  A critical decision process in data acquisition for mineral and energy
resource exploration is how to efficiently combine a variety of sensor types
and to minimize total cost. We propose a probabilistic framework for
multi-objective optimisation and inverse problems given an expensive cost
function for allocating new measurements. This new method is devised to jointly
solve multi-linear forward models of 2D-sensor data and 3D-geophysical
properties using sparse Gaussian Process kernels while taking into account the
cross-variances of different parameters. Multiple optimisation strategies are
tested and evaluated on a set of synthetic and real geophysical data. We
demonstrate the advantages on a specific example of a joint inverse problem,
recommending where to place new drill-core measurements given 2D gravity and
magnetic sensor data, the same approach can be applied to a variety of remote
sensing problems with linear forward models - ranging from constraints limiting
surface access for data acquisition to adaptive multi-sensor positioning.
