multiscale representation and analysis of spherical … · 2017-11-07 · multiscale representation...

MULTISCALE REPRESENTATION AND ANALYSIS OF SPHERICAL

DATA BY SPHERICAL WAVELETS∗

TA-HSIN LI†

SIAM J. SCI. COMPUT. c© 1999 Society for Industrial and Applied MathematicsVol. 21, No. 3, pp. 924–953

Abstract. Classic wavelet methods were developed in the Euclidean spaces for multiscale repre-sentation and analysis of regularly sampled signals (time series) and images. This paper introducesa method of representing scattered spherical data by multiscale spherical wavelets. The methodextends the recent pioneering work of Narcowich and Ward [Appl. Comput. Harmon. Anal., 3(1996), pp. 324–336] by employing multiscale rather than single-scale spherical basis functions andby introducing a bottom-up procedure for network design and bandwidth selection. Decompositionand reconstruction algorithms are proposed for efficient computation. An analytical investigationconfirms the localization property of the resulting spherical wavelets. The proposed method is illus-trated by numerical examples. It is also employed to analyze and compress a real-data set consistingof the surface air temperatures observed on a global network of weather stations.

Key words. approximation, climatology, data compression, function estimation, multiresolu-tion, objective analysis, Poisson kernel, regression, spatial statistics, spherical harmonics

AMS subject classifications. 62G07, 62H11, 41A45, 65D10, 86A32, 86A10

PII. S1064827598341463

1. Introduction. The past decade has witnessed a phenomenally rapid devel-opment of wavelet theory and methods in mathematical, statistical, biomedical, andengineering communities [1, 6, 8, 31, 32]. Wavelets in the Euclidean spaces have beenproven particularly powerful for compressing images [9, 23, 29], for detecting transientpatterns and singularities [1, 24], for estimating signals of complex structures fromnoisy measurements [10, 11], and for multiscale dynamic modeling and forecasting oftime series [20].

Similar applications, including data compression, singularity detection, functionestimation, and multiscale space-time modeling and forecasting, also demand waveletmethods for handling spherical data that frequently occur in climatology and envi-ronmental sciences. This paper introduces a general spherical wavelet (SW) methodfor the representation and analysis of spherical data. The method is particularlyapplicable to scattered data that are irregularly distributed on the sphere.

Among the existing methods of spherical data analysis, spherical harmonics havean important use in climatology as part of the numerical scheme in many generalcirculation models (GCMs) [4, 36]. They also constitute a convenient basis set forrepresenting and archiving data and for detecting and analyzing spatial trends [21, 30].In addition to having meaningful physical interpretations, spherical harmonics are ef-fective in representing large-scale phenomena and optimal for analyzing homogeneousfields that comprise stationary global waves. However, in handling multiscale phe-nomena and nonhomogeneous fields, spherical harmonics become less effective. Forexample, regional anomalous activities of a field may take many spherical harmonics

∗Received by the editors July 6, 1998; accepted for publication (in revised form) April 13, 1999;published electronically November 23, 1999. This work was supported in part by NSF grant DMS-9817552 and by a grant from the Global Program of the National Oceanic and Atmospheric Admin-istration.

http://www.siam.org/journals/sisc/21-3/34146.html†Department of Statistics and Applied Probability, University of California, Santa Barbara, CA

93106-3110 ([email protected]). Current address: Department of Mathematical Sciences, IBM T.J.Watson Research Center, Yorktown Heights, NY 10598-0218.

924

MULTISCALE DATA ANALYSIS BY SPHERICAL WAVELETS 925

to represent, and slight local perturbations of a field can affect all coefficients in itsspherical harmonic (SH) representation, making it difficult to detect the local sin-gularities by a spectral analysis. For nonhomogeneous fields, effective modeling andanalysis often require multiple spatial scales. The need for both national and localweather forecasts is just such an example. Spherical harmonics, being global wavesin nature, are ineffective in representing fields made up from components of differentspatial scales.

An optimal method for representing nonhomogeneous fields is to employ the eigen-functions of the fields’ covariance kernel when the fields are viewed as stochasticprocesses. This method, known as Karhunen–Loeve (KL) representation, is efficientbecause it provides the best approximation of the field (in mean square) for any givendegree of freedom. If the covariance kernel is estimated from a training data set, asin many practical applications, the resulting eigenfunctions are known as empiricalorthogonal functions (EOFs). The EOFs have been successfully used in climate mod-eling, analysis, and change detection (e.g., [16]). However, widespread applicationof the KL method is limited by its heavy computational burden. For example, ahigh-dimensional eigenvalue problem has to be solved to obtain the EOFs; the KLcoefficients are also burdensome to compute as compared to the SH coefficients, whichcan be efficiently computed by the FFT algorithms. Another problem with the EOFsis that they may suffer from abrupt changes in response to the statistical variation indifferent training data.

SWs provide an alternative method for the representation and analysis of non-homogeneous fields. Although not necessarily as efficient as the KL method or asphysically interpretable as spherical harmonics, the SW representation possesses anumber of advantages over the SH and KL representations. First, SWs can be madespatially well localized, so that regional activities are more efficiently represented andlocal anomalies more easily detected by spherical wavelets than by spherical harmon-ics. Second, unlike the SH method, which decomposes a field into global meridionaland zonal waves, the SW method organizes the field’s activities in terms of their spa-tial scales and locations. In so doing, it becomes particularly powerful for extractingpatterns of different scales at different locations. The multiscale, or multiresolution,concept of the wavelet method has already shown great promise in helping to improvethe accuracy of climate modeling and prediction [3]. Third, the coefficients in anSW representation can be efficiently computed, thanks to the existence of recursivealgorithms. The computational efficiency is attractive especially for forecasting anddata retrieval.

The proposed SW method is based on the pioneering work of Narcowich andWard [25], in which they introduced the spherical basis functions for representingscattered spherical data. In this paper, the original idea of Narcowich and Ward(NW) is reformulated in the context of regression analysis, which leads to a moreappropriate interpretation of the resulting SW decomposition and a more explicitprocedure of constructing SWs. Then, as one of the major contributions of the paper,a new multiscale SW method is introduced that overcomes the single-scale problemof the NW method and truly represents spherical fields of multiscale structure. Re-cursive algorithms are provided, with simulation examples, for efficient computationof the multiscale SW decomposition and reconstruction. An analytical investigationjustifies the localization property of the multiscale spherical wavelets. Finally, theproposed method is demonstrated with a real-data example concerning the repre-sentation, analysis, and compression of surface air temperature observed by a global

926 TA-HSIN LI

longitude (x 180)

latit

ude

(x 1

80)

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Networks Organized by BUD (L=1-4) = 171+80+37+14; Total = 302

Fig. 1.1. A global network of 302 weather stations.

network of weather stations.

A different approach of SW analysis was recently taken by Freeden and Wind-heuser [12, 13], in which a continuous wavelet transform needs to be approximatedby discretization schemes on regular gridpoints. Motivated by data compression forcomputer graphics, Schroder and Sweldens [27] also proposed an algorithm for SWanalysis on the basis of regular gridpoints. Our approach, like that of NW, assumesscattered data that may not be regularly distributed on the sphere. Such scattereddata are often encountered in global environmental studies, based either on groundstations or on satellites [15, 17, 22, 33]. An example of scattered weather stations isshown in Figure 1.1. Furthermore, by using the least-squares (LS) principle ratherthan interpolation, the proposed method is able to produce reliable representationsfrom noisy data. The interested reader is referred to [19] for a more comprehensivereview of different SW methods.

The main focus of this paper is on the development of general methodology. Themathematical analysis of the localization property provides a necessary justificationof the proposed method. Due to the limited space, other theoretical issues, includ-ing the error bounds and condition numbers of the SW representation for differentclasses of spherical functions and the statistical characteristics of wavelet coefficientsfor different models of spherical stochastic fields, will be addressed elsewhere. For thesame reason, practical issues on the implementation of the method are explored bythe author in [19] with great detail.

2. SWs for scattered data. In applications such as climatology, the earth isoften regarded as a sphere of unit radius so that a meteorological variable T (n), suchas the surface air temperature, can be treated as a spherical field. In this expression,n := [cosφ cos θ, cosφ sin θ, sinφ]T denotes the unit vector that points to a location onthe earth from the center of the sphere, with φ and θ being the latitude and longitudeof the location. Throughout the paper, we assume a simple mathematical model forT (n), namely, T (n) belongs to the family of square-integrable functions on the sphere.

In many applications, the field T (n) is observed only at a finite number of ob-


serving sites. If the observations, denoted by (Tj ,nj)Jj=1, are free of measurementerror, then Tj = T (nj). A more realistic model is Tj = T (nj)+ǫj , where ǫj representsthe additive noise. In the following, the additive noise is ignored because it does notaffect the presentation of the proposed methodology. Estimation in the presence ofadditive noise is discussed in [19]

Given the observed samples, which are often irregularly distributed with datavoids of various sizes, an important problem is to use the data to represent the fieldat any location on the sphere where no observations are made. In climatology, thisproblem is known as the objective analysis [7]. A solution to the problem providesthe essential input required by all applications discussed in the introduction.

2.1. Least squares with spherical harmonics. A simple way of representingthe field at an unobserved location is to approximate it by a linear combination ofspherical harmonics Yℓm(n) [2, 21],

T (n) :=

L∑

ℓ=0

ℓ∑

m=−ℓ

ξℓmYℓm(n).(2.1)

In this expression, the coefficients ξℓm can be found by LS, for example. It is easy toshow that the LS estimator of T (n) can be expressed as

T (n) =

J∑

j=1

T(

nj

)

G(

n,nj

)

,(2.2)

where

G(

n,nj

)

:= yH(

nj

)(

YHY)−1

y(

n)

,(2.3)

y(n) := vecYℓm(n), and Y := [y(n1), . . . ,y(nJ)]H , with superscript H standingfor the Hermitian transpose. Equations (2.2) and (2.3) comprise an SH representationof T (n).

As can be seen, the observation at nj is extrapolated in (2.2) by a sphericalfunction G(n,nj). This function does not necessarily decay rapidly as n moves awayfrom nj , but the particular structure of (2.2) points to a direction for constructingspatially localized representations.

