Quality investigation of vertical
datum connection
Kyra van Onselen
DEOS Report
no 98.5
Quality investigation of vertical datum connection
Bibliotheek TU Delft
1111111 IIIII
c
30372208500
696G
Quality investigation of vertical datum connection
Kyra van Onselen
Published and distributed by: Delft University Press P.O. Box 98 2600 MG Delft The Netherlands Telephone: +31 152783254 Telefax: + 31 152781661 E-mail: DUP@DUP.TUDelft.NL ISBN 90-407-1823-7 Copyright 1998 by DEOS
All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, inclu-ding photocopying, recording or by any information storage and retrieval system, without written permission from the publisher: Delft University Press.
Contents
Acknowledgements . Summary .. . . . . General introductionPart
A
tion
Theoretical background to vertical datum
connec-1 2 3 4 5 6 Introduction Definition of parameters . . . . . . Solution of the geodetic boundary value problem 3.1 Only satellite derived potential coefficients 3.2 Only terrestrial gravity measurements. . .3.3 Combination of potential coefficients and surface gravity data 3.4 Kemel modification . . . .
Least squares solution of vertical datum connection 4.1 Observation equations
4.2 4.3
Precision Reliability
A-priori covariance matrices
5.1 A-priori variance-covariance matrix for orthometric heights 5.2 A-priori variance-covariance matrix for geometric heights 5.3 A-priori variance-covariance matrix for geoid heights Other considerations . . . .
6.1 The NI-term . .. . .. . 6.2
6.3
Choice of reference system Variation in time . . .
Part B
Error propagation in vertical datum connection for a
vii Vlll 13
5 6 8 10 11 12 13 14 15 17 18 19 19 21 21 26 26 29 30stylized world
33
1 Introduction .. . . .. . . 35 2 Gravity info. solely from satellite derived potential coefficients 36 3 Combination of potential coefficients and surface gravity data 383.1 Molodenskii versus modified Meissl/Wong&Gore truncation
coeffi-cients . . . .. . 39
3.2 JGM2 model versus model 39
4 Distribution of stations 42
vi
5 6
7
4.2 Three stations / zone . . . . Oceanic islands as separate datum zones The N1-term . . . 6.1 Effect of neglecting N1-term 6.2 Estimation of N1-term . . . Conclusions and recommendations
Contents 47 50 55 55 57 59
Part C
Recommendations for vertical datum connection and
application to North Sea area
61
1 Introduction . . . 63 2 "Standard scenario" . . . 64 3 More realistic precision for terrestrial measurements 68 3.1 More realistic precision for geometrie heights 68 3.2 More realistic precision for orthometric heights 68
3.3 More realistic model for propagated error 69
3.4 Results based on improved a-priori covariance matrices 70 4 "Ideal" cap size for terrestrial gravity measurements 74 5 Influence of the number of stations . . . . . . 76
6 Non-uniform precision of gravity information 82
7 Stations on off-shore platforms . . . .. 84
8 Effect of the first term degree term . . . . . . 88 8.1 "Reference scenario"; 28 stations, N1 not considered 89 8.2 Effect of neglecting N1 term . . . . . . . . . . . 92 8.3 Estimation of N1 term without a-priori information 93 8.4 Estimation of N1 term with a-priori information 96
9 Choice of reference system . . . . . 98
10 Conclusions and recommendations 98
References 104
Appendix 1 Standard deviations for potential differences between the
da-tum zones 105
1 Results based solely on satellite derived potential coefficients . . . . . . . . 106 2 Results based on a combination of potential coefficients and terrestrial
grav-ity data . . . . . . . . . . 107 3 Results based on two stations/zone .. . . .. . .. .. . . . 108 4 Results based on th ree stations/zone .. . . 109 5 Results for the data configuration in which oceanic islands are considered
separate datum zones . . . . . . . . . . . . . . . . . . . . . . . 110 6 Results for the data configuration in which the effect of neglecting the N1
term is considered . . . . . . . . . . . . . . . . . . . . . . . . 112 7 Results for the data configuration in which the N1 term is estimated . . . . 114
Acknowledgements
This work is a continuation of earlier research conducted by Xu Peilang and Reiner Rum-mel. I would like to thank Martin van Gelderen for his support in the realization of this report. A major part of the ideas elaborated in this report stem from discussions with Martin on the subject of vertical datum connection. Furthermore, I would like to thank Radboud Koop and Ejo Schrama who simulated data for the proposed Gravity Explorer Mission, and Erik de Min who supplied the software routines to calculate the modified MeissljWong&Gore truncation coefficients. Finally, I would like to thank Martin Jutte for producing some of the figures presented in this report.
viii
Summary
In most countries or regions a vertical datum has been defined by averaging sea level observations at one or more fundament al tide gauges. This mean sea level at the funda-mental tide gauge is used as a reference for local orthometric height measurements. Since mean sea level is not an equipotential surface, different vertical datums refer to different equipotential surfaces. As a result, constant off-sets exist between different height datum zones.
Various purposes require the establishment of a global, or regional, vertical height datum. Space techniques such as SLR, VLBI or GPS yield a direct connection of vertical datums. However, they can only establish a connection in the sense of geometric heights while many purposes require a connection in orthometric height. Adjacent vertical datums can be connected by means of levelling which yields the difference in orthometric height between the fundamental stations in the datum zones. However, for larger areas the precision and reliability of levelling is inadequate. Therefore, an indirect method for connecting vertical datums has to be used.
The presented method for connecting vertical datums requires that every datum zone con-tains a fundamental station and at least one station for which the geometrie height relative to a selected reference ellipsoid and the orthometric height relative to the fundament al station is measured. From these measurements of geometric and orthometric height the geoid height, or to be more precise, the height of the equipotential surface through the fundamental station above the adopted reference ellipsoid, can be determined.
