  The Rezzolla-Zhidenko (RZ) framework provides an efficient approach to
characterize spherically symmetric black-hole spacetimes in arbitrary metric
theories of gravity using a small number of variables [L. Rezzolla and A.
Zhidenko, Phys. Rev. D. 90, 084009 (2014)]. These variables can be obtained in
principle from near-horizon measurements of various astrophysical processes,
thus potentially enabling efficient tests of both black-hole properties and the
theory of general relativity in the strong-field regime. Here, we extend this
framework to allow for the parametrization of arbitrary asymptotically-flat,
spherically symmetric metrics and introduce the notion of a 11-dimensional
(11D) parametrization space $\Pi$, on which each solution can be visualised as
a curve or surface. An $\mathscr{L}^2$ norm on this space is used to measure
the deviation of a particular compact object solution from the Schwarzschild
black-hole solution. We calculate various observables, related to particle and
photon orbits, within this framework and demonstrate that the relative errors
we obtain are low (about $10^{-6}$). In particular, we obtain the innermost
stable circular orbit (ISCO) frequency, the unstable photon-orbit impact
parameter (shadow radius), the entire orbital angular speed profile for
circular Kepler observers and the entire lensing deflection angle curve for
various types of compact objects, including non-singular and singular black
holes, boson stars and naked singularities, from various theories of gravity.
Finally, we provide in a tabular form the first 11 coefficients of the
fourth-order RZ parameterization needed to describe a variety of commonly used
black-hole spacetimes. When comparing with the first-order RZ parameterization
of astrophysical observables such as the ISCO frequency, the coefficients
provided here increase the accuracy of two orders of magnitude or more.

General Relativity and Quantum Cosmology;Astrophysics of Galaxies;High Energy Astrophysical Phenomena;High Energy Physics - Theory