Compression of the Migration Kernel

A wavelet transform with compression (Zhan and Schuster, 2010) can be used to reduce storage costs associated with the the GDM kernel, as follows.

1. Compute the source-side $g({\bf {x}},t\vert{\bf {s}},0)$ and the receiver-side $g({\bf {x}},t\vert{\bf {r}},0)$ bandlimited Green's functions by a numerical solution to the wave equation.
2. A wavelet transform (Luo and Schuster, 1992) is applied to the Green's function $g({\bf {x}},t\vert{\bf {s}},0)$ and $g({\bf {x}},t\vert{\bf {r}},0)$ to get ${\mathcal W}[g({\bf {x}},t\vert{\bf {s}},0)]$ and ${\mathcal W}[g({\bf {x}},t\vert{\bf {r}},0)]$ , respectively. Here, ${\mathcal W}$ represents the wavelet transform with up to an order-of-magnitude reduction in storage requirements (Luo and Schuster, 1992). Then, mute all wavelet coefficients below a given threshold in the wavelet domain and only store the significant coefficients to get the compressed Green's functions ${\mathcal W}[\tilde{g}({\bf {x}},t\vert{\bf {s}},0)]$ and ${\mathcal W}[\tilde{g}({\bf {x}},t\vert{\bf {r}},0)]$ . 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 $\tilde{g}({\bf {x}},t\vert{\bf {s}},0)$ and $\tilde{g}({\bf {x}},t\vert{\bf {r}},0)$ by decompressing ${\mathcal W}[\tilde{g}({\bf {x}},t\vert{\bf {s}},0)]$ and ${\mathcal W}[\tilde{g}({\bf {x}},t\vert{\bf {r}},0)]$ . This is followed by a convolution step using a FFT to get the compressed migration kernel $\tilde{\mathcal G}({\bf {r}},{\bf {s}},{\bf {x}},t)$ in equation :
   

   $$\tilde{\mathcal G}({\bf {r}},{\bf {s}},{\bf {x}},t) = \tilde{g}({\bf {x}},t\vert{\bf {s}},0) \ast \tilde{g}({\bf {x}},t\vert{\bf {r}},0),$$ (25)

   
   where $\tilde{g}({\bf {x}},t\vert{\bf {r}},0) = {\mathcal W}^{-1} [\tilde g({\bf {x}},t\vert{\bf {r}},0)]$ . Here, tilde denotes the function after lossy compression and decompression.
4. The compressed migration operator $\tilde{\mathcal G}({\bf {r}},{\bf {s}},{\bf {x}},t)$ is a Kirchhoff-like kernel that describes, for a single shot gather, pseudo-hyperbolas of multiarrivals in ${\bf {r}}$ -$t$ coordinates. 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,
   

   $$m_{mig}({\bf {x}}) = \sum_{s}\sum_{r}\sum_{t}\tilde{\mathcal G}({\bf {r}},{\bf {s}},{\bf {x}},t) d({\bf {r}},t\vert{\bf {s}},0).$$ (26)

   
   This summation is equivalent to a dot product between the recorded shot gathers and the compressed GDM kernels, and has the advantage over Kirchhoff migration in that there is no high-frequency approximation and multiarrivals are included in the imaging.