(2) |

The symbol stands for correlation, while the operator is defined as follows: The result of

is given by

where

When the series is still sparse but the temporal
distance between non-vanishing-samples is larger than the minimum
distance still resolvable by the wavelet (that is , where *f* is the highest frequency component of the wavelet), the
operator in equation (2) will not give the exact solution.
Here however, an exact solution can still be obtained through
an iterative process. Let's consider equation (2) as
the zero-order approximation to the time series :

(3) |

The residual between the original trace and the trace predicted by this zero-order approximation for the reflectivity will be

(4) |

Because the operator in equation (2) is an approximation
to the inverse of the convolution by , we can apply it
to equation (4) to get a zero-order approximation to
:

(5) |

Therefore, the first-order approximation for the reflectivity series will be

and in general

(6) |

In Figure 1 this iterative process is applied to
a synthetic trace whose original reflectivity series is
comprised of sparse spikes. The distance between some of
the spikes is successively decreased to show how the
relaxation of the sparseness condition affects
the convergence of the method. Except for *(c)*, the
exact solution was reached in a few number of iterations, even
when the separation between adjacent spikes is reduced to
one third of the wavelet size. Because in *(c)*
the exact solution was not reached within a finite number
of iterations (though an extremely close solution was
found after a few iterations), it will be necessary to improve
the basic operator *P*_{w} if we intend to apply the method to
more complex time series.

Figure 1

1/13/1998