Continuously varying exponents in a sandpile model with dissipation near surface

We consider the directed Abelian sandpile model in the presence of sink sites whose density f_t at depth t below the top surface varies as c~1/t^chi. For chi>1 the disorder is irrelevant. For chi<1, it is relevant and the model is no longer critical for any nonzero c. For chi=1 the exponents of the avalanche distributions depend continuously on the amplitude c of the disorder. We calculate this dependence exactly, and verify the results with simulations.