  We modified the modal expansion, which is the traditional method used to
calculate thermal noise. This advanced modal expansion provides physical
insight about the discrepancy between the actual thermal noise caused by
inhomogeneously distributed loss and the traditional modal expansion. This
discrepancy comes from correlations between the thermal fluctuations of the
resonant modes. The thermal noise spectra estimated by the advanced modal
expansion are consistent with the results of measurements of thermal
fluctuations caused by inhomogeneous losses.
