Multiscale Modeling From Eigen-1s, Eigen-2, Eigen-grace01s,

8/12/2019 Multiscale Modeling From Eigen-1s, Eigen-2, Eigen-grace01s,

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MULTISCALE MODELING FROM EIGEN-1S, EIGEN-2, EIGEN-GRACE01S,GGM01, UCPH2002 0.5, EGM96

Martin J. Fengler(1), Willi Freeden(1) and Martin Gutting(1)

(1) Geomathematics Group, University of Kaiserslautern,

P.O.Box 3049, 67653 Kaiserslautern, GermanyM. Gutting will present these results at the workshop.

Abstract

Spherical wavelets have been developed by the Geomathematics Group Kaiserslautern for several years andhave been successfully applied to georelevant problems. Wavelets can be considered as consecutive band-pass filters and allow local approximations. The wavelet transform can also be applied to spherical harmonicmodels of the Earths gravitational field like the most up-to-date EIGEN-1S, EIGEN-2, EIGEN-GRACE01S,GGM01, UCPH2002 0.5, and the well-known EGM96. Thereby, wavelet coefficients arise. In this paper itis the aim of the Geomathematics Group to make these data available to other interested groups. Thesewavelet coefficients allow not only the reconstruction of the wavelet approximations of the gravitationalpotential but also of the geoid, of the gravity anomalies and other important functionals of the gravitational

field. Different types of wavelets are considered: bandlimited wavelets (here: Shannon and Cubic Polynomial(CuP)) as well as non-bandlimited ones (in our case: Abel-Poisson). For these types wavelet coefficients arecomputed and wavelet variances are given. The data format of the wavelet coefficients is also included.

Keywords: Multiscale Modeling, Wavelets, Wavelet Variances, Wavelet Coefficients, Gravitational Field ModelConversion

1 INTRODUCTION

During the last years spherical wavelets have been brought into existence (cf. e.g. [3], [4], [5] and the referencestherein). It is time to apply them to well-known models in order to offer easy access to the multiscale methods.Therefore, the spherical harmonics models EIGEN-1S, EIGEN-2, GGM01, UCPH2002 0.5, EGM96 and alsoEIGEN-GRACE01S are transformed into bilinear wavelet models (see [3], [4] or [6]) and the coefficients of these

models are made available by the Geomathematics Group via the worldwide web. Moreover, the very samecoefficients enable a multiscale reconstruction of the geoid, the gravity disturbance, the gravity anomalies andthe vertical gravity gradients by special types of reconstructing wavelets.Our wavelets are constructed from so-called scaling functions, i.e. kernels that depend on a scale j , of the form

j(x, y) =n=0

n=1

j(n)2n + 1

4R2

R2

|x||y|

n+1Pn

x

|x|

y

|y|

, (x, y) extR

extR , (1)

where extR denotes the outer space of the sphere R of radius R, Pn is the Legendre polynomial of degree nand j(n) is called the symbol of the scaling function. This symbol is a sequence of numbers that determinesthe shape of the scaling function and has special properties (for details see [2], [3] or [4]). These properties makethese kernels an approximation of the Dirac distribution that converges to it for j tending to infinity. Wavelets

are defined by taking the difference of two consecutive scales which is performed by the refinement equation forthe symbols:

j(n)j(n) = 2j+1(n)

2j(n) , n= 0, 2, 3, . . . . (2)

The primal and dual wavelets j , j are obtained by either taking the square root or applying the thirdbinomial formula to equation 2. Generally, they look the following way:

j(x, y) =

n=0

n=1

j(n)2n + 1

4R2

R2

|x||y|

n+1Pn

x

|x|

y

|y|

, (3)

j(x, y) =n=0

n=1

j(n)2n + 1

4R2

R2

|x||y|

n+1Pn

x

|x|

y

|y|

. (4)

