  Hardy space on the polydisk provides the setting for a global description of
scattering in piecewise-constant layered media, giving a simple qualitative
interpretation for the nonlinear dependence of the Green's function on
reflection coefficients and layer depths. Using explicit formulas for
amplitudes, we prove that the power spectrum of the Green's function is
approximately constant. In addition we exploit a connection to Jacobi
polynomials to derive formulas for computing reflection coefficients from
partial amplitude data. Unlike most approaches to layered media, which
variously involve scaling limits, approximations or iterative methods, the
formulas and methods in the present paper are exact and direct.
