Internal multiple prediction using high-order born modeling for LSRTM

Abstract

In least squares migration (LSM), multiples are usually a type of noise. Although they contain information about underground structures, they also cause artifacts in imaging. Therefore, multiple attenuation is an important way to reduce these artifacts in LSM images. Reweighted least squares reverse time migration (RWLSRTM) can use the weighting matrix and the predicted multiples to eliminate artifacts. Because the LSM provides a high resolution model, we can predict the internal multiples by using high-order Born modeling. The method is based on the inverse scattering series (ISS), and the difference is that it forwards the modeling of the internal multiples in the time domain; the model is constructed by the RWLSRTM. Because this method does not require performing as many Fourier transforms as the ISS method, it requires less calculation. We have applied the predicted multiples in the RWLSRTM to remove the artifacts caused by the multiples. The RWLSRTM image can also serve as a parameter of multiple predictions and can make the results of multiple predictions more accurate. The results of numerical tests using synthetic data show that this method can remove artifacts of internal multiples well. A comparison with the ISS method shows that our method can reduce the calculation.

Appendix

In the ISS method, the third term of Eq. 17 is processed in the frequency domain to satisfy the condition $z_{1} < z_{2} ,z_{3} < z_{2}$. Assuming that the actual medium varies only in depth, the internal multiple attenuation in a 1D Earth can be expressed as:

$$b_{1} (k) = D(w)$$

$$b_{3} (k) = \int_{ - \infty }^{\infty } {dz_{1} e^{{{\text{ikz}}_{1} }} b_{1} (z_{1} )} \int_{ - \infty }^{{z_{1} - \varepsilon_{2} }} {dz_{2} } e^{{ - {\text{ikz}}_{2} }} b_{1} (z_{2} )\int_{{z_{2} + \varepsilon_{1} }}^{\infty } {dz_{3} e^{{{\text{ikz}}_{3} }} b_{1} (z_{3} )}$$

$$b_{3} (t) = {\text{FFT}}^{ - 1} [b_{3} (k)],$$

where $k = 2w/c_{0}$ is the vertical wavenumber and $\varepsilon$ is the parameter that is small and positive to assure $z_{1} < z_{2} ,z_{3} < z_{2}$. This equation shows that the first-order multiple is computed by combining three sets of data using convolution and cross-correlation in the wavenumber domain.