Earthquake location from interphase delays ($\Delta t=K\sqrt{(x-x_0{)}^2+z_0^2}$)

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.