2.2. Spherical basis functions. Spatial localization can be achieved by usingSWs. An example of SWs, introduced by NW [25], is derived from spherical basis

functions (SBFs). Unlike G(n,nj) in (2.3), SBFs take the form G(n,nj) = G(n ·nj),where n ·nj is equal to the cosine of the angle between n and nj . With the SBFs inplace of G(n,nj), the extrapolation becomes

T (n) =

J∑

j=1

T(

nj

)

G(

n · nj

)

.(2.4)

We refer to (2.4) as an SBF representation of T (n). A major difference between(2.4) and (2.2) is that the SBFs in (2.4) depend solely on the angles between thelocation n and the observing sites nj , so that the SBF representation is invariant toany rotations of the spherical coordinate system, thus avoiding the problem causedby the artificial poles associated with the SH representation. The formulation in (2.4)

928 TA-HSIN LI

angle (x pi)

SB

F

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0.6

0.8

1.0

Spherical Basis Functions from Poisson Kernel

ETA = 0.4ETA = 0.6ETA = 0.7ETA = 0.8

Fig. 2.1. Plot of normalized Poisson kernel G(n ·m; η) := (1− η)2(1 + η)−1G(n ·m; η), whichsatisfies G(1; η) = 1, against the angle between n and m. The parameter η controls the bandwidth:the nearer η is to unity, the smaller the bandwidth.

is also instrumental to achieving spatial localization. For example, one can choose afunction G(n ·nj) that decays to zero rapidly as the angle between n and nj grows,so the influence of the field’s activities near nj in the resulting SBF representationwill be effectively kept within its neighborhood of a predetermined size.

Another major difference between (2.4) and (2.2) is that the SBFs in (2.4) areall related to each other by rotation and induced from a single function, G(x), whichis independent of the observing network. The function from LS, on the other hand,depends not only on nj but also on the location of other stations, as can be seen from(2.3). The network independence may or may not be an advantage; it certainly makesthe extrapolation easier to control and compute. But in many cases efficiency can beimproved by treating sparse sites differently from dense sites. This latter concernleads to a multiscale approach that will be discussed later.

A simple and useful example of SBF is the generating function of Legendre poly-nomials,

G(x; η) :=1 − η2

(1 − 2ηx + η2)3/2=

∞∑

n=0

(2n + 1)ηnPn(x),(2.5)

where η ∈ (0, 1) is a bandwidth parameter and Pn(x) is the Legendre polynomial of

degree n, satisfying∫ 1

−1P 2n(x) dx = 1. We refer to G(x; η) as the Poisson kernel. Plots

of the Poisson kernel are given in Figure 2.1 for various values of η. Figure 2.2 showstwo Poisson-kernel-based SBF representations of a simulated data set observed onthe 5-by-10 network shown in Figure 2.3. See Appendix A for an interesting physicalinterpretation of the Poisson kernel.

More examples of SBF are provided by NW [25] and by Freeden and Windheuser[13]. In principle, the SBFs can be any square-integrable functions that satisfy theconditions discussed in Appendix A and decay sufficiently rapidly with desired theo-retical properties such as smoothness (see section 5 for more comments).


theta (x pi)

phi (

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SBF Representation (ETA=0.6)

(a)

theta (x pi)

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SBF Representation (ETA=0.8)

(b)

Fig. 2.2. Two Poisson-kernel-based SBF representations obtained from simulated data on a5-by-10 regular grid. (a) η = 0.6. (b) η = 0.8. The spatial bandwidth decreases with the increase ofη ∈ (0, 1). In these plots, φ ∈ (−π

2, π

2] is latitude and θ ∈ (−π, π] is longitude.

In addition to (2.4), we also consider a more general SBF representation

T (n) =

J∑

j=1

βjG(

n · nj

)

.(2.6)

The major difference between (2.6) and (2.4) is that the coefficients βj in (2.6) neednot be the observed data. This generalization enhances the flexibility of the SBFrepresentation and enables one to approximate T (n) by different methods. As anexample, consider the LS fit of the observed data t := vecT (nj) by T (n) in (2.6).The solution, with β := vecβj = (G′G)−1G′t = G−1t, where G := [G(ni · nj)],gives rise to an SBF representation that passes through all data points at the observingsites. In this case T (n) in (2.6) becomes an interpolating function. Other methods ofobtaining the βj are discussed by the author in [19].

2.3. A multiresolution analysis. The multiresolution analysis considered byNW was derived from the problem of characterizing the loss in an SBF representationas more and more stations, and hence more and more SBFs, are progressively removedfrom the original network in the representation.

To reformulate the idea, let N1 := njJj=1 be the original network employed to

represent T (n), and let N2 := njKj=1, (K < J), be a smaller network obtained byremoving, for example, the last J −K stations from N1. Given the observed data, let

T1(n) :=

J∑

j=1

β1jG(

n · nj

)

= βT1 g1(n)(2.7)

be an SBF representation of T (n), where β1 := vecβ1jJj=1 and g1(n) := vecG(n ·nj)Jj=1. By freely varying the β1j , (2.7) can generate a collection of spherical func-tions, V1 := spanG(n · nj) : nj ∈ N1. Similarly, the SBF representation based onthe smaller network N2 is

T2(n) :=

K∑

j=1

β2jG(

n · nj

)

= βT2 g2(n),(2.8)

930 TA-HSIN LI

theta (x pi)

phi (

x pi

)

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An Illustrative Network of Stations (Geometric Distortions Ignored)

Fig. 2.3. A 5-by-10 regular grid and its partition into L = 3 subnets: stations in M3 := N3

are represented by squares; stations in M2 by circles; and stations in M1 by triangles. Note thatN2 = M2 ∪N3 and N1 = M1 ∪N2 = M1 ∪M2 ∪M3.

where β2 := vecβ2jKj=1 and g2(n) := vecG(n · nj)Kj=1. It is clear that V2 :=spanG(n · nj) : nj ∈ N2 ⊂ V1.

Since T1(n) is known throughout the sphere, one can choose T2(n) to best approx-

imate T1(n) by minimizing∫

|T1(n)−∑Kj=1 β2jG(n ·nj)|2 dΩ(n), where the integral

is over the entire sphere. This LS method leads to a solution T2(n) with

β2 = A−12 b2,(2.9)

where, with G(n,m) := G(n ·m), the matrices A2 and b2 are given by

A2 := [〈G(·,ni), G(·,nj)〉]Ki,j=1, b2 := vec〈T1(·), G(·,nj)〉Kj=1.(2.10)

The inner product in (2.10) is defined as 〈U(·), V (·)〉 :=∫

U(n)V (n) dΩ(n) for anysquare-integrable functions U(n) and V (n). Note that A2 = [G ∗ G(ni · nj)]

Ki,j=1,

where the asterisk stands for spherical convolution (see Appendix A).With β2 given by (2.9), T2(n) becomes the projection of T1(n) in V2. If the

difference (or residual) field is denoted by D1(n) := T1(n)− T2(n) and the collectionof D1(n) is denoted by W1, then V1 = V2 ⊕W1. The space W1 represents the newinformation (or innovation) in V1 that is not contained in the smaller space V2; it alsorepresents the information that is lost by using the smaller network N2 instead of N1

to describe T1(n).Any field T1(n) ∈ V1 can be decomposed as

T1(n) = T2(n) + D1(n),(2.11)

where T2(n) ∈ V2 and D1(n) ∈ W1. Equation (2.11) was regarded by NW as a“multiresolution” analysis of T1(n) perhaps for the reason that T1(n) stems from alarger (possibly denser) network and thus has a higher “spatial resolution” than T2(n).Based on this interpretation, D1(n) may be regarded as the lost “high-resolution”detail when representing T1(n) by T2(n).

We provide an alternative interpretation that seems more appropriate, especiallyfor scattered data. Due to the structure of the networks and the localization property


theta (x pi)

phi (

x pi

)

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Global Component (L=2)

(a)

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Local Component (L=1)

(b)

Fig. 2.4. Decomposition of the field shown in Figure 2.2(a). (a) Global component T2(n). (b)Local component D1(n), rescaled in magnitude to show details. The field in Figure 2.2(a) is equalto T2(n) + D1(n).

of the SBFs, we interpret D1(n) as the local, or regional, activities of T1(n) near theremoved stations that cannot be accounted for by the activities represented by T2(n),i.e., the global activities of T1(n) extrapolated from the remaining stations. Based onthis new interpretation, we refer to D1(n) as the local component of T1(n) and referto T2(n) as the global component of T1(n).

Extracting local anomalies from global trends by progressively removing stationsfrom a network is an important problem in climatology and environmental sciences.It provides a solution not only for a sampling mission planner who is interested inthe lower limit on the number of stations required to effectively represent a field inquestion, but also for a climate analyzer who wants to detect the regional activitiesof a meteorological variable in a target area that are “uncorrelated” with those inother areas. It also helps a climate modeler who needs to model and forecast ameteorological variable in different scales and to compress the enormous amount ofdata at minimal loss of fidelity.

Figure 2.4 shows a decomposition of the field in Figure 2.2(a). In this example, N1

consists of the 5-by-10 gridpoints shown in Figure 2.3 and N2 is obtained by removingthe triangles. As can be seen, some moderate local activities are revealed in D1(n).

The decomposition (2.11) can be easily generalized to a nested sequence of net-works

N1 ⊃ N2 ⊃ · · · ⊃ NL.(2.12)

The corresponding spaces of spherical functions are also nested: V1 ⊃ V2 ⊃ · · · ⊃ VL,so that Vℓ = Vℓ+1 ⊕Wℓ for ℓ = 1, . . . , L− 1. Any field Tℓ(n) ∈ Vℓ can be decomposedas a global component Tℓ+1(n) ∈ Vℓ+1 and a local component Dℓ(n) ∈ Wℓ such that

Tℓ(n) = Tℓ+1(n) + Dℓ(n).(2.13)

In particular,

T1(n) = Tℓ(n) + Dℓ−1(n) + · · · + D1(n),(2.14)

where ℓ = 2, . . . , L.

