  An accurate and detailed field map is important for cyclotron beam dynamics
studies. During the long history of cyclotron studies, many techniques have
been developed by cyclotron pioneers for the treatment of median plane field
map. In this paper, we take the TRIUMF 500 MeV cyclotron as an example to study
the asymmetric field resulting from the imperfect median plane symmetry. The
``Gordon approach'' and a highly accurate compact finite differentiation method
are used to investigate the historical field survey data. The redundancy in the
survey data is revealed by the expansion method, which also makes it possible
to correct the error in the measurement. Finally, both the azimuthal field
$B_\theta$ and the axial gradient of the axial field $dB_z/dz$ in the median
plane are corrected using the radial field map $B_r$. The influence of the
correction is examined by recalculating the equilibrium orbit properties of the
TRIUMF cyclotron. The result shows significantly increased vertical centering
errors of the closed orbits. Further simulation study suggests that these
centering errors can be reduced to below 1.5 cm by adjusting the trim coils'
$B_r$ field within the output limits of our trim coils' power supplies. The
error in the measurement field data may explain why the calculated trim coils'
settings during the cyclotron commissioning in 1974 encountered difficulty.
