  Axisymmetric, time-independent accretion disc models are necessarily
mean-field theories when applied to turbulent discs since axisymmetry and
stationarity apply only upon averaging. Observations with short exposure times
represent members in the turbulence ensemble, and comparing with theoretical
mean values can falsify the latter only if the theory is additionally supplied
with a quantified precision. We previously identified "intrinsic" and
"filtering" errors as two contributions to the precision of mean-field
theories. Here we generalize to cases where the turbulent scales vary in space.
In the geometrically thin, optically thick accretion disc models, disc emission
at a given frequency has non-local contributions from the entire disc, and we
develop a framework to propagate local fluctuations to the integrated spectrum.
In addition, the nonlinear relation between the disc temperature and overall
spectrum contributes a systematic "mismatch error" that has not been previously
identified. The total error in the calculated disc spectrum is also sensitive
to the choice of binning used to compare with data from a telescope of given
temporal and spectral resolving powers. For the thin disc example, we show that
precision errors are generally small, but vary with emission frequency. In
general this means that accurately assessing falsifiability would require a
frequency-dependent likelihood function. The thin disc exemplifies our more
important purpose: to highlight the need and provide a framework for
calculating the imprecision of a mean-field theory.