932 TA-HSIN LI

-1

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0

0.5

1

theta (x pi)

-0.4

-0.2

0

0.2

0.4

phi (x pi)

-5 0

510

wav

elet

Spherical Wavelet (L=1) [7](ETA=0.6;SDV=0.471)

(a)-1

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0

0.5

1

theta (x pi)

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0

0.2

0.4

phi (x pi)

-5 0

510

wav

elet


(b)

Fig. 2.5. Examples of SWs derived from the Poisson kernel with η = 0.6 and from removingthe stations represented by triangles in Figure 2.3. (a) Wavelet at station 7. (b) Wavelet at station23. For all these wavelets, σk = 0.471.

2.4. SW representation. Now consider the composition of Dℓ(n). For sim-plicity, let L = 2 again. Recall that T2(n) takes the form (2.8) with β2 given by (2.9).Let

A1 :=[

G ∗G(

ni · nj

)]J

i,j=1=

[

A2 B1

BT1 C1

]

,(2.15)

where A2 is given by (2.10). Then, b2 = [A2,B1]β1, and hence, β2 = [I,A−12 B1]β1.

If g1(n) and β1 are similarly partitioned such that g1(n) = vecg2(n),h1(n)and β1 := vecα2,γ1, then D1(n) = γT

1 w1(n) =∑J

k=K+1 γ1kW1(n,nk), where

w1(n) := vecW1(n,nk)Jk=K+1 := [−BT1 A−1

2 , I]g1(n) and γ1 := vecγ1kJk=K+1.Since the space of D1(n) can be expressed as W1 = spanW1(n,nk) : nk ∈ M1,where M1 := N1 \ N2, the W1(n,nk) completely characterize W1 = V1 ⊖ V2. Wedefine these basis functions as SWs.

In the general case of L > 2, let Mℓ := Nℓ \ Nℓ+1 denote the subnet consistingof the stations removed from Nℓ to obtain Nℓ+1. Then, Wℓ = Vℓ ⊖ Vℓ+1 can becharacterized by SWs Wℓ(n,nk), where nk ∈ Mℓ. Note that the SWs do not dependon the observed data; they depend solely on the SBF and the nested networks. Alsonote that for a given level ℓ, the number of SWs Wℓ(n,nk) is equal to the number ofstations in Mℓ.

As an example, Figure 2.5 shows two of the 25 SWs derived from the Poissonkernel with η = 0.6 and from the networks that produced the decomposition in Fig-ure 2.4, with M1 consisting of the triangles in Figure 2.3. As can be seen, the SWsare well localized around the corresponding stations. The local component D1(n) inFigure 2.4(b) is a linear combination of these wavelets.

According to (2.14), the field T1(n) = T (n) can be decomposed as

T (n) = TL(n) + DL−1(n) + · · · + D1(n),(2.16)

where TL(n) ∈ VL corresponds to the smallest network and the Dℓ(n) are linearcombinations of the SWs. We refer to (2.16) as an SW representation of T (n).


3. Multiscale SW representation. The decomposition (2.13) was regardedby NW as multiresolution analysis probably because Tℓ+1(n) employs fewer (possiblysparser) stations than Tℓ(n). However, with the Poisson kernel (2.5), their multires-olution interpretation produces a rather paradoxical result: the wavelets Wℓ(n,nk)always have the same “size” that depends solely on the predetermined bandwidthparameter η regardless of the intended “resolution level” ℓ or the distribution of thestations.

In the following, we propose a method that overcomes this “single-scale problem.”The method leads to SWs that have true multiresolution interpretations.

3.1. The single-scale problem of NW wavelets. To see the single-scale prob-lem of the NW wavelets, consider the case of L = 2 without loss of generality, andassume

∫

|W1(n,nk)|2dΩ(n) = 1 so that |W1(n,nk)|2 can be regarded as a prob-ability density function defined on the sphere. As in NW [25], let the location ofW1(n,nk) be measured by the “mean” vector

mk :=

∫

n|W1(n,nk)|2dΩ(n),(3.1)

and let the concentration of W1(n,nk) around its mean be measured by the “variance”

σ2k :=

∫

‖n−mk‖2|W1(n,nk)|2dΩ(n).(3.2)

For the Poisson kernel (2.5), it can be shown ([25]; see also section 3.5 of this paper)that

mk =2η

1 + η2nk, σ2

k =

(

1 − η2

1 + η2

)2

.(3.3)

This result, first, confirms what has been observed from Figure 2.5: the Poisson-kernel-induced wavelets are indeed located near the corresponding stations, and thedegree of localization is determined by the closeness of η to unity.

However, the result in (3.3) also reveals that the variance σ2k is constant and

independent of the networks. In fact, even if the density of Nℓ decreases with theincrease of ℓ, the resulting NW wavelets may become increasingly separated in space,but still maintain the same size as measured by their constant variance. This problemof the NW wavelets can be easily observed by comparing Figure 2.5 with Figure 3.1.The latter contains 2 of the 10 level-2 wavelets obtained by further removing thestations in M2 (i.e., the circles in Figure 2.3).

Because the NW wavelets have a constant variance, all local activities in Dℓ(n),and hence in the decomposition (2.14), are described by a fixed-size neighborhoodregardless of where the stations are located, how closely they are spaced, or what theintended resolution level is. This defect of the NW method can be seen by comparingD1(n) in Figure 2.4 with D2(n) in Figure 3.2. The latter is obtained from a furtherdecomposition of T2(n) in Figure 2.4 by removing the circles in Figure 2.3; it is alinear combination of ten level-2 wavelets, two of which are shown in Figure 3.1.

3.2. Multiscale spherical basis functions. The single-scale problem of theNW wavelets contradicts the purpose of multiscale representation, which is to describea field with components of variable scales. The problem can be attributed to the factthat the SBFs in (2.4) and (2.6), as rotated versions of a single function, have the

934 TA-HSIN LI

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Fig. 3.1. Examples of SWs derived by further removing the stations represented by circles inFigure 2.3. (a) Wavelet at station 12. (b) Wavelet at station 32. These wavelets have the same“size” as those shown in Figure 2.5.

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Fig. 3.2. Further decomposition of T2(n) shown in Figure 2.4(a). (a) Global component T3(n).(b) Local component D2(n), rescaled in the same way as D1(n) in Figure 2.4(b). The field inFigure 2.4(a) is equal to T3(n) + D2(n) and the field in Figure 2.2(b) is equal to T3(n) + D2(n) +D1(n).

same spatial bandwidth for all stations. An advantage of this choice is the computa-tional simplicity. However, real spherical fields, such as the surface air temperature,are often observed by stations of variable density (e.g., [17, 22]). In these cases, asingle bandwidth becomes inadequate for representation and analysis because a largebandwidth is needed for sparse stations, whereas a small bandwidth is required fordescribing local activities in data-rich areas. Moreover, the fields themselves can bemade up from components of variable sizes. All these considerations demand a repre-sentation with multiple bandwidths rather than a single bandwidth. In the following,we propose a true multiscale method that overcomes the single-scale problem of theNW wavelets.

Instead of a single function, we propose to employ a set of SBFs with different

bandwidths that are properly adapted to the stations. To be more specific, let the


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Multi-Scale SBF Representation

Fig. 3.3. A simulated field that has the multiscale SBF representation (3.4). The networksof increasing density shown in Figure 2.3 are employed together with Poisson kernels of decreasingbandwidth: η3 = 0.4 for M3, η2 = 0.6 for M2, and η1 = 0.8 for M1.

nested networks in (2.12) be denoted by Nℓ := nj : j = 1, . . . , Nℓ, where 0 <NL < · · · < N2 < N1 := J . To estimate T (n), we propose to use the multiscale SBFrepresentation of the form

T (n) =

L∑

ℓ=1

Nℓ∑

j=Nℓ+1+1

βjGℓ

(

n · nj

)

,(3.4)

where NL+1 := 0. In this expression, the same SBF, Gℓ(x), is employed by all stationsin the ℓth subnet Mℓ = nk : Nℓ+1 < k ≤ Nℓ, but different SBFs (i.e., SBFs withdifferent bandwidths) are allowed for the L subnets.

In general, the networks can be designed such that the density of Nℓ, definedin terms of the smallest distance among the stations in Nℓ, decreases with the in-crease of ℓ; the multiscale SBFs can be chosen accordingly, so that the bandwidth ofGℓ(x), defined by the standard deviation of |Gℓ(x)|2 when normalized properly andviewed as a probability density, increases as ℓ is increased (and the density of Nℓ isdecreased). These requirements can be achieved by the following bottom-up design

(BUD) procedure:(a) Start with a sparse network ML := NL.(b) Choose a suitably large bandwidth for GL(x) so that VL has sufficient spatial

coverage (i.e., there exists at least one function in VL whose support is equalto the entire sphere).

(c) Add stations in Mℓ to NL successively to obtain Nℓ = Nℓ+1 ∪ Mℓ for ℓ =L− 1, . . . , 1, so that the density of Nℓ increases with the decrease of ℓ.

(d) Choose progressively reduced bandwidths for the SBFs so that the bandwidthof Gℓ(x) decreases as the density of Nℓ is increased.

As a major benefit of the BUD procedure, the index ℓ in (3.4) becomes a true scaleparameter. Indeed, if the scale of an SBF representation is defined by the smallestbandwidth of the SBFs employed, then, by following the BUD procedure, an increasein ℓ corresponds to an increase in the SBF bandwidth and thus an increase in thescale of the SBF representation.

Figure 2.3 shows a simple example of networks designed by the BUD procedure,which starts with the sparsely located squares and then sequentially generates twolarger (nested) networks of increasing density by first adding the circles and then thetriangles. If the SBF bandwidth decreases with the increase of the network density,

936 TA-HSIN LI

the multiscale SBF representation (3.4) is able to produce multiscale fields like theone shown in Figure 3.3.

In practice, the BUD procedure can be implemented with the help of suitably par-titioned regular grids on the sphere. The bandwidth selection should be well adaptedto the network density. One of the objectives of the BUD procedure is to ensurethat (3.4) is a stable representation, i.e., that λ1‖β‖2 ≤ ‖T (·)‖2 ≤ λ2‖β‖2 ∀β :=vecβjJj=1, where λ2 ≥ λ1 > 0 are constants. The stability is guaranteed if thesmallest eigenvalue of A := [Gℓ ∗ Gℓ′(ni · nj)] stays away from zero. This requiresthat the bandwidths of closely spaced SBFs be sufficiently small or different. Due tothe limited space, these topics cannot be elaborated here. Further discussion of thepractical implementation of the BUD procedure can be found in [19].

