As detailed in proxy (climate), borehole temperature measurements at a series of different depths can be effectively "inverted" (a mathematical formula to solve a matrix equation) to help estimate historic surface temperatures.
where n is dimenison of A, and the sum is taken over s and the sets of all kl ≥ 0 satisfying the linear Diophantine equation
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--Surely this was already known by the end of the 19th century.--> who used it for the inversion of geodetic matrices, and Tadeusz Banachiewicz (1937), who generalized it and proved its correctness.
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