3-D kinematic for the focusing analysis

Depth focusing analysis is a methodology for velocity estimation based on the fact that the total wave propagation time is invariant: the total traveltime from the real reflection point obtained with the real propagation velocity is equivalent to the total traveltime obtained from the focus reflection image obtained with a wrong propagation migration velocity plus a residual time related with the error in velocity. This residual time, in a horizontal stratified media case with constant velocity V_r (see Figure 2), is equal to a vertical time ${\delta t}$ associated with a depth error ${\delta}$ a paraxial approximation MacKay and Abma (1992):  

$$\delta t = \frac{(Z_{f}-Z_{m})}{2V_{m}}.$$ (6)

 

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Figure 2 The residual time obtained when the data is migrated with a migration velocity, V_m, greater than the real propagation velocity V_r.

Figure 3 presents the migration of a dipping reflector with the media velocity and with a wrong velocity V_m. Keeping the notation for the migration velocity V_m, real velocity V_r, and the downward continued point with the migration velocity to the imaging condition m=(x_m,y_m,Z_m), the real position of the dipping reflector point is r=(x_r,y_r,z_r) and the focusing point is f=(x_f,y_f,Z_f). The total time t_T can be related to the focus time t_f by the following equation MacKay and Abma (1992):  

$$t_{T}=t_f+\delta t .$$ (7)

This time equation in 2-D is rewritten using the offset, h, and the dipping angle, $\theta$, in the law of cosines for the triangle obtained by a image source (see Figure 3):

$$t^2_{T}={(2\frac{z_{r}}{V_{r}})^2+(2\frac{h_{r}}{V_{r}})^2-8z_{f}\frac{h_{r}}{(V_{r})^2}cos(\theta_{r})}$$ (8)

 

$$t^2_{f}=(2\frac{z_{f}}{V_{m}})^2 + (2\frac{h_{f}}{V_{m}})^2 - 8z_{f}\frac{h_{f}}{(V_{m})^2} cos(\theta_{m}).$$ (9)

Taking into account the previous formulas, we can conclude that in a subsurface with dipping reflectors and lateral gradient velocity, the position of the focusing point (x_f,Z_f) is not between the real position and the migrated position, as it is in a lateral homogeneous media (i.e., constant velocity media or horizontal layered media). Therefore, in heterogeneous media the depth error panel needs to be corrected using a residual zero-offset migration to move the depth error panel energy to its correct position Audebert and Diet (1993). However, there is still error in depth. Another way to understand this mispositioning of the focusing point is to consider in the normal ray as between the focusing imaging and CMP point at the surface. This normal ray is different from the normal ray obtained with the real velocity field, thus it has a different surface incident angle.

The focusing analysis methodology for 3D data is equivalent to 2-D methodology. The depth error panel is built by extracting the zero offset trace from every downward continuation step of prestack data recorded at the surface. In this paper, the downward operator is a 3-D phase-shift in CMP domain operator.

It is important to point out, that in 3-D the focusing point has associated a depth error and lateral position error in in-line and cross line directions. Therefore, to apply equation (6) and (2), it is necessary to perform a residual migration of depth error gather, then the focusing point, the migrated point and the real point are along the same normal ray.

 

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Figure 3 The residual time obtained when the data is migrated with a migration velocity, V_m, greater than the real propagation velocity V_r.