3.3. True multiresolution analysis. Given the networks Nℓ, let us define thenested spaces Vℓ by

Vℓ := span

Gℓ′(n · nk) : nk ∈ Mℓ′ ; ℓ ≤ ℓ′ ≤ L

,(3.5)

where ℓ = 1, . . . , L. In words, the space Vℓ is formed by linear combinations of theSBFs whose scale indices are greater than or equal to ℓ. With the nested spaces sodefined, the orthogonal decompositions (2.13), (2.14), and (2.16) can be obtained bya procedure similar to the single-scale case, but the real meaning of the resultingdecompositions is fundamentally different.

The difference lies in the structure of Vℓ and thus the corresponding interpretation.Because the BUD procedure ensures that the scale index ℓ represents the actual scaleof the SBF representation, the space Vℓ defined by (3.5) contains spherical fieldswhose scales are smaller (resolutions higher) than any field in Vℓ+1; the complementspace Wℓ thus contains the smaller scale (higher resolution) information on Vℓ thatcannot be described by Vℓ+1. Because of this interpretation, the decompositions(2.13), (2.14), and (2.16) become a true multiresolution analysis, which is consistentwith the classic wavelet theory (e.g., [8]). We use the adjective “true” to distinguishthe multiresolution analysis based on the multiscale SBFs from that proposed by NW[25]. For the same reason, we refer to (2.16) as a multiscale SW representation ifderived from the multiscale SBFs.

Figure 3.4 presents a multiscale SW representation for the field shown in Fig-ure 3.3. The decomposition employs the nested networks illustrated in Figure 2.3and the Poisson kernels of variable bandwidth: η1 = 0.8 for M1, η2 = 0.6 for M2,and η3 = 0.4 for M3. Observe the resolution increase as one sequentially examinesthe component fields T3(n), D2(n), and D1(n). Summing up these components ofincreasing resolution gives rise to the original multiscale field in Figure 3.3.

3.4. Decomposition and reconstruction. The decomposition (2.16) can beefficiently computed by a recursive algorithm. To describe the algorithm, let gℓ(n)denote the vector formed by the SBFs associated with Nℓ, i.e.,

gℓ(n) :=

hL(n)...

hℓ(n)

=

[

gℓ+1(n)hℓ(n)

]

,(3.6)

where hℓ(n) := vecGℓ(n · nk)Nℓ

k=Nℓ+1+1 is formed by the SBFs (of identical band-

width) associated with Mℓ. Furthermore, let

Aℓ := 〈gℓ(·), gTℓ (·)〉 =

[

Aℓ+1 Bℓ

BTℓ Cℓ

]

, Eℓ := A−1ℓ+1Bℓ,(3.7)


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Fig. 3.4. A multiscale decomposition of the field shown in Figure 3.3. (a) Global componentT2(n). (b) Local component D1(n). (c) Global component T3(n). (d) Local component D2(n). Thefield shown in Figure 3.3 is equal to T3(n) + D2(n) + D1(n).

where Bℓ := 〈gℓ+1(·),hTℓ (·)〉 and Cℓ := 〈hℓ(·),hT

ℓ (·)〉. With this notation, we havethe following theorem; see Appendix B for a proof.

Theorem 3.1. Let βℓ := vecβℓjNℓ

j=1 be formed by the SBF coefficients of

Tℓ(n) ∈ Vℓ so that

Tℓ(n) = βTℓ gℓ(n).(3.8)

Then, for ℓ = 1, . . . , L− 1, any field Tℓ(n) ∈ Vℓ can be decomposed as (2.13) with

Tℓ+1(n) = βTℓ+1gℓ+1(n), Dℓ(n) = Tℓ(n) − Tℓ+1(n) = γT

ℓ wℓ(n),(3.9)

where

βℓ = vecαℓ+1,γℓ,(3.10)

βℓ+1 = αℓ+1 + Eℓγℓ,(3.11)

wℓ(n) = hℓ(n) − ETℓ gℓ+1(n).(3.12)

Among these formulas, (3.9) and (3.12) compute the component fields, (3.10) and

(3.11) determine the SBF and SW coefficients in (3.9).

938 TA-HSIN LI

The recursion needed to obtain the βℓ and γℓ from β1 can be depicted as follows:

β1 −→ β2 −→ β3 −→ · · · −→ βL

ց ց ց ցγ1 γ2 γL−1

(3.13)

Note that only the first line in (3.13) involves computation, which is given by (3.11),and γℓ = vecβℓkNℓ

k=Nℓ+1+1 can be obtained from βℓ without computation. Also notethat if observations are available on N1, then the recursion can start simply withβ1 = t. Other (better) methods of obtaining β1 from the data are discussed in [19].

If (3.13) is regarded as an analysis procedure that decomposes the data vector t

into the multiscale components γ1, . . . ,γL−1,βL by gradually removing local ac-tivities from global trends, then the procedure can be reversed to reconstruct, orsynthesize, the data t from these components by sequentially adding the local activi-ties back to the global trends. The synthesis procedure is depicted in the following.

βL −→ · · · −→ β3 −→ β2 −→ β1

ր ր ր րγL−1 γ2 γ1

(3.14)

It is easy to see from (3.11) that αℓ+1 can be computed by

αℓ+1 = βℓ+1 − Eℓγℓ (ℓ = L− 1, . . . , 1).(3.15)

Equations (3.10) and (3.15), together with (3.8), constitute the recursive algorithmfor reconstruction.

The computationally most demanding step in the decomposition and reconstruc-tion algorithms is that of calculating the “filters” Eℓ := [eℓ(j, k)], which requires us tosolve the normal equations Aℓ+1Eℓ = Bℓ. Because most of the SBFs decay rapidly,Aℓ+1 and Bℓ are, or can be approximated by, sparse matrices. Therefore, the nor-mal equations can be solved by more efficient algorithms that take advantage of thesparse structure. In applications such as meteorology, the same networks are oftenused repeatedly for analyzing different sets of observations. In this case, the Eℓ, be-ing independent of the observations, need to be computed and saved only once. Inthe worst-case scenario where the sparseness is not utilized, the complexity of the re-cursive algorithm (3.13) is approximately O(J3), whereas that of directly computingthe βℓ from β1 is approximately O(J3 log J). The recursion is clearly more efficient,especially for large J .

3.5. Multiscale SWs. As can be seen from (3.9), the space Wℓ is characterizedby wℓ(n) in (3.12). Let the components of wℓ(n) be denoted by Wℓ(n,nk) so that

wℓ(n) = vec

Wℓ(n,nk)Nℓ

k=Nℓ+1+1,(3.16)

where ℓ = 1, . . . , L − 1. These functions completely determine Wℓ, and we refer tothem as multiscale SWs. According to (3.12), the multiscale SWs can be expressedas

Wℓ(n,nk) = Gℓ(n · nk) −∑

(ℓ′,j)∈Iℓ+1

eℓ(j, k)Gℓ′(

n · nj

)

,(3.17)

where Iℓ+1 is defined as in Appendix B.


Equations (3.12) and (3.17) describe a Gram–Schmidt procedure that orthogo-nalizes hℓ(n) with respect to gℓ+1(n), the former being the set of SBFs employed inTℓ(n) but not in Tℓ+1(n), and the latter being the set of SBFs employed in Tℓ+1(n)only. These two groups of SBFs now have different scales. The orthogonalizationprocedure removes all lower resolution components from hℓ(n) using gℓ+1(n), so theresulting wavelets contain only the higher resolution residuals (of level ℓ) that can-not be attributed to the lower resolution SBFs. Because the spatial resolution ofWℓ(n,nk) is determined by the bandwidth of Gℓ(x), as will be further justified insection 3.6, the multiscale SW not only take into account the local activities near thestations in Mℓ, as do the NW wavelets, but also express the locality of the activitieswith the appropriate scales. Furthermore, the Gram–Schmidt procedure makes themultiscale wavelets orthogonal across the scales:

Wℓ ∗Wℓ′(ni · nk) = 0 ∀ni ∈ Mℓ,nk ∈ Mℓ′ , ℓ 6= ℓ′.

Such wavelets are also known as prewavelets [5].

Figure 3.5 shows four multiscale SWs generated by the networks in Figure 2.3for the same stations as employed in Figures 2.5 and 3.1. The SBFs that induce themultiscale SWs are Poisson kernels of three bandwidths: η1 = 0.8 for M1, η2 = 0.6for M2 (the same value is used in Figures 2.5 and 3.1), and η3 = 0.4 for M3. As canbe seen, the multiscale SWs in Figure 3.5 reside near the corresponding stations withwell-localized peaks. But unlike the NW wavelets, the multiscale SWs with differentscale index ℓ indeed have different sizes that are consistent with the choice of η: thewavelets in Figures 3.5(a)–(b) have higher resolution than those in Figures 3.5(c)–(d)due to the choice of η1, which is closer to unity than η2.

The classic one-dimensional Haar wavelets (e.g., [8]) can be constructed by theprocedure described above. To see this, consider the case of L = 2 and assume that N2

consists of all even integers 2i for i = 0, 1, . . . , 12J − 1. By the BUD procedure, let us

add to N2 the set M1 that consists of all odd integers 2i + 1 for i = 0, 1, . . . , 12J − 1.

This yields a denser set of points N1 = N2 ∪ M1 that contains all integers j =0, 1, . . . , J−1. Further, by the BUD procedure, let us assign the point 2i ∈ M2 := N2

with an indicator function G2(x − 2i) of length 2 and assign the point 2i + 1 ∈ M1

with an indicator function G1(x− (2i+1)) of length 1, where Gℓ(x) := I[0,2ℓ−1)(x) forany real number x. It is easy to see that V2 := spanG2(x−xk) : xk ∈ M2 coincideswith the space of all functions defined on [0, J) that are piecewise constant on thesubintervals [2i, 2i+2). More importantly, V1 := spanGℓ(x−xk) : xk ∈ Mℓ; ℓ = 1, 2coincides with the space of all functions defined on [0, J) that are piecewise constanton the subintervals [i, i+1). Therefore, the complement space W1 := V1⊖V2, definedon L2[0, J), is formed by W1k(x) := G1(x− (2i+ 1))− 1

2G2(x− 2i), which is equal to− 1

2 on [2i, 2i+ 1), 12 on [2i+ 1, 2i+ 2), and zero elsewhere. These basis functions are

nothing but the Haar wavelets centered at xk := 2i + 1 ∈ M1.

