If we now consider f(a,b)=ab, we can write (10) as:

 at bt-s=bt(at-at+1)+at(bt-bt+1) (11)

Now we can again use Burg's observation (1998, personal communication) and use (11) to factorize cross-spectra written as polynomials in :
 (12)
After dividing both sides by , we obtain the equation that enables us to find both the causal and the anticausal part of a cross-spectrum with the Wilson-Burg algorithm:
 (13)

Next: Comparison of Wilson-Burg and Up: Theory Previous: Minimum phase factors
Stanford Exploration Project
7/5/1998