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Multiresolution Tectonic Features Over the Earth Inferred from a Wavelet Transformed Geoid

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Multiresolution Tectonic Features Over the Earth Inferred from a Wavelet Transformed Geoid Vis Geosci (2003) 8: 26–44 DOI 10.1007/s10069-003-0008-8 ORIGINAL PAPER Ludeˇ k Vecsey Æ Catherine A. Hier Majumder David A. Yuen Multiresolution tectonic features over the Earth inferred from a wavelet transformed geoid Received: 25 September 2002 / Revised: 18 November 2002 / Accepted: 18 November 2002 / Published online: 1 April 2003 Ó Springer-Verlag 2003 Abstract Geoid signals give information about the un- structures that can only be picked up visually with much derlying density structure and can be used to locate the higher resolution spherical harmonic gravity data. We source depthof themass anomalies. Wavelet analysis have also looked at the wavelength at which the maxi- allows a multiresolution analysis of the signal and per- mum signal occurs over a range of scales. This method, mits one to zoom into a specific area bounded by a known as E-max and k-max, is especially effective for particular lengthscale. Theability of wavelets to resolve detecting plate tectonic boundaries and ancient suture the geoid signal into individual wavelength components zones along withareas of strong non-isostatic gravita- without losing the spatial information makes this tional potential due to high differential stress. These method superior to the more common spherical har- areas are likely to be at high risk of earthquakes. These monic method. The wavelet analysis allows one to zoom methods will be especially useful to future studies of the into a specific area and look at the regional geology. We geoid potentials of other planets, such as Mars and have used a wavelet transform of the geoid to study the Venus, since they will allow careful studies of the regional geology of Japan and the Philippine Plate, regional geology variations withgeoid data of the SouthAmerica, Europe, NorthAmerica, East Africa resolution available from satellites. and the Middle East, India and the Himalayas, China and Southeast Asia, and Australia. By filtering the Keywords Geoid Æ Wavelets Æ Tectonics Æ Plate Earth’s geoid anomalies with 2-D Gaussian wavelets boundaries at various horizontal length scales, one can detect the subduction zones along SouthAmerica, theAleutians, and the western Pacific; the Himalayas; the Zagros Mountains; the Mid-Atlantic ridge; and the island Introduction chains of the mid-Pacific. We have processed geoid data witha horizontal resolution down to approximately Geoid anomalies are undulations of the Earth’s equi- 200 km. Using an adjustable wavelet, one can detect potential surface withrespect to a reference ellipsoid and have been used to constrain the viscosity structure of the entire mantle in the past two decades (Hager 1984; Reviewed by Prof. Shcherbakov and Prof. Bergantz. Ricard et al. 1984; Forte and Mitrovica 1996; King 1995; L. Vecsey Cˇ adek et al. 1995; Kido and Cˇ adek1997; Cˇ adek and Geophysical Institute, Academy of Sciences of the Czech Republic, Fleitout 1999). There are various mechanisms for pro- Prague, CzechRepublic ducing geoid anomalies, depending on the horizontal L. Vecsey wavelength, such as the rotation of the Earth, which can Department of Geophysics, Faculty of Mathematics and Physics, cause sea-level fluctuations and operates over long Charles University, Prague, Czech Republic horizontal wavelengths (Sabadini et al. 1990). Longer C. A. Hier Majumder (&) Æ D. A. Yuen wavelength geoid anomalies have sources in the lower Department of Geology and Geophysics, University of Minnesota, mantle (Chase 1979) and shorter wavelength anomalies 310 Pillsbury Dr. SE, Minneapolis, MN , 55455-0219, USA are caused by heterogeneities in the lithosphere (Hager E-mail: [email protected] Tel.: +1-612-6246730 1983; Le Stunff and Ricard 1995). One can use the geoid Fax: +1-612-6255045 locally to study a particular area (Calmant and Cazen- C. A. Hier Majumder Æ D. A. Yuen ave 1987) or globally to probe the deep mantle (Cazen- Minnesota Supercomputing Institute, University of Minnesota, ave et al. 1989). The gravity dataset continues to grow Minneapolis, MN, USA with the current GRACE and CHAMP missions, which 27 should allow measurements of the gravity field to demonstrate that, although this geoid model may not spherical harmonic degree and order 160 from satellite have the highest resolution, the wavelet-filtered version data alone withsurface data being needed only for will show some impressive fine features. First, we will higher degrees and orders (Gruber et al. 2000). New discuss the mathematical aspects of the wavelet filter. techniques are needed to datamine and analyze the large Then we display the filtered geoid for various length amount of data being produced by these missions. scales. We then zoom into the signal at short length scales Wavelet analysis is a relatively new mathematical to