Ottolini's radial trace dip moveout

Ordinarily we regard a common-midpoint gather as a collection of seismic traces, that is, a collection of time functions, each one for some particular offset h. But this (h,t) data space could be represented in a different coordinate system. A system with some nice attributes is the radial-trace system introduced by Turhan Taner. In this system the traces are not taken at constant h, but at constant angle. The idea is illustrated in Figure 17.

 

otto Figure 17 Inside the data volume of a reflection seismic traverse are planes called radial-trace sections. A point scatterer inside the earth puts a hyperbola on a radial-trace section.

Besides having some theoretical advantages, which will become apparent, this system also has some practical advantages, notably: (1) the traces neatly fill the space where data is nonzero; (2) the traces are close together at early times where wavelengths are short, and wider apart where wavelengths are long; and (3) the energy on a given trace tends to represent wave propagation at a fixed angle. The last characteristic is especially important with multiple reflections. But for our purposes the best attribute of radial traces is still another one.

Richard Ottolini noticed that a point scatterer in the earth appears on a radial-trace section as an exact hyperbola, not a flat-topped hyperboloid. The travel-time curve for a point scatterer, Cheops' pyramid, can be written as a ``string length'' equation, or a stretched-circle equation. Making the definition  

$$\sin \, \psi \eq { 2\,h \over v\,t}$$ (13)

and substituting into equation () yields  

$$v\,t \eq 2\ \sqrt{ { z^2 \over \cos^2 \psi } \ +\ (y\ -\ y_0 )^2 }$$ (14)

Scaling the z-axis by $\cos \, \psi$ gives the circle and hyperbola case all over again! Figure 18 shows a three-dimensional sketch of the hidden hyperbola.

 

ottohyp
Figure 18 An unexpected hyperbola in Cheops' pyramid is the diffraction hyperbola on a radial-trace section. (Harlan)

We will see that the radial hyperbola of Figure 18 is easier to handle than the flat-topped hyperboloid that is seen at constant h. Refer to the equations in Table 7.2.

 

$$\omega \rightarrow k_z k_y^2 + k_z^2\cos^2\psi = 4\omega^2/v^2$$ migration

$$k_z\rightarrow\omega_0 k_y^2 + k_z^2 = 4\omega_0^2/v^2$$ zero-offset diff.

$$\omega\rightarrow\omega_0 .25v^2 k_y^2\sin^2\psi + \omega_0^2\cos^2\psi = \omega^2$$ DMO+NMO

$$\omega\rightarrow\omega_s .25v^2 k_y^2\sin^2\psi + \omega_s^2 = \omega^2$$ radial DMO

$$\omega_s\rightarrow\omega_0 \omega_0\cos\psi = \omega_s$$ radial NMO

The second equation in Table 7.2 is the usual exploding reflector equation for zero-offset migration. It may also be obtained from the DSR by setting $H\,=\,0$.As written it contains the earth velocity, not the half velocity. Equation (14) says that hyperbolas of differing $\psi$ values are related to one another by scaling the z-axis by $\cos \, \psi$.According to Fourier transform theory, scaling z by a $\cos \, \psi$ divisor will scale k_z by a $\cos \, \psi$ multiplier. This means the first equation in Table 7.2 can be used for migrating and diffracting hyperbolas on radial-trace sections. Eliminating k_z from the first and second equations yields the middle equation $\omega \to \omega_0$ in Table 7.2. This middle equation combines the operation of migrating all offsets (really any radial angle) and then diffracting out to zero offset. Thus the total effect is that of offset continuation , i.e. NMO and DMO. The last two equations in Table 7.2 are a decomposition of the middle equation $\omega \to \omega_0$ into two sequential processes, $\omega \to \omega_s$ and $\omega_s \to \omega_0$.These two processes are like DMO and NMO, but the operations occur in radial space. Radial NMO is a simple time-invariant s tretch; hence the notation $\omega_s$.

Unlike the constant-offset case, dip moveout is now done before the stretching, velocity-estimating step. Let us confirm that the dip moveout is truly velocity-independent. Substitute (13) into the radial DMO transformation in Table 7.2 to get the equation for transformation from time to stretched time:  

$${h^2 \over t^2} \ k_y^2 \ \ +\ \ \omega_s^2 \eq \omega^2$$ (15)

We observe that the velocity v has dropped out of (15). Thus dip moveout in radial coordinates doesn't depend on velocity. Dip-moveout processing $\omega \to \omega_s$does not require velocity knowledge. Radial coordinates offer the advantage that this comparatively costly process is done before the velocity is estimated $\omega_s \to \omega_0$.

The dip-moveout process, $\omega \to \omega_s$,can be conveniently implemented with a Stolt-type algorithm using (15).

The foregoing analysis has assumed a constant velocity. A useful practical approximation might be to revert to a v(z) analysis after the dip moveout, just before conventional velocity analysis, stack, and zero-offset migration.

Both the radial-trace method and Hale's constant-offset method handle all angles exactly in a constant-velocity medium, but neither method treats velocity stratification exactly nor is it clear that this can be done--since neither method is rooted in the DSR. Yilmaz [1979] rooted his DMO work in the DSR, so his method can be expected to be exact for velocity stratification, but Yilmaz could not avoid angle-dependent approximations. So there remains theoretical work to be done.