GRAVITY FIELD DETERMINATION USING

MULTIRESOLUTION TECHNIQUES

Michael Schmidt�����

, Oliver Fabert�����

, C.K. Shum�����

, Shin-Chan Han�����

�����Deutsches Geodatisches Forschungsinstitut, Marstallplatz 8 80539 Munchen, Germany,

�����Laboratory for Space

Geodesy and Remote Sensing, Ohio State University, 2070 Neil Av., Columbus, Ohio 43210, USA

AbstractIn this paper we present results for modeling the Earth’s gravitational field using spherical wavelets and applying method-ologies for the estimation of the corresponding coefficients. The observation types in our techniques could either begravity gradient tensor measurements from the Goce gradiometer, or other gravity mapping mission data such as theGrace low-low intersatellite KA-band range-rate, or Champ high-low intersatellite GPS phase data, or a combination ofall the data types. Our approach allows both, either a wavelet-only solution or a combination of a spherical harmonicspart with an corresponding spherical wavelet part. Using appropriate techniques for the solution of the resulting normalequation system, series coefficients up to a certain detail level can be estimated. Finally, we provide a demonstration ofthe developed methodology using simulated data.

1 General ConceptCurrent gravity models of the Earth, such as EGM 96, are based on a series expansion in terms of spherical harmonics. Inorder to create a new Earth gravity model this paper deals with an alternative representation of the geopotential, namely amultiresolution representation based on wavelet theory. Spherical harmonics are global basis functions and therefore notappropriate for regional or local representations. Wavelet functions, however, are characterized by their ability to localizeboth in the spatial and in the frequency domain. Hence, they are appropriate to model global, but also regional and evenlocal structures; for details see e.g. [1], [2].

Now we assume, that the real-valued signal (function) � ��� ( ��������� geocentric position vector, ����� ��� , ��� unitvector) is assumed to be harmonic, i.e. it fulfills the Laplacian differential equation ���������! outside a sphere "$# withradius % . The solution of Dirichlet’s problem for the outer space of " # can be expressed by the spherical harmonicsrepresentation

� ���&� '()+*-,)(./*102)43 )65 .879)65 . ���;: (1)

wherein 7 )<5 . ���� are the solid spherical harmonics of degree = and order > . The Fourier coefficients 3 )65 . are charac-terized by an optimal frequency (degree) localization, but they do not have any spatial information. Hence, we speak ofglobal parameters.

The computation of the Fourier coefficients requires preferable homogeneously distributed global data sets. Whereassatellite data is almost globally distributed, terrestrial and airborne observations are always restricted to certain regions orlocal areas. Consequently, we are interested in an alternative representation such as

� ���&� � ���@?BA( C * �@DC-E��F:G�

C�-?IH� ���KJ (2)

Herein L���� means a global trend model, e.g. a spherical harmonics expansion up to a low degree. In opposite to the globalparameters 3 )65 . the coefficients D

Care point parameters, since they are related to the M computation points N

C�ONP �

C� .

To be more specific they represent the regional and local information extracted from the data by means of the ”two-point”function

E �Q:R�

C� . Furthermore, in Eq. (2) H����� stands for the remaining unmodeled signal parts.

In order to compute the point parameters DC

of the alternative representation (2) we first replace the signal values �����by the observations ST��VUF� , measured in the W observation points NP �XUF� , e.g. Goce measurements along the satelliteorbit. Next, we choose an appropriate kernel function such as the Shannon kernel or the reproducing kernel as well as an

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

appropriate system of points NC O

NP��C� with �@���F:XJ JXJ :GM for the computation of the parameters D

Ccollected in the M����

vector � . This way we obtain the Gauss-Markoff model� ?������ with � �/��� ��� 0 � : (3)

which is possibly not of full rank [3]. Herein � means the observation vector, � the vector of the corresponding measure-ment errors, � � the variance of unit weight and

�the positive definite weight matrix. Additional operators, e.g. for the

second radial derivative, have to be implemented into the coefficient matrix .

Generally the least-squares solution of model (3) reads�� � ��� � ������ � � with � ����&��� � ��� � ��� : (4)

wherein � � � � � means the pseudoinverse of the normal equation matrix. The estimator�� can be used to calculate

both, estimators of the Fourier coefficients 3 )65 . up to a certain degree = � =�� and an estimation of any sphericalconvolution

E � T� ��� with a band-limited kernel. Hence, an estimation of (2) may read�� ��� � � � ���@?8 �E � T� ����� )��()+*-,)(.;*�02) �3 )<5 . 7 )<5 . ����@? A( C * � �D

C-E �F:G�

C�;: (5)

wherein the kernel

E �F:G�

C� can be expanded as Legendre seriesE

��F:G�C��� )�� �()+*-) � � �

=�?��!#" % � $ % ���� C&% ) ��&' =@��N ) �(� � C � (6)

in terms of the Legendre polynomials N ) �� � � C � of degree = . The Legendre coefficients

' =@� reflect the frequency-behavior of the kernel.

