Two-step solution for fold equalization

Using the definition of data-space pseudo inverse, Chemingui and Biondi (1997) presented a new technique to invert for reflectivity models while correcting for the effects of irregular sampling. The final reflectivity model is a two step solution where the data is equalized in a first stage with an inverse filter and an imaging operator is then applied to the equalized data to invert for a model.

We start from the definition for the data-space inverse solution

$$\bold m = \bold L^T( \bold L \bold L^T)^{-1} \bold d, \EQNLABEL{equ3}$$ (6)

then considering an irregularly sampled input of n seismic traces and letting $\bf L_{m,d_i}$ be the operator that maps trace d_i into the model space ${\bf m}$, we write the cross-product matrix $\bf L\bf L^T$ as

$$\bf L\bf L^T = \left[ \begin{array} {cccc} \left[ \bf L_{(... ...\bf L^T_{(m,d_n)}\right] \end{array} \right] \EQNLABEL{equ6}$$ (7)

Each inner product $\left[ \bf L_{(m,d_i)} \bf L^T_{(m,d_j)}\right]$ is therefore a reconstruction of a data trace with input offset h_i as a new trace with offset h_j. We recognize this mapping as an AMO transformation. We name this cross product filter A, and we write it in terms of its AMO elements as

$${\bf A}= \left[ \begin{array} {ccccc} \bf I & \bf A_{(h_1,... ...h_n,h_3)} &...... & \bf I \end{array} \right] \EQNLABEL{equ7}$$ (8)

where $\bf A_{(h_i,h_j)}$ is AMO from input offset h_i to output offset h_j and, $\bf I$ is the identity operator (mapping from h_i to h_i). Conforming to the definition of AMO Biondi et al. (1996), $\bf A_{(h_i,h_j)}$ is the adjoint of $\bf A_{(h_j,h_i)}$; therefore, the filter ${\bf A}$ is Hermitian with diagonal elements being the identity and off-diagonal elements being AMO transforms.