Magnetic measurements alone on spherical tori should allow a very good separation of the poloidal beta beta p, from the internal self-inductance li/2 and should even permit an accurate estimate of the current density jphi 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 rin 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 Ip 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 jphi profile is extremely well reconstructed in peaked, flat and even hollow cases. The effect of an uncertainty +or- delta rin upon the spectroscopic identification of the plasma inboard boundary on the equatorial plane rin is assessed