In this paper we describe the analysis of quasi-static stress-strain data using a Preisach-Mayergoyz (PM) Preisach, 1935; Mayergoyz, 1985> space picture for the elastic behavior of rock. In contrast to the traditional analytic approach to stress strain (an energy density as a function of the strain invariants), the PM space picture reproduces hysteresis and discrete memory seen in the data. In addition, the PM space picture establishes a relationship between experimental data and a number density &rgr; of microscopic mechanical units within the rock. The density &rgr; allows us to make quantitative predictions of dynamic elastic properties. Determining &rgr; from quasi-static stress-strain data requires us to solve a highly undetermined inverse problem. We explore the following three methods of solving the inverse problem: simulated annealing, normal modes, and exponential decay. All three methods are tested on a Berea sandstone data set and found to give an excellent description of stress versus strain. Choosing one method, the normal mode method, we analyze quasi-static stress-strain curves on two additional sandstones, namely, another sample of Berea and a sample of Castlegate sandstone. From the density &rgr; for each sample we predict the dynamic modulus as a function of pressure and the nonlinear elastic constants. For each of these cases the agreement between the predictions based on &rgr; and experiment is quite good. We establish that PM space provides a quantitative description of the elastic response of a rock and that PM space may be found by a variety of inversion methods.¿ 1997 American Geophysical Union