spherical slepian wavelet_

Spherical- Slepian wavelets and application in global Geophysics

Ashraf Mohamed RatebPhD. Student

Engineering Geomatics, [email protected]

Agenda

Wednesday, April 12, 2023

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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.

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 .

2.0.Basis I: spherical harmonics (1/4)


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For potential fields gravity/magnetic

Laplace's Equation



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Solid spherical harmonics of degree L and order m



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


72.4.Basis I: spherical harmonics 4/4


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2.0.Theory of Slepian Functions

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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Corresponding time/spatial domain

4a

4b

5b

5a

6b



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Sum of concentration of eigenvalues

7a

7b

8a



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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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2.1.Spatiospectral Concentration in the Cartesian Plane 2D



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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

2.2. Spatiospectral Concentration on the Surface of a Sphere in 3D





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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

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

Figure 4. Gravity signature of the Sumatra earthquake



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.

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

Thanx for attention

Q&A


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