  Permanent magnets with high-permeability yokes have been widely used in watt
balances for supplying a robust and strong magnetic field at the coil position.
Subjected to the mechanical realization, only several $B_r(z)$ (radial magnetic
field along the vertical direction) profiles can be measured by coils for field
characterization. In this article, we present an algorithm that can construct
the global magnetic field of the air gap based on $N(N\geq1)$ additional
measurements of $B_r(z)$ profiles. The proposed algorithm is realized by
polynomially estimating the $B_z(r)$ function with analysis of basic relations
between two magnetic components in air gap, i.e., $B_r(r,z)$ and $B_z(r,z)$,
following the Maxwell's equations. The algorithm, verified by FEM simulations,
can characterize the three-dimensional contribution of the magnetic field for a
watt balance magnet with acceptable accuracy, which would supply basic field
parameters for alignment and misalignment corrections.
