Best basis construction and comparison of trial functions for ill-posed inverse problems in Earth sciences - studied at the examples of global-scale seismic tomography and gravitational field modelling
Geodesy, Photogrammetry, Remote Sensing, Geoinformatics, Cartography
Mathematics
Final Report Abstract
Amongst the challenges in geomathematics is the resolution of ill-posed inverse problems related to the Earth. Here, we consider approximating a) the gravitational potential at the Earth’s surface from satellite data (downward continuation) and b) the velocity field of seismic waves in the Earth’s interior from travel time data of these waves (travel time tomography). In previous DFG projects, the Geomathematics Group Siegen developed, amongst others, the (L)RFMP ((Learning) Regularized Functional Matching Pursuit) to solve such linear(ized) ill-posed inverse problems. The aim here is: with respect to a) to improve its efficiency in order to process big data sets and with respect to b) to further develop the method to make it applicable to travel time tomography. The LRFMP relies on a dictionary: a basically arbitrary set of trial functions. Every system of basis functions has its own advantages and disadvantages. For this reason, the intended benefit of the LRFMP is the union of diverse bases to a dictionary, which yields a mutual compensation of the disadvantages. In practice, this is realized by a combination of global and localized functions: spherical harmonics (orthogonal polynomials on the sphere) on the one hand and radial basis functions and wavelets on the other hand for the downward continuation as well as orthogonal polynomials on the ball and f inite element hat f unctions (FEHFs) for the travel time tomography. Then the algorithm approximates the desired signal by a linear combination of weighted dictionary elements. These elements and their weights are iteratively determined via the minimization of a Tikhonov–Phillips functional. The more novel learning add-on enables the use of an infinite dictionary with continuously selectable parameters and a significant gain of efficiency. Three aspects enabled us to use high-dimensional data: first, we developed an efficient dictionary. Next, we optimized the source code and determined its bottleneck. For the terms that cause the bottleneck, we then were able to derive a closed form. With these advancements, we now run the algorithm with roughly 500 000 data points given at kinematic orbit rays. The results show that the LRFMP can be tailored for upscaled experiments and yields good results. This is an important step towards experiments with real satellite data. For applying the method to travel time tomography, also three aspects were relevant: the choice of the dictionary elements, of the regularization norm and of an appropriate test data set. Due to the use of finite elements, we opted to utilize a classical Sobolev norm for the regularization term. Our primary objective was a proof of concept regarding the applicability to travel time tomography and the collection of new experiences for the associated realization. For this reason, we used travel time delays from a contrived underlying model in order to be able to quantify the error in the calculated approximation. To obtain nevertheless a scenario which is as realistic as possible, we generated those data for the source-receiver constellations of real earthquakes. As expected, even small experiments already had high computational costs due to the lack of efficient formulae for the forward calculation (in contrast to gravitational field modelling). Thus, we developed an additional divide-and-conquer strategy and adjusted the accuracy parameters in order to consider a suitable number of data. This way, we were able to obtain first results which show that the LRFMP is applicable to travel time tomography. In particular, we observed that the approximation quality can be locally adapted to the data covering: large errors were limited to regions which had been hardly illuminated by earthquakes.
Publications
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(L)IPMP source code for gravitational field modelling, v2-dc-2023. Zenodo, 2023
N. Schneider
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(L)IPMP source code for travel time tomography, v3-tt-2023. Zenodo, 2023.
N. Schneider
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A matching pursuit approach to the geophysical inverse problem of seismic traveltime tomography under the ray theory approximation. Geophysical Journal International, 238(3), 1546-1581.
Schneider, N.; Michel, V.; Sigloch, K. & Totten, E. J.
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High-dimensional experiments for the downward continuation using the LRFMP algorithm. GEM -International Journal on Geomathematics, 16(1).
Schneider, N.; Michel, V. & Sneeuw, N.
