Processing

I applied three different processing sequences to each data set in order to compare the resulting amplitudes from angle-domain common-image gathers with the amplitudes from offset-ray parameter images.

1.

Prestack wave-equation migration with output in offset domain + conversion to angle gather by Fourier transform.

2.

Prestack wave-equation migration with output in offset ray parameter domain.

3.

Iterative execution of prestack wave-equation migration and inversion with output in offset ray parameter domain (10 iterations).

Figure 3 shows the resulting CIG gathers for each processing sequence in model 1 and Figure 4 shows the resulting CIG gathers in model 2.

 

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Figure 3 CIG result of applying the 3 processing sequences to model 1. Left: Prestack wave-equation migration with output in offset domain + conversion to angle gather by Fourier transform. Middle: Prestack wave-equation migration with output in offset ray parameter domain. Right: Iterative execution of prestack wave-equation migration and inversion with output in offset ray parameter domain.

 

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Figure 4 CIG result of applying the 3 processing sequences to model 2. Left: Prestack wave-equation migration with output in offset domain + conversion to angle gather by Fourier transform. Middle: Prestack wave-equation migration with output in offset ray parameter domain. Right: Iterative execution of prestack wave-equation migration and inversion with output in offset ray parameter domain.

We can see some artifacts that are stronger in cases 2 and 3. These artifacts could affect the amplitudes values in the near and far offset. Also, for cases 2 and 3, the output is in the offset ray parameter domain which is not directly the angle of incidence; the offset ray parameter, p_hx, is related to the aperture angle $\theta$, the dip $\phi$ along the in-line direction, and the velocity function $V\left(z,{\bf m}\right)$,as follows Prucha et al. (1999):  

$$p_{hx}= \frac{2 \sin\theta \cos\phi}{V\left(z,{\bf m}\right)}$$ (1)

Therefore, for cases 2 and 3, I calculated the conversion to the angle domain using the relation from equation (1). Figure 5 shows the picked amplitudes at the flat interface in the migrated CIG gather for model 1.

 

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Figure 5 Picked amplitudes at the flat reflector for model 1: before migration (left), after migration and angle gather conversion (middle left), after migration (middle right) and conversion to angle domain, after migration/inversion and conversion to angle domain (right).

Figure 6 shows the picked amplitudes at the flat interface in the migrated CIG gather for model 2.

 

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Figure 6 Picked amplitudes at the flat reflector for model 2: before migration (left), after migration and angle gather conversion (middle left), after migration (middle right) and conversion to angle domain, after migration/inversion and conversion to angle domain (right).