Angle-domain CIGs

Most current migration velocity analysis methods Clapp (2001); Sava and Biondi (2003) are based on the curvature information from Angle-Domain CIGs(ADCIGs), which are created from the migration cube Sava and Fomel (2003). When the migration velocity is the true velocity, the ADCIG at a reflection point is a flat line. When the migration velocity is inaccurate, the curvature parameters estimated from ADCIGs can be back-projected and inverted for velocity updates. Biondi and Symes (2003) demonstrates that in an ADCIG cube, the image point lies on the line normal to the apparent reflector dip, passing through the point where the source ray intersects the receiver ray. Figure shows the ADCIGs at reflection points of different dip angles. Under the stationary-raypath assumption, the shift of the image point along the normal direction is Biondi and Symes (2003)  

$$\Delta {\bf n}=\frac{1-\rho}{1-\rho(1-\cos\alpha)} \frac{\sin^2\gamma}{\cos^2\alpha-\sin^2\gamma}z_0 {\bf n},$$ (1)

where $\rho$ is the constant scaling factor of the slowness, $\alpha$ is dip angle of the reflector, $\gamma$ is the opening angle, ${\bf n}$ is the normal direction vector of the reflection point, and z₀ is the depth at the reflection point. For flat reflectors, the shift (1) reduces to  

$$\Delta {\bf n}=(1-\rho)\tan^2\gamma z_0 {\bf n}.$$ (2)

Equations (1) and (2) can be used to estimate the curvature parameters for velocity analysis. However, to estimate the curvature parameters caused by local velocity perturbation, we don't consider the effect of depth z₀.

 

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Figure 1 ADCIGs with different dip angles. ${\bf n}$ is the normal direction vector, and $\gamma$ is the opening angle.