Glaciers’ mass balance and melting patterns can be monitored through the study of their facies. Whilst using Synthetic aperture radar (SAR) remote sensing data facilitates glaciology observations as it can be used under all weather conditions, the systemic backscatter intensity decay due to incidence angle (IA) variation makes classification even more challenging on such complex terrain. We investigate the classification accuracy of glacier facies using SAR data through a supervised learning algorithm that incorporates class-dependent local incidence angle correction. Focusing on the Holtedahlfonna and Kongsvegen glacier complexes in northeast Svalbard, we pre-process SAR data from Sentinel-1 using the SNAP toolbox. We then compare three Bayesian classifiers: one without any IA correction, one with a common IA slope correction, and the third incorporating a class-dependent IA slope correction. Our results show that per-class IA slope correction on training regions improves the models by around 30% compared to the naive one and around 10% from the common IA slope correction. When tested on the glaciers, their firn line could be mapped from 2017 to 2023 and a general retreat of 400-500 m is observed, changing to 3-4 km in some regions of Holtedalhfonna. However, when looking at regions of lower altitudes, regions with crevasses are largely misclassified. To aid crevasse classification, we finish this study by providing some insights on potential texture features, using either standard deviation or spatial Fourier transforms. In all, this work explores the extent to which class-dependent IA corrections should be included in SAR data analysis, contributing to enhanced glacier monitoring and climate research.

This thesis delves into glacier dynamics using the 2D Shallow Ice Approximation enriched by satellite remote sensing data on glacier surface velocities and ice thickness, aiming to refine empirical laws for better predicting glacier movements. The integration of such data has been pivotal, markedly enhancing model calibration despite challenges like steady-state assumptions necessitated by data scarcity. This underscores the critical role of high-quality, temporally resolved data in modeling glacier dynamics accurately.
A significant advancement was the implementation of spatial stratification, which notably improved model performance—reducing Root Mean Square Error (RMSE) by up to 30% and elevating the coefficient of determination (R²) by 0.2 to 0.4 across different regions. This highlights the potential of fully distributed inversions to capture the complex variability of glaciers. Employing Julia for its computational efficiency proved effective for large-scale modeling tasks, setting a promising foundation for future research aimed at understanding and predicting glacier responses to climate change. It is recommended to utilize geostatistical interpolation methods for inverting glacier characteristics from sparse data, in order to acquire these characteristics across the entire glacier area.

Ultrasound images are typically generated using the Delay-And-Sum (DAS) method, which assumes a homogeneous propagation medium. When an aberrating layer is situated between the sensor array and the imaging target, this assumption does not hold, and DAS is replaced with model-based methods. These methods are computationally expensive and require to accurately model the aberrations caused by the layer. This thesis investigates novel methods for image formation and aberration estimation. The effect of the layer is described using a set of transfer functions from the sensor array to a virtual array placed after the layer. In the first part, we assume the transfer functions are known, and we propose a new method for image formation. The transfer functions allow to map the signal from the sensor array to the virtual array, and the DAS method is used on the virtual array signal. This technique is equivalent to model-based matched filtering in terms of image quality, without requiring expensive matrix computations. In the second part, the transfer functions are unknown, and a novel technique is introduced for their estimation. Using pulse-echo data, a focus-quality metric is computed to quantify the accuracy of the transfer function estimate. The transfer functions are modeled using a dictionary and the dictionary coefficients are iteratively updated to increase the defined metric. The optimization leads to improved focus quality and sharper images. In the case the layer model requires a limited dictionary, the proposed algorithm generates an accurate estimate of the transfer functions.

