The effects of horizontal heat conduction due to radioactivity variations are investigated within the framework of the linear relation discovered by Birch and colleagues: Q,, = Q, + D . Ao! We study the relationship between the apparent depth-scale which is deduced from this relation and the true depth-scale for uniform and exponential vertical distributions of heat production.There are two different effects of horizontal heat transfer, both related to the smoothing of deep radioactivity variations. The first is the reduction of the apparent depth-scale, as reported by England, Oxburgh & Richardson. The second is the averaging of contributions from neighbouring formations. Because the exponential distribution extends to much greater depths than the uniform distribution for the same value of D , it enhances significantly the magnitude of these effects.We show that the two effects are very important for the large values usually found for D (around 10 km). This leads to two contradictions. First, surface heat flow integrates the heat production distribution over many different geological terrains and it is surprising to obtain a single and good linear correlation with local radioactivity over a whole province. Furthermore, the large thicknesses suggested imply a great sensitivity to formation shape and size which is not reflected in the data. Secondly, once a proper correction for horizontal conduction is made, the true values of D are unrealistically high and in disagreement with the crustal structure deduced from other geophysical studies in the State of New Hampshire and the Sierra Nevada (USA).This paradox can only be resolved if the true depth-scale of radioactive enrichment is smaller than what is indicated by the linear relation. This is probably due to the effects of alteration and weathering which deplete near-surface rocks in uranium and lead in compensation to an artificial increase of depth-scale. Even when these effects are corrected for, we find that the exponential distribution is too sensitive to size, shape and interunit distance to be realistic. The linear relation is made possible by two mechanisms. The first may 412 C Jaupart be the redistribution of radioelements by fluid circulation at the time of granite emplacement. The second is simply horizontal conduction which creates a rough alignment in a Qo versus A. plot whose slope is some complex average of the various thicknesses involved. Alteration then produces another apparent alignment with a larger slope which bears little resemblance to any physical dimension in the province. The linear relation remains a simple useful tool in the absence of more complete data coverage. Q, represents a large-scale average of heat flow which is free from the shallowest radioactivity contrasts and which allows the calculation of representative geotherms. It integrates the heat production distribution in the lower crust over a lateral distance as large as 350km. To compute the mantle heat flow beneath continents, it is therefore necessary to use a model of crus...