____________________________________________Proc. Second International GOCE User Workshop GOCE, The Geoid and Oceanography,ESA-ESRIN, Frascati, Italy, 8-10 March 2004 (ESA SP-569, June 2004)


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This construction allows an approximation of level Jof a potential F:

FJ = (2)J F =

(2)J0

F+

J1j=J0

j (j F) = J0 (J0 F) +

J1j=J0

j (j F), (5)

where denotes the convolution in L

2

(R). These convolution integrals can be discretized for numericalevaluation by methods presented in [1] or [4]. For the wavelet coefficients that we offer to other groups we chosethe equiangular grid discussed in e.g. [1].The important examples that we used for the computation of our multiscale representations are the Shannontype, the CuP type, and the Abel-Poisson type wavelets:

1.1 Shannon Wavelets

In the case of Shannon scaling functions the symbol j(n) reads as follows

SHj (n) =

1 for n [0, 2j)0 for n [2j ,),

(6)

and for the corresponding wavelets we choose the P-scale version to resolve the refinement equation (2), i.e.

SHj (n) = SHj (n) =

SHj+1(n)

2

SHj (n)2

. (7)

1.2 Cubic Polynomial (CuP) Wavelets

In the CuP case the symbol takes the following form:

CPj (n) =

(1 2jn)2(1 + 2j+1n) for n [0, 2j)

0 for n [2j,) (8)

and for the corresponding wavelets we apply again the P-scale version.

1.3 Abel-Poisson Wavelets

For the Abel-Poisson scaling function the symbol takes the following form APj (n) = e2jn ,n [0,), with

some constant >0. We choose = 1. SinceAPj (n)= 0 for all n Nthis symbol leads to a non-bandlimitedkernel. It should be noted that the Abel-Poisson scaling function has a closed form representation which allowsthe omission of a series evaluation and truncation, and when constructing bilinear Abel-Poisson wavelets wewant to keep such a representation as an elementary function. Thus, we decide to use M-scale wavelets whosesymbols are deduced from the refinement equation (2) by the third binomial formula:

APj (n) =

APj+1(n) APj (n)

, APj (n) =

APj+1(n) +

APj (n)

. (9)

Since the Abel-Poisson scaling function and its corresponding wavelets are non-bandlimited we obtain just agood approximation by the numerical integration method based on an equiangular grid (we choose the parameter

of polynomial exactness sufficiently large enough).

2 MULTISCALE REPRESENTATION OF THE GRAVITATIONAL POTEN-TIAL

The Earths gravitational potential V in a point x of the outer space of R, i.e. the gravity potential Wwithout the part caused by centrifugal force, possesses the following representation by convolutions withscaling functions and wavelets:

V(x) =

R

(j0 V)(y)j0(x, y)d(y) +

Jmax1j=j0

R

W Tj(V; y)j(x, y)d(y), (10)


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whereJmax is some suitably chosen maximal level of approximation and W Tj(V; y) = (j V)(y) denotes thewavelet transform ofVat scale j in the point y . In discrete form we get:

V(x) =GM

R

Nj0i=1

wj0i vj0i j0(x, y

j0i ) +

GM

R

Jmax1j=j0

Nji=1

wji vji j(x, y

ji ) . (11)

The weights of the integration are named wj0i ,wji and the corresponding knots are y

j0i ,y

ji . The scaling function

coefficientsvj0i and the wavelet coefficientsv

ji result from the convolutions (12):

GM

R v

j0i = (j0 V)(y

j0i ) ,

GM

R v

ji =W Tj(V; y

ji ) = (j V)(y

ji ). (12)

Fig. 1 and 2 show examplarily a part of the multiscale resolution of (10) where the details (Fig. 2) are addedto the approximation of scale 7 (Fig. 1). By subtracting the non-centrifugal part of the ellipsoidal normal

Figure 1: Approximation at Scale 7 of V fromEIGEN2 using the CuP scaling function, [m2/s2].

