Subspace identification techniques have gained widespread acceptance as a method of obtaining a low-order model from data. These are based on using the singular-value decomposition as a means of estimating the underlying system order and extracting a basis for the extended observability space. In the presence of noise rank determination becomes difficult and the low rank estimates lose the structure required for exact realizability. Furthermore the noise corrupts the singular values in a manner that is inconsistent with physical noise processes. These problems are addressed by an optimization based approach using a nuclear norm minimization objective. By using Hankel matrices as the underlying data structure exact realizability of the low rank system models is maintained. Noise in the data enters the formulation linearly, allowing for the inclusion of more realistic noise weightings. A cumulative spectral weight is presented and shown to be useful in estimating models from data corrupted via noise. A numerical example illustrates the characteristics of the problem.