Modeling volume power spectra for collections of spheres in a finite container

When modeling the volume power spectrum for a collection of randomly positioned scatterers in an impedance map, the assumption is usually made that the incoherent component of the spectrum is much larger than the coherent component. With this assumption, scattering form-factor models such as the Gaussian, fluid-filled sphere, etc. can be fitted to the impedance map volume power spectrum for the purpose of estimating parameters such as the scatterer size. The accuracy of the assumption concerning incoherent and coherent component spectrum magnitudes was studied using simulated collections of spheres in a finite container. A collection of spheres with a specific number density was simulated and the volume power spectrum was computed by taking the 3D Fourier Transform and squaring the modulus. Realizations of spheres with random positions within a cubical or spherical container were simulated and the resulting volume power spectra were averaged to study the expected value of the power spectrum versus different number densities of spheres. These results were compared to the volume power spectrum for a single sphere (ie., the fluid-filled sphere form factor). A single sphere volume power spectrum is expected if no coherent component in the spectrum exists. For low number density, the volume power spectrum for a collection of spheres matched the single sphere power spectrum. As volume fraction increased, the volume power spectrum became biased for low values of ka compared to the power spectrum for a single sphere. The source of this bias can be attributed to the interaction of sphere positions. As more spheres are placed in the container, their positions become less random, increasing the effect of the coherent component relative to the incoherent component. The simulations also indicated that the container shape affected the shape of the volume spectrum. When using a cube as the container for the collection of spheres, asymmetry was observed in the volume power spectrum. Specifically, peaks were present on-axis in k-space that were not present off-axis nor in the power spectrum for a single sphere. When changing the container to a sphere, the volume power spectrum was symmetric and the on-axis peaks observed for the cube container were not present. The results of this study indicate that when modeling the effects of scattering using single scattering approximations, a large number density may yield a significant coherent component in the power spectrum and the shape of the scattering volume must be taken into account.