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Recent applied literature introduces the Stockwell transform (S-transform) as a new approach to time-frequency analysis. It is the purpose of this letter to encourage the interaction between the wavelet and the Stockwell communities by demonstrating that-up to minor modifications-the S-transform is a special case of the well-known continuous wavelet transform via a Morlet-type mother wavelet, with the features of a linear frequency scale, and an amplitudeand modulation adjustment in phase space. The extensive research and applications obtained for the continuous wavelet transform can therefore be directly applied to the Stockwell domain.

Reflection and transmission of waves in piecewise constant layered media are important in various imaging modalities and have been studied extensively. Despite this, no exact time domain formulas for the Green's functions have been established. Indeed, there is an underlying combinatorial obstacle: the analysis of scattering sequences. In the present paper we exploit a representation of scattering sequences in terms of trees to solve completely the inherent combinatorial problem, and thereby derive new, explicit formulas for the reflection and transmission Green's functions.

A family of orthogonal polynomials on the disk (which we call scattering polynomials) serves to formulate a remarkable Fourier expansion of the composition of a sequence of Poincaré disk automorphisms. Scattering polynomials are tied to an exotic riemannian structure on the disk that is hybrid between hyperbolic and euclidean geometries, and the expansion therefore links this exotic structure to the usual hyperbolic one. The resulting identity is intimately connected with the scattering of plane waves in piecewise constant layered media. Indeed, a recently established combinatorial analysis of scattering sequences provides a key ingredient of the proof. At the same time, the polynomial obtained by truncation of the Fourier expansion elegantly encodes the structure of the nonlinear measurement operator associated with the finite time duration scattering experiment.which if w ∈ D n are automorphisms of the Poincaré disk. (Note that if w j ∈ T is on the disk boundary then Ψ w j z j (v) = z j w j collapses to a constant function.) We fix notation for the polar *

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