EXTENDED STOLT MIGRATION

The normal Stolt migration algorithm for the dispersion relation of the scalar wave equation

$${\omega^2\over{v^2}}= k_x^2+k_z^2$$ (1)

is implemented using the following equation to map from the ``data'' to the ``image'' space:

$$image(x,z) = \int\int {v \vert k_z\vert\over{\sqrt{k_x^2+k_z^2}}} P(k_x, \omega(k_z)) e^{i k_x x} e^{i k_z z} dk_x dk_z$$ (2)

where P(k_x,w(k_z)) is the fourier transformation of the data recorded at the surface. This equation can easily be transformed into the time domain by using the relationship $k_\tau = v_z k_z$ between two-way traveltime $\tau$, depth z and vertical velocity v_z and can be extended into:  

$$image(x,\tau) = \int \int J(k_x,k_\tau) P(k_x,\omega(k_\tau))e^{i k_x x} e^{i \gamma(k_x,\omega) \tau} dk_x dk_\tau$$ (3)

where

$$k_\tau = \gamma(k_x,\omega)$$

is a function of k_x and $\omega$ and $J(k_x,k_\tau)$ is the new Jacobian depending now on $k_\tau$ instead of k_z. Any desired dispersion relation can be put into this new generalized equation by simply deriving the appropriate Jacobian and $k_\tau$ which can be substitited in equation (3).

The Stolt migration algorithm for the extended imaging equation (3) can then be represented as:

$$p(x,z=0,t) \rightarrow P(k_x,w) \rightarrow Q(k_x,k_\tau)=J(k_x,k_\tau) P(k_x,\omega(k_\tau)) \rightarrow q(x,\tau)$$

In order to determine the value $Q(k_x,k_\tau)$ from $P(k_x,\omega(k_\tau))$a mapping from the $\omega$ to the $k_\tau$ axis has to be performed. In calculating $Q(k_x,k_\tau)$, we require the value of $P(k_x,\omega(k_\tau))$which is calculated by a frequency-domain interpolation. In this study, an exact interpolation scheme by Rosenbaum 1981 is implemented in the migration algorithm:

$$C'(n+\delta n) = \sum_{m=0}^{N-1} C(m) e^{-\pi i[(n+\delta n) - m]} sinc[(n + \delta n) - m]$$ (4)

where N is the number of given points and the point to be interpolated lies at the point $n+\delta n$.In this interpolator the weights are the product of a sinc function and a corkscrew function $e^{-\pi i[(n+\delta n) - m]}$. Popovici et al. 1993 compare Rosenbaums technique with a slow Fourier transformation in time for an irregular range of frequencies followed by an inverse Fourier transformatio. This has been implemented on a parallel computer by Blondel and Muir 1993.