The geoid height for a station can not only be determined from its geometric and orthometric height, but also through a geodetic boundary value problem. Depending on wh at kind of gravity information is used, different solutions for the geodetic boundary value problem are found. In this report solutions for three different sources of gravity information are discussed, i.e, 1) satellite derived potential co efficients , 2) agiobal dis-tribution of terrestrial gravity measurements, and 3) a combination of satellite derived potential coefficients and terrestrial gravity measurements in a spherical cap around the stations of interest.
The sol ut ion of the geodetic boundary value problem for the geoid height of aspecific station contains two unknown datum connection parameters, i.e., 1) difference in potential value between the equipotential surface of the datum zone in which this station is situated and some kind of reference surface, and 2) the potential anomaly of this reference surface. By comparing the two sets of equations for the geoid heights of the stations the datum connection parameters can be solved for by means of a least squares adjustment. A least squares adjustment is used since this allows the verification of the precision of the derived datum connection parameters and the reliability of the involved measurements and datum connection parameters.
Error propagation in a stylized global data configuration consisting of seven datum zones, containing all together 63 space stations, showed that the accuracy for the derived datum connection parameters is not very good if the solution of the geodetic boundary value problem was solely based on potential coefficients as given by the JGM2 model. Standard deviations for the potential differences ranged between 36 and 84 cm. If a higher degree
potential coefficient model from a proposed Gravity Explorer Mission, henceforth denoted as GOCE model, becomes available, these potential differences can be determined with standard deviations ranging between 14 and 32 cm.
Better results can be achieved if the solution of the geodetic boundary value problem for the geoid heights of the stations is based on a combination of potential coefficients and terrestrial gravity measurements in a spherical cap around the stations. If both potential coefficients from the JGM2 model and terrestrial gravity measurements in a spherical of two degrees around the stations are used, potential differences can be determined with standard deviations ranging between 10 and 24 cm. Based on the GOCE model, potential differences can be determined with standard deviations ranging between 2 and 5 cm. To verify what kind of quality is feasible for connecting vertical datums, the proposed method is also applied to a regional datum con neet ion problem. A regional approach is used since this better allows the use of realistic error descriptions for the different measurement types, i.e, based on existing measuring networks. In the North Sea region three datum zones have been defined, containing all together 30 stations. State of the art measurements are assumed for the geometrie heights, orthometric heights and the terrestrial gravity information.
Error propagation in this regional datum connection problem shows that potential differences, based on potential coefficients for the JGM2 model can be estimated with standard deviations ranging between 8 and 12 cm. If the GOCE model becomes available these potential differences between the datum zones could be determined with standard deviations in the order of 1 cm.
The conclusions derived in these report are that the accuracy of derived datum connection parameters based on the JGM2 model is not very good. In order to be able to estimate potential differences between datum zones with standard deviations well below the 5 cm level, a potential coefficient model like the GOCE model is needed. These standard deviations can also be attained based on the JGM2 model if terrestrial gravity information of high quality is available in a very large area around every station, i.e., in spherical caps of at least seven to ten degrees.
Concerning the required number of stations in the datum zones, the following remarks can be made. At least two stations are needed in every datum zone in order to detect out-liers in the measurements. Having only two stations in every datum zone gives relatively good results for the precision of the datum connection parameters based on the GOCE
potential coefficient model. However, for a sufficient reliability of these datum connection parameters, datum zones should contain at least three stations.
General introduction
Over the last years greenhouse warming and the expected resulting rise in mean sea level has become a point of major concern. Especially for low-Iying countries like the Netherlands, even a small, but incessant rise of mean sea level will have a tremendous impact on society. Therefore, it is important to determine, as soon as possible, what order of sea level rise is to be expected in the near fut ure:
On agiobal scale changes in sea level can accurately be detected by means of satellite altimetry. However, results derived for global variations of mean sea level cannot be applied directly to specific regions. Due to various processes sea level rise is far from uniform on a regional scale. For example, due to all kind of local oceanographic and atmospheric processes, the effects of a changing climate on the mean sea level in the North Sea may differ significantly from the effects on agiobal scale.
Local sea level is traditionally measured using tide gauges. Tide gauges relate changes in sea level to a 10 cal tide gauge benchmark. Consequently, changes in the height of the tide gauge bench mark, e.g., due to subsidence, directly affect the measured sea levels.
To verify the height of tide gauge benchmarks they are usually connected to a, preferably stabie, local reference system.
If ti de gauge benchmarks are connected to alocal reference frame this allows the measured sea level variations for various tide gauges along the same coast to be compared. Due to local effects, results derived for a single tide gauge cannot be used to predict variations in sea level along the whole coast. For example, due to the freshwater flow with a relative low density tide gauge situated close to a river mouth will measure relatively high sea levels. As aresuit, to make an accurate prediction of the expected variation in sea level aling a coast, measurements from various tide gauges have to be compared. This requires that the tide gauges benchmarks are all known in height relative to the same reference surface.
The problem arises when we want to compare sea level measurements from tide gauges which are not situated in the same country. Although the heights of all involved tide gauge bench marks might be known, in general they will not be measured relative to the same reference surface. This can be explained as follows.
To be able to compare results of various tide gauges, their orthometric heights have to be known. Orthometric heights are heights relative to the geoid. Since this geoid is a reference surface which cannot be used directly, in most countries some kind of approximation has been introduced. Of ten, a reference surface for local orthometric height measurements is defined by averaging sea level observations at one or more fundamental tide gauges, thereby creating alocal vertical datum.