2 Multiresolution RepresentationThe basic idea of the multiresolution representation (MRR) is to split a given input signal into a smoothed version anda certain number of detail signals by successive low-pass filtering. An MRR can be achieved in two steps, namely thedecomposition of the signal into level-dependent coefficients (analysis) and the (re)construction by means of the detailsignals (synthesis). Both the decomposition process and the (re)construction process can be illustrated by means of filterbanks [4]. Filter banks can be implemented efficiently applying down- and upsampling strategies (pyramid schemes) [5].

In order to establish an MRR we identify the kernel

E �Q:R�

C� in the Eqs. (2) and (6) with the scaling function )+* � � ��F:R�

C�

of resolution level (scale) ,$?�� , namely

) * � � �Q:R�C��� )�� � * �.-0/21 0 �()+*-) � � �

= ?��!#" % � $ % ���� C % ) ��43 * � � �=@�1N ) ��(� �

C�/J (7)

Hence, the alternative representation (2) can be rewritten as an MRR, namely

� ���&� � ���@?8�)5* � � � T� ���L? H� ���&� )6�()F*T,)(.;*�02)43 )65 . 7 )65 . ����@? *(7 * 7 �&8 7 ����L?IH� ���/J (8)

Herein on the right hand side the detail signals 8 7 ����� :9; 7 �=< 7 �V ��� are computed by the wavelet coefficients

< 7 ��?>X��� ; 7 � T�V����&�A@2BDC�� ��� ; 7 ��F:G�?> � D6E # ���GF AIH( C * � D 7 5C ; 7 �:> :R� 7 5 C �/: (9)

which are the components of the MRR. These scale-dependent point parameters are the counterpart to the Fourier coef-ficients of the spherical harmonics expansion (1). The Legendre coefficients J 7 =@� and 9J 7 �=@� of the wavelet function ofresolution level (scale) K , i.e.

; 7 �Q:R� C ��� � H /L1 0 �()F*-) � � � = ?��!�" % � $ % ��L� C&% ) �

� J 7 �=@��N ) �� � � C �;: (10)

and the dual function 9; 7 �Q:R� C � are computed via the two-scale relation J 7 =@��9J 7 =@�/� 3 7 � � =@��� 3 7 �=@� [6]. In general,a spherical harmonic wavelet function is always globally but mostly ”quasi-compactly” supported. In our computationswe use the Blackman function as scaling function ) 7 �F:G� C � defined by the Legendre coefficients3 7 �=@���

��� �� � for = �!= � ?��F: JXJ J : 7 0 � � �� J ! �4 J �F �� ��� ��� )� H�� ? J ����������� � )� H��� �

for =�� 7 0 � :XJ J JX: 7 � � for = � 7 : JXJ J :��� ��� J (11)

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i = 0 i = 1 i = 2 i = 3 i = 4 i = 5

a) spatial localization ( one - dimensional )

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b) spatial localization ( two - dimensional )

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c) frequency localization

i = 0 i = 1 i = 2 i = 3 i = 4 i = 5 i = 6 i = 7 i = 8 i = 9

Figure 1: Blackman wavelet for different scale indices K

Note, that this function is derived from the defini-tion equation of the Blackman window often used insignal analysis [5]. The Blackman wavelet is illus-trated in Figure 1 for different scale values K . In thefrequency domain these wavelets are compactly sup-ported or strictly band-limited, respectively, i.e. only afew Legendre coefficients are not equal to zero (Figure1c).

3 Estimation ConceptThe computation of the M *�� � vector � �"!#� * � D * 5

C�

of scaling coefficients D * 5C

of highest resolution level ,from the observation vector � by parameter estima-tion methods according to the Gauss-Markoff model(3) means the initial step of the decomposition algo-rithm mentioned before. To be more specific we as-sume that the general solution reads�� *$�$# * �

with �� * �&�%# *� � � ��# �* : (12)

wherein the matrix # * may include the pseudoinverseof the normal equation matrix according to Eq. (4). Inthe following three steps we first compute the M 7 ��vector � 7 � D 7 5

C� of level- K scaling coefficients from

the corresponding vector of level K/? � by low-passfiltering�� 7 �'& 7 �� 7 � � �%# 7 �

with �� 7 �&�'# 7 � � ��# �7 J (13)

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0 60 120 180 240 300 360longitude

Figure 2: Level- ( Reuter grid with M*)�� �+ � points. Reutergrids are non-hierarchical but equidistributed point systems onthe sphere, i.e. the integration weights are independent on theposition.

