This paper presents a model predictive control (MPC)-based spatio-temporal optimization strategy that is applied to the problem of optimizing the altitude of an airborne wind energy (AWE) system. Altitude optimization for AWE systems represents a challenging problem under which the wind speed at the operating altitude dictates the net power produced by the system. The wind speed varies with both time and altitude and is typically only instantaneously observable at the operating altitude of the AWE system. The MPC strategy proposed in this work avoids the need for a computationally expensive Markov process model for characterizing the wind speed and is structured in a way that the need for instantaneous power maximization (termed exploitation) is balanced with the need to maintain an accurate map of wind speed vs. altitude (termed exploration). The MPC strategy is calibrated through data-driven statistical characterizations of the wind profile and is validated through real wind speed vs. altitude data.