A general scheme for constructing orthogonal wavelets on locally compact Abelian groups is suggested. This scheme can be realized in the cases related to some generalizations of the Schoenberg interpolating theorem and the Shannon–Whittacker–Kotel'nikov sampling theorem (see [1]). For one of these cases we obtained a generalization of the classical Littlewood–Paley decomposition.

For searching precursors of strong earthquakes, a concept of wavelet-aggregated signal is proposed. Qualitatively, an aggregated signal could be defined as such a scalar signal which accumulates in its own variations only those components, which are presented simultaneously in each scalar time series of the multi-dimensional signal to be analyzed. The main purpose of constructing the aggregated signal is to make clearer common tendency in data flow in geophysical networks, which indicate an increasing of collective behavior. The procedure of aggregated signal construction includes combining of methods of canonical correlations and principal components applied to wavelet coefficients of initial time series (see [2]).

The library of the subroutines for compression of the seismic information is developed. This library has the following functions:

• the preservation of compressed seismic trace in a file,
• change of factors of compression of the seismic trace,
• reception of the service information about compressed seismic trace (factor of compression, trace length, volume of the information in the compressed and not compressed data and so on).

References

1. Farkov Yu. A. Orthogonal wavelets on locally compact Abelian groups // Funkts. Anal. i Prilozhen. 1997. Vol. 31. No. 4. P. 86–88 (in Russian). English translation: Funct. Anal. Appl. 1997. Vol. 31. No. 4.
2. Lyubushin A. A. (Jr.) Wavelet-aggregated signal and synchronous peaked fluctuations in problems of geophysical monitoring and earthquake prediction // Fizika Zemli. 2000. No. 3. P. 20–30 (in Russian). English translation: Izvestiya, Physics of the Solid Earth. 2000. Vol. 36.

W. F r e e d e n (University of Kaiserslautern, Germany)
Regularization wavelets and multiresolution

Many problems arising in technology can be formulated as compact operator equation of the first kind. Due to the ill-posedness of the equation, a variety of regularization methods are being considered for an approximate solution, where particular emphasis must be put on balancing the data and the approximation error.

In this lecture the interest lies in an efficient algorithmic realization of a special class of regularization methods. More precisely, we implement regularization methods based on filtered singular-value decomposition as a wavelet analysis. This enables us to perform, for example, Tikhonov–Philips regularization as a multiresolution analysis. In other words, we are able to pass over from one regularized solution to another by adding or subtracting so-called detail information in terms of wavelets. It is shown that regularization wavelets as proposed here are economically applicable in future satellite problems, e.g. satellite gravity gradiometry. Furthermore, a convergence rate and a stop strategy in terms of regularization wavelets are mentioned for the regularized solution from (error-affected) gradiometer data. Finally, the multiscale concepts are illustrated for the numerical computation of the Earth's gravitational field from satellite data.

А. И. Г л у ш ц о в (Белорусский государственный университет, Минск, Белоруссия)
Применение всплесков к решению задач дифракции

Как известно, строгие методы решения дифракционных задач в случае произвольного рассеивателя состоят в получении интегральных уравнений первого или второго рода относительно плотности наведенного тока, последующей их дискретизации и решении возникающих систем алгебраических уравнений. Для хорошей аппроксимации требуется порядка десяти точек разбиения контура на длину волны, что в случае коротких волн или больших поперечных размерах рассеивателя приводит к системам из нескольких сотен уравнений с матрицами общего вида, прямые методы решения которых требуют значительных вычислительных ресурсов. Обсуждаемый в докладе метод состоит в предварительном применении к системе уравнений каскадного алгоритма, эквивалентного разложению плотности тока по базису всплесков Добеши [1]. Матрица преобразованной системы обладает замечательным свойством: значительное число ее элементов близко к нулю, что связано с наличием у всплесков Добеши нулевых моментов. Заменив нулями матричные элементы, не превосходящие по модулю некоторого порогового значения, получаем систему уравнений с разреженной матрицей, которая эффективно решается итерационными методами. Суммарная сложность такого алгоритма пропорциональна квадрату размерности системы алгебраических уравнений, т. е. на порядок меньше, чем в методе Гаусса. В работе рассматривается применение всплесков к плоским и пространственным задачам дифракции.

