K-correction dependence of redshift trends, luminosity-distance offsets, and surface-brightness detection thresholds. (a) k-correction in the r band as a function of redshift. The blue scale shows the histogram of the distribution, while the black dots indicate the mean values in redshift bins. The error bars represent the standard deviation within each bin. The red dashed line labels the k_r(z)∼z relation. (b) Redshift from the maximum luminosity distance versus the difference between values without and with the k-correction. The vertical orange line indicates z = 0.3. (c) Maximum redshift set by the surface brightness limit (z_μ) versus redshift. Solid lines show z_μ obtained with the k-correction, assuming k(z)≈z for illustration, while dashed lines show z_μ computed without k-correction. Different colors indicate results for galaxies with different surface brightnesses. For this test, we set μ_lim = 29. The horizontal gray line indicates z = 0.3, the maximum redshift limit we impose on our sample.