= Solution
In the three-dimensional <Earth>, a network identifies P- and S-wave arrivals and fits source coordinates $(x_0,y_0,z_0)$ and origin time $t_0$ using a travel-time model. Time differences constrain path lengths without requiring a common absolute clock zero; calibrated absolute times then fix the origin time. Real inversions trace refracted rays through a layered or heterogeneous velocity model and use many stations, azimuths, and phases, rather than literal <circles> at fixed <wave speed>. Residuals and the sensitivity <matrix> quantify uncertainty and expose poor station geometry or a poor velocity model.
\b[Depth and origin time are often the weakest pair], especially for a shallow event observed by distant surface stations. In the homogeneous example,
$$
t_i=t_0+\frac{\sqrt{r_i^2+z_0^2}}{v},\qquad
\delta t_i=\delta t_0+\frac{z_0}{v\sqrt{r_i^2+z_0^2}}\delta z_0.
$$
If the depth sensitivities are small or nearly alike across the network, a deeper source's longer <travel time> can be offset by an earlier origin time. In contrast, horizontal displacement often gives opposite-sign residuals on opposite sides of an event and is better constrained by good azimuthal coverage. Nearby stations and depth-sensitive reflected phases can break the <earthquake depth–origin-time trade-off>; the ranking is not universal for every network or event depth.
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