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We still have a data sequence *y*(*t*) ().
For each time *T* and order *k*, we define the forward and backward residuals
and *r*_{k}(*T*); a single reflection coefficient
*K*_{k} and Burg's recursions are used:
| |
(9) |

To compute the reflection coefficients *K*_{k}, we minimize the weighted
energy of the forward and backward residuals of order *k*. The weights
are centered on the particular time *T*:
It is important to notice that the residuals depend here on the whole
set of data, since the energy is a summation on the whole
time window. However, this expression of the energy induces an adaptive
formalism, since more emphasis is put on the residuals near the time *T*
of interest than of the residuals far from this time. Minimizing this
expression with respect to *K*_{k} leads to a time-dependent reflection
coefficient:

| |
(10) |

Then Hale (1981) showed that the reflection coefficients *K*_{k,T} can be easily
computed, if the numerator and denominator of expression (11)
are split between past and future summations. For example, the numerator
becomes:

In the same way, we can split the denominator between *D*^{-}_{k}(*T*) (summation
up to ), and *D*^{+}_{k}(*T*) (summation from *T* to *T*_{max}).
Then, it is straightforward to show that:
| |
(11) |

These recursions, together with the recursions (10) and the
expression (11), form the adaptive version of Burg's algorithm.

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Stanford Exploration Project

1/13/1998