Imaging by inversion of acoustic or electromagnetic wave fields have applications in a wide variety of areas, such as non-destructive testing, biomedical applications, and geophysical explorations. Unfortunately, each modality suffers from its own application specific limitations, typically being difficulties in distinguishing different materials or tissues from each other in the case of acoustic wave fields and a low spatial resolution in the case of electromagnetic wave fields. To exploit the advantages of both imaging modalities, methods to combine them include image fusion, usage of spatial priors and application of joint or multi-physics inversion methods. The latter can be based on empirical relations between acoustic and electromagnetic medium properties or on structural similarity. In this work, two joint inversion algorithms based on structural similarity are presented. To account for the structural similarity the error-functional of standard Born inversion is extended with an additional penalty term. This additional term is either based on the L2-norm of the cross-gradient (CG), i.e. the cross product of the gradients of the acoustic and electromagnetic contrasts or on the L2-norm of the gradient difference (GD), i.e. the difference between the normalized gradients of both contrasts. To test the proposed methods, two synthetic models are considered; one with the gradients of the contrasts pointing in the same direction and one where the gradients point in opposite directions. Results show that the GD constraint significantly improves the resolution for the electromagnetic reconstruction compared to separate BI. The mean square errors (MSE) of the reconstructed profiles for the separate BI are 0.12 for the acoustic and 0.51 for the electromagnetic case, and for the joint GD inversion, 0.09 for the acoustic and 0.46 for the electromagnetic case. The joint GD inversion fails when using the model with the gradients of the contrasts pointing in opposite directions. The joint CG inversion does not enhance the reconstructed images, but shows similar performances for the different models. In conclusion, joint inversion based on structural constraints is shown to improve the electromagnetic resolution, especially using the GD constraint. Further research needs to be conducted to extend the functionality of the GD constraint to acoustic and electromagnetic contrasts with opposite contrast gradient directions.

Current research on osteoporosis detection and transcranial imaging suggests that it is necessary to find a method to calculate the speed of sound in curved bones. A popular method to find the speed of sound in bones is by the bidirectional axial transmission technique. Currently, this method assumes that the arrival time of the head wave depends linearly on the source-receiver distance. For curved bones this assumption is not valid. The arrival time of the head wave as a function of the source-receiver distance follows a curved trajectory. In this report, we attempt to find an extension to the current method that also works for curved bones, and evaluate how accurate the results of this extended method are.

This extension is based on the semblance method, that quantifies how well a certain trajectory agrees with the received data. In the first part of our project, theoretical trajectories are calculated for flat and semicircular bones. Furthermore, a numerical method is discussed to find these trajectories for arbitrary geometries. Comparing the trajectories corresponding to different velocities using the semblance methods allows for the determination of the speed of sound.

The accuracy of this extended method is evaluated by computer simulations. In these simulations, a Philips P41 Cardiac Sector Probe consisting of 96 elements was used to perform measurements on three different geometries. These geometries consisted of three different single interfaces between water (simulating soft tissue) and bone. The first interface consists of a straight interface between a 2 mm thick layer of water and a 5 mm thick layer of bone (that continues until the bottom edge of the simulation). The second simulation is the same, but the interface is rotated 2 degrees around its midpoint. In the third and fourth simulation, the interface is a semicircle with a 50 mm and 40 mm radius, respectively. The speed of sound in the bone was taken to be 3 mm/s. The received data was interpolated by a cubic interpolation with a refinement of 4. Both the bidirectional and the extended,
trajectory-based method were applied on the raw and interpolated data.

For the flat interface, the bidirectional method performed slightly better than the trajectory-based method (0.13% and 0.30% error, respectively). The same is true for the tilted interface (0.16% and 0.23% error). Interpolation did not affect the results here. For the 50 mm semicircular interface, the trajectory-based method performed much better than the bidirectional method: it had a 0.10% error compared to a 2% one. This was also true when the interpolation was applied: then the errors were 0.3% and 1.97%. Similar
improvements were seen for the 40 mm radius semicircle. The higher error of the trajectory-based method for the flat and tilted interfaces can be explained by small errors in the additional assumptions that this method makes.