Due to various processes, e.g., wind forcing, variations in salinity, etc., mean sea level is not an equipotential surface. Therefore, different vertical datums refer to different equipotential surfaces. As a result, constant off-sets exist between the different height datum zones.
In this report, an indirect method for connecting vertical datums will be discussed. In particular, the quality which might be expected from a connection of vertical datums, for all kind of data configurations and based on measurements of varying precision, will be
2
investigated.
First, in part A, the method for connecting vertical datums will be explained. The theoretical background for the method will be given, required formulas will be derived, and it will be indicated what kind of data is needed to solve the datum connection problem. Next, in part B, the presented method for datum connection will be applied to a stylized global data configuration. Based on error propagation, the quality of the the datum connection will be investigated. In part C, a number of aspects of the datum connection problem will be worked out in greater detail. Applied to a regional approach, it will be examined what kind of quality for connecting vertical datums is feasihle. Furthermore, it will he tried to make some recommendations concerning the required quality of the various measurements and the numher of stations.
Part A
Theoretical background to vertical
datum connection
1
Introduction
The direct connection of vertical datums as used in different regions can be performed using space techniques such as SLR, VLBI or GPS. The disadvantage of this method is that only a purely geometric con ne ct ion can be established. To connect orthonormal heights the geoid is required.
In case of adjacent vertical datums a direct connection could be performed by level-ling, however, for larger areas the precision and reliability of th is method is inadequate. Consequently, for establishing a regional or global vertical reference system, and especially for trance oceanic connections, an indirect method of datum connection is needed. Various authors have proposed a number of indirect methods for vertical datum connec-tion. The method used in this report is based on the method of Rummei and Teunissen (1988), which has been further developed by Xu anà Rummei (1991). This method can be summarized as follows:
• Geoid heights are determined for a number of stations in every datum zone, based on measurements of geometric and orthometric heights. These geoid heights are determined relative to the local datum, i.e, they are a combination of the actual geoid height of the station and the height of the equipotential surface through the fundamental station above the adopted reference el!ipsoid.
• For these stations, geoid heights can also be determined by solving a geodetic boundary value problem.
• From these two equations for the geoid heights the datum connection parameters can be solved for by a least-squares adjustment.
For high quality connection of vertical datums, the solution of the geodetic boundary value problem should be based on a (scalar) Molodenskii approach. As an alternative, an approach based on gravity ratios and ratios of radial distances could be used; for more details see Baarda (1979) and Baarda (1995). However, in this report it is not tried to find an optimal method for connecting vertical datums in which all possible contributions are incorporated. The purpose of the is report is to gain insight in the influence of the various geodetic measurements on the quality of the derived datum connection parameters. Consequently, model errors wil! not be considered and error propagation can be based on a more simple solution of the geodetic boundary value problem. Therefore, to investigate error propagation in vertical datum connections, the solution of the geodetic boundary value problem wil! be based on a Stokes approach.
In the next section, a simple example wil! be given to explain the method of datum con-nection under consideration. Furthermore, some parameters used for solving the datum connection problem wil! be explained, and relations on which the geode tic al boundary value problem is based wil! be derived. In section 3 the solution of the geodetical bound-ary value problem wil! be given for three different sources of gravity information, i.e., 1) satellite derived potential coefficients, 2) agiobal distribution of terrestrial gravity mea-surements, and 3) a combination of satellite derived potential coefficients and terrestrial gravity measurements in a spherical cap around the stations of interest. Next, in section 4, the solution of the datum connection problem by means of a least-squares adjustment wil!
6 Part A. Theoretical background to vertical datum connection STATION 1
STATION 2
GEOID
REFERENCE ELLIPSOID
Figure 1: Vertical datums A and E are connected by potential difference CAB = HAB· T' which can be determined from a combination of geometrie, orthometric, and geoid heights.
be discussed. In section 5, the a-priori covariance matrices for the different measurement types wil! be derived. Finally, in section 6, some other considerations wil! be discussed, i.e., the effect of the first degree term, the choice of the reference system, and variation in time of vertical datum systems.
2
Definition of parameters
First, with a simple example the method of datum connection under consideration wil! be explained. Figure 1 shows two vertical datums (equipotential surface A and
equipo-tential surface E) defined by a reference station, fundamental station A and fundamental station E, in the two datum zones. To determine the difference in height between the two equipotential surfaces, we need the orthometric heights of both reference stations above
the geoid. As can be se en from figure 1, the orthometric height of the fundamental station above the geoid can be derived if in the datum zone at least one station is available for which the geometrie height above the selected reference surface is measured, the ortho-metric height relative to the fundament al station is determined based on e.g., level!ing, and the geoid height is estimated through sol ving a geodetic boundary value problem.
In this example, we have used the difference in orthometric height between the funda-mental stations in the datum zones to conneet the two datums. Since orthometric heights
are subject to assumptions concerning topography and cru st density, in the fol!owing we wil! use the potential difference (or to be more precise, dynamic heights) between the
equipotential surfaces for connecting vertical datums.
This simple example shows that only potential differences between datum zones can be determined using the proposed method. Consequently, to con neet vertical datums, some kind of reference equipotential surface has to be selected. How this reference surface is selected depends on the purpose for which the datum connection is performed.
be-2 Definition of parameters
tween the datum zones themselves are important, not the potential differences between
the datum zones and the adopted reference equipotential surface. As aresuIt, the
se-lection of the reference equipotential surface is not of major importance. Therefore, the
equipotential surface defined by the fundamental station in an arbitraryly selected datum
zone can be used.
