Emission-weighted effective density from a line ratio

For optically thin lines with local ratio $R(n_e,T)$, their integrated observed ratio is $\int j_1(s)R(n_e(s),T(s))ds/\int j_1(s)ds$. Inverting a uniform-plasma calibration yields an effective diagnostic density. It is generally not the volume mean, nor the density obtained by simply averaging the local densities, because emissivity weighting and the ratio response are nonlinear.