Using parametricBsplines to fit specular reflectivities

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Abstract
Parametric B-spline curves offer a flexible and appropriate mathematical description of scattering length density profiles in specular reflectivity analysis. Profiles combining smooth and sharp features can be defined in low dimensional representations using control points in the density-depth plane which provide graded local influence on profile shape. These profiles exist in vector spaces defined by B-spline order and parameter knot set, which can be systematically densified during analysis. Such profiles can easily be rendered as adaptive histograms for reflectivity computation. B-spline order can be chosen to accommodate the asymptotic (large-Q) behavior indicated by reflectivity data. We describe an interactive fitting strategy in which the Nelder and Mead simplex method is used in the B-spline control point space to guide the discovery of profiles that can produce given reflectivity data. Examples using actual and simulated spectra are discussed.