are set out in @CalGoo08aa, @Cal09ab, and @NewCal10aa.

Figure \[fig:BVP\] displays the results of such a calculation for one of the cases considered above: $B_0=1$ kG, $\nu=5$ mHz, $k_h=1$ $\rm Mm^{-1}$. Travel time perturbations (top left) are calculated according to the prescription of @Cal09ab. These are computed based on the phase of the reflected fast wave when it exits the box at the bottom, compared to the same for quiet Sun. No full seismic skip is constructed, but this assessment of the phase change through the surface layers amounts to the phase change per skip. It is therefore directly comparable to the time-distance results calculated above based on standard TD methodology, and indeed, the correspondence between the results of the two approaches is striking. Experiment with higher magnetic field strength (not shown) indicates that peak travel time perturbations scale roughly linearly with $B_0$, as expected from our Fig. \[fig:tau\] and from Fig. 7 of @Cal09ab. This is sufficient by $B_0=2$ kG for the phase difference $\delta\varphi$ to fold over the $360^\circ$ ambiguity, so that one would not be sure whether it were advanced or retarded. Since $\delta\tau=-\omega^{-1}\delta\varphi$, travel time perturbations may therefore artificially appear positive.

Figure \[fig:BVP\] also displays the 