# APPENDIX

The proposition I want to prove is that the vector perpendicular to the constant-offset isochron bisects the angle between the ray coming from the source and the one returning to the receiver. This is the direction of the zero-offset ray, as the incident angle is equal to the reflected angle and the zero-offset ray bisects both angles. The gradient to a constant-offset isochron is a vector

where

The constant-offset travel-time field can be decomposed into the sum of the travel-time from the source (Ts) and the travel-time to the receiver (Tr). The vector can be therefore as the sum of vectors

where Tsx, Tsz, Trx, Trz are the partial derivatives of the travel-time from source and receiver, respectively. If we note

then

which are two vectors: along the ray from the source and along the ray from the receiver. The gradient of the constant-offset travel-time field is a sum of the two vectors and , and because the two vectors have equal length (from the eikonal equation), the vector bisects the angle between the vectors and .