 Origin-time and speed-ratio regression

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.