Least-squares imaging and deconvolution using the hybrid norm conjugate-direction solver |

where r(t) is the reflectivity series, s(t) is the source wavelet, n(t) is random noise, and d(t) is the seismic traces (we assume a certain kind of amplitude compensation has already been applied).

Intrinsically, this is an under-determined problem, because both
and
are unknown. Further assumptions
about the reflectivity series
are needed in order to get a deterministic answer.
In the L2 scenario, the underlying assumption is that the
reflectivity model is purely random (i.e., has a *white
spectrum*). As mentioned before, the model may in fact be spiky,
which is better matched by an L1 type inversion. Therefore the hybrid
result should
outperform the L2 result.

For simplicity, we also assume that source wavelet is minimum phased. The conventional spiking deconvolution can be defined as an inverse problem,

where is the data convolution operator, and is the filter. In this formulation, the filter is the only unknown, and in theory the data residual itself is the reflectivity model.

To incorporate the model regularization into the inversion framework, we generalize the formulation above by posing the deconvolution problem as such inversion problem:

in which is the data convolution operator, is the filter, and is the reflectivity model. The parameter indicates the strength of the regularization. Since the source wavelet is assumed to be a minimum-phase wavelet, ideally the inversion gives the exact inverse of the source wavelet to .

The first equation in (4) (data fitting) implies that after convolving the data with the filter, we should get the reflectivity model; any values that cannot fit the reflectivity model are considered as noise in data. The second equation in (4) is the spiky regularization of the model; thus we apply the hybrid norm.

Least-squares imaging and deconvolution using the hybrid norm conjugate-direction solver |

2010-05-19