In an ideal application of sequential simulation, parameters are simulated one at a time conditioned to all previously simulated parameters. This requires that marginal distributions of all dimensions (used to derive the conditional distributions) from the random field can be extracted and used for the simulation. However, in practice, only incomplete information from limited-size marginals is used for sequential simulation due to, e.g., computational unwieldiness or to ensure adequate pattern statistics. In this paper, we start out by addressing the problem of how to reconstruct an unknown random field that is consistent with known limited-size marginals. This problem turns out to be highly underdetermined (i.e., infinitely many solutions exist). Therefore, we describe possible additional constraints to supplement the marginals in order to reconstruct well-defined random fields. Secondly, we investigate which random field (out of infinitely many) that is sampled by sequential simulation algorithms using limited-size marginals. We show that sample distributions of such algorithms may depend on the sampling sequence and, sometimes, are inconsistent with the known marginals. We reviewed a formulation of a Markov random field that provides a well-defined solution to the underdetermined problem. Finally, we investigate the relation between marginal-size and information content of reconstructed random fields.