One drawback of the trajectory-based method is that there can be problems interpreting the results. This can be seen in the semblance plots, which plot the semblance coefficient (how well the trajectory fits the data) against the test velocity. For the bidirectional method, these plots clearly have a single biggest peak. For the trajectory-based methods the plot contains multiple large peaks. Furthermore, for the semicircular bone this plot contains a flat peak, which can cause multiple test velocities to have approximately the same semblance. This is a problem, since it means that small changes to the experimental setup can cause big changes in the calculated velocity. Indeed, this can be seen for the trajectory-based method on the 40 mm radius semicircular bone, which has a 1.6% error without interpolation (0.43% with interpolation). Further research could examine solutions to this problem, such as increasing the window size of the semblance method, taking the average velocity over flat peaks, or using an alternative to the semblance method. Finally, we conclude that the trajectory-based method represents an improvement over the bidirectional method when examining curved bones.

Magnetic Resonance Fingerprinting (MRF) is a relatively new approach for simultaneously estimating multiple quantitative maps in one acquisition. Sequence optimisation for MRF can be a powerful tool in increasing the accuracy an precision of the quantitative results. Multi-component analysis in the MRF framework can distinguish multiple different tissues in one voxel such as myelin water and white matter which play an important role in monitoring progressive diseases such as multiple sclerosis. Using the estimation theoretic Cramér-Rao bound, optimisations of the acquisition sequences can be performed, that increase the precision of the resulting tissue maps. The effect of this optimisation has been confirmed using numerical simulations. Speed-ups in MRF are generated using significant undersampling of the k-space information. This results in spatially coherent undersampling artefacts, that generally is the dominating error source for regular $T_1$ and $T_2$ mapping. The undersampling artefacts can be predicted using a mathematical model leveraging on techniques from perturbation theory. Numerical simulations suggested that optimisations of the acquisition parameters are effective in reducing the undersampling error. This was confirmed using in vivo scans. The optimisations resulting from these two different models are easily implemented in future clinical practice.

This essay shows a two dimensional implementation of the finite element method for the Westervelt equation. To do this the finite element method is first applied to the linear wave equation, then to non-linear diffusion and finally to the Westervelt equation. Both an element by element and a faster vectorized implementation are given for the finite element method. To verify the numerical solution two analytical solutions are used. The first is a one dimensional wave and the second a circularly symmetric wave.

We found that the two-dimensional implementation was successful in computing the Westervelt equation. The error of the solution scales with the mesh size with a power of around 1.7. It was also found that the time step used to compute the solution needs to be small enough for the implementation to converge.

A model is designed for solving gravitational profiles of self-gravitating and rotating planets via the use of Poisson's equation for total gravity, i.e., the sum of the gravitational and rotational potential. Poisson's equation is a partial differential equation that is solved with the usage of Green's functions. This is a major advantage because Earth's surface is an equipotential, which leads to a Dirichlett boundary condition. We apply this methodology to a spherical Earth, where the Green's function is closed. It is demonstrated that, when this Green’s functions based methodology is applied to a homogeneous density, a linear spherical density and the PREM density profiles (where all density profiles are spherically symmetric), then discontinuities occur in the accelerationof gravitation across the surface of the Earth, in the range of 0.01 %- 0.13 %. Such an effect is a symptom of incompatibility with the laws of physics for geostaticalequilibrium, and it is certainly significant as compared to the numerical accuracy of the solution. Moreover, the discrepancy is dependent on geographical latitude, as was to be expected, because it somehow reflects the centrifugal effect. The conclusion is made that this method is indeed capable of detecting, and even quantifying, the incompatibility of spherically symmetric mass distributions with the fundamental laws of physics for self-gravitating and rotating planets. This opens a way -and this is one of the main results- to computing corrections to spherically symmetric mass distributions, such as the PREM model. Based on this method, a further way forward to this end is suggested.

