Clayton's cosine corrections

A tendency exists to associate the sine of the earth dip angle with Y and the sine of the shot-geophone offset angle with H. While this is roughly valid, there is an important correction. Consider the dipping bed shown in Figure 30.

 

clay Figure 30 Geometry of a dipping bed. The line bisecting the angle $2 \beta$ does not pass through the midpoint between g and s. (Clayton)

The dip angle of the reflector is $\alpha$, and the offset is expressed as the offset angle $\beta$.Clayton showed, and it will be verified, that

$$\begin{eqnarray} Y \ \ \ &=&\ \ \ \sin \, \alpha \ \ \cos \, \beta \\ H \ \ \ &=&\ \ \ \sin \,\beta \ \ \cos \, \alpha\end{eqnarray}$$ (65) (66)

For small positive or negative angles the cosines can be ignored, and it is then correct to associate the sine of the earth dip angle with Y and the sine of the offset angle with H. At moderate angles the cosine correction is required. At angles exceeding 45$^\circ$ the sensitivities reverse, and conventional wisdom is exactly opposite to the truth. The reader should be wary of informal discussions that simply associate Y with dip and H with velocity. ``Larner's streaks'' were an example of mixing the effects of dip and offset. Indeed, at steep dips the usual procedure of using H to determine velocity should be changed somehow to use Y.

Next, (65) and (66) will be proven. The source takeoff angle is $\gamma_s$, and the incident receiver angle is $\gamma_g$.First, relate $\gamma_s$ and $\gamma_g$ to $\alpha$ and $\beta$.Adding up the angles of the smaller constructed triangle gives

$$\begin{eqnarray} ({\pi \over 2}\ -\ \gamma_s\ -\ \alpha )\ +\ \beta \ +\ {\pi \... ... 2 }\ \ \ &=&\ \ \ \pi \nonumber \\ \gamma_s \eq \beta \ -\ \alpha\end{eqnarray}$$ (67)

Adding up the angles around the larger triangle gives  

$$\gamma_g \eq \beta \ +\ \alpha$$ (68)

To associate the angles at depth, $\alpha$ and $\beta$, with the stepouts dt/ds and dt/dg at the earth's surface requires taking care with the signs, noting that travel time increases as the geophone moves right and decreases as the shot moves right. Recall from equations (46), (47), (49), (50), (51) and (52), the definitions of apparent angles Y and H,

$$\begin{eqnarray} Y \ -\ H \ &=& S \ =\ {v\,k_s \over \omega }\ \ =\ \ v \ {dt \... ...\ +\ \sin \, \gamma_g \ \ = \ \ \sin ( \alpha \ +\ \beta ) \quad\end{eqnarray}$$ (69) (70)

Adding and subtracting this pair of equations and using the angle sum formula from trigonometry gives Clayton's cosine corrections (8):

$$\begin{eqnarray} Y &=& {1 \over 2 }\ \sin ( \alpha \ +\ \beta ) \ +\ {1 \over 2... ...\ \sin ( \alpha \ -\ \beta ) \ \eq \ \sin\, \beta \ \cos \, \alpha\end{eqnarray}$$ (71) (72)