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

Discussion

It is not always true that wavelet can be extracted from the seismic data, in this case we have to perform blind deconvolution. To overcome the difficulty brought by non-minimum phase wavelet, we turn back to the original non-linear convolution model (3), and solve the non-linear inversion problem directly.

There are two ways to linearize this model. The first one is to use model perturbation and neglect the non-linear higher order terms in the following:

$$\mathbf {(s+\Delta s) * ( m+\Delta m) \approx s*m + s*\Delta m + m*\Delta s = d},$$

in which $\mathbf{m, s}$ are the initial model and source wavelet respectively. $\bold{\Delta m, \Delta s}$ are the pertubation of them, the linearized inversion will output $\bold{\Delta m, \Delta s}$ . The other way of linearization is a two-stage linear least squares formulation; i.e. alternately fixing one term (m or s) and inverting for the other one. First use an initial wavelet $\bold s$ , keep $\bold s$ unchanged and invert for model m

$${\bf Sm = d},$$ (6)

and then use the updated m to invert for wavelet s

$${\bf Ms = d}.$$ (7)

Repeat this process (6) and (7) for several iterations.

As is in all non-linear inversion problems, the difficulty in these methods is to find a good starting model. Another issue is to add proper constrain on the wavelet $\bold s$ , for example, the wavelet should have constant energy during inversion, but this constrain does not fit the linear inversion framework.

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