THE PSPM OPERATOR IN CONSTANT VELOCITY MEDIA

Prestack partial migration is a mapping of prestack data to zero-offset data. This mapping can be subdivided in two steps:

1.

Full prestack migration to a depth model. An input data point in a constant offset section, is migrated into an ellipse. In other words a single impulse on a trace can be generated by a reflection in any point on the reflecting elliptic surface.

2.

Zero-offset modeling. Each point on the ellipse can be modeled into a hyperbola, which is obtained by considering the chosen point as a point diffractor.

As a result, an impulse in a constant offset section will be moved in a zero offset section accounting also for the effect of the dipping reflector. The element which fixes the position of the migrated point on the ellipse is the dipping angle $\alpha$. The coordinates of the points on the ellipse are functions of the dip angle $\alpha$.

The mapping of the input point (t_h,x_h) from a constant offset section into the point (t₀,x₀) in a zero-offset section can be expressed as

$$(t_h,x_h) = migration(\bf {v},h,\alpha) \Longrightarrow (z_m,x_m;\alpha)$$

$$(z_m,x_m;\alpha) = modeling(\bf {v},\alpha) \Longrightarrow (t_0,x_0;\alpha).$$

Given the geometry in Figure 1, the coordinates of the point P can be expressed as a function of the semiaxes of the ellipse (a and b ) and the dip angle $\alpha$.The semiaxis are a = t_hV and b $=\sqrt{t^2_hV^2 - h^2}$, where V is half the earth velocity. The equation of the ellipse is:

$${x^2_m\over a^2}+{z^2_m\over b^2} = 1$$ (1)

Which can be rewritten as

z²_ma² = a²b² - x²_mb²

The equation of the tangent to the ellipse is $z_m={\bf m}x_m+{\bf c}$ where ${\bf{m}}=\tan\alpha$and ${\bf c}=\sqrt{a^2\tan^2{\alpha} + b^2}$ .

$$z_m = x_m \tan\alpha + \sqrt{a^2\tan^2{\alpha} + b^2}$$ (2)

Squaring equation 2 and multiplying both sides with ${\bf a^2}$ we obtain:

$$z^2_ma^2 = a^2(x^2_m\tan^2{\alpha} + 2x_m\tan{\alpha} \sqrt{a^2\tan^2{\alpha}+b^2} + a^2\tan^2{\alpha} + b^2)$$

As the intersection of the tangent to the ellipse should satisfy both equation (1) and (2), by equalizing both sides of z_m²a² we obtain:

$$x^2_m(b^2 + a^2\tan^2{\alpha}) + 2x_ma^2\tan{\alpha}\sqrt{a^2\tan^2{\alpha}+b^2} + a^4\tan^2{\alpha} + a^2b^2 = a^2b^2$$ (3)

$$(x_m\sqrt{a^2\tan^2{\alpha}+b^2} + a^2\tan{\alpha})^2 = 0$$ (4)

From the previous equation (4) we obtain the value for x_m, and introducing this value in equation 1, we obtain z_m.

$$x_m = -{a^2\tan{\alpha}\over \sqrt{a^2\tan^2{\alpha}+b^2} }$$ (5)

$$z_m = {b^2 \over \sqrt{a^2\tan^2{\alpha}+b^2}}$$ (6)

Each point on the ellipse corresponding to a certain parameter $\alpha$ (dipping angle), can be considered a point diffractor and therefore generate a hyperbola. For a zero-offset section the point will be situated on the modeling hyperbola with the following coordinates:

$$t_0 = {z_m \over{V\cos{\alpha} }}$$ (7)

$$x_0 = x_m + z_m\tan{\alpha}$$ (8)

Inserting equations (5,6) in (7,8) we obtain:

$$t_0 = {b^2 \over V} {\sqrt{1\over{a^2\sin^2{\alpha}+b^2\cos^2{\alpha} }}}$$ (9)

$$x_0 = {-h^2\sin{\alpha}\sqrt{1\over{a^2\sin^2{\alpha}+b^2\cos^2{\alpha} }}}$$ (10)

which are the parametric equations (function of the dip angle $\alpha$)of the well known DMO ellipse.

In the end the DMO operator contoured for only half space will look like in Figure 3. We demonstrate in Appendix A that the operator follows the envelope of the hyperbolas.