A severe incident with the cargo ship MSC Zoe, which lost hundreds of containers near the Dutch coast, shows how important it can be to detect and localise hidden objects at sea quickly. Also in defence applications localisation of hidden objects in water is important. If the objects are made of magnetic materials, they disturb the Earth magnetic field. This disturbance is called the object's magnetic signature. An aerial drone that carries magnetic field sensors could in principle be able to locate magnetic objects autonomously. Another defence-related application of such a drone would be estimating magnetic signatures of ships under water, using measurements above water - signature translation. For both applications, algorithms need to be developed that process magnetic field data, and computes estimates. In this thesis, data-driven techniques are applied to the localisation problem and the magnetic signature translation problem. For the localisation problem, an algorithm is proposed to localise a magnetic dipole using a limited number of noisy measurements from a sensor array forming a horizontal grid. The algorithm is based on the theory of compressed sensing. In the algorithm, a number of sensors is chosen which perform each a measurement of the three magnetic field components. The sensors can be chosen randomly from the sensor array, but also optimal placement using QR pivoting is considered. Using the obtained field measurements, a sparse representation in the location domain is computed using L1-optimisation. Based on the resulting sparse representation, a classification is produced, which consists of estimates on its location and on its magnetic moment magnitude and orientation. The possibility of performing iterations is explored, where the basis and chosen sensors improve after every location estimate. Using results from simulations as well as experiments, the algorithm is shown to be effective in localising magnetic dipoles. To solve the signature translation problem, two approaches are investigated. One algorithm was developed using the Gappy POD technique, and one using neural networks. Using only a few measurements (a gappy measurement), a full signal can be reconstructed. This method is adapted to also be able to translate from a field above a ship to a field below a ship. Tikhonov regularisation is applied in the process to obtain better results. Second, different neural networks are created, trained using data above and below a ship. Linear and non-linear networks are compared, and standard loss functions are compared to physics-guided losses. Both the Gappy POD based algorithm and linear neural networks are shown to give good magnetic signature estimates. The Gappy POD method is greatly improved by applying Tikhonov regularisation. Results show to improve when more basis modes are considered, and when more sensors are used for measurements. Linear neural networks show the same results as Gappy POD using Tikhonov regularisation, indicating that some regularisation is done while training the neural network. Non-linear neural networks show no improvement over linear ones, and a physics-guided loss function is not needed for a resulting field that obeys known laws of magnetism.