  We present Frequency Marching, FM, an algorithm that refines
three-dimensional electron density distributions from solution X-ray scattering
data in both the small- and wide-angle regimes. This algorithm is based on a
series of optimization steps, marching along the frequency (reciprocal) space
and refining detailed periodic structures with the corresponding real-space
resolution. Buffer subtraction and excluded volumes, key factors in extracting
the signatures of the biomolecule of interest from the sample, are accounted
for using implicit density models. We provide the numerical and analytical
basis of the FM algorithm. We demonstrate this technique by application to
structured and unstructured nucleic acid systems, where higher resolution
features are carved out of low resolution reconstructions as the algorithm
marches into wider angles.
