 Seasonal source depth for a reversible heat pump

For a sinusoidal ground-temperature model whose coldest surface day is $t=0$, source temperature on that day is $T_0-T_A{e}^{-x}\cos x$, where $x=z/\delta$. The first and largest thermal maximum occurs at $x=3\pi /4$, not at infinite depth. This maximizes the winter coefficient of performance for a reversible pump with a fixed indoor temperature. With a heating load even about that day, the same optimum minimizes annually integrated reversible heating work. Excavation cost, finite heat extraction and real heat-pump behaviour can change the optimum.