Magnetic measurements alone on spherical tori should allow a very good separation of the poloidal beta beta _p, from the internal self-inductance l_i/2 and should even permit an accurate estimate of the current density j_phi profile. However, the reduced space allowed for magnetic sensors near the central conductor in a spherical tokamak, and the possibility of producing flux core spheromak configurations without a central conductor, could imply that magnetic probes are not present in the cavity of the spherical torus. The fluxes and fields of a variety of calculated spherical torus configurations, all endowed with a single or double null separatrix, are analysed in terms of spherical multipolar moments obtained from simulated magnetic measurements located only upon a sphere surrounding the spherical plasma. The solution to the problem of the absence of magnetic measurements in the cavity of the spherical torus is to fix from non-magnetic measurements (e.g., spectroscopy) the plasma inboard boundary r_in on the equatorial plane. This constraint is added to the constraints of matching the spherical multipolar expansion in an iterative solution of the Grad-Shafranov equation, on the basis of a spherical geometry. The convergence of the spherical reconstructive equilibrium code is extremely fast and gives an error on the total plasma current I_p of less than 1% at an aspect ratio A=1.2, an error on the position of the plasma boundary of less than 2% of the radius of the plasma sphere, an error on beta _p of at most 15% and, finally, the j_phi profile is extremely well reconstructed in peaked, flat and even hollow cases. The effect of an uncertainty +or- delta r_in upon the spectroscopic identification of the plasma inboard boundary on the equatorial plane r_in is assessed