Geometric interpretation of 3-D ADCIG$\left(p_{x_h},p_{y_h}\right)$

 

cig-simple-v3
Figure 2 A schematic of the geometry of an ADCIG gather in 2-D. Depending on the context, the angles can be either the angles formed by the propagation direction of the rays, or those formed by the propagation direction of the associated plane waves. The arrows indicate positive angles; that is, $\beta_s$, $\beta_r$, and $\alpha_x$are negative and $\gamma$ is positive. This sign convention is consistent with upward propagating rays (plane waves).

Regardless of whether they have been computed using source-receiver migration or shot-profile migration, ADCIGs computed before imaging have the same geometrical interpretation. Strictly speaking, they are functions of the offset ray parameters (p_xh,p_yh) and not of the reflection opening angle $\gamma$and the reflection azimuth $\phi$.However, the offset ray parameters (p_xh,p_yh) are easily related to $\gamma$ and $\phi$ by simple trigonometric relationships. We now derive these relationships, starting from the simpler 2-D case. The understanding gained from the 2-D analysis will help the analysis of the more complex 3-D case.

The schematic in Figure 2 defines the angles relevant to 2-D ADCIGs: the reflection opening angle $\gamma$ and the geological dip angle $\alpha_x$,which is defined as the angle formed by the vertical direction and the normal to the apparent geological dip ${\bf n}$,both oriented in the upward direction. The box in Figure 2 signifies that for the purpose of our discussion the geometric relationships are local around the imaging point $\bf{\bar{I}}$.

The opening angle and the dip angle are related to the source and receiver rays (plane waves) propagation angles $\left(\beta_s,\beta_r\right)$by the following simple relationships:  

$$\gamma=\frac{\beta_r-\beta_s}{2}, \;\;\;\;\;\ {\rm and} \;\;\;\;\;\ \alpha_x=\frac{\beta_s+\beta_r}{2}.$$ (6)

The source ray parameter p_xs and the receiver ray parameter p_xg are related to the midpoint ray parameters p_xm and offset ray parameter p_xh by the following relationships:  

$$p_{x_m}= p_{x_g}+p_{x_s}, \;\;\;\;\;\ {\rm and} \;\;\;\;\;\ p_{x_h}= p_{x_g}-p_{x_h}.$$ (7)

In 2-D the source and receiver ray parameters (p_xs, p_xg) are functions of the source and receiver ray propagation angles $\left(\beta_s,\beta_r\right)$and of the velocity at the reflector $v\left(z,x\right)$as follows:  

$$p_{x_s}=\frac{\sin\beta_s}{v\left(z,x\right)} \;\;\;\;\;\ {\... ...nd} \;\;\;\;\;\ p_{x_g}=\frac{\sin\beta_r}{v\left(z,x\right)}.$$ (8)

Substituting the relationships in (8) into equation (7), then using the angular relationship expressed in equation (6) and applying fundamental trigonometric identities, we obtain the desired relationship:  

$$p_{x_h}= \frac{\sin\beta_r-\sin\beta_s}{v\left(z,x\right)}=... ...\right)}= \frac{2 \sin\gamma \cos\alpha_x }{v\left(z,x\right)},$$ (9)

which directly links the offset ray parameter p_xh to the reflection opening angle $\gamma$.

The relationship in equation (9) is also valid in 3-D when the reflection azimuth $\phi$is equal to zero. But in the general 3-D case we need to add another relationship linking the cross-line offset ray parameter p_yh to p_xh and the angles $\gamma$ and $\phi$ at the reflection point. To derive this relationship, we impose the constraint that the source and the receiver rays must become coplanar before meeting at the reflection point. We will thus refer to this relationship as the coplanarity condition.

The coplanarity of the two rays also simplifies the 3-D geometry of ADCIGs and enables us to generalize the 2-D geometric interpretation in a straightforward manner. The schematic in Figure 3 is the 3-D generalization of the 2-D schematic shown in Figure 2. The plane that is shared by the two rays in 3-D corresponds to the vertical plane in 2-D. As for Figure 2, all the geometric relationships represented in Figure 3 are local in a neighborhood of the image point $\bf{\bar{I}}$.This locality implies that the source and receiver rays (plane waves) need to satisfy the coplanarity condition only at the image point, not during propagation in the overburden.

 

cig-3d-v4
Figure 3 A schematic of the geometry of an ADCIG gather in 3-D. The geometry is analogous to the 2-D case illustrated in Figure 2. In contrast with the schematic of Figure 2, the plane of coplanarity is not vertical but it is tilted by $\alpha_y{'}$and rotated by $\phi$ with respect to the horizontal coordinates. Notice also that the angles $\delta_{s'}$ and $\delta_{r'}$ formed by the rays (plane waves) propagation directions with the ``true'' vertical axis are different from the angles $\beta_{s'}$ and $\beta_{r'}$formed by the rays with the tilted vertical axis.

