Dip and offset together

Constant offset migration

Considering $h$ in equation (8.6) to be a constant, enables us to write a subroutine for migrating constant-offset sections. Forward and backward responses to impulses are found in Figures 8.4 and 8.5.

Cos1 Figure 4. Migrating impulses on a constant-offset section. Notice that shallow impulses (shallow compared to $h$) appear ellipsoidal while deep ones appear circular.

Cos0 Figure 5. Forward modeling from an earth impulse.

It is not easy to show that equation (8.5) can be cast in the standard mathematical form of an ellipse, namely, a stretched circle. But the result is a simple one, and an important one for later analysis. Feel free to skip forward over the following verification of this ancient wisdom. To help reduce algebraic verbosity, define a new $y$ equal to the old one shifted by $y_0$. Also make the definitions

$$t v = 2 A$$ (7)

$$\ \alpha = z^2 + (y + h)^2$$

$$\beta = z^2 + (y - h)^2$$

$$\alpha - \beta = 4 y h$$

With these definitions, (8.5) becomes

$$2 A \eq \sqrt \alpha + \sqrt \beta$$

Square to get a new equation with only one square root.

$$4 A^2 - (\alpha + \beta) \eq 2 \sqrt{ \alpha \beta }$$

Square again to eliminate the square root.

$$\begin{eqnarray*} 16 A^4 - 8 A^2 (\alpha + \beta) + (\alpha... ...a + \beta) + (\alpha - \beta)^2 &=& 0 \end{eqnarray*}$$

Introduce definitions of $\alpha$ and $\beta$.

$$16 A^4 - 8 A^2 [ 2 z^2 + 2 y^2 + 2 h^2 ] + \ 16 y^2 h^2 \eq 0$$

Bring $y$ and $z$ to the right.

$$A^4 - A^2 h^2 = \ A^2 ( z^2 + y^2 ) - y^2 h^2$$

$$A^2 ( A^2 - h^2 ) = \ A^2 z^2 + ( A^2 - h^2 ) y^2$$

$$A^2 = {z^2 \over 1 - {h^2 \over A^2}} + y^2$$ (8)

Finally, recalling all earlier definitions and replacing $y$ by $y-y_0$, we obtain the canonical form of an ellipse with semi-major axis $A$ and semi-minor axis $B$:

$${(y - y_0)^2 \over A^2} + {z^2 \over B^2} \eq 1 ,$$ (9)

where

$$A \eq {v t \over 2}$$ (10)

$$B \eq \sqrt{A^2 - h^2}$$ (11)

Fixing $t$, equation (8.9) is the equation for a circle with a stretched $z$-axis. The above algebra confirms that the ``string and tack'' definition of an ellipse matches the ``stretched circle'' definition. An ellipse in earth model space corresponds to an impulse on a constant-offset section.

Dip and offset together