Geological discontinuities commonly occur in nature. Sharp jumps such as distinct layering or formation of localized bodies in the subsurface can occur due to various geological processes. However, the bandwidth of the geophysical signals such as seismic or electromagnetic probing the Earth often produces smooth representations of the model via imaging or inversion. Introducing informative priors through regularization process has been shown to produce discontinuous model. Among the various classes of the regularization operators that preserve discontinuity are total variation norm, compactness constraints and Lp norm such as sparse spike solutions. In this paper we focus on a Bayesian hyper model formulation originally developed for image processing applications to geophysical inversion of the data. Both the locations and the magnitude of the discontinuities are considered unknowns which are commonly encountered in practical applications. Using a series of examples we compare this approach with other well known approaches such as total variation and L1 inversion in the wavelet domain.