Next, the estimator� , 7 of level- K wavelet coefficients< 7 � > � follows from band-pass filtering� , 7 �.- 7 �� 7 �%/ 7 �

with � � , 7 �&�'/ 7 � ��/ �7 J (14)

Finally the estimator of the level- K detail signal vector0 7 is computed from� 0 7 � 9- 7 � , 7 �$1 7 �with � 0 7 ���%1 7 � � ��1=�7 J (15)

The procedure providing the results of Eqs. (12) to(15) is characterized by the advantage that the compu-tation of the estimators is completely embedded into

geodetic parameter estimation techniques. Thus, besides the estimators all the corresponding covariance matrices arecomputable. Additional statistical investigations like the estimation of confidence intervals and the testing of hypothesisfor the parameters and for outliers can be performed easily.

4 Numerical ExampleThe estimation concept, presented in the last section, was applied to a simulated data set based on the EGM 96 gravitymodel. We computed disturbing potential values � ��� �"!�� ��� up to degree = � = � � ����( on a standard longitude-latitude grid at satellite altitudes randomly distributed between 450 km and 500 km. Furthermore, additional noise within� J � m

���s�

was considered. These values are collected in the observation vector � of the Gauss-Markoff model (3). Adiagonal weight matrix

�was chosen with purely latitude-dependent elements.

In order to compute an MRR of the given disturbing potential data we set ,�� � (see Figure 1c) and estimate ���"!#��� � D � 5C� from the model (3) by means of the pseudoinverse according to (4). To be more specific, the coefficients D � 5

Cwith�-� �F:XJ J J :GM�� are related to a level- � Reuter grid consisting of M� �'� � �Q points on the sphere "K# with radius % ���( + �

km. Figure 2 shows the level-3 Reuter grid with altogehter M ) � �+ � points; see e.g. [1].

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EGM 96 disturbing potential [100 m2/s2] up to level i = 5 at Earth’s surface Blackman wavelet , level-5 Reuter grid with 5180 points , pseudoinverse solution

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synthesis at Earth’s surface

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Figure 3: Detail signals of EGM 96 disturbing potential data for K ��F:XJ JXJ : � and synthesis on the sphere " # with radius % ���( + � km.

Figure 3 shows the five detail signals�8 7 ��� ac-

cording to Eq. (8) with K � � � on the sphere" # . The sum of the corresponding five detailsignal vectors 0 7 of levels K � �Q: JXJ JV: � yieldsan approximation of the disturbing potential onthe sphere ";# (synthesis).

Basically a regional or local multiresolutionrepresentation can be estimated in the samemanner. For this purpose both, the observa-tion vector � and the coefficient vector � ofthe Gauss-Markoff model (3) are decomposedinto two subvectors, namely � ��� � � � �� and� ��� � � � �� . If e.g. the vector � � standsfor the coefficients related to the region of in-terest, the reduced normal equation system canbe established. The neglect of the observationsoutside the region of interest, collected in thesubvector � � , leads to edge effects. However,due to the localization property of the chosenscaling and wavelet functions these undesiredeffects are relatively small.

References1. Freeden, W., Gervens, T. and M. Schreiner(1998): Constructive Approximation on theSphere (With Applications to Geomathemat-ics). Clarendon Press, Oxford.

2. Schmidt, M., Fabert, O. and C.K. Shum(2004): Towards the Estimation of Multi-Resolution Representation of the Gravity Field Based on Spherical Wavelets.IUGG XXIII General Assembly, Sapporo, Japan 2003, in press.

3. Koch, K.R. (1990): Parameter Estimation and Hypothesis Testing in Linear Models. Springer, Berlin.

4. Schmidt, M., Fabert, O. and C.K. Shum (2002): Multi-Resolution Representation of the Gravity Field Using SphericalWavelets. Weikko A. Heiskanen Symposium, The Ohio State University, Columbus.

5. Mertins, A. (1999): Signal Analysis. Wiley, Chichester

6. Freeden, W. (1999): Multiscale Modelling of Spaceborne Geodata. Teubner, Stuttgart.