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We present an approach for forming sensitivity maps (or sensitivites) usingensembles. The method is an alternative to using an adjoint, which can be verychallenging to formulate and also computationally expensive to solve. The mainnovelties of the presented approach are: 1) the use of goals, weighting theperturbation to help resolve the most important sensitivities, 2) the use oftime windows, which enable the perturbations to be optimised independently foreach window and 3) re-orthogonalisation of the solution through time, whichhelps optimise each perturbation when calculating sensitivity maps. These novelmethods greatly reduce the number of ensembles required to form the sensitivitymaps as demonstrated in this paper. As the presented method relies solely onensembles obtained from the forward model, it can therefore be applied directlyto forward models of arbitrary complexity arising from, for example,multi-physics coupling, legacy codes or model chains. It can also be applied tocompute sensitivities for optimisation of sensor placement, optimisation fordesign or control, goal-based mesh adaptivity, assessment of goals (e.g. hazardassessment and mitigation in the natural environment), determining the worth ofcurrent data and data assimilation.  We analyse and demonstrate the efficiency of the approach by applying themethod to advection problems and also a non-linear heterogeneous multi-phaseporous media problem, showing, in all cases, that the number of ensemblesrequired to obtain accurate sensitivity maps is relatively low, in the order of10s.

Goal-based sensitivity maps using time windows and ensemble perturbations