Angle-domain illumination gathers by wave-equation-based methods

Scattering-angle-domain illumination

In a locally constant velocity medium (Figure 1), the midpoint ray parameter ${\bf p}_m$ and the subsurface offset ray parameter ${\bf p}_h$ can be expressed as follows:

$${\bf p}_m = {\bf p}_s+{\bf p}_r$$ (7)

$${\bf p}_h = {\bf p}_r-{\bf p}_s,$$ (8)

where ${\bf p}_s$ and ${\bf p}_r$ are the source and receiver ray parameters, respectively. Using trigonometric relations, we can further express the horizontal and vertical components of ${\bf p}_m$ and ${\bf p}_h$ as functions of the scattering angle $\gamma$ that bisects the incident and the scattered rays (plane waves) and the corresponding dip angle $\alpha$ as follows:

$${\bf p}_m = \left( \begin{array}{c} p_{m_x} \\ p_{m_z} \end{arra... ...\cos{\gamma}\sin{\alpha} \\ -2 s \cos{\gamma}\cos{\alpha} \end{array} \right),$$ (9)

and

$${\bf p}_h = \left( \begin{array}{c} p_{h_x} \\ p_{h_z} \end{arra... ... \sin{\gamma}\cos{\alpha} \\ 2 s \sin{\gamma}\sin{\alpha} \end{array} \right),$$ (10)

where $s$ is the slowness at the reflection point. Dividing $p_{h_x}$ by $p_{m_z}$ yields

$$\tan{\gamma} = -\frac{p_{h_x}}{p_{m_z}} = -\frac{k_{h_x}}{k_{m_z}},$$ (11)

where $k_{h_x}$ and $k_{m_z}$ are vertical-subsurface-offset wavenumber and depth wavenumber, respectively. Equation 11 converts the sensitivity kernel from the subsurface-offset-domain into the scattering-angle domain. The action of the angle-domain sensitivity kernel to the data residual $r({\bf x}_r,{\bf x}_s,\omega)$ gives the angle-domain reflectivity image. Since the Fourier-domain mapping is a linear operator and is independent from the data residual, the sources, receivers and frequencies, it can be postponed after the subsurface-offset-domain reflectivity image is obtained. Therefore, to obtain the angle-domain reflectivity image more efficiently, we perform the mapping on the image $I_{h}$ after stacking over sources, receivers and frequencies, instead of the sensitivity kernel $L_{h}$ for each source, receiver and frequency (Sava and Fomel, 2003).

local-phase-relation Figure 1. Geometric relations between ray vectors at a reflection point in a locally constant velocity medium. [NR]

We obtain the scattering-angle-domain Hessian by correlating the sensitivity kernels $L_{\gamma}$ as follows:

$$H_{\gamma}({\bf x},{\bf x}',\gamma,\gamma') = \sum_{\omega}\sum_{... ...}_r,{\bf x}_s,\omega) L_{\gamma} ({\bf x}',\gamma',{\bf x}_r,{\bf x}_s,\omega).$$ (12)

The diagonal of equation 12, or the scattering-angle-domain illumination is obtained when ${\bf x}={\bf x}'$ and $\gamma = \gamma'$

$$H_{\gamma}({\bf x},\gamma) = \sum_{\omega}\sum_{{\bf x}_s}\sum_{{... ...r,{\bf x}_s)\vert L_{\gamma}({\bf x},\gamma,{\bf x}_r,{\bf x}_s,\omega)\vert^2.$$ (13)

Contrary to the case of computing angle-domain reflectivity image, postponing the Fourier-domain mapping after stack becomes less obvious for the angle-domain Hessian computation due to the correlation term inside the summation loop in equations 12 and 13. In the following numerical examples, we take a more straight forward way that directly transforms the sensitivity kernel from the subsurface-offset domain to the scattering-angle-domain to compute the angle-domain Hessian or illumination. The steps can be summarized as follows:

• for fixed $x$ , ${\bf x}_s$ , ${\bf x}_r$ and $\omega$ , apply 2-D Fourier transform along axes $z$ and $h_x$
  

  $$L_{h}(x,z,h_x,{\bf x}_r,{\bf x}_s,\omega) \rightarrow L_{h}(x,k_z,k_{h_x},{\bf x}_r,{\bf x}_s,\omega);$$

