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Spherical- Slepian wavelets and application in global Geophysics
Ashraf Mohamed RatebPhD. Student
Engineering Geomatics, [email protected]
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Agenda
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1) Introduction
2) Basics of Spherical Harmonics
3) Theory of Slepian functions
a) Spatiospectral localization in 1-D
b) Spatiospectral localization in 2-D
c) Spatiospectral localization in 3-D
4) Multiscale Trees of Slepian Functions
5) Conclusions
6) References.
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Introduction
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Mathematical function• Time function Information theory• Space function
In
Geosciences Spectral Model “Spatiospectral” / “Localization” Cosmology
Functions cannot have finite support in the temporal (or spatial) and spectral domain at the same time (Slepian 1983)
Slepian and Pollak1961; Landau andPollak1961,
Daubechies1988,1990;1992 Flandrin1998;Mallat1998 .
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2.0.Basis I: spherical harmonics (1/4)
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For potential fields gravity/magnetic
Laplace's Equation
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2.0.Basis I: spherical harmonics (2/4)
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Solid spherical harmonics of degree L and order m
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62.3.Basis I: spherical harmonics (3/4)
Background• Spherical Harmonics:– Orthogonal basis
functions on a sphere– Represent data on a sphere in a globally
continuous manner– Well understood; e.g. useful for spectral analysis
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72.4.Basis I: spherical harmonics 4/4
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2.0.Theory of Slepian Functions
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2.0.Theory of Slepian Functions (1/9)
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2.1. Spatiospectral Concentration on the Surface , 1D
Maximum Optimally concentration of signal
1
2
3
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2.0.Theory of Slepian Functions (2/9)
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Corresponding time/spatial domain
4a
4b
5b
5a
6b
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2.0.Theory of Slepian Functions (3/9)
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Sum of concentration of eigenvalues
7a
7b
8a
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2.0.Theory of Slepian Functions (5/9)
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Figure 1.
Band limited
Eigenfunctions
g1,g2,...,g3 that are
optimally concentrated
within the Columbia
Plateau, a
physiographic region
in the United States
centered on 116.02○ W
43.56○N (near Boise
City, Idaho) of
areaA≈145×103 km2
.
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2.0.Theory of Slepian Functions (4/9)
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2.1.Spatiospectral Concentration in the Cartesian Plane 2D
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2.0.Theory of Slepian Functions (6/9)
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Fig. 2.Bandlimited eigenfunctions gα(r,θ)that are optimally concentrated within a Cartesian disk of radius R=1. The dashed circle denotes the region boundary. The Shannon number N 2D =42. The eigenvalues λα have been sorted to a global ranking with the best concentrated eigenfunction plotted at the top left and the 30th best in the lower right
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2.2. Spatiospectral Concentration on the Surface of a Sphere in 3D
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152.0.Theory of Slepian Functions (7/9)
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2.0.Theory of Slepian Functions (8/9)
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Fig. 3.Bandlimited L=60
eigenfunctions g1,g2,...,g12that
are optimally concentrated
within Antarctica. The
concentration factors
λ1 ,λ2,...,λ12 are indicated; the
rounded Shannon number is
N3D =102
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3. Multiscale Trees of Slepian Functions
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Figure 5: The binary tree
subdivision
scheme and associated dictionary
D. define the top-level region R as
R
(1)and the generic subsets R'as
R(j).R,L
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Figure 4. Gravity signature of the Sumatra earthquake
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182.0.Theory of Slepian Functions (9/9)
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Slepain function (Summary).
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Slepian functions, focusing on the case of spherical geometry.
Convenient and easily obtained doubly-orthogonal mathematical basis to represent geographically localized data, or incompletely (and noisily) observed.
Much better suited than the traditional Fourier or spherical harmonic bases,
More “geologically intuitive” than wavelet bases in retaining a firm geographic footprint and preserving the traditional notions of frequency or spherical harmonic degree.
performant as data tapers to regularize the inverse problem of power spectral density determination from noisy and patchy observations, provided are themselves scattered within a specific areal region of study.
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4-)Conclusions
• Spherical Harmonic functions can be limited by:– Data gaps– Very high degrees have small valued coefficients
– Differing survey data sampling density
So its poor choice for represent and analyze geophysical process without global coverage
• Slepianfunctions:– Concentrate energy into a region of interest (minimise effects of data gaps)– Allow a trade‐off between spatial and/or spectral fidelity– Locally and globally orthogonal.
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References
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Eugene Brevdo., Efficient Representations of Signals in Nonlinear Signal Processing with Applications to Inverse Problems., PhD thesis2011 ., Electrical Engineering. Princeton University.
Dahlen, F.A. & F.J. Simons, Spectral estimation on a sphere in geophysics and cosmology, Geophys. J. Int., 2008, 174 (3), 774–807.
Simons, F.J. & F.A. Dahlen, A spatiospectral estimating potential fields on the surface of a sphere from noisy, incomplete data taken at satellite altitudes, Proc. of SPIE, 2007, 6701 (670117).
Simons, F.J. & F.A. Dahlen, Spherical Slepian functions and the polar gap in geodesy, Geophys. J. Int., 2006, 166
Simons, F.J., F.A. Dahlen & M.A. Wieczorek, Spatiospectral concentration on a sphere, SIAM Review, 2006, 48 (3), 504–53
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Thanx for attention
Q&A
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