Gravity and magnetic data resolve Earth models with variable spatial resolution, and Earth structure exhibits both discontinuous and gradual changes. Therefore, model parametrization complexity should address such variability, for example, by locally adapting to the spatial resolution of the data. The reversible-jump Markov chain Monte Carlo (rjMcMC) algorithm can be employed to explore Bayesian models with variable spatial resolution that is consistent with data information. To address non-uniqueness in joint inversion of potential field data, I employ novel hierarchical models that are based on irregular spatial partitioning and incorporate geological constraints about the subsurface as prior information. Spatial partitioning choices include nested Voronoi cells, linear interpolation and alpha shapes, and Voronoi cells and planes. These parametrizations partition the subsurface in terms of rock types, such as sedimentary rocks, rock salt and basement rocks. Therefore, meaningful prior information can be included in the inversion which reduces non-uniqueness. In addition, nonoverlapping prior distributions are used for density contrast and susceptibility between rock types. Another significant challenge for potential field data is unknown noise characteristics. In particular, poorly estimated noise characteristics can significantly change model spatial resolution and complexity. I consider empirical and hierarchical approaches to noise estimation that include theory and measurement errors. The empirical estimation of full data covariance matrices is based on residuals and an iterative scheme. The hierarchical approach employs a trans-D autoregressive noise model that quantifies the impact of spatial noise correlations on geophysical parameters. Both 1-D and 2-D spatial correlations are considered for 2-D and 3-D inversions, respectively. The method is applied to gravity and magnetic data to study salt and basement structures. This thesis demonstrates that meaningful partitioning of the subsurface into sediment, salt, and basement structures is achieved by these methods without requiring regularization. Multiple simulated- and field-data examples are presented. Simulation results show a clear delineation of salt and basement structures while resolving variable length scales. The field data results are consistent with observations made in the simulations. This work resolves geologically plausible structures with varying length scales and clearly differentiates salt structure and basement topography.