Minimum-structure inversions using L 2-norm measures have been widely applied to geophysical exploration problems. However, the smeared-out models resulting from L 2-norm inversions are not always consistent with the real or expected geological structures, especially in regions where distinct interfaces between different rock units exist. To obtain sharp boundaries and blocky features, non-L 2 inversions have been used successfully in geophysical imaging problems. In reality, however, both smooth and blocky features can be present in the subsurface physical properties or interfaces to be recovered. To deal with this situation, we develop a new method for adaptively recovering both smooth and blocky features in the constructed model from geophysical inversions. This method first detects different regions of the smoothness or blockiness in a model based on a sequence of inversions and then adaptively applies appropriate L p model norm with different p values at different locations to complete the final inversion. We present two synthetic examples from basement inversion using gravity data and crosswell seismic traveltime tomography before demonstrating our method on a field data example at the U.S. Geological Survey Fractured Rock Research Site in central New Hampshire.