Inverse scattering theory and almost periodic media

We investigate several inverse methods to determine the dielectric profile of an almost periodic structure composed of several periodicities or tones. For widely spaced tones a resonance method provides the most useful solution while for closely spaced tones the Gelfand-Levitan-Marchenko theory appears to be the most promising. In this way these two methods complement each other so that a wide range of almost periodic media can be investigated. Criteria for characterizing resonant (widely spaced tones) and analytic (closely spaced tones) reflection coefficients are given and the appropriate inversion methods are demonstrated.