Characterization of volume scattering power spectra in isotropic media from power spectra of scattering by planes

Abstract

Relations between the power spectrum of scattering by a plane and by a volume of a statistically isotropic random medium are developed from two basic expressions. One gives the power spectrum of scattering by an n-dimensional isotropic medium as a one-dimensional transform of a rotationally symmetric correlation function of medium variations. The other describes the power spectrum of scattering by an (n-k)-dimensional cross section as a projection of the power spectrum in n-dimensional space. The results are used to determine the power spectrum of scattering by a volume from the power spectrum of scattering by a plane within the volume and also to relate the moments of power scattered in spaces of various dimensions. To illustrate the relations, calculations of power spectra and selected moments for volume scattering are made from power spectra having analytic forms as well as form power spectra computed in two dimensions from images of closely packed spheres and of pig liver sections. Numerical results obtained by processing power spectra having closed form expressions for volume scattering are in close agreement with theoretical values. Calculations for packed spheres show general concentrations of power explained by theory but required substantial smoothing of data in two dimensions to obtain consistent results because the images were not true cross sections. Results from sections of pig liver agree wth expected relationships. The theoretical expressions and numerical data indicate how quantitative information about the three-dimensional structure can be obtained from two-dimensional sections when the basic isotropy conditions are satisfied.