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This paper considers a robotic sensor network, deployed in an environment of interest, that takes successive measurements of a spatial random field. Taking a Bayesian perspective on the kriging interpolation technique from geostatistics, we design the distributed kriging algorithm to estimate the distribution of the random field and of its gradient. The proposed algorithm makes use of a novel distributed strategy to compute weighted least squares estimates when measurements are spatially correlated. This strategy results from the combination of the Jacobi overrelaxation method with dynamic consensus algorithms. The network agents use the information gained on the spatial field to implement a gradient ascent coordination algorithm, whose convergence is analyzed via stochastic Lyapunov functions in the absence of measurement errors. We illustrate our results in simulation.