Wave front healing and the evolution of seismic delay times

Using a simple Gaussian beam solution to the one-way scalar wave equation, we derive analytical expressions for the evolution of phase and group delay after a wave passes through a Gaussian-shaped heterogeneity of half width L. As a function of distance x, there are two clearly separated regimes, de pending upon the wavelength lambda of the wave. In regime I, when x/L much less than pi L/lambda, the absolute magnitude of the phase delay decreases approximately linearly with x, and the anomaly does not widen appreciably e xcept by developing small sidelobes with delays of opposite sign. Tomograph ic inversions of such delays will be damped but are theoretically well pose d. In regime II, when x/L much greater than pi L/lambda, the absolute delay decreases toward zero as 1/x, most markedly on the ray itself, and the cro ss-path shape of the wave front bears little resemblance to the original an omaly. Tomographic inversions of delay times in this regime are ill posed. Group delay times show a similar behavior in the two regimes. Although thei r rate of decrease with distance is slower in regime I, they develop more d isturbing sidelobe behavior off the central ray. The effects of wave front healing for surface waves traveling in two dimensions are less severe than those for body waves in three dimensions; as a result, surface wave inversi ons will commonly be in regime I. Short-period body wave group delays are a lso in regime I; nevertheless, the damping of delays in this regime is like ly to contribute significantly to the scatter of observed travel time anoma lies. Tomographic inversions of long-period body waves, which fall at the l imit of regime I, or even in regime II, face perceptible limitations in the oretical resolving power. Finally, we show that there is an asymmetry in th e evolution of positive versus negative travel time anomalies.