Creating accurate engineering models for predicting the response of large structures often requires geotechnical modelling of soil behaviour. Soil material damping is an important input parameter for modelling the structural response and energy dissipation, especially for embedded and lightly damped structures, as in particular for offshore wind turbines. Non-invasive in-situ measurements like the multichannel analysis of surface waves, offer cost-effective solutions and estimations over a wider volume of soil. This research aims to demonstrate that MASW technique can be used to generate a reliable in-situ estimation of the soil damping in marine environments in the depth range of 0-30m.
A unique and high-quality dataset has been made available through a collaboration with the Norwegian Geotechnical Institute (NGI). Shear wave data are rarely available for offshore site investigations and have not been used before for a damping inversion from real surface waves measurements. Moreover, also the horizontal soil response was recorded in this measurements which allow for reducing the uncertainties of the results obtained by using the Scholte wave model. Therefore, two forward waves models are implemented in Matlab to reproduce both the in-line and cross-line measurements.
A reliable shear wave velocity profile is derived via the stiffness inversion method, after which the damping inversion is performed. A modal damping inversion method is developed and it uses a novel modified random search algorithm in order to estimate the material damping profile over depth. The attenuation coefficient is chosen as the reference parameter for the damping inversion and the misfit function for the damping inversion is defined as the normalized difference per frequency of the attenuation curves. The measured attenuation coefficient is extracted based on a modified half-power bandwidth method. The modelled attenuation curve is retrieved as the imaginary part of the complex wavenumber. The wavelet compression technique is employed to reduce the number of roots used in the inversion for both the experimental and the theoretical curves. The identification requires solving an inverse problem with a global optimization method. To get a better understanding of the model and computational time, a combination of sensitivity studies, behaviours of the phase damping ratio curves and layer reduction were performed. Then, synthetic
inversions are run to verify the validity of the proposed technique. Finally, the
method is applied to the aforementioned collected data and two damping profiles (Scholte and Love models) are computed and they show good agreement in terms of both trend and magnitude.
The frequency dependence of the material damping ratio of the soil is analysed
from the use of measured surface wave data. A novel technique is proposed for retrieving this relation starting from the results of the damping inversion and it shows that the material damping ratio is independent from frequency. Then, two new alternative approaches are proposed, which are based on a simplification of the traditional dependency relation. The first confirms the frequency independence of damping while, although it seems very promising, the second one cannot be tested due to computational limitations. The demonstrated frequency independence of the material damping is only valid based on the frequency ranges available in the dataset and cannot be generalized to cover all scenarios.

This work determines whether the amount of frequency components present in the data can be reduced, whilst still retaining image quality, whereas most efforts in seismological research are done in reducing spatial sampling. It is shown using a PCA on the frequency spectra of several data sets that indeed a large redundancy in frequency content is present in onshore seismic data, and an attempt is made to generate a distribution of frequencies in order of importance. Given this redundancy in the frequency spectrum of onshore seismic data, it has been attempted to reconstruct the missing frequencies by applying the Fourier transformation iteratively to the data. However, this transform does not take spatial sampling into account, which is aimed at to compensate for the missing frequencies. Therefore it has been elected to use a linear Radon transformation instead, which keeps components which are connected in space-time connected in the transform domain. A CGNE scheme has been set up to reconstruct the data, which performs very well along the almost linear asymptots in the shot records, up to a reduction of 70% of frequency components. This scheme iteratively applies the linear Radon transform to a shot record, weighing the data in the transform domain with an amplitude based norm. The energy that was spread out due to aliasing because of the missing frequencies is refocused to the main reflectors, especially along the asymptots of the reflection hyperbola. Missing frequencies are reconstructed, up to a scaling factor, and band gaps of up to 6Hz get filled in very well. Next, it is attempted in this work to give quantitative quality metrics, to make comparison between seismic images easier and based on data, rather than subjective visual inspection. Treating migration as a black box, several quality metrics have been devised for the migrated sections: correlation to the ground truth, contrast within an image, average length of found lines, and local SNR. Contrast is not a very good metric to compare between images as its average across an image is almost constant with reduction percentage. The other parameters are good metrics and show a clear trend that the fewer frequency components present in the shot records, the worse the quality of the final image. An increase in deterioration of image quality is observed around 70% reduction, which is in correspondence with the earlier found value for the shot records.