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

f(a_{t})=s+f'(a_{t})(a_{t}a_{t+1})

(9) 
This is clearly nothing but a first order Taylor expansion around
(s,a_{t+1}).
We can write a similar relation for a twodimensional function as:

f(a_{t},b_{t})=s+f^{(a)}(a_{t},b_{t})(a_{t}a_{t+1})+f^{(b)}(a_{t},b_{t})(b_{t}b_{t+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:

a_{t} b_{t}s=b_{t}(a_{t}a_{t+1})+a_{t}(b_{t}b_{t+1})

(11) 
Now we can again use Burg's observation (1998, personal communication)
and use (11) to factorize crossspectra 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
crossspectrum with the WilsonBurg algorithm:
 
(13) 
Next: Comparison of WilsonBurg and
Up: Theory
Previous: Minimum phase factors
Stanford Exploration Project
7/5/1998