F. A. Dahlen

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 χ, there are two clearly separated regimes, depending upon the wavelength λ of the wave. In regime I, when χ/L ≪ πL/λ, the absolute magnitude of the phase delay decreases approximately linearly with χ, and the anomaly does not widen appreciably except by developing small sidelobes with delays of opposite sign. Tomographie inversions of such delays will be damped but are theoretically well posed. In regime II, when χ/L ≫ πL/λ, the absolute delay decreases toward zero as 1/χ, 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. Tomographie 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.

Diffraction tomography using multimode surface waves

A new method is described that makes it feasible to include scattered and converted surface waves into waveform inversions for the three-dimensional (3-D) structure of the Earth. The single scattering (Born) approximation forms the basis of the method. In order to minimize the amplitude of the scattered wave field, the background model is first adapted to correct for nonconverted, forward-scattered wave energy. We then perform Born inversion of the difference between the measured and synthetic waveforms, including a suite of Love and Rayleigh modes. The Born approximation yields linear equations of the form Aδγ = δu Bor n, which allow the determination of the three-dimensional perturbations γ to the background model from the scattered wave field δu Born . This procedure is followed separately for each source-receiver pair to allow for optimized background models for each signal, as well as to minimize the computational burden. We winnow the data vector for each path by performing singular value decomposition using a diagonalization of AA T . In a realistic example we found that each vertical component seismogram yields 30-40 linear constraints on the 3-D Earth, significantly more than with conventional pure-path (WKBJ) inversions. In a synthetic test, one seismogram is shown to be able to image a simple model of a point scatterer off the great circle. As a spin-off of the formulation of the multimode inverse scattering problem, we not only obtain a series of eigenvectors that rank the sensitivity of a seismogram to Earth structure in a series of geometrical patterns, we also can compute the surface wave equivalent of a Fresnel zone.

The Berry Phase of a Slowly Varying Waveguide

A wavetrain in a slowly varying waveguide has, apart from the familiar dynamical phase or eikonal, an additional variation in phase which fixes the location of crests and troughs. This additional slow variation in phase is a multidimensional analogue of the Berry phase of an adiabatic quantum system. Both the quantum geometric phase and the additional variation in phase in a classical waveguide can be obtained from a variational principle based upon a slowly varying lagrangian.