Task-dependent timing admissibility in digital seismology: forward propagation of structured timing errors into phase, polarisation, and array measurements

Abstract. Accurate timing is essential in digital seismology, yet a single clock-accuracy specification does not determine whether a recording system is adequate for a given seismic measurement. Here I formulate timing quality as a task-dependent admissibility problem in which the temporal evolution, spatial or channel topology, and covariance of timing errors are propagated through the measurement operator of the observable of interest. Analytical and numerical formulations are developed for spectral phase, differential delay, three-component polarisation, array slowness and backazimuth, and coherent beamforming. The framework distinguishes absolute, differential, intercomponent, spatially structured, and common-mode timing errors and expresses admissibility through deterministic or probabilistic performance criteria. Controlled synthetic experiments, a perturbation benchmark using a public three-component seismic waveform, and a controlled benchmark based on a public compact-array geometry demonstrate that nominally identical timing uncertainties can produce markedly different measurement errors. At 10 Hz, a 95 % single-path phase criterion of 15° admits a timing standard deviation of approximately 2.13 ms, whereas corresponding representative benchmark limits are approximately 6.44 ms for intercomponent polarisation and 2.99 ms for compact-array backazimuth; an independent-error beam-coherence criterion C_B ≥ 0.90 gives approximately 5.17 ms. These values are not interchangeable specifications because they act on different timing-error variables. Timing covariance is equally important: common-mode errors can cancel from relative observables, whereas differential errors remain observable, and spatially linear timing gradients can be indistinguishable from slowness perturbations under a plane-wave model. Extension to empirically estimated, non-Gaussian, and non-stationary clock processes would allow the same framework to support operational task-specific timing assessment. The results show that timing performance should be evaluated forward from the structured timing process to the intended seismic observable, rather than inferred from a universal millisecond threshold.