The multiscale SWs are determined solely by the multiscale SBFs and the nestednetworks of stations. To compute the multiscale SWs in (3.17) and the decompositionsin (3.9), one only needs the matrix A1, which is defined in (3.7). This matrix iscomposed of all inner products among the multiscale SBFs. The inner products canbe expressed as

aℓ,ℓ′(i, j) := Gℓ ∗Gℓ′(

ni · nj

)

.(3.18)

For the Poisson kernel, it is easy to show that if one defines Gℓ(x) = G(x; ηℓ) and

940 TA-HSIN LI

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MS Spherical Wavelet (L=1) [7](ETA1=0.8; ETA2=0.6; ETA3=0.4)

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MS Spherical Wavelet (L=1) [23](ETA1=0.8; ETA2=0.6; ETA3=0.4)

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MS Spherical Wavelet (L=2) [32](ETA2=0.6; ETA3=0.4)

(d)

Fig. 3.5. Examples of the proposed multiscale SWs derived from the Poisson kernels: η = 0.4for M3, η = 0.6 for M2, and η = 0.8 for M1, where the subnets are shown in Figure 2.3. (a)Multiscale SW of scale 1 at station 7. (b) Multiscale SW of scale 1 at station 23. (c) MultiscaleSW of scale 2 at station 12. (d) Multiscale SW of scale 2 at station 32.

Gℓ′(x) = G(x; ηℓ′), then

aℓ,ℓ′(i, j) = 4π√

2 G(

ni · nj ; ηℓηℓ′)

.(3.19)

In other words, the Poisson kernel is invariant under spherical convolution. Thisproperty, together with its closed-form expression (2.5), makes the Poisson kernelcomputationally attractive.

3.6. Localization property. An important requirement for wavelets is the lo-calization property: a wavelet should decay sufficiently rapidly from its “center.” Asin the single-scale case discussed by NW [25], the localization property of the proposedmultiscale spherical SWs can be assessed in terms of their mean and variance, whenviewed as probability densities.

For any (square-integrable) spherical function, the mean m and the variance σ2,


as defined by (3.1) and (3.2), are related to each other such that σ2 = 1 − ‖m‖2. Ittherefore suffices to compute the mean of |Wℓ(n,nk)|2, which can be expressed as

mℓk :=1

λℓ(k, k)φℓ(k, k),(3.20)

where

φℓ(k, k) :=

∫

n|Wℓ(n,nk)|2 dΩ(n),(3.21)

λℓ(k, k) :=

∫

|Wℓ(n,nk)|2 dΩ(n).(3.22)

The variance of |Wℓ(n,nk)|2 is given by σ2ℓk = 1−‖mℓk‖2. Note that λℓ(k, k) is equal

to the kth diagonal element of Λℓ := [λℓ(i, j)]Nℓ

i,j=Nℓ+1+1 := 〈wℓ(·),wTℓ (·)〉. It is easy

to show from (3.12) that

Λℓ =[

− ETℓ , I

]

Aℓ

[

−Eℓ

I

]

= Cℓ − ETℓ Bℓ,(3.23)

which can be easily computed. The major task in the following is to evaluate φℓ(k, k).To that end, let us define

ξℓ,ℓ′(

ni,nj

)

:=

∫

n Gℓ

(

n · ni

)

Gℓ′(

n · nj

)

dΩ(n).(3.24)

Furthermore, for p = 1, 2, 3, let ξℓ,ℓ′,p(ni,nj) be the pth coordinate of ξℓ,ℓ′(ni,nj),and for fixed ℓ and p, let Ξℓ,p := [ξℓ,ℓ′,p(ni,nj)] be formed by ξℓ,ℓ′,p(ni,nj) in thesame way as Aℓ is formed by aℓ,ℓ′(i, j). One can show by using (3.17) that the pthcoordinate of φℓ(k, k), denoted by φℓ,p(k, k), is equal to the kth diagonal element of

Φℓ,p := [φℓ,p(i, j)]Nℓ

i,j=Nℓ+1+1, where

Φℓ,p :=[

− ETℓ , I

]

Ξℓ,p

[

−Eℓ

I

]

.(3.25)

As can be seen, the vectors ξℓ,ℓ′(ni,nj) are crucial to the computation of φℓ(k, k), andhence, of mℓk. The following propositions are helpful to the evaluation of ξℓ,ℓ′(ni,nj).

The first proposition provides an expression of ξℓ,ℓ′(ni,nj) in terms of the con-volution of the SBFs. A proof of this proposition is given in Appendix C.

Proposition 3.2. Let Gℓ(x) and Gℓ′(x) be multiscale SBFs and let ξℓ,ℓ′(ni,nj)be defined by (3.24). Then, for any ni and nj,

ξℓ,ℓ′(

ni,nj

)

= Uℓ,ℓ′(

xij

)

ni + Uℓ′,ℓ

(

xij

)

nj ,(3.26)

where xij := ni · nj,

Uℓ,ℓ′(x) :=

(

1 − x2)−1

Hℓ ∗Gℓ′(x) − xGℓ ∗Hℓ′(x)

if x 6= ±1,12Hℓ ∗Gℓ′(x) = 1

2xGℓ ∗Hℓ′(x) if x = ±1,

and Hℓ(x) := xGℓ(x). Note that Uℓ,ℓ′(x) = xUℓ′,ℓ(x) if x = ±1.Alternatively, the next proposition expresses ξℓ,ℓ′(ni,nj) in terms of Legendre

series. A proof of this result is given in Appendix D.

942 TA-HSIN LI

Proposition 3.3. Let gℓ,n and hℓ,n be the Legendre coefficients of Gℓ(x)and Hℓ(x), respectively, as defined by (A.1). Then,

ξℓ,ℓ′(

ni,nj

)

=

(

1 − x2ij

)−1∞∑

n=0

Pn

(

xij

)(

uℓ,ℓ′,nni + uℓ′,ℓ,nnj

)

if xij 6= ±1,

∞∑

n=0

Pn

(

xij

)

vℓ,ℓ′,nni if xij = ±1,

where xij := ni · nj,

uℓ,ℓ′,n = vℓ,ℓ′,n − 4π√

2

n

(2n− 1)2gℓ,n−1hℓ′,n−1 +

n + 1

(2n + 3)2gℓ,n+1hℓ′,n+1

,

vℓ,ℓ′,n =4π

√2

2n + 1hℓ,ngℓ′,n, hℓ,n =

n

2n− 1gℓ,n−1 +

n + 1

2n + 3gℓ,n+1, gℓ,−1 := 0.

In principle, the mean vector mℓk can be evaluated numerically by using theexpressions of ξℓ,ℓ′(ni,nj) given by Propositions 3.2 and 3.3 together with (3.20)–(3.23) and (3.25). However, for simple analytical results that help to improve ourunderstanding of the localization property of the multiscale SWs, it is necessary tospecify the SBFs. The Poisson kernel is an easy choice for this purpose.

3.7. The Poisson case. In the following, let Gℓ(x) = G(x; ηℓ), Gℓ′(x) =G(x; ηℓ′), Fℓ(x) := F (x; ηℓ), and Fℓ′(x) := F (x; ηℓ′), where G(x; η) is given by (2.5)and F (x; η) is defined by

F (x; η) :=1

1 − 2ηx + η2=

∞∑

n=0

ηnPn(x).(3.27)

In addition, let pℓ := ηℓ/(1 + η2ℓ ), rℓ,ℓ′ := 1

2 (ηℓ − ηℓ′)2/ηℓηℓ′ , sℓ,ℓ′ := 1

2 (η2ℓ − η2

ℓ′)/ηℓηℓ′ ,and tℓ,ℓ′ := pℓ′ − pℓ. With this notation, an exact expression of ξℓ,ℓ′(ni,nj) can beobtained in the following; see Appendix E for derivation.

Proposition 3.4. For the Poisson kernel, the vector ξℓ,ℓ′(ni,nj) can be ex-

pressed as (3.26) with

Uℓ,ℓ′(x) =

pℓGℓ ∗Gℓ′(x) + Rℓ,ℓ′(x) if x 6= ±1,12 (1 + x)pℓGℓ ∗Gℓ′(x) + Rℓ,ℓ′(x) if x = ±1.

In these expressions,

Rℓ,ℓ′(x) :=

[

rℓ,ℓ′(

pℓ′ − xpℓ)

+ tℓ,ℓ′(1 + x)]

Gℓ ∗Gℓ′(x)

+sℓ,ℓ′(

pℓ′ + xpℓ)

Gℓ ∗ Fℓ′(x)

(

1 − x2)−1

if x 6= ±1,

12

[

rℓ,ℓ′pℓ′ + tℓ,ℓ′(1 + x)]

Gℓ ∗Gℓ′(x)

+sℓ,ℓ′pℓ′Gℓ ∗ Fℓ′(x)

if x = ±1.

Note that Rℓ,ℓ(x) = 0 and hence Uℓ,ℓ(x) = pℓGℓ ∗Gℓ(x).Proposition 3.4 enables us to investigate two “asymptotic” cases. In the first case,

the bandwidth is assumed to increase gradually with the scale index ℓ. More precisely,let the bandwidth parameters satisfy

1 > η1 ≥ · · · ≥ ηL > 0(3.28)


and

∆ℓ := ηℓ − ηL ≪ 1.(3.29)

Then, the following theorem can be obtained; see Appendix F for a proof.Theorem 3.5. Under the assumptions (3.28) and (3.29), it follows that

mℓk =2ηℓ

1 + η2ℓ

nk + O(

∆ℓδ−2ℓ,ℓ+1

)

,

σ2ℓk =

(

1 − η2ℓ

1 + η2ℓ

)2

1 + O(

∆ℓδ−2ℓ,ℓ+1δ

−2ℓ,ℓ

)

for k = Nℓ+1 + 1, . . . , Nℓ and ℓ = 1, . . . , L− 1, where δℓ,ℓ′ := 1 − ηℓηℓ′ .Theorem 3.5 indicates that if the bandwidth parameters are not dramatically

different, for example, if ηL = η1 − O((1 − η1)4) so that ∆ℓ = O(δ2

ℓ,ℓ+1δ2ℓ,ℓ) for all ℓ,

then the resulting multiscale SWs would be located near the corresponding stationsand the bandwidths of the multiscale SWs would be proportional to 1−ηℓ. Moreover,the formulas in Theorem 3.5 are exact (without the O terms) if the bandwidths areidentical. This last result was also obtained by NW [25].

