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I wonder how deep does the ground stop becoming colder and begins to get warmer? Moreover, are there any maps or methods to determine this for regions with different elevations and temperatures that does not involve drilling?

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There are some maps for geothermal temperature gradient but all the ones I know of are smaller scale local maps. As a general guide temperature rises by 25-30°C for every kilometre of depth in the continental crust, faster in areas of high geothermal/volcanic activity. This gradient is totally dominant below the 200-300m mark, above that annual temperatures at time of deposition play a role in the temperature of sedimentary rocks and latent heat from emplacement may play a role in igneous rocks.

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