The multidimensional potential-energy "landscape" formalism offers useful insights into the properties of supercooled liquids and glasses. However, its mathematical fundamentals present formidable subtlety and complexity. In the interests of developing a useful approximation for the statistical mechanics of landscapes, we have developed a simple family of models describing the energy-depth distribution of landscape basins. Our analysis begins with the "Gaussian" model that has been advocated in the recent literature, a physically appealing and thermodynamically rather accurate description that straightforwardly predicts a positive-temperature ideal glass transition. Careful enumeration of low-lying basins reveals however that the Gaussian model requires modification in the form of a logarithmic correction. Consequently, we have carried out algebraic and numerical analyses of a logarithmically modified Gaussian model, including depth dependence of the mean intrabasin vibrational free energy. The logarithmic modification has the effect of eliminating the positive-temperature ideal glass transition of the precursor pure-Gaussian model. Nevertheless, it is sufficiently similar to that unmodified model at and above any kinetic glass transition temperature to be able to represent measurable calorimetric data with reasonable accuracy.