Method

The modified RTM algorithm that emphasizes imaging of multiple scattered events is described as follows.

1. Compute the source-side $g({\bf {s}},t\vert{\bf {x}},0)$ and the receiver-side $g({\bf {r}},t\vert{\bf {x}},0)$ bandlimited Green's function by a numerical solution to the wave equation, for a source at trial image point ${\bf {x}}$ and the recording position at source position ${\bf {s}}$ and receiver position ${\bf {r}}$ , respectively. This assumes an accurate velocity model.

2. Wavelet transform (Luo and Schuster, 1992) the Green's functions $g({\bf {s}},t\vert{\bf {x}},0)$ and $g({\bf {r}},t\vert{\bf {x}},0)$ to get $\mathcal{W}[g({\bf {s}},t\vert{\bf {x}},0)]$ and $\mathcal{W}[g({\bf {r}},t\vert{\bf {x}},0)]$ , respectively. Then mute all zero values below a given threshold in the wavelet domain and only store those nonzero coefficients to get the compressed Green's functions $\mathcal{W}[\tilde{g}({\bf {s}},t\vert{\bf {x}},0)]$ and $\mathcal{W}[\tilde{g}({\bf {r}},t\vert{\bf {x}},0)]$ . Save these compressed Green's functions on the disk.

3. After reading these compressed Green's functions from disk, an inverse wavelet transform is performed to reconstruct the Green's functions $\tilde{g}({\bf {s}},t\vert{\bf {x}},0)$ and $\tilde{g}({\bf {r}},t\vert{\bf {x}},0)$ by decompressing $\mathcal{W}[\tilde{g}({\bf {s}},t\vert{\bf {x}},0)]$ and $\mathcal{W}[\tilde{g}({\bf {r}},t\vert{\bf {x}},0)]$ . This is followed by a convolution step to get the compressed migration kernel:

$$\tilde{\mathcal G}({\bf {r}},{\bf {s}},{\bf {x}},t) = \tilde{g}({\bf {s}},t\vert{\bf {x}},0) \ast \tilde{g}({\bf {r}},t\vert{\bf {x}},0).$$ (12)

Here, tilde denotes the function after lossy compression and decompression.

4. The compressed migration kernel $\tilde{\mathcal G}({\bf {r}},{\bf {s}},{\bf {x}},t)$ is a Kirchhoff-like kernel that describes, for a single shot gather, pseudo-hyperbolas of multiarrivals in ${\bf {r}}$ -$t$ space. The reflection energy in a recorded shot gather $d({\bf {r}}, t\vert{\bf {s}},0)$ is then summed along such pseudo-hyperbolas to give the migration image,

$$m_{mig}({\bf {x}}) = \sum_{s}\sum_{r}\sum_{t}\tilde{\mathcal G}({\bf {r}},{\bf {s}},{\bf {x}},t)~d({\bf {r}},t\vert{\bf {s}},0).$$ (13)

This summation is equivalent to a dot-product between the recorded shot gathers and the compressed migration kernel. It is just like Kirchhoff migration except there is no high-frequency approximation and multiarrivals are included in the imaging (Schuster, 2002); this method will be denoted as generalized diffraction-stack migration (GDM).