Wave front healing and the evolution of seismic delay times

Abstract

Abstract. 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, depending upon the wavelength h of the wave. In regime I, when x/L << •rL/h, the absolute magnitude of the phase delay decreases approximately linearly with x, and the anomaly does not widen appreciably except by developing small sidelobes with delays of opposite sign. Tomographic inversions of such delays will be damped but are theoretically well posed. In regime II, when x/L • •rL/h, the absolute delay decreases toward zero as l/x, most markedly on the ray itself, and the cross-path shape of the wave front bears little resemblance to the original anomaly. Tomographic inversions of delay times in this regime are ill posed. Group delay times show a similar behavior in the two regimes. Although their rate of decrease with distance is slower in regime I, they develop more disturbing 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 inversions will commonly be in regime I. Short-period body wave group delays are also in regime I; nevertheless, the damping of delays in this regime is likely to contribute significantly to the scatter of observed travel time anomalies. Tomographic inversions of long-period body waves, which fall at the limit of regime I, or even in regime II, face perceptible limitations in theoretical resolving power. Finally, we show that there is an asymmetry in the evolution of positive versus negative travel time anomalies.

Helioseismology of sunspots: defocusing, folding, and healing of wavefronts

“…Dahlen & Tromp 1998). Inside the caustics, multiple arrivals contribute to the cross-correlation travel-time measurements and the interpretation of travel time becomes ambiguous under the raytheoretical picture (Nolet & Dahlen 2000;Hung et al 2001). The location of the caustics depends on the perturbation strength and the sunspot geometry.…”

“…This phenomenon, called wavefront healing (Nolet & Dahlen 2000;Hung et al 2001), is inevitable because the amplitude of the scattered wave (a "circular" wave) decreases with distance from the sunspot more rapidly than the amplitude of the direct wave (a plain wave) by a factor proportional to the square root of distance. Wavefront healing is a finite-wavelength effect (Nolet & Dahlen 2000). In particular, single-ray theory cannot explain wavefront healing but instead gives a constant travel-time perturbation for x > R .…”

“…In this study, we extend the observations to modes with different radial orders and wavelengths that are closer to the sunspot radius, in order to investigate finite-wavelength effects (e.g. Nolet & Dahlen 2000;Hung et al 2001). We measure not only traveltime shifts but also wave amplitude perturbations with respect to the quiet Sun.…”

Evaluation of inversion approaches for guided wave thickness mapping

“…If a cross-correlation approach is used for this, then the resolution is close to the width of the first Fresnel zone √ Lλ, where L is the distance between the furthest two transducers in the array and λ is the wavelength [16,40]. Nolet & Dahlen [41] provide a physical explanation of the causes of this resolution limit; in summary, diffraction causes 'wavefront healing' to occur, which smooths the wavefronts once they have interacted with any features and hence reduces the resolution of any reconstruction from the arrival times. By focusing on arrival time pickers which extract the earliest arrivals rather than the centre of the wavepacket (as is the case with the cross-correlation approach), the result will be less affected by the diffraction-type scattering and it may be possible to improve on this resolution limit to an extent.…”

The Spectral-Element Method, Beowulf Computing, and Global Seismology

“…29 NOVEMBER 2002 VOL 298 SCIENCE www.sciencemag.org er, they may be of limited validity when we wish to recover velocity variations on length scales smaller than approximately 1000 km (11,12), close to the resolution provided by the most recent 3D models. To assess the quality of a 3D model, i.e., to evaluate the misfit between the data and the synthetic seismograms, the same approximate methods that were used to construct the model are employed.…”