An inversion method based on rejection sampling is developed, with the objective of obtaining posterior distributions of the P-wave velocity, S-wave velocity and density. The inversion algorithm is based on the exact expression of the reflection coefficient from the Zoeppritz equation. The results of inverting directly on the reflection coefficient is compared with inversion algorithms based on the two- and three term Taylor approximation to the Zoeppritz equation, allowing larger contrasts across the reflecting interface. At first, the inversions are compared for a broad limited prior model and subsequently different constraints are applied for a better resolution of the posterior distribution of the P-wave velocity, S-wave velocity and density, including acoustic- and shear impedance and Poisson's ratio. The inversions are tested for a subsurface three layer model, assuming ideal conditions. Reference values of the P-wave velocity, S-wave velocity and density is interpreted from well logs in the NINI oil field of the North Sea.