Figure 2: Wavelet detail at scale 7 ofV from EIGEN2using the CuP wavelet, [m2/s2].

potentialVell = U from V the disturbing potential T = V Vell can be obtained (see [7], [8] or [9]) and

this subtraction can be performed for the coefficients vj0i , v

ji in order to obtain a multiscale representation

of the disturbing potential similar to (11), but with coeffients tj0i andt

ji that are related tov

j0i , v

ji by the

equations (13):

GM

R tj0i =

GM

R v

j0i (j0 Vell)(y

j0i ) ,

GM

R tji =

GM

R v

ji (j Vell)(y

ji ) (13)

A more detailed derivation can be found in [2].

3 FUNCTIONALS OF THE DISTURBING POTENTIAL

By virtue of the Bruns formulaN=T /(cf. [7], [8] or [9]) a multiscale representation of thegeoid undulations

can be computed from the multiscale decomposition of the disturbing potential, i.e. from the coefficients tj0i

andtji . Thereby, the normal gravityis taken spherically as=

GMR2

. Thus, the geoid heights Nare describedby

N(x) = R

Nj0i=1

wj0i tj0i j0(x, y

j0i ) + R

Jmax1j=j0

Nji=1

wji tji j(x, y

ji ) . (14)

In Fig. 3 and 4 parts of a multiresolution ofN are presented (see [2] for a full multiresolution).


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Figure 3: Multiscale geoid heights Nat Scale 7 fromEGM96 using the CuP scaling function, [m].

Figure 4: Wavelet detail at scale 7 of geoid heights Nfrom EGM96 using the CuP wavelet, [m].

By definition the gravity disturbancescorrespond to the negative first radial derivative ofTwhich leads to the

multiscale representation (15):

g(x) = GM

R

Nj0

i=1

wj0i tj0i

rxj0(x, y

j0i ) +

Jmax1j=j0

Nji=1

wji tji

rxj(x, y

ji )

= GM

R|x|

Nj0i=1

wj0i tj0i

gj0

(x, yj0i ) +GM

R|x|

Jmax1j=j0

Nji=1

wji tji

gj (x, y

ji ) (15)

with

gj0 (x, yj0i ) = |x|

rxj0(x, y

j0i ) =

n=0

n=1

gj (n)2n + 1

4R2

R2

|x||yj0i |

n+1Pn

x

|x|

yj0i|yj0i |

(16)

where gj (n) = (n+ 1)j(n) and the corresponding reconstructing waveletsgj are constructed by applying

the symbol gj (n) = (n + 1)j(n).

Analogously, the multiscale descriptions of the gravity anomalies g = g 2|x|T and the vertical gravity

gradientsgr = 2Tr2

assume the following shape:

g(x) = GM

R|x|

Nj0i=1

wj0i tj0i

gj0

(x, yj0i ) +GM

R|x|

Jmax1j=j0

Nji=1

wji tji

gj (x, y

ji ) , (17)

gr(x) = GM

R|x|2

Nj0

i=1wj0i t

j0i

grj0

(x, yj0i ) + GM

R|x|2

Jmax1

j=j0Nj

i=1wji t

ji

grj (x, y

ji ) , (18)

where

gj0 (x, yj0i ) =

n=0

n=1

gj (n)2n + 1

4R2

R2

|x||yj0i |

n+1Pn

x

|x|

yj0i|yj0i |

, (19)

grj0 (x, yj0i ) = |x|

2 2

r2xj0(x, y

j0i ) =

n=0

n=1

grj (n)2n + 1

4R2

R2

|x||yj0i |

n+1Pn

x

|x|

yj0i|yj0i |

(20)

with the symbols

gj (n) = gj (n) 2j(n) = (n 1)j(n) ,

grj (n) = (n + 1)(n + 2)j(n) , (21)


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and corresponding symbols gj (n) andgrj (n) for the respective wavelets. As an example we show in Fig. 5

and 6 for the gravity anomalies gagain scale and detail 7 with CuP scaling functions or wavelets, respectively.

