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.

For surface stations at similar distances, the travel-time derivatives with respect to depth can be small or nearly equal. Their arrival-time shifts are then difficult to distinguish from a common change in earthquake origin time. A deeper source can be paired with an earlier origin. Nearby stations, broad distance coverage, and depth-sensitive phases help separate the two parameters.

For two phases travelling the same path length $R$ at constant wave speeds $v_A>v_B$, let $r=v_A/v_B>1$, $t_A=t_0+R/v_A$ and $\Delta t=t_B-t_A$. ThenThe intercept of this geometric relation is earthquake origin time $t_0$ and its slope is $1/(r-1)$.

A common station clock offset $c_i$ added to both phase arrivals cancels from $\Delta t$, but remains in $t_A$. Ordinary linear regression can then estimate the relation under zero-mean offsets uncorrelated with the true delays. A common systematic clock bias instead shifts the intercept, and unrestricted fixed station offsets destroy identifiability of the speed ratio.

Phase-specific arrival-picking errors are a different issue. If independent errors ${\epsilon }_A,{\epsilon }_B$ affect the two picks, the predictor error is ${\epsilon }_B-{\epsilon }_A$ and the response error contains ${\epsilon }_A$. Their covariance is $-\mathrm{var}({\epsilon }_A)$, so even individually unbiased picks do not produce an ordinary regression with error-free predictor and independent response noise. A justified errors-in-variables model or a joint arrival-time likelihood must account for that shared measurement error. More stations alone do not remove this structural bias.

In a homogeneous medium with surface stations along coordinate $x$, an hypocenter at horizontal position $x_0$ and depth $z_0>0$ has straight-ray distance $R=\sqrt{(x-x_0{)}^2+z_0^2}$. For two phases along that path, set $K=1/v_B-1/v_A>0$. Their arrival delay satisfieswith $A=K^2$, $B=-2K^2{x}_0$ and $C=K^2(x_0^2+z_0^2)$. Completing the square yieldsThe ratio of intercept to curvature gives squared depth only after centering at the fitted vertex: $(\Delta t{)}^2=A(x-x_0{)}^2+A{z}_0^2$. The raw uncentered ratio $C/A$ includes $x_0^2$ and generally overestimates squared depth.

The fitted curvature also determines $K=\sqrt{A}$, but it does not separately identify both wave speeds. Shared clock offsets cancel in the interphase delay, while a wrong homogeneous velocity model or phase-picking errors still affect the result. Squaring noisy delays changes their error probability distribution; practical inference should model that transformation or fit the unsquared delay relation directly.

The delay between corresponding S wave and P wave arrivals. For a common uniform path of length $r$, $\Delta t=r(1/V_s-1/V_p)$, cancelling earthquake origin time. Matched phase identification and an appropriate velocity model are needed to infer distance.

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.