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We now introduce a more flexible
form to the Needleman-Wunsch algorithm Karp (2000).
Let the two input strings, (*x*,*y*), given by
| |
(1) |

| (2) |

where (*m*,*n*) need not be equal.
The subscript *i* denotes the row direction, and *j* denotes columns.
We write the scoring function *V*(*i*,*j*) as the equation set
| |
(3) |

| (4) |

| (5) |

| (6) |

| (7) |

| (8) |

where the indices range and .In this form, the function is the similarity
matrix and it is calculated on the fly rather than precomputed.
Further it can be customized to reflect different weights
associated with matches and mismatches.
In biological applications an element on each string either
matches or does not, and this fact is represented in the
choice of a similarity measure, for example
| |
(9) |

in which a value of 1 is awarded for a match, and
all other cases are are equally awarded .
In the seismic case we do not expect or need an exact
amplitude match, rather it is important to reward small
amplitude differences and penalize large ones.
We capture this idea in a similarity function as

| |
(10) |

where *c* is a constant chosen to keep and *abs*[] is the absolute value.
Computational complexity and cost of this algorithm applied to
two strings of length *n* and *m* is *O*(*n***m*).

** Next:** Modification to the seismic
** Up:** METHODOLOGY
** Previous:** Needleman-Wunsch algorithm
Stanford Exploration Project

11/11/2002