# User:Zhennan/InverseScatteringSeries

The scattering theory relates the difference between the wavefield ${\displaystyle P}$ in the actual medium and the wavefield in the reference medium, to the difference between the properties of actual medium and reference medium. Wave equations that govern seismic wave propagation in actual medium and reference medium are respectively[1],

${\displaystyle LP=\delta }$

${\displaystyle L_{0}G_{0}=\delta }$

where ${\displaystyle L}$ and ${\displaystyle L_{0}}$ represent the differential operator in the actual medium and reference medium, respectively. And ${\displaystyle G}$ and ${\displaystyle G_{0}}$ are the corresponding actual Green's function and reference Green's function, respectively. ${\displaystyle \delta }$ represents an impulsive source. The perturbation operator ${\displaystyle V}$ and the scattered field operator ${\displaystyle \Psi _{s}}$ are defined as follows,

${\displaystyle V\equiv L_{0}-L}$

${\displaystyle \Psi _{s}\equiv G-G_{0}}$

The Lippmann-Schwinger equation as the fundamental equation of scattering theory relates ${\displaystyle \Psi _{s}}$, ${\displaystyle G_{0}}$, ${\displaystyle V}$ and ${\displaystyle G}$,

${\displaystyle \Psi _{s}=G-G_{0}=G_{0}VG}$

Expanding this equation in an infinite series through a substitution of higher order approximations for ${\displaystyle G}$ in the right hand side, the forward scattering series can be derived,

${\displaystyle \Psi _{s}=G-G_{0}=G_{0}VG_{0}+G_{0}VG_{0}VG_{0}+...=(\Psi _{s})_{1}+(\Psi _{s})_{2}+...}$

where ${\displaystyle (\Psi _{s})_{n}\equiv G_{0}(VG_{0})^{n}}$ is the portion of ${\displaystyle \Psi _{s}}$ that is ${\displaystyle n}$th order in ${\displaystyle V}$. The relationship above provides a Geometric forward series rather than a Taylor series. In general, a Taylor series doesn't have an inverse series, whereas a Geometric series has an inverse series. This enables inverse scattering series to provide a direct method to solve for ${\displaystyle V}$, the subsurface information, from the recorded reflection data at the earth surface ${\displaystyle D=(\Psi _{s})_{m.s.}}$. To do that, first write the perturbation ${\displaystyle V}$ as a series,

${\displaystyle V=V_{1}+V_{2}+V_{3}+...}$

where ${\displaystyle V_{i}}$ is the portion of $V$ that is ${\displaystyle i}$th order in the data ${\displaystyle D=(\Psi _{s})_{m.s.}}$.

Therefore we have,

${\displaystyle (\Psi _{s})_{m.s.}=D=(G_{0}V_{1}G_{0})_{m.s.}}$

${\displaystyle (G_{0}V_{2}G_{0})_{m.s.}=-(G_{0}V_{1}G_{0}V_{1}G_{0})_{m.s.}}$

${\displaystyle (G_{0}V_{3}G_{0})_{m.s.}=-(G_{0}V_{1}G_{0}V_{1}G_{0}V_{1}G_{0})_{m.s.}-(G_{0}V_{1}G_{0}V_{2}G_{0})_{m.s.}-(G_{0}V_{2}G_{0}V_{1}G_{0})_{m.s.}}$

${\displaystyle ......}$

Since ${\displaystyle (\Psi _{s})_{m.s.}}$ is the measured scattered wavefield and ${\displaystyle G_{0}}$ can be calculated from the reference medium, ${\displaystyle V_{1}}$ can be solved directly from the first equation, giving the linear portion of the perturbation in terms of data. With ${\displaystyle V_{1}}$, the second equation can be solved for ${\displaystyle V_{2}}$, and likewise for the higher terms ${\displaystyle V_{i}}$.

In summary, through equations above, the perturbation ${\displaystyle V=V_{1}+V_{2}+V_{3}+...}$ can be found directly using the scattered wavefield on the measurement surface and the information of reference medium, without appealing to earth model type (acoustic, elastic, anelastic, ...) and subsurface information[2].

## References

1. [1] Weglein, A.B., Araújo, F.V., Carvalho, P.M., Stolt, R.H., Matson, K.H., Coates, R.T., Corrigan, D., Foster, D.J., Shaw, S.A. and Zhang, H., 2003. Inverse scattering series and seismic exploration. Inverse problems, 19(6), p.R27.
2. [2] Arthur B. Weglein, Fernanda Araújo Gasparotto, Paulo M. Carvalho, and Robert H. Stolt (1997). ”An inverse‐scattering series method for attenuating multiples in seismic reflection data.” GEOPHYSICS, 62(6), 1975-1989. https://doi.org/10.1190/1.1444298.