Migration Algorithm

The migration algorithm used for this project uses a variation of the split-step Fourier domain migration method, developed by Stoffa Stoffa et al. (1990). Simple Fourier domain migration based on the single square root operator (SSR) is also known as phase-shift migration Gazdag (1978). In phase-shift migration, a wavefield at the surface is downward-continued, assuming the wavefield is generated by the subsurface reflectors exploding Claerbout and Black (1997). The single square root operator marches the wavefield down, by shifting in the $\omega-k_x$ domain:

$$e^{ik_z \Delta z} \; = \; e^{-i \frac{2\omega}{v} \sqrt{1- \frac{v^2 k_x^2}{4\omega ^2}} \Delta z} \;$$ (2)

The imaging principal sums over all $\omega$, giving the wavefield at t=0, which is the migrated image Yilmaz (1987). The split-step methods developed by Stoffa use an approximation of the SSR, where v is replaced by an average velocity of a given depth step, v_ref. A new term is added to correct for the difference of the average velocity and the actual medium velocity v_m. The equation looks like this variation of equation (2) (k_z is shown):

$$k_z \; = \; \sqrt{\frac{4\omega ^2}{v_{ref} ^2} - k_x ^2} + (\frac{2\omega}{v_m}-\frac{2\omega}{v_{ref}}) \;$$ (3)

In equation (3), the first term is calculated in the $\omega-k_x$ domain, the second in the $\omega-x$ domain. So at every depth step, a 2-D FFT transforms the wavefield into $\omega-k_x$ space. The wavefield is downward-propagated using the phase-shift operator and v=v_ref. A 1-D IFFT transforms into $\omega-x$ space, where the correction term is applied. Once again, summing over all frequency gives the image. This method is exact if v_ref=v_m, or if layers are flat. If lateral velocity variations are significant or units are dipping steeply, a better approximation of the SSR equation is needed. To handle these conditions, we use the double square root equation (DSR). The DSR equation is a higher order approximation of the SSR equation. The DSR equation as displayed in IEI Claerbout (1984) is

$$\frac{dU}{dz} \; = \; -\frac{i\omega}{v}[\sqrt{1-(Y+H)}+\sqrt{1-(Y-H)}] U \;$$ (4)

where,

$$Y = \frac{v^2{K_y}^2}{4\omega^2}$$ (5)

and

$$H = \frac{v^2{K_h}^2}{4\omega^2}$$ (6)

The DSR equation inherently includes both migration and NMO. Therefore, with input CMPs and a velocity function, a good image is created. The actual migration program used in this project is one developed by Biondo Biondi that uses a variation of the split-step method with the DSR equation Biondi (1998). The DSR equation can better handle lateral velocity variations as well as steeper dips (as compared to the SSR equation). The use of the split-step method with the DSR makes the algorithm more effective. In this way, multiple reference velocities can be used, and sharp lateral velocity variations, such as salt in contact with sediment, can be handled. For this project, three reference velocities were used. Biondi's program also has an input salt velocity, so that the reference velocities in the sediment will not be anomalously high if the salt is at the same depth step. So for each depth step, three wavefields were generated, then an interpolation in the space domain was used as needed.

 

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Figure 5 Zero-Offset Section

 

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Figure 6 Prestack Migrated Section Using P-Wave Velocity Function