The statistical analysis of surfaces is an important issue of Image Analysis, especially in Computational Anatomy. In [23], Vaillant and Glaunès proposed to handle surfaces through some mathematical currents defined as linear forms on a space of mappings from the space into itself. In this paper, we extend this deterministic representation of surfaces using some random linear forms inspired from generalized stochastic processes. Then, we set an observation model where observed surfaces are viewed as random variations of a mean representative of a population (called the template). This observation model accounts not only for the variability of surfaces within a homogeneous population but also for errors due to acquisition. Within this model, we construct an estimate of the template and establish its consistency.