If the reference equipotential surface is to be used as some kind of global vertical
reference system, some kind of weighted sum of the equipotential surfaces defined by the
various fundamental bench marks might be used instead, for more details see Rapp and
Balasubramania (1992) and Balasubramania (1994) and section 6.2.
To determine the potential differences between various datum zones, first the potential difference between the zones and a reference equipotential surface is determined. If the
potentialof the reference equipotential surface is denoted by Wo, the potential difference between this reference surface and any fundamental benchmark, P, is given by:
CM
(R)l+
l
Cpo
=
Wo - W(P)=
Wo -RL -:;:-
C1mYim(P) - Z(P) l,rn P(1) with Z(P) the centrifugal potential.
In order to linearize equation 1 approximate coordinates for the stations and an approx
-imate gravity field are needed. A reference ellipsoid is introduced to determine these
approximate coordinates. It is assumed that this reference ellipsoid is the ellipsoid on
which geometric heights determined by space geodetic techniques like GPS are based.
The approximate, or normal, gravity field can be defined as:
CM
(R)I+1
U(P)
=
Z(P) +-R!
L
-:;:-
C?mYim(P)l,m P
(2)
in which C Mo and
cg
n are values corresponding to the adopted reference field, and ap-proximate coordinates of point Pare assumed available.
Next, we linearize the (unknown) potential value of the reference equipotential field relative to the reference ellipsoid. We wil! only consider the vertical component which yields for the potential value of the reference equipotential:
TU
=
W(R )=
U(P.')oU(P~)
N"
OU(P~)
tJ.CYY 0 - 0 0
+
0+
L.. oC lmn l,m lm
(3) in which tJ.C1m
=
Ctm - CPm and P~ is the approximate point on the reference ellipsoid, see also figure 2.If coefficients are determined in a spherical, constant radius approximation ~~ = -
,.
Consequently, the reference potential anomaly, tJ.Wo, can be written as:
(4)
The sum on the right si de of equation 4 gives the anomalous potential T, i.e.,
(5)
8 Part A. Theoretical background to vertical datum connection P surface o ______________ Po level surface - - - through 0 ...::::..:::::.::::::.:::::::::::.-T---_~P~~reference ellipsoid
Figure 2: Definition of points P, Po, and P~. Reproduced from RummeI and Teunissen (1988)
in which Ta is the zero degree term of the disturbing potential, i.e., Ta = GM ~GMQ, end Tl is the first degree term of the disturbing potential, i.e., the effect of the difference between the center of mass of the adopted reference ellipsoid and the center of mass of the earth. i.e,
Substituting equation 5 into equation 4 and some rearranging yields Bruns' formula,
N
=
T - D.Wa'Y
(6)
This equation yields "geoid heights" relative to the selected reference equipotential surface
(Wo). .
To illustrate the parameters which have to be solved for in a datum con ne ct ion problem, figure 3 shows the separation between a selected reference equipotential, the adopted reference ellipsoid, and a number of other equipotential surfaces corresponding to local vertical datums. The separation between the reference equipotential and the geoid is given by Bruns' formula, i.e., equation 6. The separation between datum zone i and the reference zone is given by~. In order to solve the datum connection problem, these separations have to be determined.
In the following section, the solution of the geodetic boundary value problem for the geoid height of a point P will be discussed for three different sources of gravity information. In section 4 it will be discussed how the separations between the surfaces as shown in figure 3 can be estimated by means of a least-squares adjustment.
3
Solution of the geodetic boundary value problem
In this section we will derive the solution of the geodetic boundary value problem for the geoid height of a point P in a specific datum zone. We will assume three different sources of gravity information, i.e., 1) satellite derived potential coefficients, 2) agIobal distribution of terrestrial gravity measurements, and 3) a combination of satellite derived potential coefficients and terrestrial gravity measurements in a spherical cap around the stations of interest.
3 Solution of the geodetic boundary value problem datum zone 1 REFERENCE EQUIPOTENTIAL ----~---L---._---W=WO W=W2---da-t-um-zone 2 T-flWO y -REFERENCE ELLIPSOIO
u=u
oFigure 3: Separation between reference ellipsoid, adopted reference equipotential surface and other equipotential surfaces as defined by local bench marks.
zones worldwide, we wil! first assume that there exists one, unified global datum, i.e., 6.Wo
is constant. Substituting the equation for the disturbing potential, i.e., equation 5, into Bruns' formula yields the solution for the geodetic boundary value problem based on one, uniform, datum. The geoid height for a point P can be determined from:
(7)
or, if all terrestrial gravity measurements have been reduced to the geoid (or to be more precise, to the unified global datum which is used as an approximation of the geoid), in spherical approximation as
N(P) = No
+
NI+
4R1
St(1j;PQ)6.g(Q)d(JQ1f, (J
(8) In these equations St(1j;PQ) is Stokes' integral function, NI is the first degree term, i.e.,
the deviation between the center of mass of the reference ellipsoid and the center of mass of the earth, and No the zero-degree term, i.e.,
No
=
_
(Wo - Uo)+
GM - GMo=
_
6.Wo+
8(GM)I TI I R, (9)
By sol ving the datum connection problem, the zero degree term is solved for as weil. However, it is not possible to estimate ~ as a separate unknown. Only the combi-nation 6.Wo
+
8(GM) can be solved for. Fortunately, the geocentric gravitational co n-stant of the reference ellipsoid, G Mo can be determined rather accurately from satellite tracking measurements. For example, according to Ries et al. (1992), the value forGM
= 398600.4415km
3 jsec2 (including the mass of the atmosphere) with an estimated uncertainty of 0.0008km3 jsec2. Therefore, we will assume that, with a high enoughaccu-racy, 8(GM)
=
0, i.e., the "true" geocentric gravitational constant, GM, is equal to the geocentric gravitational constant of the reference ellipsoid.For now, the first degree term wil! be neglected as weil, i.e, it is assumed that the center of mass of the reference ellipsoid and the center of mass of the earth coincide. In section 6.1 the first degree term wil! be considered.
