The method of measuring the r-band magnitude distribution of true counterparts. Top panel: magnitude distribution of SDSS objects within 10 arcsec of all SPIRE positions (solid lines) divided into galaxies and quasars (magenta) and stars (blue), and the total (heavy black line). The background magnitude distributions of SDSS (normalized to the same search area) are given by the dashed lines (the background distribution of stars closely follows that of SPIRE centres). Middle panel: the magnitude distribution of true counterparts q(m) is given by the difference between SPIRE and random distributions above, normalized to Q₀. Bottom panel: the ratio of the magnitude distribution of true counterparts to that of background objects is used in the calculation of LR in equation (5). At magnitudes m < 14.0 (galaxies) and m < 21.5 (stars), the value of q(m)/n(m) is fixed at the average within this range, as shown by the horizontal dashed lines.