Дальнейшее уменьшение вычислительной сложности рассматриваемых алгоритмов связано с адаптивным учетом в базисе лишь тех всплесков, которые вносят существенный вклад в решение. В случае длинных волн, когда поле близко к стационарному, можно заранее оценить величину матричных элементов преобразованной системы и провести пороговое обнуление малых элементов без их вычисления [2].

Литература

1. Goswami J. C., Chan A. K. Fundamentals of wavelets: theory, algorithms and applications. Texas A&M University, John Wilay & Sons, 1999.
2. Beylkin G., Coifman R., Rokhlin V. Fast wavelet transforms and numerical algorithms // Comm. Pure and Appl. Math. 1991. Vol. 44. P. 141–183.

T. M a i e r (University of Kaiserslautern, Germany)
A vector wavelet approach to iono- and magnetospheric geomagnetic satellite data

Though mankind is not able to sense the geomagnetic field, it is directly influencing life on Earth. The magnetic field deflects the solar wind for example, thus providing useful protection from highly energetic charged particles which otherwise would harm most living beings. Measurements show, however, that the main part of the geomagnetic field is decreasing and that this effect seems to be speeding up recently. In order to keep track of this situation and the resulting effects on the solar wind, precise knowledge of the time and space dependent behavior of the Earth's magnetic field is needed.

The standard technique of geomagnetic field modelling is known as Gauss–representation, i.e. the spherical harmonic expansion of a scalar geomagnetic potential. The expansion coefficients are chosen in a way, that the gradient of the potential fits – in the sense of the L₂-metric – the given vectorial data as good as possible. To guarantee the existence of such a geomagnetic potential, one assumes the corresponding magnetic field to be curl-free which, in connection with Maxwell's equations, means that no electric current densities must be present at the place where the measurements are taken. For Earth-bound or low-atmosphere surveys this is valid, but satellite missions, like MAGSAT or the new CHAMP, acquire their data in the ionosphere where significant electric current densities can be found. Therefore, the magnetic field, as measured by satellites, cannot be considered to be a gradient field anymore but also contains magnetic contributions from currents on the satellite's track. From a theoretical point of view, this problem can be resolved by using the so-called Mie–representation, i.e. by splitting the magnetic field into poloidal and toroidal parts. The poloidal fields can be shown to be due to purely tangential current densities, while the toroidal field is created by radial current densities crossing the satellite's orbit.

There remains the question of how to numerically obtain the Mie–representation of a given set of vectorial data. [4] introduced a method based on the spherical harmonic analysis of scalar functions which are closely related to the poloidal and toroidal vector fields. This technique, however, involves the evaluation of spherical harmonics which, due to the polynomial character of the harmonics, is numerically disadvantageous. We will present here so-called spherical vectorial wavelets (e.g. [1]) which, completely circumventing the computation of spherical harmonics, enable us to directly approximate a given vectorial data set and immediately yield a decomposition into the poloidal as well as the toroidal field contributions (see e.g. [2], [3]). Imbedded into a vectorial multi-resolution background, the wavelets show – in contrast to the spherical harmonics – strong localization properties in the space domain and therefore additionally give us the possibility of local reconstructions as well as efficient data compression.

References

1. Bayer M., Beth S., Freeden W. Geophysical field modelling by multiresolution analysis // Acta Geod. Geoph., Hung. 1998. Vol. 33. No. 2–4. P. 289–319.
2. Bayer M., Maier T., Freeden W. A vector wavelet approach to iono- and magnetospheric geomagnetic satellite data // J. Atmos. Terr. Phys. 2001. Vol. 63/6. P. 581–597.
3. Maier T. Multiscale Analysis of the Geomagnetic Field. Diploma Thesis, Geomathematics Group, University of Kaiserslautern. 1999.
4. Olsen N. Ionospheric F Region Currents at Middle and low Latitudes Estimated From Magsat Data // J. Geoph. Res. 1997. Vol. 102. P. G4563–4576.