To estimate the seismic hazard to underground facilities or operations in the environs of a mining-induced tremor or a natural earthquake, it is useful to be able to relate locally recorded seismic waveforms to peak ground velocity and slip at the causative fault. For this purpose, far-field S wave pulses are analyzed to define the faulting slip D and near-fault peak ground velocity D/2 that give rise to the most significant ground motion. This most intense region of faulting, an assumed circular asperity, has radius r within a broader source zone of radius r₀, which is traditionally calculated from the corner frequency of the S wave spectrum. In developing relationships between peak far-field velocity v and peak acceleration a, and the source processes of the asperity, D and D, as well as its radius r, the key model assumption is that r = kβ/ω, where ω is the angular frequency of the sinusoidal velocity pulse of maximum amplitude, β is the sheaf wave speed, and k is a constant. Observations in deep-level gold mines of fault slip and slip velocity as well as laboratory observations of slip rate as a function of stress drop for stick-slip failure support a choice of about k = 2.34, the value commonly used for estimating r₀ using the Brune model. In particular, observations of fault slip up to 410 mm for mining-induced tremors in the moment magnitude range 4–5 are consistent with D = 8.1 Rv/β, where R is hypocentral distance. Moreover, estimates based on underground damage of near-fault ground velocities ranging up to 3.5 m/s are in accord with D/2 = 1.28(β/μ)_ρRa, where μ is the modulus of rigidity and ρ is the density. Alternatively, the average slip velocity 〈D〉 can be expressed in terms of the stress drop Δσ_a of the asperity as 〈D〉 = 0.51 β Δσ_a/μ, and the agreement of this relationship with measurements made during stick-slip failure in the laboratory is good. To the extent that seismic slip exterior to the asperity is a consequence of preevent suppression of slip due to the asperity, the broader-scale(r₀) slip can be related to that of the asperity. Just as the asperity radius r can be estimated from r = 2.34 βv/a, an alternative estimate for r₀ is given by r₀ = ρRaM₀/[75.8ρμ(Rv)²], the results of which are generally in good agreement with estimates based on the spectral corner frequency method.