The amplitude spectra displaying in the previous spectrum were created using a ``direct'' slant stack. The transformation to the slant stack domain was performed using the traditional ``sum along lines'' operator plus a rho filter.

Where the rho filter is defined as .
The problems with using this approach have been well documented
Kostov (1990). The limited aperture and sampling in the
*x*-*t* domain cause artifacts. Irregular sampling in x is not well
handled by the direct stacking procedure. The alternative is to
perform a least squares inverse slant stack. This method assumes that
we can exactly perform the mapping from to *x*-*t* and casts
the mapping from *x*-*t* to as an inverse problem. This is most
conveniently calculated in the frequency domain.

The operator that maps from to is *L*,

The operator (*L*^{H}*L*) has a Toeplitz form when the data is regularly
sampled in *p*. This makes the inversion cheap to solve using Levinson
recursion. A problem with this method is that the operator *L*^{H}*L*
becomes singular when the data is aliased. Kostov suggests two ways to
overcome this problem, ``Aliasing can be overcome by using prior
information, either in the form of an interval of wavenumbers smaller
than twice the Nyquist wavenumber, or by introducing ``a priori''
information via a model-covariance matrix.'' (I would suggest that
the first statement can be modified to allow multiple wavenumber
intervals as long as the total width does not exceed twice the Nyquist
wavenumber and no two intervals occur at positions that are the alias
of one another.)

My first solution to the problem is to constrain the solution to
be non-zero only in those places that I have defined as unaliased. I
define a masking operator, *M*, that is one where the data is not
aliased and zero where it is aliased. I then solve the least squares
problem,

single-lsq
spectrum of the least squares
transform using the masked operator.
Figure 5 |

Figure 6

11/18/1997