All Earth System Models (ESMs) have climatological biases relative to the observed historical climate. The quality of a model and, more importantly, the accuracy of its predictions are often associated with the magnitude and properties of its biases. For more than a decade, new strategies have been developed to empirically reduce such biases in the model components of ESMs during their execution. The present study considers a cyclostationary class of empirical runtime bias corrections to a climate model, referred to here as ERBCs. Such ERBCs are state independent and designed to reduce biases in the climatological annual cycle of the model. We present a new procedure for deriving such ERBCs called Climatological Adaptive Bias Correction (CABCOR). CABCOR is argued to be superior to the standard relaxation approach to defining ERBCs because it requires only a climatological, rather than a multi-year time evolving, observational reference dataset. As part of this study, we perform a novel analysis of the relaxation approach in which a mapping is made between the parameter values that define the relaxation and the biases produced by ERBCs in the corrected model. This allows us to identify the optimal bias correction produced by the relaxation approach and to additionally demonstrate that the CABCOR approach can produce bias-corrected models with smaller climatological biases.