Earthquake location by arrival times ($t_i=t_0+T_i(\mathbf{x})$)

Arrival-time location estimates a hypocenter and origin time by comparing picked phase arrivals with predicted travel times. Linearization gives residual changes $\delta t_i=\delta t_0+\nabla T_i\cdot \delta \mathbf{x}$. Poorly distinct derivative columns cause tradeoffs, while incorrect wave speeds or phase labels cause systematic bias even with many stations.

An earthquake origin time is the estimated onset time of rupture. Predicted arrivals add this time to a propagation travel time; an uncertain origin time therefore couples to uncertain source depth in some station geometries.

Differences between arrival times of nearby earthquakes at a common station largely cancel common propagation errors. Comparing observed and predicted differences constrains relative source separation and origin-time differences. Closely similar waveforms permit precise differential picks by cross-correlation. Absolute position still needs an accurate model or external anchor.