= Origin-time and speed-ratio regression
{title2=$t_A=t_0+\Delta t/(r-1)$}
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 $\Delta t=t_B-t_A$. Then
$$
\Delta t=(r-1)(t_A-t_0),\qquad t_A=t_0+\frac{\Delta t}{r-1}.
$$
The 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 $\varepsilon_A,\varepsilon_B$ affect the two picks, the predictor error is $\varepsilon_B-\varepsilon_A$ and the response error contains $\varepsilon_A$. Their <covariance> is $-\operatorname{var}(\varepsilon_A)$, 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.
Back to article page