10 Part A. Theoretical background to vertical datum connection
Instead of one, unified, global datum we now assume that there are (I
+
1) datum zones worldwide. One, arbitrarily chosen, datum zone is used as reference equipotential surface. The potential differences between the fundamental station in the reference zone and the fundamental stations, Qi, of all other datums zones are denoted CQ,o.As derived by Rummel and Teunissen (1988), for stations in the reference zone, we can still determine the geoid height from Bruns's formula. However, since N is the separation between the fundamental station and the reference ellipsoid, for other datum zones Bruns' formula has to be adapted, i.e.,
N(i)
=
T - .6. Wo+
CQ,o,
(10)Based on the adapted version of Bruns' formula the geoid height for a point P situated in datum zone i can be determined. The actual sol ut ion for the geodetic boundary value problem depends on the nature of the available gravity information. In the following sections, solutions for three different kinds of gravity information will be discussed.
The notation N(i) is used to indicate that although the term geoid height is used, we
actually mean the height of the equipotential surface through the fundamental station in zone i above the adopted reference ellipsoid.
3.1
Only satellite derived potential coefficients
If gravity information consists solely of satellite derived potential coefficients, all gravity data refers to the same datum surface. In this case, the geoid height for a point P situated in datum i can be determined as:
(11)
in which the mass of the earth's atmosphere is included in GM.
This is the same equation as equation 7. The only difference is the zero degree term. The zero degree term now not only consists of the reference equipotential anomaly, but also of the potential difference between zone i, in which the point under consideration is situated, and the reference zone, i.e.,
NJi)
= _
.6. Wo, ,
+
CQ,o(12)
Potential coefficient models only supply information up to a maximum degree and order Lo, consequently, . GM La
(R)l
1 N6')+-'L -'L
(.6.ClmCosm-À+.6.S1msinm-À)Plm(Cos8)+
(13) r, 1=2 r m=O GMf
(!!.)lt
(C1mcosm-À+
Slmsinm-À)Plm(Cos8) r, La+ 1 r m=OIn this equation, the first part of the information is obtained from a potential coefficient model, the second part, i.e., information above degree Lo, is either omitted or approxi-mated based on some kind of model.
3 Solution of the geodetic boundary value problem
The advantage of gravity information solely based on satellite derived potential coefficients is that in this case the proposed method of datum connection can be applied for connecting datums in only a specific region. Datum connection solely based on terrestrial gravity information is only possible to perform a worldwide connection of vertical datums.
3.2
Only terrestrial gravity measurements
If only terrestrial gravity measurements are available, geoid heights can be determined with an adapted version of equation 8. The problem is that this Stokes' integral requires a global coverage with gravity anomalies. Furthermore, the terrestrial gravity measurements must have been reduced to the geoid. In section 3 we assumed one, unified global datum and assumed that all terrestrial gravity measurements had been reduced to this unified global datum which was used as an approximation of the geoid.
If more than one height datum zone is used worldwide this implies that different equipotential surfaces have been used as local approximation of the geoid. Consequently, terrestrial gravity measurements in different datum zones have been reduced to different equipotential surfaces. This vertical datum effect in the gravity anomalies has to be incorporated in the solution of the geodetic boundary value problem.
Note that this problem is not averted by using a scalar Molodenskii solution. Although in this case the terrestrial gravity measurements do not need to be reduced to the geoid, the solution contains 6.. W's referring to the various potential surfaces.
Assuming that gravity measurements have been reduced to sea level by means of free air reduction gives for the gravity at point Po on the equipotential surface of zone i, see figure 2,
(i) 0, (i) 0, 1
9 (Po)
=
g(P) - -Hon
=
g(P) - -=Cpoang (14)
in which H(i) is the orthometric height referring to datum zone i, Cpo is the potential difference between the equipotential surface through the fundamental station in zone i
and point P on the surface, and 9 is the mean gravity along the plumb line.
If we linearize equation 14 with respect to the normal gravity field and the approxi-mated point
P6
and consider only the vertical component, this yields:g(Po)
=
,(P~)
+
a
,
~P~)
N+
L
aJC(P~)
6..C1mun lm u lm
(15)
In spherical (constant radius) approximation
?n
~ -~ andfn
~fr
.
Substituting these relations in equation 15 gives the well-known equation for gravity anomalies:,
2,
oT
6..g
=
g(Po) - ,(Po)=
- - N --r
or
(16)For stations in the reference zone, we can substitute Bruns' formula for the geoid height in equation 16. This gives for the gravity anomaly:
N = T - 6..Wo
=*
6..g =~6..Wo
-(~+ ~)
T, r r
or
(17)12 Part A. Theoretical background to vertical datum connection
For stations in the other datum zones the adapted version of Bruns' formula has to be
substituted for the geoid height. Consequently,
N(i) _ T - l:!.Wo
+
C QiO A (i) _2
A UT 2C(2
a)
T-
=>
ug - - UYVO - - Q'O - -+-'Y r r ' r a r
(18)
Comparison of these two equations gives the rel at ion between gravity anomalies reduced with respect to the reference datum zone and gravity anomalies reduced with respect to the other datum zones, i.e.,
l:!.g
=
l:!.g(i)+
~CQ
'
o
r ' (19)
Finally, substituting relation 19 in Stokes' integral, equation 8, gives the solution for the geodetic boundary value problem for the geoid height of a point P situated in datum zone i as
(20) The zero degree term is again given by equation 12. Since we have assumed (I
+
1) datum zones, the index j runs from 1 to I, indicating that for gravity anomalies which have been reduced to datum (j), the potential difference CQjO has to be included.Equation 20 clearly shows that in order to determine the geoid height for stations in a
specific vertical datum zone, not only the potential difference between this datum zone and the reference zone has to be estimated, but potential differences for all other datum zones used to reduce gravity data have to be estimated as weil. Since equation 20 requires
agiobal distribution of gravity information, this method of datum connection can only be applied to agiobal connection of vertical datums.