  
  
• perform the mapping
  

  $$L_{h}(x,k_z,k_{h_x},{\bf x}_r,{\bf x}_s,\omega) \rightarrow L_{\gamma}(x,k_z,\gamma,{\bf x}_r,{\bf x}_s,\omega)$$

  
  according to relation 11;
• apply inverse 1-D Fourier transform along axis $k_z$
  

  $$L_{\gamma}(x,k_z,\gamma,{\bf x}_r,{\bf x}_s,\omega) \rightarrow L_{\gamma}(x,z,\gamma,{\bf x}_r,{\bf x}_s,\omega)$$

  
  to obtain the scattering-angle-domain sensitivity kernel.
• compute either the scattering-angle-domain Hessian using equation 12 or the illumination using equation 13.

As a simple example, Figures 3 shows the real part of the scattering-angle-domain sensitivity kernel, converted from its subsurface-offset-domain counterpart (Figure 2). The results are obtained by using a constant velocity model ($2000$ m/s), and only one source ($-600$ m), one receiver ($600$ m) and one frequency ($19$ Hz) are computed. Figure 4 shows the corresponding single-frequency scatter-angle-domain illumination for the given velocity model and acquisition configuration. Since there are only one source and one receiver, each subsurface point is illuminated by only one scattering angle.

The scattering-angle-domain illumination is useful for point scatterers, it, however, fails to accurately predict the illumination strength for planar reflectors, where the scattered waves have preferred orientations according to the local dips of the reflectors. The reason behind this is that the transformation (equation 11) is dip-independent, the resulting angle-domain illumination implicitly averages over all dip angles and measures the overall scattering-angle illumination from all dips illuminated, instead of the illumination from one particular dip of the actual planar reflector. This point is further illustrated by Figures 5 and 6, where the computed scattering-angle-domain illumination (Figure 6(d)) accurately predicts the illumination for the point scatterer (Figures 5(c) and 6(c)), instead of the horizontal (Figures 5(a) and 6(a)) and the $-30^{\circ}$ dipping reflectors (Figures 5(b) and 6(b)).

const-kernel-odcig-1s1r-1w
Figure 2. The real part of the subsurface-offset-domain sensitivity kernel for a constant velocity model ($2000$ m/s), one source at $-600$ m, one receiver at $600$ m and one frequency ($19$ Hz). Panel (a) shows the kernel at zero-subsurface-offset; (b), (c) and (d) show the kernel for different spatial locations at $-1000$ m, 0 m and $1000$ m, respectively. [CR]

const-kernel-adcig-1s1r-1w
Figure 3. The real part of the scattering-angle-domain sensitivity kernel after conversion from the subsurface offset domain (Figure 2). Panel (a) shows the sensitivity kernel for a constant scattering angle $18.75^{\circ }$ ; (b), (c) and (d) show the kernel for spatial locations at $-1000$ m, 0 m and $1000$ m, respectively. [CR]

const-hess-adcig-1s1r-1w
Figure 4. The single-frequency scattering-angle-domain illumination obtained using the sensitivity kernel shown in Figure 3. Panel (a) shows the illumination for scattering angle $18.75^{\circ }$ ; (b), (c) and (d) show the illumination angle gathers for spatial locations at $-1000$ m, 0 m and $1000$ m, respectively. [CR]

const-imag-stack
Figure 5. Migrated zero-subsurface-offset images (stacked images) for (a) a horizontal reflector, (b) a dipping reflector ($-30^\circ$ ) and (c) a point scatterer. All images are obtained by migrating only one shot located at $-600$ m, where $401$ receivers spread from $-2000$ m to $2000$ m with a $10$ m spacing. [CR]

const-imag-adcig Figure 6. The scattering-angle-domain image gathers extracted at spatial location 0 m for (a) the horizontal reflector, (b) the dipping reflector and (c) the point scatterer. Panel (d) shows the scattering-angle-domain illumination gather extracted at the same spatial location. [CR]

Angle-domain illumination gathers by wave-equation-based methods