Alternatively, changing geometrical bases can be viewed as estimating
missing data from known data.
Subscripts _{m} denote missing and _{k} denote known data or model values.
Identifying with the known portion and with the
unknown portion
of the data, we see directly that equation (3) is an estimation
of unknown data (at the new geometry).

(4) |

(5) |

Given a line in *x*-*t* with a gap in the middle, it is easy to
demonstrate the null space problem. We run into that problem
when the domain is not
finely enough sampled in *p*, or in other words the model space is
underdetermined.

Figure shows a line with a bandlimited spike. The least squares forward transform incorporates the information about the gap into the operator. The slant stack domain shows clearly a single spike for the properly sampled data.

Figure 1

12/18/1997