The thermal history modelling of basins is essential for petroleum systems analysis. The methodology involves the derivation of the heat flow history by calibration using a kinetic model to match the predicted and observed vitrinite reflectance values in wells. Although this method is almost universally used by basin modellers, a simple check using two independent methods (with tectonic-subsidence history being the second method) shows that the thermal histories derived from the subsidence history do not match those predicted by the vitrinite reflectance method. For any point in a basin, there can only be one thermal history, and the derivation of two different thermal histories indicates that there is a problem that has been ignored by geochemists and basin modellers. The thermal history must predict the tectonic-subsidence history and this heat flow history must also account for the maturation data, e.g. vitrinite reflectance. Sediments subside under the action of forces, and energy (e.g. heat) is a function of force. During subsidence the downward force exceeds the upward force and the heat flow supplied to the basin must be reduced as the upward force is reduced, while force and heat flow must be increased during uplift. Applying this relationship to the Central Graben, North Sea, a heat flow history is derived that accounts for the rifting and subsequent subsidence (tectonic history) of two wells, the British 23/26-2z and the Danish Lulu-1. The vitrinite reflectance predicted by this thermal history and the Easy%Ro kinetic vitrinite reflectance model are higher than the measured values in both wells. Both wells are overpressured and the problem with the vitrinite reflectance-calibration method is that the kinetic parameters used do not correctly account for the retardation produced by pressure (Le Chatelier's principle). Without the pressure retardation effect incorporated into the kinetic parameters, the modeller has to use lower temperatures to simulate retardation. Using the PresRo(R) vitrinite reflectance model which incorporates pressure retardation as required by thermodynamics, the thermal history derived from the rifting-subsidence history can also be used to model the vitrinite reflectance data in the British 23/26-2z and Danish Lulu-1 wells. This method of deriving an unknown (e.g. thermal history) using two independent models is widely used in physical sciences as an essential check on any prediction, and this scientific rigor has to be followed by geoscientists modelling the thermal history of basins. (C) 2003 Elsevier Science B.V. All rights reserved.