The sol ut ion for the geodetic boundary value problem solely based on terrestrial in-format ion is only given from a theoretical point of view. It will not be used in the error
propagation studies as performed in part Band part C due to the fact that: 1) the
required global coverage of terrestrial gravity information is not realistic, and 2) this method requires that potential differences between all vertical datums used globally are
incorporated.
3.3
Combination of potential coefficients and surface gravity
data
The third possibility is a combination of potential coefficients and terrestrial gravity data in a spherical cap ab out the station of interest. In this case the geoid height for a point P situated in datum zone i consists of three parts, i.e.,
(21)
No is the zero degree term, which is again given by equation 12, which is repeated below:
NJi)
= _
l:!. Wo+
C QiO3 Solution of the geodetic boundary value problem
Nt is the contribution of the terrestrial gravity measurements in the spherical cap around
the station. This contribution is given by an adapted version of equation 20 in which the integration is performed up to the size of the spherical cap 'lj;c, this yields,
R
l
"'
c
1
21\" [ ( ) 2 ]Nt
=
- 4 St ('lj;PQ) flg/+ -RCQ
;O sin 'lj;d'lj;da 'Tr'Y "'=0 <>=0(22) The terrestrial gravity measurements are assumed to have been corrected to contain the earth's atmosphere.
Ns is the contribution of the potential coefficients, i.e.,
in which information above degree Lo is either omitted or estimated based on some kind of model, and flgt is the I-th degree Laplace harmonie of flg((}, À), and Qt is the truncation coefficients as given by Molodenskii et al. (1962), i.e.,
(24)
In practice, it is easier to use recursive formulas for the computation of the truncation
coefficients. The recursive procedure used in this paper is one developed by Paul (1973).
3.4 Kernel modification
In the preceding, "norm al" Molodenskii truncation coefficients corresponding to Stokes' function have been used. In order to minimize omission errors, and/or the total error,
Stokes' kemel can be modified, leading to different truncation coefficients. Following de Min (1996), the combination of terrestrial gravity data and potential coefficients can
in general be written as:
(25)
in which flgt are the terrestrial gravity anomalies and !:::.gs are anomalies derived from
the potential coefficient model. In general it should hold that Str
+
St;=
St.For the combination as described in the preceding, within the spherical cap only gravity
information from the terrestrial measurements is used, the gravity information supplied
by the potential coefficients is neglected. Outside the spherical cap, terrestrial gravity
information is solely based on the potential coefficient model. For St l and St2 this implies:
Stl('lj;)
=
0=
St('lj;) St2('lj;) = St('lj;) =0 if 0 ::; 'lj; ::; 'lj;c if 'lj;c ::; 'lj; ::; 'Tr if 0::; 'lj; ::; 'lj;c if 'lj;c ~ 'lj; ~ 'Tr 1314 Part A. Theoretical background to vertical datum connection (1-1)/2. al (t.Aeissl/Wong&Gore) {1-l)/2 • al (Molodenskii) ~
.,
,~\ ,'. ,'.
. ' ~ ~ 100 200 300 400 500 600 700Figure 4: Spectral weights for truncation coefficients; Molodenskii dotted line, modified MeissljWong&Gore fullline
Modified truncation coefficients combine, for the area of the spherical cap, terrestrial gravity information with gravity information from the potential coefficients. According to de Min (1996) probably the best modification is a combined MeissljWong&Gore mod-ification, which implies the following expressions for Stl and St2:
{
Lm ( Lm )St!jWG('Ij;) St('Ij;) - ~~Pt(coS'lj;) - St('Ij;c) - ~~~ll Pt(cos'lj;c) 0:::; 'Ij;
::q16)
0 'lj;c :::; 'Ij; :::; 7r
{
Lm Lm
St~WG('Ij;) = L~l~ll 1=2 P1(cos'lj;)
+
St('Ij;c) - L~l~ll 1=2 Pt(cos'lj;c)o :::;
'Ij; :::; 'Ij;(27)St ('Ij;) 'lj;c :::; 'Ij; :::; 7r
in which Lm is the maximum degree of modification. This maximum degree of modifi-cation implies, approximately, that up to degree Lm the solution is mainly based on the potential coefficients.
Figure 4 shows the spectral weights, 1-;1
Q;,
for both Molodenskii and modified MeissljWong & Gore truncation coefficients. The "ringing" effect displayed by the Molodenskii weights is caused by the discontinuity of the Stl function. If modified MeissljWong&Gore trunca-tion coefficients are used, the discontinuity between the inner zone (containing the terres
-trial data) and the outer zone is removed. Consequently the "ringing" effect disappears; for more details see de Min (1996).
4
Lea
s
t squares solution of vertical datum connection
To connect vertical datums, the potential differences between the fundamental stations in these datum zones have to be determined. In section 2 we have introduced the parameters involved, i.e, the potential differences CQiO between the I datum zones and the selected
4
Least squares solution of vertical datum connectionreferenee zone and, in addition, the potential value of the seleeted referenee zone (or to be more preeise the reference potential anomaly) ~Wo. These datum connection parameters
can be determined as follows.
