Least squares imaging condition

Changing deconvolution for convolution, a different imaging condition can be stated for each time in terms of the following fitting goal:

109#109 (45)

where 110#110 is a convolution matrix in which columns are downshifted versions of the downgoing wavefield 9#9.

The least squares solution to this problem is A damped solution is usually used to guarantee 112#112 to be invertible as in where 114#114 is a small positive number to guarantee no zeroes in 112#112 diagonal. This is equivalent to the fitting goal

115#115 (46)

where 116#116 is the identity matrix that is used here as the regularization operator. Using this regularization scheme we are adding to the denominator a constant value where it is needed and where it is not.

As it is our intention to use the previous knowledge of how the image should be, we could choose an smarter way to fill the zero values off 112#112 diagonal. We can substitute the regularization operator for one constructed with a priori information, using

117#117 (47)

where our regularization operator 8#8 could be a steering filter (). Steering filters can efficiently guide the solution toward a more geologically appealing form. This type of filter has been used with success to smooth existing reflectors and fill shadow zones in least squares inversion ().