Reflection points and surface points1

A simple expression ties the midpoint of a surface spread to reflection points on a dipping plane. If we use two coordinate systems, an unprimed one with a z-axis perpendicular to the surface and a primed one with a z-axis perpendicular to the reflector, we have where θ is the dip angle, φ is the profile angle, X is the source-to-receiver separation, and D is the depth of the reflector. The reflection point is (x, y_p, D) and the surface midpoint is (x_c, y_c, 0). Using the expression, I show that if complete azimuthal coverage is required at a CMP position, the reflection points lie on an ellipse. Similarly, a fixed reflection point generates a circle of surface midpoints. A circle of CMP positions for fixed θ and φ becomes an ellipse of reflection points and a circle of reflection points becomes an ellipse of midpoints. A user can easily find the shape and location of the reflection area generated by a surface aperture.