Statistical Estimates of Amplitude and Phase Corrections

In seismology one frequently meets the problem of finding amplitude and phase corrections at one station relative to another, when there are errors of observation at both. This is a particular example of the following more general problem. We write H = r(ω) exp [iθ(ω)] and show how to obtain maximum likelihood estimators for r and θ when observed values of s(t) and σ(t), accompanied by noise, are used as data. We further show that these maximum likelihood estimators of r and θ are asymptotically unbiased, whereas conventional estimators are biased.