We consider a model of a floodplain evolving with channel avulsion, deposition, and erosion. Avulsion is modeled as a random process in space and time. Sediment transport is modeled by the diffusion equation (Culling's model). The power spectrum of variations in the topographic profiles predicted by the model S(k) is proportional to k-2 (where k is the wavenumber). This is the Brown noise often observed in topography. The power spectrum of variations in the local elevation in time is proportional to f-3/2 (where f is the frequency). The model prediction of Brown noise floodplain topography is roughly consistent with spectral analyses of microtopography measured with laser altimetry. We inferred Brown noise paleotopography by comparing the pair correlation function of showing wells in the Denver and Powder River basins with a synthetic oil field based on a caprock with Brown noise topography. Topographic control of variations in the grain size of deposited sediment suggests that porosity variations may exhibit the scale invariance predicted for the topographic profile. To illustrate vertical scale invariance in porosity variations, we computed the power spectrum of vertical porosity well logs in 15 offshore wells in the Gulf of Mexico. At spatial scales above 3 m we find an average power spectral exponent of -1.4, close to our model prediction of -1.5. ¿ American Geophysical Union 1996