We define the reflection azimuth angle $\phi$ as the angle formed by the in-line axis x with the line defined by the intersection of the plane of coplanarity with a constant depth plane. After rotation by $\phi$,the horizontal coordinates x and y become x' and y', respectively. Once rotated by $\phi$,the plane of coplanarity is tilted with respect to the vertical by the cross-line dip angle $\alpha_y{'}$.All the angles in Figure 2 that are formed between the ray (plane wave) directions and the true vertical axis z ($\beta_{s'}$, $\beta_{r'}$, and $\alpha_x{'}$)are now relative to the tilted vertical axis z'; these angles have the same meaning on the tilted plane as they have on the vertical plane in 2-D. Furthermore, the horizontal ray parameters $({\bf {p_{s'}}},{\bf {p_{g'}}})$ are not affected by the tilt, since they depend on the angles formed by the rays with the x axis. In contrast, the angles $\beta_{s'}$, $\beta_{r'}$, $\alpha_x{'}$, and $\alpha_y{'}$are affected by the rotation of the horizontal axes by $\phi$;we include the prime in the notation to indicate the angles after rotation.

Imposing the coplanarity of the source and receiver rays along a plane rotated by $\phi$ with respect to the original coordinate system provides us with the needed relationship that constraints the cross-line offset ray parameter p_y'h. For the azimuth of the plane of coplanarity to be $\phi$,the ``rotated'' source and receiver ray-parameters p_x's, p_y's, p_x'g, p_y'g must be related by the following expression:  

$$\left(p_{y{'}_g}- p_{y{'}_s}\right) = \left(p_{y{'}_g}+ p_{y... ...{'}_g}^2}+ \sqrt{ \frac{1}{v^2({{\bf s'},z})} - p_{x{'}_s}^2}}.$$ (10)

This relationship was derived by Biondi and Palacharla (1996) to define common-azimuth migration, and it is demonstrated in the Appendix of their paper.

So far we have interpreted the coplanarity condition in terms of source and receiver rays, but it has a similar interpretation in terms of plane waves. The phase vectors of two arbitrary plane waves are coplanar by construction, since they meet at the origin. The tilt angle $\alpha_y{'}$and the azimuth $\phi$ of the plane of coplanarity for the incident and the reflected plane waves are implicitly determined by the coplanarity condition expressed in equation (10). The cross-correlation of the two plane waves does not define an imaging point but an imaging line that is orthogonal to the plane of coplanarity. The tilted plane shown in Figure 3 is an arbitrary plane parallel to the plane of coplanarity and the imaging point $\bf{\bar{I}}$is the intersection of the imaging line with this plane. The source and receiver rays are defined as the lines parallel to the phase vectors of the corresponding plane waves and passing through the imaging point $\bf{\bar{I}}$.

Using the relationships among the in-line ray parameters expressed in equations (7) (and their analogues along the cross-line direction y), and equations (8), and after applying trigonometric identities, we can rewrite equation (10) as a function of the local velocity $v\left(z{'},x{'},y{'}\right)$and the angles $\alpha_x{'}$,$\alpha_x{'}$,$\gamma$ and $\phi$ as follows:  

$$p_{y{'}_h}= \frac {\sin \alpha_y{'}\left(\cos \beta_r- \cos ... ...{'}\tan \gamma\tan \alpha_x{'}} {v\left(z{'},x{'},y{'}\right)}.$$ (11)

The in-line offset ray parameter from equation (9) can be rotated by the azimuth $\phi$to give the following expression:  

$$p_{x{'}_h}= \frac{2 \sin\gamma \cos\alpha_x{'} }{v\left(z{'},x{'},y{'}\right)},$$ (12)

which together with equation (11), can be used to map the image from the offset ray parameter domain $\left(p_{x_h},p_{y_h}\right)$into the angle domain $\left(\gamma,\phi\right)$.

This mapping of the ray parameters into angles depends on both the local velocity and the geological dips. When the spatial velocity variations are significant and the geological structure is complex, unraveling this dependence might be a challenge because of the well known difficulties in reliably measuring rapidly changing local dips. For some applications this dependency does not cause difficulties. For example, in a ray-based MVA Clapp and Biondi (2000), there is no need to perform the mapping from ray parameters to angles, since ray parameters are adequate initial conditions for ray tracing. For other applications, the reflection angles are needed, and thus the mapping must be performed. When this mapping is difficult, it might be convenient to use the ADCIGs computed after imaging, which are presented in the next section. This kind of ADCIG circumvents the need to estimate local dips by transforming the image directly into angles.