his thesis investigates the modelling of a ship’s magnetic field, by means of an equivalent source model defined on an arbitrarily shaped surface enclosing the source. This setup allows for the use of measurements close to the enclosing surface, such as on the seabed in shallow waters. A multipolar basis is introduced to reduce the dimensionality of the problem, which allows for a convenient mapping to the Decreasing Spherical Harmonic Expansion to describe the far field with low computational costs. It is found that the method performs well in determining a source model and predicting the magnetic field in new locations, provided that enough well-distributed measurements are available. The method remains accurate in the presence of noise, although measurements very close to the enclosing surface reduce performance, a reduction attributed to an approximation that may be improved. Bayesian inference is used to stabilise the method when a low number, badly distributed and/or noisy measurements are available. We find that this regularisation indeed enables the use of the method in such situations. In addition, it is investigated whether orthogonality of the basis is required. It is found that it is not, and that a non-orthogonal basis can yield slightly better results. This suggests that the discretisation of the surface and the choice of solution space of the equivalent source are the main limiting factors in improving the solutions. ... Read more

Magnetostatics play a crucial role in the detection and localisation of naval vessels. Also, minimising a vessel’s magnetic signature is essential to reduce the risk posed by naval mines, which often rely on magnetic detection. This research aims to improve the calculation of magnetic signatures using the Method of Moments (MoM) by implementing it in Julia, a high-performance programming language. A simplified version of TNO’s current MATLAB-based approach is implemented in Julia to establish a baseline for the accuracy and efficiency. Linear basis functions and automatic differentiation (AD) are incorporated into the methodology to explore potential improvements. These extended methods are compared to the baseline to evaluate their performance.
Results show that Julia can be of great value, since it significantly improves the assembly time of the interaction matrix. Point matching is not a suitable approach when using linear basis functions. The Galerkin method shows promising results, though its computational performance remains a significant drawback. Also, using AD shows potential to simplify the implementation of the MoM by eliminating the need for analytical integral expressions. However, AD disappoints in terms of computational performance. Moreover, the AD implementation relies on a mesh-dependent parameter.
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Centuries ago, navigators used compasses to traverse oceans, and compasses remain part of modern Inertial Navigation Systems (INS). Although Global Navigation Satellite Systems (GNSS) are widely used today, they are not always available, for example underground, indoors, in tunnels, or in conflict zones where GNSS can be jammed or spoofed. This motivates research into GNSS-independent navigation methods. Magnetic field-based navigation is a promising alternative, as the Earth’s magnetic field is globally present, relatively stable, and only weakly affected by environmental conditions or human activity at large scales.

Magnetic maps are also used in applications such as resource exploration, archaeology, and geophysical studies. The Earth’s magnetic field consists of contributions from both core and crustal sources. Global magnetic maps are commonly represented using spherical harmonics, which model large-scale fields originating from the Earth’s core. However, at regional scales these models become insufficient due to crustal and near-surface variations. In theory, infinite spherical harmonic expansion could represent the field, but this is not feasible in practice.

To address regional mapping, local extensions of global models are used. Techniques include interpolation methods, dipole approximations, and Equivalent Layer methods. Equivalent Layer formulates a linear inverse problem in which magnetic dipoles below the surface are fitted to measurements. While effective, it requires a priori assumptions on dipole placement. Upward continuation is another key technique, allowing estimation of the magnetic field at higher altitudes using measurements at a lower altitude by exploiting harmonic properties of the field.

This thesis advances magnetic map-making by providing a complete overview of the pipeline, from theory to applications. It reviews magnetic models, their limitations, and spatial resolution effects. It derives the Equivalent Layer formulation from first principles, extending from single dipole cases to multiple measurements. A novel method based on Anderson functions is introduced, enabling magnetic field reconstruction without prior knowledge of source locations and allowing dipole depth estimation. An orthonormalized wavelet extension is also developed.

A Python framework, MagMap, is developed to benchmark mapping techniques on simulated magnetic fields, comparing interpolation and extrapolation performance. The methods are further validated on real-world data, highlighting practical challenges such as noise and measurement distortions from ferromagnetic platforms.

The research is structured around understanding magnetic maps, improving reconstruction techniques, and evaluating their performance under realistic conditions. Key research questions address magnetic map definitions, existing methodologies, dipole depth estimation, interpolation accuracy, noise effects, and applications in navigation and exploration. The work demonstrates that magnetic maps are a viable candidate for regional-scale GNSS-independent navigation, particularly for aeromagnetic applications. ... Read more

The world’s reliability on the Global Position System (GPS) is experiencing more and more vulnerabilities. Not just environmental factors are responsible for GPS inaccuracies, but blocking or spoofing location signals has become more commonly available worldwide. This raises interest in other methods to navigate without 3rd party connections. Using an Inertial Navigation System (INS) is such a method. In this paper an algorithm is created based on the sensor fusion input of acceleration and orientation data. In order to correct itself from integration drift the algorithm makes use of Kalman filtering. Multiple simulations and real-world experiments have been done with the use of a tablet. This has promising results in the computer simulations but showcases some real difficulties when used in practice. This report gives a basis of research in this field, and many more recommendations of extensions are made. ... Read more

Eddy currents are currents which are induced in a conducting object by a varying magnetic field. These currents generate a magnetic field of their own. Modelling this phenomenon constitutes a diverse and challenging set of problems, for which many applications exist. One such application is the degaussing system of a naval ship. To guarantee safety on missions, a ship uses a degaussing system to reduce its magnetic signature. Being able to model the effect of eddy current fields is necessary to improve the accuracy of future degaussing systems.

