  Based on magnetohydrodynamic turbulence simulations, we generate synthetic
synchrotron observations to explore the scaling slope of the underlying MHD
turbulence. We propose the new $Q$-$U$ cross intensity $X$ and
cross-correlation intensity $Y$ to measure the spectral properties of magnetic
turbulence, together with statistics of the traditional synchrotron $I$ and
polarization $PI$ intensities. By exploring the statistical behavior of these
diagnostics, we find that the new statistics $X$ and $Y$ can extend the
inertial range of turbulence to improve measurement reliability. When focusing
on different Alfv{\'e}nic and sonic turbulence regimes, our results show that
the diagnostics proposed in this paper not only reveal the spectral properties
of the magnetic turbulence but also gain insight into the individual plasma
modes of compressible MHD turbulence. The synergy of multiple statistical
methods can extract more reliable turbulence information from the huge amount
of observation data from the Low-Frequency Array for Radio astronomy and the
Square Kilometer Array.