In the second “asymptotic” case, we assume that the bandwidth of the SBFs isadapted to the density of the networks, as determined by the BUD procedure. Moreprecisely, define

dℓ := min

1 − ni · nj : ni ∈ Mℓ,nj ∈ Nℓ+1

,

which can be regarded as the smallest “distance” between the removed stations andthe remaining ones at scale ℓ. Then, the assumption in the second case is

dℓ = O(

δǫℓ,ℓ+1

)

, ǫ ∈ (0, 1].(3.30)

This assumption, combined with (3.28), implies that dℓ = O(δǫℓ,ℓ′) for all ℓ′ > ℓ; it

is also equivalent to assuming that δℓ,ℓ+1 = O(dλℓ ), where λ := ǫ−1 ∈ [1,∞). In thiscase, the following results can be obtained; see Appendix G for a proof.

Theorem 3.6. Under the assumptions (3.28) and (3.30), it follows that

mℓk =2ηℓ

1 + η2ℓ

nk + O(

∆ℓδ−2ǫℓ,ℓ+1δ

2ℓ,ℓ

)

,

σ2ℓk =

(

1 − η2ℓ

1 + η2ℓ

)2

1 + O(

∆ℓδ−2ǫℓ,ℓ+1

)

for k = Nℓ+1 + 1, . . . , Nℓ and ℓ = 1, . . . , L− 1. These expressions also hold for ǫ = 1if dℓ = 0.

To see the implication of Theorem 3.6, let us assume that the bandwidth increaseswith ℓ such that ηℓ+1 < ηℓ and 1 − ηℓ+1 = O((1 − ηℓ)

ρ) for some ρ ∈ (0, 1). Sinceδℓ,ℓ+1 = 1 − ηℓ + ηℓ(1 − ηℓ+1) = O((1 − ηℓ)

ρ), it follows from Theorem 3.6 thatthe bias in mℓk (i.e., the O term) and the variance σ2

ℓk both can be expressed asO((1 − ηℓ)

2−2ρǫ). If ηℓ is near unity, then the resulting multiscale SWs would residenear the corresponding stations and their bandwidth (standard deviation or rootmean-squared error) would be proportional to (1 − ηℓ)

1−ρǫ. As another example, let1 − ηℓ = r(1 − ηℓ+1) for some r ∈ (0, 1). In this case, δℓ,ℓ+1 = r−1(r + ηℓ)(1 − ηℓ) =O(1 − ηℓ), so the bias in mℓk and the variance σ2

ℓk take the form O((1 − ηℓ)2−2ǫ).

944 TA-HSIN LI

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Observations: 1967 (J=302;Min=-46.9;Max=29.5)

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Multiscale SBF Representation: PLS-GCV (Min=-37.8;Max=28.4)

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(b)

Fig. 4.1. (a) Observed surface air temperature data (in C). (b) A multiscale SBF representa-tion of the temperature data.

This implies that the bandwidth of the resulting multiscale SWs is proportional to(1 − ηℓ)

1−ǫ.As can be seen, the bandwidth of the scale-ℓ multiscale SWs is a power function

of 1 − ηℓ. Therefore, the degree of localization of the wavelets can be controlled byproperly selecting the bandwidth parameter ηℓ. Generally, Theorem 3.6 suggests thatthe multiscale SWs should have the desired localization property, as shown by theexamples in Figures 3.4 and 3.5, if the networks and the bandwidth parameters aredesigned properly by following the BUD procedure.

4. An application example. In this section, the proposed method of mul-tiscale analysis is applied to a real data set that contains the average surface airtemperatures (in C) observed by a network of weather stations during the period ofDecember 1967 to February 1968. See [15] and [22] for more information about thedata. The objectives of this analysis include (a) constructing a multiscale SBF repre-sentation of the scattered data; (b) decomposing the constructed field into multiscalecomponents to discover local anomalies at different scales; (d) compressing the databy approximation using a subset of wavelet coefficients.

From the original network of size 939, the BUD procedure selects stations thatfall into a gradually reduced neighborhood of, and are nearest to, the gridpointsgenerated by progressively bisectioning the longitude and latitude. Some excessivestations at high latitudes are removed manually from the selected network to overcomethe unequal coverage problem of the partitions. Figure 1.1 shows the resulting 302stations grouped into L = 4 subnets. Figure 4.1(a) shows the observed temperatures


at these stations. The remaining stations are ignored in this analysis. Note that theproposed multiscale approach is necessitated by the highly irregular distribution ofthe selected (as well as the complete) stations.

With the networks so designed, Figures 4.1(b) and 4.2 show a multiscale SBFrepresentation and its multiscale decompositions. The bandwidth parameters aretaken to be η1 = 0.88, η2 = 0.77, η3 = 0.57, and η4 = 0.10, which approximatelycorrespond to spatial resolutions of 15, 30, 60, and 120, respectively. A penalizedLS method, known as ridge regression, is employed to obtain β1. For a given penaltyparameter λ > 0, ridge regression minimizes

∑ |T (nj) − βT1 g1(nj)|2 + λβT

1 β1. Thegeneralized cross-validation (GCV) technique is employed to select an optimal λ thatbalances the goodness of fit and the stability of the SBF representation; see Wahba [35,Chap. 4] or Hastie and Tibshirani [14, section 3.4] for discussions of GCV. The optimalvalue of λ in this experiment turns out to be λ = 1, with which the LS fit yieldsan R-square of 97.5% and a root mean-squared error of 3.51C, with an effectivedegree of freedom equaling 105. For other statistical methods of obtaining the SBFrepresentation, see [19].

The decomposition shown in Figure 4.2 reveals a number of strong local anomaliesat each scale of the analysis. For example, notice the extreme cold in Central Siberiaand Central Canadian Shield as shown in Figure 4.2(f). Also observe the regionalanomalies (cooler temperatures) in the Antarctic (Figure 4.2(d)) and in the Andes ofSouth America (Figure 4.2(b)), both due to high elevation. The global componentat the coarsest level (Figure 4.2(e), with 14 stations) sufficiently captures the generaltrend of global temperature: winter in the northern hemisphere and summer in thesouthern hemisphere.

This example of anomaly detection serves only for the purposes of demonstration;it should not be considered as a formal analysis of the temperature data. Furtheranalysis should incorporate geographical information and meteorological constraintsinto the BUD procedure so as to enhance the interpretability of the decomposition.

The pure geometric approach employed here is still useful for exploratory analysisand for data compression. An example of data compression is shown in Figure 4.3. Forillustration purposes, the results are obtained by a simple technique of data compres-sion: retaining a given percentage of the largest wavelet coefficients γℓk within eachscale and discarding the remaining ones. The retained wavelet coefficients, togetherwith βL, are saved instead of the original data. To reconstruct the SBF representation,the discarded coefficients are replaced by zero in the reconstruction algorithm (3.13).This technique of “lossy” compression intends to preserve most significant featuresof the uncompressed data while tolerating a certain loss of fidelity. The compressionratio is defined here as the size of the uncompressed data (i.e., J) divided by thenumber of retained coefficients. The root mean-squared error values in Figure 4.3(a)are computed in terms of the difference between the compressed and uncompressedSBF representations. As can be seen, the field in Figure 4.3(b), with compressionratio 2:1 and root mean-squared error 0.43 C, is able to preserve the major features,in both large and small scales, of the uncompressed field in Figure 4.1(b).

5. Concluding remarks. We have proposed a general wavelet method for therepresentation and analysis of scattered spherical data. The multiscale SW derived inthis paper possess the localization property with intended scales and thus overcomethe single-scale problem of the SW suggested by [25]. Together with the multiscaleSBF representation, we have also proposed a bottom-up procedure for the design andselection of the analysis networks and the bandwidth parameters. In addition, we have

946 TA-HSIN LI

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2

4

6

8

(d)

Fig. 4.2. A multiscale decomposition of the temperature field shown in Figure 4.1. Globalcomponents: (a) ℓ = 2, (c) ℓ = 3, (e) ℓ = 4; local components: (b) ℓ = 1, (d) ℓ = 2, (f) ℓ = 3. Thecrosses identify the stations involved in the corresponding components.


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Fig. 4.2 (Cont.)

developed recursive algorithms to efficiently decompose a given field into multiscaleSW components and to reconstruct a field from its wavelet coefficients. Althoughsimilar to those of Schroder and Sweldens [27] in appearance, the proposed algorithmsare different in essence, not only because they do not require regular gridpoints butalso because they are based on the LS principle rather than interpolation, and theresulting decompositions are orthogonal.

An analytical study of the localization property has confirmed the simulationresults and shown that the proposed Poisson-kernel-based wavelets are indeed spa-tially localized, with the bandwidth easily controlled by a bandwidth parameter. Theanalytical study not only serves as a justification of the proposed method but alsoprovides the necessary foundation for future research on the statistical properties ofthe wavelet coefficients and the utility of the SWs for detecting regional singularitiesfrom global trends, for smoothing and compression of spherical data, and for multi-scale space-time modeling and forecasting of spherical fields. The potential usefulnessof the proposed method is further enhanced by the real-data example.

Just as no simple rules exist for determining the smoothing kernels in nonpara-metric regression problems, simple rules that tell a practitioner how to select the SBFsare yet to be developed. In addition to rapid decaying, smoothness is often considereda requirement. Some SBFs, such as spherical splines [35], have interesting optimiza-tion properties. Others, such as the Poisson kernel employed in this paper, havedesirable computational properties and meaningful physical interpretations. How toselect the SBFs so that the resulting wavelets satisfy a given set of requirements is aninteresting theoretical issue that deserves further investigation.

948 TA-HSIN LI

Compression Ratio (log10)

RMSE

(deg

ree

C)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

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0.8

1.0

Rate-Distortion Function

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(b)

Fig. 4.3. (a) Rate-distortion function of the simple data-compression method. (b) A com-pressed multiscale SBF representation with compression ratio 2:1 ( 50%) and root mean-squarederror 0.43 C. Compression is applied to ℓ = 1, 2.

