The geometry of curvelets, and their higher dimensional analogue in dimensions greater than 2, is based upon a parabolic scaling of phase space: at frequency scale of lambda, the curvelets are localized in frequency to angular width of square root of lambda.
It was shown by Fefferman, and later Seeger-Sogge-Stein, that this parabolic localization was well suited to decompose oscillatory integrals (Fourier integral operators) such as arise in wave evolution operators, in that such a scale is the largest upon which the oscillations can be linearized with suitable error.
We discuss this work, as well as our own work showing that the parabolic localization is also well adapted to studying the propagation of waves in rough media, where the wave speed function is twice continuously differentiable.
We conclude by considering examples where a finer (cubic) scaling is required, in particular problems involving refraction, and media where the wave speed function is only once differentiable.
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