We consider the initial state observability of linear time-invariant dynamical networks based on arbitrary noisy observations (in time) from spatial locations. We assume that the observation times are Lebesgue measurable and show that the estimation problem is feasible if a properly defined continuous frame exists. Moreover, the quality of the estimation depends on the spectra of a matrix, which can be explicitly calculated using the space-time observation strategy. It turns out that total observation time dictates a fundamental limit on the best achievable estimation quality. Next, we illustrate how to synthesize observation strategies for a stable estimation. Finally, we use a randomized method that efficiently sparsifies the observation strategies, while providing a performance guarantee.