Another theoretical question is how to quantify the condition number of the SWrepresentation. This question was addressed recently by Narcowich, Sivakumar, andWard [26] for the single-scale case. Extension of their results to the multiscale cases iscertainly an interesting subject for future research. In principle, a stable representa-tion with bounded condition numbers requires that the SBF bandwidth be judiciouslyselected to avoid the multicollinearity problem. This requirement can be satisfied byusing SBFs of small bandwidth. But a bandwidth that is too small, as comparedwith the density of stations, loses the ability of representation. A trade-off thus takesplace. For practical purposes, the cross-validation techniques, such as GCV, can beemployed, together with the BUD procedure, to help select optimal bandwidths thatbalance the stability and the goodness of fit. Elaborate discussions along these lines,with examples, are given in [19].

Appendix A. Further remarks on the SBFs. In addition to the square-integrability, the SBFs also have to satisfy other conditions. To motivate a usefulconstraint, consider the SH representation (2.2). With G(ni,nj) given by (2.3), wehave G := [G(ni,nj)]

Ji,j=1 = Y(YHY)−1YH . Under the full-rank assumption, G is

positive definite. In order for an SBF G(x) to have this property, the coefficients gn


of the Legendre series

G(x) =

∞∑

n=0

gnPn(x)(A.1)

must be absolutely summable and strictly positive for all but finitely many of themthat may be equal to zero [28]. According to this definition, the spherical splinesconsidered by Wahba [34] can also be regarded as SBFs. A drawback of the sphericalsplines is that their bandwidth lacks the desirable flexibility (as compared to thePoisson kernel, for example).

A useful way of generating SBFs is via spherical convolution. If G(x) and H(x)are SBFs, having Legendre series expansions of the form (A.1) with coefficients gnand hn, then their spherical convolution, defined as

G ∗H(n ·m) :=

∫

G(n · ν)H(ν ·m) dΩ(ν),(A.2)

with the integration computed over the entire sphere, is also an SBF. This result canbe easily justified upon noting that (A.2) has a Legendre series expansion

G ∗H(x) =

∞∑

n=0

4π√

2

2n + 1gnhnPn(x),(A.3)

where the coefficients are absolutely summable and strictly positive except for possiblyfinitely many that are equal to zero. Equation (A.3) also shows that the convolutionis a symmetric operator: G ∗H(n ·m) = H ∗G(n ·m) = G ∗H(m · n).

The Poisson kernel (2.5) is attractive computationally because of its closed-formexpression and its invariance under spherical convolution, as shown in (3.19). ThePoisson kernel also has an interesting physical interpretation in potential theory. Tosee this, let us assume that a point mass M := (1 − η2)η−1 is located at ηnj (withinthe unit sphere). Then, the (gravitational) Newtonian potential at n for the pointmass takes the form Φ(n) := M(1− 2η(n ·nj) + η2)−1/2. Since n ·nj is equal to theprojection of n onto nj , it follows that

∂Φ(n)

∂nj=

Mη

(1 − 2η(n · nj) + η2)3/2= G(n · nj ; η).

In words, the Poisson kernel G(n · nj ; η) can be interpreted as the gradient of theNewtonian potential at n in the direction of nj . Furthermore, since Φ(n) satisfiesthe Laplace equation ∇2Φ(n) = 0, so does the Poisson kernel: ∇2G(n · nj ; η) =∂(∇2Φ(n))/∂nj = 0. Therefore, the corresponding SBF representations and SWsalso satisfy the Laplace equation.

Appendix B. Proof of Theorem 3.1. Any field Tℓ(n) ∈ Vℓ can be expressedas

Tℓ(n) =∑

(ℓ′,j)∈Iℓ

βℓjGℓ′(

n · nj

)

= βTℓ gℓ(n),

where Iℓ := (ℓ′, j) : ℓ ≤ ℓ′ ≤ L;Nℓ′+1 < j ≤ Nℓ′. Since Tℓ+1(n) = βTℓ+1gℓ+1(n)

is the projection of Tℓ(n) in Vℓ+1, it can be shown that βℓ+1 = [I,Eℓ]βℓ. Therefore,

Dℓ(n) = βTℓ+1gℓ+1(n) − βT

ℓ gℓ(n) = γTℓ wℓ(n), where wℓ(n) := [−ET

ℓ , I]gℓ(n). The

950 TA-HSIN LI

assertion can be proved by rewriting these expressions according to the partitionsdefined by (3.6) and (3.10).

Appendix C. Proof of Proposition 3.2. Only a sketch is provided here. De-tails can be found in [18]. The proof closely follows [25] in showing that ξℓ,ℓ′(ni,nj) =ani + bnj for some constants a and b.

To determine a and b, it is helpful to note that ni · ξℓ,ℓ′(ni,nj) = a + bxij

and ξℓ,ℓ′(ni,nj) · nj = axij + b. On the other hand, it follows from (3.24) thatni · ξℓ,ℓ′(ni,nj) = Hℓ ∗ Gℓ′(xij) and ξℓ,ℓ′(ni,nj) · nj = Gℓ ∗ Hℓ′(xij). Combiningthese results leads to

a + bxij = Hℓ ∗Gℓ′(

xij

)

, axij + b = Gℓ ∗Hℓ′(

xij

)

.(C.1)

Assuming xij 6= ±1, one obtains a = Uℓ,ℓ′(xij) and b = Uℓ′,ℓ(xij), where the expressionfor b follows from the fact that Hℓ′ ∗Gℓ(x) = Gℓ∗Hℓ′(x) and Gℓ′ ∗Hℓ(x) = Hℓ∗Gℓ′(x).If xij = ±1, then ni = ±nj and hence ξℓ,ℓ′(ni,nj) = (a± b)ni. It follows from (C.1)that a± b = Hℓ ∗Gℓ′(±1) = ±Gℓ ∗Hℓ′(±1); the second equality can also be verifiedfrom (A.2) upon noting that Hℓ(x) = xGℓ(x) and Hℓ′(x) = xGℓ′(x).

Appendix D. Proof of Proposition 3.3. The assertion follows from Proposi-tion 3.2 and (A.3), combined with the identity xPn(x) = (2n+1)−1(n+1)Pn+1(x)+nPn−1(x) and the fact that if xij = ±1, then ξℓ,ℓ(ni,nj) = 2Uℓ,ℓ′(xij)ni = Hℓ ∗Gℓ′(xij)ni.

Appendix E. Proof of Proposition 3.4. It can be shown [25] that Hℓ(x) :=xG(x; ηℓ) = 1

2p−1ℓ Gℓ(x)−q−1

ℓ Fℓ(x), where pℓ := ηℓ(1+η2ℓ )

−1 and qℓ := ηℓ(1−η2ℓ )

−1.

Therefore, Hℓ ∗Gℓ′(x) = 12p−1

ℓ Gℓ ∗Gℓ′(x)− q−1ℓ Fℓ ∗Gℓ′(x). Using (2.5) and (3.27),

together with (A.2), one can show that Fℓ ∗ Gℓ′(x) = 4π√

2F (x; ηℓηℓ′). This result,combined with (3.19), yields

Hℓ ∗Gℓ′(x) = 2π√

2

p−1ℓ G

(

x; ηℓηℓ′)

− q−1ℓ F

(

x; ηℓηℓ′)

= pℓ′Qℓ,ℓ′(x),

where

Qℓ,ℓ′(x) := 4π√

2

(1 + x)G(

x; ηℓηℓ′)

+ rℓ,ℓ′G(

x; ηℓηℓ′)

+ sℓ,ℓ′F(

x; ηℓηℓ′)

= (1 + x)Gℓ ∗Gℓ′(x) + rℓ,ℓ′Gℓ ∗Gℓ′(x) + sℓ,ℓ′Gℓ ∗ Fℓ′(x).(E.1)

The expression of Uℓ,ℓ′(x) for x = ±1 can be obtained from (E.1) upon noting thatUℓ,ℓ′(x) = 1

2pℓ′Qℓ,ℓ′(x). For x 6= ±1, since Gℓ ∗Hℓ′(x) = Hℓ′ ∗Gℓ(x) = pℓQℓ′,ℓ(x), itfollows that

Uℓ,ℓ′(x) =(

1 − x2)−1

pℓ′Qℓ,ℓ′(x) − xpℓQℓ′,ℓ(x)

.

The final expression of Uℓ,ℓ′(x) can be obtained by adding and subtracting pℓ(1 +x)Gℓ ∗ Gℓ′(x) in the braces of the foregoing expression and then using (E.1). Theproof is thus complete.

Appendix F. Proof of Theorem 3.5. Note that Gℓ ∗ Gℓ′(1) = O(δ−2ℓ,ℓ′) and

Gℓ ∗Gℓ′(x) = O(1) if x 6= 1. Since the same is true for Gℓ ∗Fℓ′(x), and since ηℓ−ηℓ′ ≤∆ℓ and δℓ,ℓ′ > δℓ,ℓ+1 for all ℓ′ > ℓ under (3.28), it follows that Rℓ,ℓ′(1) = O(∆ℓδ

−2ℓ,ℓ+1)

and Rℓ,ℓ′(x) = O(∆ℓ) if x 6= 1, for ℓ′ > ℓ; the same expression holds for Rℓ′,ℓ(x).Therefore, by Proposition 3.4 and (3.18),

ξℓ,ℓ′(ni,nj) =

pℓaℓ,ℓ(i, j)(

ni + nj

)

if ℓ = ℓ′,

pℓaℓ,ℓ′(i, j)(

ni + nj

)

+ O(

∆ℓδ−2ℓ,ℓ+1

)

if ℓ′ > ℓ.(F.1)


It is not surprising that ξℓ′,ℓ(ni,nj) has the same “asymptotic” form.To proceed with the proof, let ℓ = 1 and L = 2 without loss of generality. It is

easy to show from (F.1) that

Ξ1,p =

[

p2

(

N2A2 + A2N2

)

p1

(

N2B1 + B1M1

)

p1

(

M1BT1 + BT

1 N2

)

p1

(

M1C1 + C1M1

)

]

+ O(

∆1δ−212

)

,

where M1 := diagnipJi=K+1 and N2 := diagnipKi=1, with nip denoting the pthcoordinate of ni. Substituting this expression in (3.25) yields

Φ1,p = p1

(

M1C1 + C1M1

)

+ p2

(

ET1 N2B1 + BT

1 N2E1

)

−p1

(

ET1 N2B1 + ET

1 B1M1

)

− p1

(

M1BT1 E1 + BT

1 N2E1

)

+ O(

∆1δ−212

)

= p1

(

M1Λ1 + Λ1M1

)

+ O(

∆1δ−212

)

,

where the second equality results from cancellation and from (3.23). Thus, φ1(i, i) =2p1λ1(i, i)ni + O(∆1δ

−212 ). The proof is complete upon noting (3.20).

