A parametric level-set method for partially discrete tomography

This paper introduces a parametric level-set method for tomographic reconstruction of partially discrete images. Such images consist of a continuously varying background and an anomaly with a constant (known) grey-value. We represent the geometry of the anomaly using a level-set function, which we represent using radial basis functions. We pose the reconstruction problem as a bi-level optimization problem in terms of the background and coefficients for the level-set function. To constrain the background reconstruction we impose smoothness through Tikhonov regularization. The bi-level optimization problem is solved in an alternating fashion; in each iteration we first reconstruct the background and consequently update the level-set function. We test our method on numerical phantoms and show that we can successfully reconstruct the geometry of the anomaly, even from limited data. On these phantoms, our method outperforms Total Variation reconstruction, DART and P-DART.Comment: Paper submitted to 20th International Conference on Discrete Geometry for Computer Imager

Adaptive spectral decompositions for inverse medium problems

Inverse medium problems involve the reconstruction of a spatially varying unknown medium from available observations by exploring a restricted search space of possible solutions. Standard grid-based representations are very general but all too often computationally prohibitive due to the high dimension of the search space. Adaptive spectral (AS) decompositions instead expand the unknown medium in a basis of eigenfunctions of a judicious elliptic operator, which depends itself on the medium. Here the AS decomposition is combined with a standard inexact Newton-type method for the solution of time-harmonic scattering problems governed by the Helmholtz equation. By repeatedly adapting both the eigenfunction basis and its dimension, the resulting adaptive spectral inversion (ASI) method substantially reduces the dimension of the search space during the nonlinear optimization. Rigorous estimates of the AS decomposition are proved for a general piecewise constant medium. Numerical results illustrate the accuracy and efficiency of the ASI method for time-harmonic inverse scattering problems, including a salt dome model from geophysics

Extrapolated full waveform inversion with deep learning

The lack of low frequency information and a good initial model can seriously affect the success of full waveform inversion (FWI), due to the inherent cycle skipping problem. Computational low frequency extrapolation is in principle the most direct way to address this issue. By considering bandwidth extension as a regression problem in machine learning, we propose an architecture of convolutional neural network (CNN) to automatically extrapolate the missing low frequencies without preprocessing and post-processing steps. The bandlimited recordings are the inputs of the CNN and, in our numerical experiments, a neural network trained from enough samples can predict a reasonable approximation to the seismograms in the unobserved low frequency band, both in phase and in amplitude. The numerical experiments considered are set up on simulated P-wave data. In extrapolated FWI (EFWI), the low-wavenumber components of the model are determined from the extrapolated low frequencies, before proceeding with a frequency sweep of the bandlimited data. The proposed deep-learning method of low-frequency extrapolation shows adequate generalizability for the initialization step of EFWI. Numerical examples show that the neural network trained on several submodels of the Marmousi model is able to predict the low frequencies for the BP 2004 benchmark model. Additionally, the neural network can robustly process seismic data with uncertainties due to the existence of noise, poorly-known source wavelet, and different finite-difference scheme in the forward modeling operator. Finally, this approach is not subject to the structural limitations of other methods for bandwidth extension, and seems to offer a tantalizing solution to the problem of properly initializing FWI.Comment: 30 pages, 22 figure

Application of machine learning for the extrapolation of seismic data

Low frequencies in seismic data are often challenging to acquire. Without low frequencies, though, a method like full-waveform inversion might fail due to cycle-skipping. This thesis aims to investigate the potential of neural networks for the task of low-frequency extrapolation to overcome aforementioned problem. Several steps are needed to achieve this goal: First, suitable data for training and testing the network must be found. Second, the data must be pre-processed to condition them for machine learning and efficient application. Third, a specific workflow for the task of low-frequency extrapolation must be designed. Finally, the trained network can be applied to data it has not seen before and compared to reference data. In this work, synthetic data are used for training and evaluation because in such a controlled experiment the target for the network is known. For this purpose, 30 random but geologically plausible subsurface models were generated based on a simplified geology around the Asse II salt mine, and used for finite-difference simulations of seismograms. The corresponding shot gathers were pre-processed by, among others, normalizing them and splitting them up into patches, and fed into a convolutional neural network (U-Net) to assess the network’s performance and its ability to reconstruct the data. Two different approaches were investigated for the task of low-frequency extrapolation. The first approach is based on using only low frequencies as the network’s target, while the second approach has the full bandwidth as target. The latter yielded superior results and was therefore chosen for subsequent applications. Further tests of the network design led to the introduction of ResNet blocks instead of simple convolutions in the U-Net layers, and the use of the mean-absolute-error instead of the mean-squared-error loss function. The final network designed in this way was then applied to the synthetic data originally reserved for testing. It turned out that the chosen method is able to successfully extrapolate low frequencies by more than half an octave (from about 8 to 5 Hz) given the experimental setup at hand. Although the results start to deteriorate in the low-frequency band for larger offsets, full-waveform inversion will overall benefit from the application of the presented machine learning approach