This thesis will examine how eddy current effects can be modeled, and how such a model can be validated. An analytical solution for a sphere is derived and investigated. A boundary element method (BEM) is implemented, which is able to numerically approximate the electromagnetic fields in terms of the modified magnetic vector potential A* and the reduced magnetic scalar potential. The approximation using the BEM is compared to the analytical solution. The BEM shows promising results, being able to model the shape of the magnetic signature of a sphere accurately. The model is also applied to more realistic geometries resembling naval ships, making it a strong candidate for further development and potential implementation in a future degaussing system. ... Read more

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. ... Read more

In this thesis, an algorithm to model the magnetic perturbation field caused by ships is designed and implemented. A systematic description of methods used for modelling the magnetic signature of ships is given. The algorithm fits coefficients of a prolate spheroidal harmonic expansion of the scalar potential of the magnetic field using a least angle regression method (LARS) modified to implement Lasso regularisation. A Monte Carlo method with model selection based on Akaike's information criterion (AIC) is used to select optimal parameters specifying the prolate spheroidal coordinate system centred on the ship. Furthermore, a method to restrict the degree and order of the harmonic expansion is presented and an extension of the scikit-learn module in Python is given. The predictive power of the model was verified using simulated test data, which showed that the designed model is able to make adequate predictions, but improvements are needed. Different analyses on the inputs of the model showed that the model is succesful for low levels of noise, but is susceptible to overfitting for higher levels of noise. Several recommendations for further research are made. ... Read more

In this thesis a model for interpolation of magnetic fields is constructed using Gaussian processes. This model takes the curl- and divergence-free properties of magnetic fields into account. The Gaussian process regression, or kriging, is tested on both simulated and real data. It is also attempted to reconstruct the magnetization of objects based on measurement data. ... Read more

Magnetic anomaly detection (MAD) is a widely used method for detecting ferromagnetic targets, particularly hidden objects. A common way to localize a magnetic target is to look at the maximum of the magnitude of the magneticeld gradient. In this thesis, we consider a threeaxis total eld gradiometer which measures the gradient of the magnitude of the magnetic eld along three orthogonal axes. Based on the measured data from the three-axis total eld gradiometer, we decompose the inversion problem into two linear systems. By solving those two systems, we get an approximation for the location and moment that we are looking for. For the considered gradiometer, we can improve the parameters if the detailed geometry of the gradiometer is taken into account. ... Read more

This thesis explores different methodes to optimally guide a swarm of drones towards a steel shipping container. It does this using techniques such as Simulated Annealing and Magnetic Anomaly Detection. The drones will use Simulated Annealing in combination with Magnetic Anomaly Detection to estimate where the steel shipping container is located. To optimally share information between the different drones in the swarm four methods are proposed and analysed. ... Read more

Reliable and efficient modelling of magnetic hysteresis in inhomogeneous and aniso-tropic media is an important step in developing a state-of-the-art closed-loop degaussing system for naval ships and submarines, to be developed by TNO and to be used by the Royal Netherlands Navy in an updated generation of naval vessels and submarines. Different models have been proposed to describe the nonlinear and history-dependent nature of ferromagnetic hysteresis at a material level. With a focus on three key differing aspects of models, namely linear versus nonlinear(hysteresis), isotropic versus anisotropic and homogeneous versus inhomogeneous, we attempt to discriminate between the performance of models on the basis of these criteria. More specifically, with increasing model complexity, we have combined Maxwell's equations with four different hysteresis models within the context of a prolate steel ellipsoid, whose ferromagnetic properties evolve under the influence of a uniform applied background field. Among other aspects, the hysteresis models differ in terms of physical motivation, complexity and parameter spaces. In this research, we have analysed four hysteresis models in more detail: The Induced - Permanent magnetization model, The Rayleigh} model, the Jiles-Atherton model and an Energy-Variational model, based on energy balances. The thus derived forward models have subsequently been inverted in order to estimate material hysteresis parameters. With increasing complexity also, twin experiments have been performed. This increasing complexity \textit{temporally} stems from the fact that the hysteresis models named previously, are stated in increasing order of complexity, and can all be modified in order to model anisotropic material by generalizing model parameters to tensors. Spatially, the increase in complexity is caused by the fact that in special cases, namely of uniform ellipsoid magnetization, an analytical formula relating the magnetic field, the background field and the ellipsoid magnetization exists by solving the Poisson partial differential equation on an infinite domain using direct computation with Green's functions. ... Read more

This research presents a data-driven model for the magnetic signature of an object, consisting of linearly reacting isotropicmaterial. From magnetostatics mathematical-physical model is derived for the linear behaviour of the induced magnetization. Data-driven updates for the permanent magnetization are computed from comparisons of the computed magnetic field with measurements from onboard sensors, in order to describe magnetic hysteresis. In order to improve the solutions for ill-posed inverse problems, the Tikhonov regularization method is studied. Furthermore, the performance of the model is examined by a number of twin experiments.
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For my bachelor’s graduation project, I did an internship at TNO, working on a project for the Ministry of Defence. This thesis describes and discusses the project that looked for a prediction model for the magnetic background field on a naval ship. The prediction model that was developed during this project separates the magnetic background field from other existing, observed magnetic fields. Determination of the magnetic background field on naval ships helps the reduction of detection risk. Further research and fine tuning of this model might be of use to The Royal Dutch Navy in order to avoid any risk of detonation of naval mines in threat areas. ... Read more