study the regional geology of an area. Finally, we dis- technique in image processing (Antoine et al. 1993). It is cuss the results and the prospects of this blossoming field. especially powerful for simultaneous detection of both the position and shape of a signal (Antoine et al. 1993). It is basically a local spectral analysis, controlled by a specified position and a scale parameter, which acts to The continuous Gaussian wavelet transform contract or magnify the image, like a microscope. A wavelet basis exists on many scales of resolution A continuous and isotropic wavelet transform, Y, can be (Tymczak et al. 2002), which means that the same viewed as a convolution of a spatial function, f, witha wavelet basis can be used to represent features that given wavelet function, w, which is shifted in the spatial range from small-scale microscopic phenomenon to domain by b and equally dilated in eachdirection by the large-scale flow in the Earth’s mantle. Lifting techniques scale, a. The continuous wavelet filter is applied to the have been developed that allow wavelet bases to be built discrete data of the geoid model at discrete scales. In two quickly in any dimension for any lattice withany num- dimension this takes the form: ber of vanishing moments (Kovacˇ evic´ and Sweldens ZLx ZLy 2000). The wavelet analysis is carried out everywhere, 1 à x À b 2 Wf ðÞ¼a; b f ðÞx w d x ð1Þ and this aspect endows it with a global character. Thus, a a the principal advantage of wavelets is the ability to view 0 0 the whole scenario at different scales. where Lx and Ly are the lengths of the Cartesian Wavelets are more satisfactory than the classical domain, and x is the position space vector. The above windowed Fourier transform because they are deter- integral is a true convolution only if: mined self-consistently within a sound mathematical wÃðÞ¼x wðÞÀx ð2Þ framework, and the window size is adjusted automati- cally according to the length scale under consideration. which is the case in our study. Since the wavelet trans- These characteristics make wavelets an excellent tool for formation (Daubechies 1992) corresponds to the con- addressing problems in many geophysical fields, in- volution of the signal, it is most efficiently computed cluding seismic tomography (Bergeron et al. 1999, in Fourier space using the Fast Fourier Transform 2000a, 2000b; Piromallo et al. 2001), convection studies (Bergeron et al. 1999). (Hier Majumder et al. 2002; Yuen et al. 2002; Vecsey The convolution kernel chosen here for the filtering is and Matyska 2001), crystal zoning (Wallace and Ber- the second derivative of the Gaussian function, called gantz 2002), and the geoid (Yuen et al. 2002; Vecsey et al. the Mexican hat (Daubechies 1992). Since it is real and 2001; Vecsey 2002). Wavelets have also been used to relatively narrow in space, it has good spatial resolution invert for crustal gravity and magnetic data in explora- withan excellent ability to isolate peaks and disconti- tion geophysics (Boschetti et al. 2001; Hornby et al. nuities in space (Torrence and Compo 1998; Antoine 1999). et al. 1993). The expression of the Mexican hat function The wavelet transform provides a multiresolution is given in Fourier space: 1 1 2 2 À jjakF representation of the function (Strang and Nguyen wðÞ¼ðakF 2pÞ2jjakF e 2 ð3Þ 1996), which allows one to examine the behavior of the function at different spatial resolutions (Kumar and where kF is the global Fourier wavenumber. Higher Foufoula-Georgiou 1994). Wavelet multiresolution edge order Gaussian wavelets, including the fourth and eighth detection has been used in the medical sciences to sep- derivative of the Gaussian, are also available in our code arate the edges of real structures from noise in magnetic (Piromallo et al. 2001). resonance images (Laine 2000) and in exploration geo- The wavelet scale, a is not necessarily equal to the physics to outline structures on gravity and magnetic horizontal Fourier wavelength. In this case, a is defined as: maps (Hornby et al. 1999). We have developed a tech- 1 nique similar to edge detection for locating bothmodern a ¼ ð4Þ and fossil plate tectonic boundaries in the geoid field. expðÞ 0:22k In this paper, we will report the geological results where k is the wavenumber or wavelet mode. The obtained by applying a 2-D continuous wavelet filter at wavelength, k, of a given scale, a, is defined as twice the discrete scales to discrete data from a 90 spherical har- diameter of the circular structure: monic degree non-hydrostatic geoid model, which is 2p obtained by recomputing and truncating the full C ¼ exp i ðÞx þ y ð5Þ model of 360 degrees (Rapp and Paulis 1990). We will k 28 with the diameter that maximizes the value of the filters are currently being developed for cases suchas the wavelet transform (Eq. 1) for that scale. The diameter of geoid (Kido et al. 2002). this circular structure gives an approximation of the size of features that give the highest wavelet values at a given scale, a.
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