This study addresses the distributed average set-membership filtering of spatially varying processes using sensor networks. The system under consideration contains sensor saturation in the presence of unknown-but-bounded process and measurement noise in the sensor network. The so-called distributed average set-membership filtering is defined to quantify bounded consensus regarding the estimation error. A sufficient condition for distributed average set-membership filtering parameter design is established in terms of a set of time-varying linear matrix inequalities. A recursive algorithm is developed for computing the estimator, controller gains and the ellipsoid that guarantees to contain the true state. Simulation results are provided to demonstrate the effectiveness of the proposed method.