Multiscale Potential Theory

Freeden, Willi; Michel, Volker

doi:10.1007/978-1-4612-2048-0

Cited by 174 publications

(55 citation statements)

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“…Additionally, the measurements are only given pointwise and are afflicted with noise. The problem of getting the potential from the satellite orbit onto the Earth's surface is the so-called downward continuation problem, which is a severely unstable and, therefore, ill-posed inverse problem (see e.g., [7,8]). …”

Section: Introductionmentioning

confidence: 99%

Study on Parameter Choice Methods for the RFMP with Respect to Downward Continuation

Gutting

Kretz

Michel

et al. 2017

Front. Appl. Math. Stat.

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Recently, the regularized functional matching pursuit (RFMP) was introduced as a greedy algorithm for linear ill-posed inverse problems. This algorithm incorporates the Tikhonov-Phillips regularization which implies the necessity of a parameter choice. In this paper, some known parameter choice methods are evaluated with respect to their performance in the RFMP and its enhancement, the regularized orthogonal functional matching pursuit (ROFMP). As an example of a linear inverse problem, the downward continuation of gravitational field data from the satellite orbit to the Earth's surface is chosen, because it is exponentially ill-posed. For the test scenarios, different satellite heights with several noise-to-signal ratios and kinds of noise are combined. The performances of the parameter choice strategies in these scenarios are analyzed. For example, it is shown that a strongly scattered set of data points is an essentially harder challenge for the regularization than a regular grid. The obtained results yield, as a first orientation, that the generalized cross validation, the L-curve method and the residual method could be most appropriate for the RFMP and the ROFMP.

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Section: Introductionmentioning

confidence: 99%

Study on Parameter Choice Methods for the RFMP with Respect to Downward Continuation

Gutting

Kretz

Michel

et al. 2017

Front. Appl. Math. Stat.

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“…As a starting point, see for example [1,[4][5][6][7]9]. Moreover, at least when low earth orbiters (such as the satellites GRACE, CHAMP, and GOCE) come into play we are confronted with millions of data distributed over dense structured grids.…”

Section: Fields Of Applicationmentioning

confidence: 99%

Locally supported spherical wavelets based on block grids

Schreiner

2007

Proc Appl Math and Mech

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This paper presents a method for approximating spherical functions from discrete data of a block-wise grid structure. The essential ingredients of the approach are scaling and wavelet functions within a biorthogonalisation process generated by locally supported zonal kernel functions. In consequence, geophysically and geodetically relevant problems involving rotationinvariant pseudodifferential operators become attackable. A multiresolution analysis is formulated enabling a fast wavelet transform similar to the algorithms known from classical tensor product wavelet theory. Main IdeaSeveral different concepts for the construction of wavelets are known in the literature. For an overview, see [8]. Within these different methods, two essential approaches can be identified: The first one uses certain grids on the sphere and tries to transfer concepts of the one-dimensional wavelet analysis to the sphere. The approaches with longitude/latitude grids or triangulations fall into this category. The second approach put the emphasis on the solution of rotation-invariant pseudodifferential equations on the sphere (as proposed, e.g., in [3]). For this concept, zonal functions are an appropriate tool. In this case, there is -at first glance -no need for a particular point distribution on the sphere.In this paper we are interested in a new compromise for the construction of spherical wavelets. In fact, our wavelets are based on zonal functions but also relate to structured grids in order to obtain fast algorithms that are easy to implement.The key points of our paper can be characterized as follows:• The wavelets and the scaling function are based on zonal kernel functions so that our approach is well-suited for the solution and regularization of rotation-invariant pseudodifferential equations.• The wavelets are locally supported so that the integration can be made efficient.• The construction is based on a biorthogonal system of zonal functions, which gives us almost all advantages of an orthogonal approach.• A new block-wise grid structure is used on the sphere: The points lie on circles of constant latitude to ensure that the construction of a biorthogonal system is not expensive. But the grid gets sparser in the polar regions so that there are not too large differences in the distances between neighboring points.• The grid is organized in blocks of size 2 p by 2 p−1 so that the algorithms can be formulated easily.• Finally, a scaling equation is established with only a few coefficients. In fact, we end up with a fast wavelet transform which is completely similar to the algorithms known from tensor product approaches of Euclidean wavelet theory.What is the prize that we have to pay for all these advantages? It is by no means dramatic: First, we do not develop a multiresolution analysis of the whole space of square-integrable functions but on finite dimensional subspaces. In fact, we have -at the beginning of the analysis -to choose a finest level of resolution. From numerical point of view, this is what has to be done for p...

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“…In order to investigate earthquake-induced coseismic changes of geoid heights associated with the Tohoku-Oki earthquake, we apply the bilinear multiscale wavelets analysis based on the Abel-Poisson scale and its corresponding wavelet functions as described in [4] to GRACE Level 2 Release 04 monthly gravity data.…”

Section: Introductionmentioning

confidence: 99%

Coseismic geoid height changes of the 2011 Tohoku-Oki earthquake in GRACE monthly gravity data

Park¹,

Hong

2013

Current Research on Material, Architecture and Civil Engineering

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Abstract. A Mw 9.0 earthquake occurred off the coast of Japan in the Pacific Ocean on March 11, 2011. At shorter time and spatial scales, mass redistribution caused by an earthquake produces local variations of the geoid reaching a few centimeters. In this paper, we apply the bilinear wavelets analysis based on the Abel-Poisson scale and its corresponding wavelet functions to GRACE monthly gravity data in order to detect coseismic changes of the geoid heights due to the Tohoku-Oki earthquake. As a result, we can clearly distinguish that multiscale geoid heights are decreased temporarily in the order about -4 meter on average throughout the region between February 2011 and March 2011 when compared with monthly geoid heights observed by GRACE satellite.

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Multiscale Potential Theory

Cited by 174 publications

References 0 publications

Study on Parameter Choice Methods for the RFMP with Respect to Downward Continuation

Study on Parameter Choice Methods for the RFMP with Respect to Downward Continuation

Locally supported spherical wavelets based on block grids

Coseismic geoid height changes of the 2011 Tohoku-Oki earthquake in GRACE monthly gravity data

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