In every datum zone we assume at least one station. As shown in the preceding seetions, for three different types of gravity information, the geoid height for these stations
can be determined by sol ving the scalar Stokes' boundary value problem. These geoid
heights are related to the equipotential surface determined by the fundamental station in
the datum zone in which the station under consideration is situated. Consequently, the solution for the geoid height of station P, situated in datum zone i, not only contains the
unknown value of ~Wo but also the unknown value of CQ,o.
For every station p(i) the geoid height cannot only be determined through a geodetie boundary value problem, but also from measurements of the geometrie height of the station and the orthometrie height of the station relative to the fundamental station in the datum zone in whieh p(i) is situated. From these two equations for the geoid height the datum con neet ion parameters can be solved for.
Summarizing, it can be stated that the (I
+
1) unknowns, i.e., ~Wo and CQ,o, i=
1,2, .. " I, can be determined if in every datum zone, including the reference zone, for at
least one station the geometrie height of the station and the orthometrie height relative to the eorresponding fundament al station is measured and the geoid height of the station
is determined through a geodetie boundary value problem. From these two equations for
every station the unknown datum connection parameters ean be determined by means of
a least-squares adjustment.
The advantage of using a least-squares adjustment is that this method allows the verification of the precision of the derived datum eonneetion parameters and the reliability
of the measurements and datum connection parameters. The preeision of the datum
connection parameters will be discussed in section 4.2. The formulas to determine the
reliability of the measurements and the derived datum connection parameters will be
shown in section 4.3. First, in the next seetion, the observation equations on which the
least squares estimation of the datum connection parameters is based will be discussed.
4.1
Observation equations
For every station p(i) we have two equations to determine the geoid height of this station relative to equipotential surfaee as defined by the fundamental station of the datum zone
in which the station is situated. The geoid height can be determined from the geocentric
height, h, and orthometric height, H(i) (with respect to the vertical datum zone i in which
the station is situated), as:
(28)
Besides, the geoid height of this station can be determined as a solution of a scalar Stokes' boundary value problem, i.e., depending on the gravity information used, by equation 13,
equation 20 or equation 21.
Comparing the two sets of equations for the geoid height for all stations allows, af ter some rearranging, the system of observation equations to be written in the form:
Y
=
AX+é
(29)16
Part A. Theoretical background to vertical datum connection in which Y is a vector containing for every station the observations, A is the design matrix,and X is the vector containing the unknown datum connection parameters.
The elements of the vector with observations Y and the design matrix A depend on the type of gravity information used in the geode tic boundary value problem. For the three types of gravity information as introduced in the preceding, the parameters as introduced in equation 29 can be described as follows.
Case 1 Gravity information consists solely of satellite derived potential coefficients: Y:the observation vector has elements YPk
=
h - H(i) - Nsat - NomA: the design matrix has elements either -1, or
+
1, or°
X: the vector with unknowns has elements
7
'
~
,
i=I,2,···,I
in which the following abbreviation has been introduced: GM La
(R)l
INsat=-L: -
L:
(~ClmCOSmÀ+~SlmsinmÀ)Ptm(cos(i)r, 1=2 r m=O
Nom = GM
f
(~)l
t
(C1mcosmÀ+
SlmsinmÀ) P1m(cos(i)r, La+1 r m=O
(30)
(31)
Strictly speaking, Nom is not an observation and is either neglected or approximated based on some kind of model.
Case 2 Gravity information consists solely of terrestrial gravity measurements: Y:the observation vector has elements YPk
=
h - H(i) - ~St(~g~j))A: the design matrix has elements -1, (1
+
2ISPkQ,) , or 2ISPkQj X: the vector with unknowns has elements7'
~,
i=
1,2,···,1in which the following abbreviations have been introduced:
(32)
(33)
Case 3 Combination of potential coefficients and terrestrial measurements in a spherical cap around each station. It is assumed that the gravity anomalies in a cap centered around point P, all refer to the same vertical datum, i.e., the vertical datum i in which point
P is situated. Consequently, there is only one vertical datum involved with the gravity anomalies. In th is case:
Y: the observation vector contains YPk
=
h - H(i) - ~StC(~g~j)) - N~at - Nom
A: the design matrix has elements either 0, -1, or (1
+
2IS'j,kQ,)4
Least squares solution of vertical datum connection in whieh the foIlowing abbreviations have been used:(34)
GM La
(R)l
IN~at = -2 -
L
(l - 1) -Q;
L
(~Clm cos m).+
~Slm sin m).)Ptm
(cosB)
,r 1=2 r m=O (35) GM 00
(R)l
I N~m= -2-L:
(1-1) -QT
L:
(C1meosm).+
Slmsinm).)Plm(eosB) ,r I=La+ 1 r m=O (36) 1l"'c
1
21\" ISPkQi=
-4 St*(1/J)d(Ji 'Ir "'=0 0=0 (37)Again, NÖm is not reaIly an observation; this term is either negleeted or approximated
based on some kind of model.
4.2 Precision
In order to solve the datum connection parameters from the system of observation equa
-tions, i.e., equation 29, we need the a-priori varianee-eovarianee matrix of the
measure-ments. FoIlowing Xu and Rummei (1991), if the observations of geometrie height,
or-thometrie height and geoid height ean be assumed uneorrelated, the varianee-eovarianee
matrix of the observations ean be determined by
(38)
in whieh Eh is the varianee-eovarianee matrix for the observations of geometrie height,
EH the varianee-eovarianee matrix for the orthometrie height measurements, and EN is
the varianee-eovarianee matrix eorresponding to the geoid heights.
