Least-squares diffraction imaging using shaping regularization by anisotropic smoothing

Synthetic Data Example

We test the approach on a synthetic data example with a reflectivity model is shown in Figure 3. The synthetic seismic data are generated by Kirchhoff modeling method with $15$   Hz peak-frequency Ricker wavelet, reflectivity distribution shown in Figure 3 and $2.0$   km/s constant velocity model, which is further utilized in both migration and inversion. Zero-offset geometry with $1$   m sampling in both inline and crossline directions is used. The synthetic data are shown in Figure 4a. Random noise with a maximum amplitude of 30% of that of the signal, filtered to the signal frequency band is added to the synthetic. After 3D Kirchhoff migration diffractions become focused and a channel-like structure with a zigzag pattern becomes prominent (Figure 4b). However, channel edges appear to be somewhat blurred and masked by the reflections. We follow the workflow described above. Data to be fit by the inversion is shown in Figure 5a. We use the following parameters for the inversion: $\lambda=100$ , $\epsilon=10$ , $N=5$ and $K=1$ . In total we use $20$ iterations - 5 inner by 4 outer. The inversion result is shown in Figure 5b. Edges are highlighted and denoised.

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Figure 3. Reflectivity model for the synthetic data example. Reflectivity corresponds to density contrasts while the velocity is kept constant at $2.0$   km/s throughout the volume.

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Figure 4. (a) Zigzag zero-offset synthetic; (b) 3D Kirchhoff time migration of the zigzag zero-offset synthetic (Figure 4a). While focusing of both reflections and edge diffractions can be observed in (b), channel edges are masked by specular energy and appear to be blurred.

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Figure 5. (a) AzPWD and path-summation integral migration applied to the zigzag zero-offset synthetic (Figure 4a) : A high diffraction probability along the channel edges can be observed; (b) Inversion result: edge diffractions are highlighted and denoised.

Least-squares diffraction imaging using shaping regularization by anisotropic smoothing