  Search sensitivity to a stochastic gravitational-wave background (SGWB) is
enhanced by cross-correlating detector signals. However, one of the most
serious concerns is the environmental noise correlated between detectors. The
global electromagnetic fields on the Earth, known as Schumann resonances,
produce the correlated noise through the instrumental magnetic couplings. In
this paper, we study the detectability of a SGWB in the presence of the
correlated magnetic noise, using the Fisher analysis based on the analytical
model of the correlated magnetic noise. We find that there is no significant
degeneracy between the SGWB and noise parameters. Marginalizing over the
correlated noise parameters degrades the constraint on each SGWB parameter by a
factor of $\sim2$ at most in the four-detector case, irrespective of the
strength of the magnetic coupling. We also confirm that the forecast results
are robust against the variation of correlated noise parameters and can vary up
to $40\%$ in the realistic range of the coupling parameters for the
second-generation detectors. However, ignoring the correlated noise in
parameter estimation generally leads to a biased constraint on the SGWB
parameters. If the coupling strength is twice as large as expected, this could
result in a serious bias.
