Two point raytracing for reflection off a 3D plane |

I make life simpler by observing that the traveltime gradient has
the same azimuth as the reflector's dip azimuth. This must be so because
translating the source-receiver pair along strike does not change the
reflection arrival time. I note that this does *not* say that
the reflection point moves along the dip azimuth when the surface arrival
point moves along the dip azimuth.

The next twist is that instead of translating the source-receiver pair along the dip azimuth, I'll translate the reflector plane along its normal direction. This implies that derivatives with respect to the reflector normal direction need to be scaled by the sine of the reflector dip, i.e. , as the surface intercept of the reflector moves a distance inversely related to the sine of the dip. Fortunately, even the zero dip case, where the reflector does not intersect the surface, is handled properly because the sine is zero in that case.

If we translate the initial reflection point by , where my convention for implies corresponds to a positive time slope, we obtain a point on the displaced reflection plane, though generally not the new reflection point . The relation of to can be ascertained as before by dotting with :

Continuing as before,

yielding

which says that does not rotate.

To compute changes in lengths (traveltimes), we have the relations

and

whence

Taking first differences, we have

and

whence

. |

With these in hand, we may differentiate the traveltime

to get

which, as remarked earlier, is then multiplied by to obtain the surface time slope.

This last expression has a simple geometric meaning. As

and

are unit vectors pointing towards the reflection point from the receiver and source respectively, their sum is necessarily parallel to their angle bisector, the normal. In particular, they sum to where is the angle of incidence or reflection. Dotting this with and multiplying by we have that the time slope is simply . Changing units from glorps back to seconds, this agrees with the well-known zero-offset result .

Two point raytracing for reflection off a 3D plane |

2012-10-29