Sparsity and irregularity of spatial sampling are common problems in seismic data. Irregularity severely limits the types of processes which may be applied to a data set, and any process will likely fail on data which are overly sparse, as data and operator aliasing become a crippling problem. Interpolation schemes seek to dealias data, but are themselves challenged by aliasing, because it is difficult for an algorithm to pick exclusively the correct dip or dips to interpolate. Transformation to velocity (or slowness) space is an attractive basis for an interpolation algorithm, because the operator is limited to 'reasonable' directions, in that it operates only along centered hyperbolas. Since data can be organized so that it is symmetric, and largely hyperbolic, in CMP gathers, this type of interpolation should greatly reduce the risk of interpolating incorrect dips. However, the velocity transform is not an exact forward/inverse-transform pair, and the smoothness and/or noisiness of the estimated velocity spectrum presents a new and serious pitfall. While the original data space may be remodeled exactly, large artifacts are likely to appear in alternate similar data spaces, such as are appropriate for interpolation or regularization. By preconditioning the inversion, the model may be made more parsimonious, resulting in improved remodeling into a new data space.