Orthorhombic anisotropy arises when a system of vertical fractures is embedded in, an otherwise, finely layered medium. Arbitrary orthorhombic elastic anisotropy is often characterized by ten independent parameters, including density. In the best of circumstances, only six of these parameters can be inverted from P-P scattered energy, reliably. Thus, we evaluate the potential for additional information from converted waves. We, specifically, remap the scattering radiation patterns from the conventional elastic coefficients, as well as what we deemed recently as the hierarchical parameterization, into the spatial wavenumber domain, to analyze the tradeoffs between and the resolution involved in illuminating the orthorhombic parameters. It turns out that P-SV waves alone can, principally, lead to the reconstruction of eight parameters - everything except for V_p and γ1. Together with P-waves, we can also invert for V_p, and finally with P-SH waves when can theoretically reconstruct all orthorhombic elastic parameters. We show how the information from different wavetypes combines in the inversion and that all ten parameters can be reconstructed if converted waves are used in the inversion together with P-waves.