Figure 5: g at Scale 7 from EGM96 using CuP,

[mgal].

Figure 6: Wavelet detail at scale 7 of gfrom EGM96

using CuP, [mgal].

4 WAVELET VARIANCES

The distribution of space-dependent signal energy of the disturbing potential is described by the scale and spacevariancesofTat scale j and pointx (see [2], [5]):

Varj; x(T) =2j; x(T) =

R

R

T(y)T(z)j(y, x)j(z, x)d(y)d(z) . (22)

Using the wavelet coefficientstji one can compute this quantity with the help of a Shannon kernelS H( , ) as

j; x(T) = GM

R j; x ,

2j; x =

Nji=1

wji tji SH(y

ji , x)

2

. (23)

The wavelet variances of the geoidal heights N are then given by j; x(N) = Rj; x; and for the gravitydisturbances, the gravity anomalies, and the vertical gravity gradients the variances can be obtained by aconvolution with special Shannon kernels similar to (16), (19) and (20):

2j; x(g) =

GM

R|x|

Nji=1

wji tji SH

g(yji , x)

2

, (24)

2j; x(g) =GM

R|x|

Nji=1

wji tj

i SHg(yji , x)2

, (25)

2j ; x(gr) =

GM

R|x|2

Nji=1

wji tji SH

gr (yji , x)

2

. (26)

Fig. 7 to 10 demonstrate the development of the energy distribution of the disturbing potential for j = 4 to 7.

5 WAVELET COEFFICIENTS

We supply to the end-user the scaling function or wavelet coefficients, vj0i orv

ji corresponding to the locations

of the equiangular grid on R as well as the integration weightswj0i ,w

ji . The wavelet coefficients are ordered as


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Figure 7: 4; x(T) (rescaled to maximal 400), [m2/s2]. Figure 8: 5; x(T) (rescaled to maximal 250), [m

2/s2].

Figure 9: 6; x(T) (rescaled to maximal 150), [m2/s2]. Figure 10: 7; x(T) (rescaled to max. 100), [m

2/s2].

a square matrix, in addition the weights form a first column. Comments are located in the first 20 lines of the fileand are indicated by a % sign. The coefficients, a detailed model description and further figures can be found anddownloaded at the following web page: http://www.mathematik.uni-kl.de/~wwwgeo/waveletmodels.html

References

[1] Driscoll J. R., Healy D. M.,Computing Fourier Transforms and Convolutions on the 2-Sphere, Adv. Appl.Math., Vol. 15, 202250, 1994.

[2] Fengler M. J., Freeden W., Gutting M., Darstellung des Gravitationsfelds und seiner Funktionale mitMultiskalentechniken, Zeitschrift fur Vermessungswesen (ZfV), 2004. (submitted)

[3] Freeden W., Multiscale Modelling of Spaceborne Geodata, BG Teubner, Stuttgart, Leipzig, 1999.

[4] Freeden W., Gervens T., Schreiner M., Constructive Approximation on the Sphere (With Applications toGeomathematics), Oxford Science Publications, Clarendon, Oxford, 1998.

[5] Freeden W., Maier T., On Multiscale Denoising of Spherical Functions: Basic Theory and NumericalAspects, Electron. Trans. Numer. Anal. (ETNA), Vol. 14, 4062, 2002.

[6] Freeden W., Michel V., Multiscale Potential Theory (with Application to Earths Sciences), Birkhauser,2004. (in print)

[7] Groten E.,Geodesy and the Earths Gravity Field I, II, Dummler, 1979.

[8] Heiskanen W. A., Moritz H., Physical Geodesy, W. H. Freeman and Company, 1967.

[9] Torge W.,Geodasie, Sammlung Goschen, de Gruyter, 1975.

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Multiscale Modeling From Eigen-1s, Eigen-2, Eigen-grace01s,