Increased melt rates under floating ice shelves around Antarctica have been suggested as a dominant cause for observed acceleration of marine ice sheets that feed these ice shelves. The associated melt rates are difficult to observe directly. We present first steps towards estimating the melt rates underneath floating ice shelves from accessible hydrography data and optimal control methods. We address to which extent ocean hydrographic observations away from the ice-ocean boundary can be used to constrain sub-ice shelf melt rates. We derive sensitivity patterns of sub-ice shelf melt rates to changes in ocean circulation underneath the Pine Island Ice-Shelf in the Amundsen Sea Embayment. The sensitivity patterns are computed with an adjoint model of a full-fledged ocean general circulation model that resolves the sub-ice shelf circulation and includes a thermodynamic melt rate parameterization. The adjoint state can be used to identify dominant water mass pathways and time scales that affect melt rates, provide guidance for oceanographic field campaigns.