Depth focusing equations

The focusing principles can be understood if we think of a point diffractor. The seismic response of a point diffractor is a hyperboloid in 3-D or a hyperbola in 2-D (Z=0). If this CMP gather is downward continuated with a migration velocity different than the real propagation velocity, the image obtained at the imaging condition (t=0s) will not be well focused. A good focused image will be obtained at a depth (Z_f), called focal depth and the energy at zero-offset trace will be maximum. Therefore, the goal of depth focusing analysis is to estimate the real propagation velocity from the best focal depth.

In order to estimate the new migration velocity we use the following equations assuming small offsets and a small error in migration velocity (Doherty and Claerbout (1974); MacKay and Abma (1992)):

 

V_rZ_r = V_mZ_f, (1)

 

$$2\frac{Z_{r}}{V_{r}} = 2\frac{Z_{m}}{V_{m}},$$ (2)

where Z_f is the focusing point depth obtained with a migration velocity V_m and Z_r is the real depth. Expressing the estimated real velocity and depth as a function of the vertical depth error ${\delta}$, we obtain the useful following equations:  

$$V_{r}=\frac{V_{m}(2\delta+Z_{m})}{Z_{m}+\delta},$$ (3)

 

$$Z_{r}=\delta+Z_{m}.$$ (4)

The vertical depth error (${\delta}$)is defined by MacKay and Abma (1992) as  

$$\delta = \frac{Z_{f}-Z_{m}}{2},$$ (5)

where Z_f is estimated from the error depth gather defined by the zero-offset trace chosen at every downward continued operator step (a negative error implies a high downward-continuation velocity; a positive depth error implies a low downward continuation velocity).

The new estimated velocity V_r is a rms velocity that needs to be interpolated and converted to interval velocity. In general, it is necessary to use a different approach as a tomographic, in order to transform the depth error to interval velocity Audebert (1996).