Explaining Extreme Events of 2012 from a Climate Perspective
Summary (3 min read)
Introduction
- Furthermore, it is also desirable that the samples on the sphere are uniformly distributed and/or well separated on the sphere [10]–[12].
- These schemes have well separated points and therefore the sampling points exhibit superior geometrical properties.
- In section IV, the authors formulate the geometrical properties and carry out the comparative analysis.
III. SAMPLING SCHEMES ON THE SPHERE
- The authors focus on the recently developed sampling schemes on the sphere which permit accurate computation of SHT of a bandlimited signal from its samples.
- The authors first review these sampling schemes before analysing their geometrical properties in the next section.
- For a signal band-limited at L, the authors use N to denote the spatial dimensionality, that is the number of samples, required by each of the sampling scheme to compute SHT or equivalently represent the band-limited signal accurately.
A. Guass-Legendre Quadrature based Sampling
- This sampling scheme is devised on the basis of the well known Gauss-Legendre quadrature on the sphere [21] and is therefore referred to as Gauss-Legendre (GL) sampling scheme.
- For a signal band-limited at L, this scheme takes samples on L iso-latitude rings with 2L−1 equiangular placed samples along longitude φ, resulting in a total requirement of NGL = L(2L − 1) samples, for the exact computation of SHT.
- The location of the rings along colatitude θ is given by the roots of the Legendre polynomials of order L as dictated by the Gauss-Legendre quadrature to discretize the integral given in (3).
- The variants of the GaussLegendre quadrature scheme have also been proposed (e.g., [22]) that require less number of samples.
- These sampling schemes do not support exact or sufficiently accurate computation of SHT.
C. Optimal-Dimensionality Sampling Scheme
- Like GL and equiangular sampling schemes, it is also an isolatitude sampling scheme of the sphere and takes L rings along each latitude.
- Let θk, k = 0, 1, · · · , L − 1 denotes the sample position of the ring along latitude, where these sample locations are chosen such that the accuracy of the computation of SHT is maximized.
- In total, the number of samples required by optimal-dimensionality sampling scheme is NO = L−1∑ k=0 (2k + 1) = L2. (7) As an example, the samples on the sphere for optimal dimensionality sampling scheme are shown in Fig.1(c) for L = 10.
D. Spherical Designs
- A set of points on the sphere is called a spherical design such that the integral of the signal of maximum spherical polynomial degree t or maximum band-limit t + 1 over the sphere can be evaluated as an average value over the samples of the signal [11].
- For the computation of SHT using the points given by spherical design, the authors first note that the SHT requires to evaluate the integral given in (3), where the integrand is the product of a signal band-limited at L and spherical harmonic Y m (θ, φ).
- Since the authors require to evaluate the integral for all < L, |m| ≤ , the maximum polynomial degree of integrand is 2L − Consequently, they require (2L − 2)-spherical design for the sampling of band-limited signal such that the SHT can be computed accurately.
E. Extremal Points on the Sphere
- For a given band-limit L, the extremal (maximum determinant) systems are sets of L2 extremal points on the sphere which, by definition, maximize the determinant of a basis matrix (see [10] for details).
- For spherical harmonic basis, extremal points are supported by interpolatory cubature rule with positive weights and therefore enables the accurate computation of SHT of a signal band-limited at L using NES = L 2 sampling points of extremal system.
- The authors analyse the accuracy of SHT computation later in the paper.
- The sampling scheme based on the points2 of the extremal system will be referred to as extremal system sampling scheme.
A. Sampling Efficiency
- The sampling efficiency, defined as a ratio of the dimensionality of the subspace formed by the band-limited signals, that is the number of coefficients required to represent a band-limited signal in the harmonic domain, to the number of samples required to accurately compute SHT, is the fundamental property of any sampling scheme.
- For a band-limit L, the authors define the sampling efficiency, denoted by EL, of any sampling scheme as a ratio of the dimension of the subspace HL formed by the band-limited signals to the number of samples, denoted by N , required to compute SHT of a band-limited signal f ∈ HL.
- It is evident that the optimal dimensionality sampling and 1The spherical designs are available at http://web.maths.unsw.edu.au/∼rsw/.
- 2We use the the points of extremal systems publicly available at http://web. maths.unsw.edu.au/∼rsw/Sphere/Extremal/New/extremal1.html.the authors.
- Extremal points attain almost the twice (exactly as L → ∞) of the sampling efficiency achieved by equiangular, GL and spherical designs sampling schemes.
B. Minimum Geodesic Distance and Packing Radius
- For a set of sampling points on the sphere, the minimum geodesic distance is defined as the minimum distance between any two points in the set.
- It is also defined as twice the packing radius on the sphere.
- For each of the sampling schemes presented in Section III, the authors plot the normalized minimum geodesic σn(S) for different band-limits 10 ≤ L ≤ 50 in Fig. 2, where it can be observed that extremal system of points, spherical design and optimal dimensionality, all have well separated points on the sphere.
- The nomralized minimum geodesic distance curves, obtained by using the points of equiangular and GaussLegendre quadrature based sampling schemes, are well below the lower bound values for all degrees 10 ≤ L ≤ 50.
D. Mesh Ratio
- Mesh ratio is the ratio of the covering to the packing radius of the identical spherical caps on the surface of a sphere.
- It can also be observed that the extremal system sampling scheme has the smallest mesh ratio.
- The authors also normalize with the sampling efficiency for a meaningful comparison.
- But, as s is increased, s-energy for both equiangular and GL sampling points goes away from the extremal system sampling scheme.
F. Discussion
- Among the geometrical properties analysed for different sampling schemes, sampling efficiency, mesh norm and Riesz s-energy encapsulates the other properties and therefore serve as the measures of the uniform distribution of sampling points.
- Analysis of geometrical properties of the sampling scheme reveals that the mesh ratio and s-energy grow with the band-limit for the equiangular and Gauss-Legendre sampling schemes which is a consequence of the fact that these sampling schemes require dense sampling at the poles.
- In contrast, the optimal dimensionality, spherical design and extremal systems sampling schemes exhibit desired geometrical properties.
- The mesh ratio achieved by optimal dimensionality is a little higher than extremal system, yet, it is very small compared to the equiangular schemes.
- Summarizing their analysis, the authors propose that the extremal system sampling scheme is suitable for applications [27], [28] where the signals have smaller bandlimits (L = 10 − 50) due to superior geometrical properties.
V. CONCLUSIONS
- The authors have carried the comparative analysis of the geometrical properties of those sampling schemes that support the accurate representation of band-limited signals on the sphere.
- These schemes included equiangular sampling, Gauss-Legendre (GL) quadrature based sampling, optimal-dimensionality sampling, sampling points of extremal systems and spherical design.
- The authors have illustrated that the optimal dimensionality, extremal system and spherical design sampling schemes have a uniform distributions and the points are well separated on the sphere.
- Equiangular and GL sampling schemes exhibit poor geometrical properties due to the dense sampling near the poles.
- Extremal system sampling scheme has superior geometrical properties, which the authors propose to use for the representation of band-limited signal at small band-limits.
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