Systematic sampling is a frequently used sampling method in surveys, because of its ease of implementation and its design efficiency.  An important drawback of systematic sampling, however, is that no direct estimator of the design variance is available.  We describe a new estimator of the model-based expectation of the design variance, under a nonparametric model for the population.  The nonparametric model is sufficiently flexible that it can be expected to hold at least approximately for many practical situations.  We prove the consistency of the estimator for both the anticipated variance and the design variance under the nonparametric model.  The approach is used on a forest survey dataset, on which we compare a number of design-based and model-based variance estimators.