Appendix G. Proof of Theorem 3.6. Let L = 2 for simplicity. By Proposi-tion 3.2, the exact expression of Ξ1,p takes the form

Ξ1,p =

[

N2U22 + U22N2 N2U21 + UT12M1

M1U12 + UT21N2 M1U11 + U11M1

]

,

where U11 := [U11(xij)]Ji,j=K+1, U12 := [U12(xij)] (1 ≤ j ≤ K < i ≤ J), U21 :=

[U21(xij)] (1 ≤ i ≤ K < j ≤ J), and U22 := [U22(xij)]Ki,j=1. It is easy to show that

Φ1,p = M1S1 + ST1 M1 + ET

1 N2R1 + RT1 N2E1,

where

S1 :=[

s1(i, j)]J

i,j=K+1:= U11 − U12E1,

R1 :=[

r1(i, j)]

:= U22E1 − U21 (1 ≤ i ≤ K < j ≤ J).

Therefore, it follows that

m1i = 2s1(i, i)

λ1(i, i)ni + 2

K∑

j=1

e1(j, i)r1(j, i)

λ1(i, i)nj .

By Proposition 3.4, U22(xij) = p2a22(i, j), so that U22 = p2A2 and hence R1 =p2B1 − U21. Also by Proposition 3.4, if xij 6= −1, then U21(xij) = p2a21(j, i) +R21(xij), so that r1(i, j) = −R21(xij); if xij = −1, then U21(xij) = R21(xij),r1(i, j) = p2a21(j, i) −R21(xij), hence r1(i, j) = O(1).

Now, for 1 ≤ j ≤ K < i ≤ J , assume that 1 − xij attains the minimum value d1

so that 1−xij = cδǫ12 for some constant c. Set ǫ = ∞ if d1 = 0. Then, G1 ∗G2(xij) =O(δ−ǫ1

12 ) and G1 ∗ F2(xij) = O(δ−ǫ212 ), where ǫ1 := max( 3

2ǫ− 1, 2) and ǫ2 := max(ǫ, 2).

Since ǫ2 ≥ ǫ1, it follows from Proposition 3.4 that r1(j, i) = O(∆1δ−2ǫ012 ), where

ǫ0 = max(ǫ, 1). This expression also dominates the other cases in which 1− xij > d1.On the other hand, a11(i, i) = O(∆−2

11 ) and e1(j, i) = O(1), so that

λ1(i, i) = a11(i, i) −K∑

j=1

e1(j, i)a21(j, i)

= O(

∆−211

)

+ O(

∆−ǫ112

)

= O(

∆−211

)

.

952 TA-HSIN LI

Furthermore, by Proposition 3.4, U11 = p1C1 and thus S1 = p1Λ1 + RT1 E1 + (p1 −

p2)BT1 E1. This, coupled with the fact that 2ǫ0 ≥ ǫ1, implies that

s1(i, i) = p1λ1(i, i) + O(

∆1δ−2ǫ012

)

+ O(

∆1δ−ǫ112

)

= p1λ1(i, i) + O(

∆1δ−2ǫ012

)

.

Combining these results yields m1i = 2p1ni + O(∆1δ−2ǫ012 δ2

11). The assertion is thusproven.

Acknowledgments. The author thanks the anonymous reviewers for helpfulcomments. He also thanks Gerald R. North, distinguished professor of meteorologyand oceanography at Texas A&M University, College Station, for insightful discussionsand suggestions.

REFERENCES

[1] M. Akay, ed., Time Frequency and Wavelets in Biomedical Signal Processing, IEEE Press,Piscataway, NJ, 1998.

[2] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed., AcademicPress, San Diego, 1995.

[3] E. J. Barron, D. Hartmann, D. Schimel, and M. Schoeberl, Earth Observing System:Science Contributions to National Goals and Interests, Tech. Report, Earth System ScienceCenter, Pennsylvania State University, University Park, PA, 1995.

[4] W. Bourke, Spectral methods in global climate and weather prediction models, in Physically-Based Modeling and Simulation of Climate and Climate Change, Part I, M. E. Schlesinger,ed., Kluwer Academic Press, New York, 1988, pp. 169–220.

[5] M. D. Buhmann and C. A. Micchelli, Spline prewavelets for non-uniform knots Numer.Math., 61 (1992), pp. 445–474.

[6] C. K. Chui, An Introduction to Wavelets, Academic Press, San Diego, 1992.[7] R. Daley, Atmospheric Data Analysis, Cambridge University Press, Cambridge, UK, 1991.[8] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math. 61,

SIAM, Philadelphia, PA, 1992.[9] R. A. DeVore, B. Jawerth, and B. L. Lucier, Image compression through wavelet transform

coding, IEEE Trans. Inform. Theory, 38 (1992), pp. 719–746.[10] D. L. Donoho, De-noising by soft-thresholding, IEEE Trans. Inform. Theory, 41 (1995),

pp. 613–627.[11] D. L. Donoho and I. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika,

18 (1994), pp. 425–455.[12] W. Freeden and U. Windheuser, Spherical wavelet transform and its discretization, Adv.

Comput. Math., 5 (1996), pp. 51–94.[13] W. Freeden and U. Windheuser, Combined spherical harmonics and wavelet expansion—A

future concept in earth’s gravitational determination, Appl. Comput. Harmon. Anal., 4(1997), pp. 1-37.

[14] T. J. Hastie and R. J. Tibshirani, Generalized Additive Models, Chapman & Hall, London,1990.

[15] P. D. Jones, S. C. B. Raper, B. S. G. Cherry, C. M. Goodess, T. M. L. Wigley, B. Santer,

P. M. Kelly, R. S. Bradley, and H. F.Diaz, An Updated Global Grid Point SurfaceTemperature Anomaly Data Set: 1851–1988, Environmental Sciences Division PublicationNo. 3520, U. S. Department of Energy, Washington, DC, 1991.

[16] K.-Y. Kim and G. R. North, EOF analysis of surface temperature field in a stochastic climatemodel, J. Climate, 6 (1993), pp. 1681–1690.

[17] S. S. Leroy, Measurement of geopotential heights by GPS radio occultation, J. GeophysicalResearch, 102 (1997), pp. 6971–6986.

[18] T. H. Li, Multiscale Representation of Spherical Data by Spherical Wavelets, Tech. Report 356,Department of Statistics and Applied Probability, University of California, Santa Barbara,1998.

[19] T. H. Li, Multiscale Wavelet Analysis of Spherical Data: Design and Estimation, Tech. Report358, Department of Statistics and Applied Probability, University of California, SantaBarbara, 1999.


[20] T. H. Li and M. J. Hinich, A Filter Bank Approach for Modeling and Forecasting SeasonalPatterns, Tech. Report 355, Department of Statistics and Applied Probability, Universityof California, Santa Barbara, 1998.

[21] T. H. Li, G. R. North, and S. S. Shen, Aliased power of a stochastic temperature field on asphere, J. Geophysical Research, 102 (1997), pp. 4475–4489.

[22] Z. Luo, G. Wahba, and D. R. Johnson, Spatial-temporal analysis of temperature usingsmoothing spline ANOVA, J. Climate, 11 (1997), pp. 18–29.

[23] S. Mallat, A theory of multiresolution signal decomposition: The wavelet representation,IEEE Trans. Pattern Analysis and Machine Intelligence, 11 (1989), pp. 674–693.

[24] S. Mallat and W. L. Hwang, Singularity detection and processing with wavelets, IEEE Trans.Inform. Theory, 38 (1992), pp. 617–643.

[25] F. J. Narcowich and J. D. Ward, Nonstationary wavelets on the m-sphere for scattered data,Appl. and Comput. Harmon. Anal., 3 (1996), pp. 324–336.

[26] F. J. Narcowich, N. Sivakumar, and J. D. Ward, Stability results for scattered-data inter-polation on Euclidean spheres, Adv. Comput. Math., 8 (1998), pp. 137–163.

[27] P. Schroder and W. Sweldens, Spherical wavelets: Efficiently representing functions on thesphere, Computer Graphics Proceedings, SIGGRAPH95, 1995, pp. 161–172.

[28] M. Schreiner, On a new condition for strictly positive definite functions on spheres, Proc.Amer. Math. Soc., 125 (1997), pp. 531–539.

[29] J. M. Shapiro, Embedded image coding using zero trees of wavelet coefficients, IEEE Trans.Signal Processing, 41 (1993), pp. 3445–3462.

[30] S. S. Shen, G. R. North, and K.-Y. Kim, Spectral approach to optimal estimation of theglobal average temperature, J. Climate, 7 (1994), pp. 1999–2007.

[31] G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley,MA, 1996.

[32] M. Vetterli and J. Kovacevic. Wavelets and Subband Coding, Prentice-Hall, Upper SaddleRiver, NJ, 1995.

[33] K. Y. Vinnikov, P. Y. Groisman, and K. M. Lugina, Empirical data on contemporary globalclimate changes (temperature and precipitation), J. Climate, 3 (1990), pp. 662–677.

[34] G. Wahba, Spline interpolation and smoothing on the sphere, SIAM J. Sci. Statist. Comput,2 (1981), pp. 5–16; Erratum, SIAM J. Sci. Statist. Comput., 3 (1982), pp. 385–386.

[35] G. Wahba, Spline Models for Observational Data, SIAM, Philadelphia, PA, 1990.[36] W. M. Washington and C. L. Parkinson, An Introduction to Three-Dimensional Climate

Modeling, Univ. Sci. Books, Mill Valley, CA, 1986.

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