Profile imaging

Let us first study the generalized image-ray method for profile imaging. Figure  shows the ray paths in a common shot experiment. The seismic energy generated by a shot at surface location x_s propagates towards a point scatterer at subsurface position (x,z). After being diffracted from the scatterer, the energy travels along the multiple ray paths connecting the scatterer to each receiver of a receiver array. On the recorded shot gather, the seismic energy is distributed in the data samples along a diffraction curve that is determined by the functional relationship between traveltime t and receiver location x_r. The diffraction curve has a hyperbolic shape when the overburden velocities of the scatterer are free of lateral variations. If we assume that the lateral velocity variations are confined to a reasonable scale, this curve usually is well fit by a hyperbola in the vicinity of its apex. Consequently, a profile time-migration can collapse the energy distributed over this portion of the diffraction curve.

 

imraycsg
Figure 1 Ray paths in a common shot experiment. The solid line shows an image ray.

If time migration is implemented by using a Kirchhoff summation, the time-image sample at (t₀,x₀) is the weighted sum of data samples along a migration curve, as follows:

 

$$t = \sqrt{t^2_0+\displaystyle{(x_s-x_0)^2 \over v^2_m}}+ \sqrt{t^2_0+\displaystyle{(x_r-x_0)^2 \over v^2_m}},$$ (3)

where v_m is a migration velocity. For a given shot location x_s, this equation defines traveltime t as a function of receiver location x_r. By taking the derivative of traveltime t in equation (3) with respect to receiver location x_r and setting the result to zero, we can find the coordinates of the apex of the migration curve for given (t₀,x₀), as follows:

 

$$\left\{ \begin{array} {lll} t_a & = & \sqrt{t^2_0+\displayst... ...x_0)^2 \over v^2_m}}+t_0 \\ \\ x_a & = & x_0.\end{array}\right.$$ (4)

Thus, the Kirchhoff time-migration focuses the diffracted energy along the diffraction curve whose apex is at (t_a,x_a) to the point (t₀,x₀) of the time-migrated image. The correct time-to-depth conversion should map the image sample at (t₀,x₀) to the true position (x,z) of the scatterer.

To find the apex position of the diffraction curve, I define image rays for profile imaging to be the propagation paths of the energy in the signals at the apex positions of diffraction curves on a common shot gather. At the apex of a diffraction curve, the derivative of the traveltime with respective to the receiver location is zero. Therefore, an image ray emerges vertically at the receiver location. Figure  shows an example of such an image ray. By tracing the image ray passing through the scatterer at subsurface position (x,z), we can find the coordinates of the apex position of the diffraction curve, as follows:

 

$$\left\{ \begin{array} {lll} t_a & = & \tau(x,z;x_s)+\tau_0(x,z) \\ \\ x_a & = & \lambda_0(x,z),\end{array}\right.$$ (5)

where $\tau(x,z;x_s)$ is the traveltime from shot location x_s to subsurface position (x,z), and $\tau_0(x,z)$ and $\lambda_0(x,z)$are the same as defined in equation (1). Combining equations (5) and (3) gives us the mapping functions of the time-to-depth conversion, as follows:

 

$$\left\{ \begin{array} {lll} t_0 =\displaystyle{[\tau(x,z;x_s... ...x_s)+\tau_0(x,z)]} \\ \\ x_0 = \lambda_0(x,z)\end{array}\right.$$ (6)

Once these two mapping functions are computed, the image-ray corrections for profile imaging is again a variable-substitution process.