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To obtain the solution (1), let us assume the
existence of a function basic , such
that the function *f* (*x*) can be represented by a linear combination
of the basis functions, as follows:
| |
(7) |

The linear coefficients *c*_{k} can be found by multiplying both
sides of equation (7) by one of the basis functions
(e.g. ). Inverting the equality
| |
(8) |

where the parentheses denote the dot product, and
| |
(9) |

gives us the following explicit expression for the coefficients
*c*_{k}:
| |
(10) |

Here refers to the *kj* component of the matrix,
inverse to . The matrix is invertible as long as the
basis set of functions is linearly independent. In the special case of
an orthonormal basis, reduces to the identity matrix:
Equation (10) is a least-square estimate of the coefficients
*c*_{k}. For a given set of basis functions, it approximates the
function *f* in formula (1) in the least-square sense.

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Stanford Exploration Project

11/11/1997