How the a-priori varianee-eovarianee matrices for the geometrie, orthometrie, and
geoid height ean be obtained will be derived in section 5. In this seetion, only relations
to verify the precision of the datum connection parameters will be shown.
The weIl-known solution for the unknown datum connection parameters, X, foIlowing
from a least squares adjustment of equation 29 is:
(39)
The first part of equation 39 consists of the a-posteriori varianee-eovarianee matrix of the
datum connection parameters,
(40)
the diagonal elements of this matrix eontain the varianees for the estimated datum
eon-nection parameters, i.e., the varianees for .ê1!:J>. and cQ;o. These varianees indieate how
weIl the unknown potential value of the refJrenee surface, and the potential differenees
between the referenee surface and the other datum surfaces ean be determined.
As already indicated in seetion 2, for the purpose of eonneeting various vertieal datum
zones it should also be estimated how wen the potential differenees between the various 17
18 Part A. Theoretical background to vertical datum connection
datum surfaces themselves can be determined. The varianee of the potential difference between two arbitrary datum zones, CPQ , can be estimated from the covariance matrix
as given in equation 40 by:
(41)
It should be noted that potential differences divided by , are estimated in order to derive more convenient results in meters. However, for reasons of simplification, of ten only potential difference is written, whereas potential difference divided by , is actually meant.
In part Band part C, results for error propagation in datum connection problems will be given. In order to improve on derived results it is useful to be able to determine which type of measurements are the major limiting factor in solving the datum connection problem. The relative effect of an error in resp. the geometrie height, orthometric height, or geoid height on the precision of the derived datum connection parameters can be computed as:
(42)
in which, again, Ex is the a-posteriori variance-covariance matrix of the datum con neet ion
parameters, A is the design matrix as introduced in the preceding section, Ey is the a
-priori variance-covariance matrix of the combined height measurements, and Eh, EH, and
EN are the a-priori variance-covariance matrices for the measurements of resp. geometrie height, orthometric height, and geoid height.
The three matrices as defined in equation 42 give the precision of the derived datum connection parameters due to the uncertainty in the measurements of respectively geo-metrie height, orthometric height, and geoid height. Combining these three matrices gives the a-posteriori variance-covariance matrix of the datum connection parameters, i.e, the precision of the datum connection parameters due to the uncertainty in the combined height measurements:
(43)
4.3 Reliability
The advantage of sol ving the datum connection problem by means of a least-squares adjustment is that not only the precision of the derived datum con neet ion parameters can be estimated but their reliability and the reliability of the involved measurements can be verified as well.
A measure of the reliability of the measurements, of ten referred to as the internal relia-bility, is the marginally detectable error, élYi. This is a measure of the error in the mea-surements that can be detected with significant level 0:" and power {j. Following Baarda
(1968), this marginally detectable error can be calculated with the following equation:
-élYi
= -
~o
5 A-priori covariance matrices
In this equation Ào is related to the test parameters a and {3. For example, for a = 0.05
and {3
=
0.8, VX;; is 2.8. The parameter Mi ean be determined from:(45) in which ei is a vector consisting of ze ros except for the i-th position corresponding to the measurement which is been tested, i.e, Ci
= (0,0
, .. ',1,0, .. "of.
Q€ is thevariance-eovariance matrix of the measurement errors, i.e., Q€ = Ey - AExAT.
The reliability of the derived datum connection parameters, of ten referred to as the ex -tema! re!iability, is the infiuence of a marginally detectab!e error ËlYj on the estimated
parameters. The infiuence of marginally detectab!e errors on the i-th datum connection
parameter is, according to Baarda (1968), given by
- - T 1
-ËlXi
=
ExA Ey ËlYJ (46)in which ËlYJ is a vector consisting of zeros except for the position corresponding to the infiuence of the measurement which is been tested, i.e, ËlYJ
=
(0,0, .. " ËlYj, 0,' . "of
.
5
A-priori covariance matrices
In the preceding section we have shown re!ations which allow the verification of the
pre-cision, re!iability and detectabi!ity of the datum connection parameters. In part Band in part C, error propagation will be performed for resp. a stylized world and aspecific region. In order to determine the accuracy of the datum connection parameters for these data configurations we need the a-priori variance-covariance matrix of the measurements. As indicated in the preceding section, assuming no corre!ation between the three types of height measurements, the a-priori covariance matrix of the measurements can be de-termined by simp!y adding the a-priori covariance matrices for the measurements of resp. geometric height, orthometric height, and geoid height.
In this section, the a-priori variance-covariance matrices for the various types of height measurements wil! be derived. First, in section 5.1, it will be exp!ained how the a -priori variance-covariance matrix for orthometric height measurements can be estimated.
Next, in section 5.2, the a-priori variance-covariance matrix for the geometric height measurements will be discussed. In section 5.3 the a-priori variance covariance matrix for the geoid heights of the stations wil! be discussed for the three kinds of gravity information
as introduced in the preceding sections.
It wil! on!y be indicated how the a-priori variance-covariance matrices can be derived.
In part Band part C, based on the data configuration under consideration, speeific a-priori
va!ues for the measurements, e.g., the precision of !evelling, will be se!ected.
5.1
A-priori variance-covariance matrix for orthometric heights
In order to determine the a-priori variance-eovarianee matrix for the orthometric heights of the stations it is assumed the the precision of orthometric height measurements is
20 Part A. Theoretical background to vertical datum connection entirely determined by the precision of the levelling measurements. Consequently, the variance-covariance matrix for the orthometric heights of the stations can be derived from the variance-covariance matrices of existing levelling networks. Although resulting in very realist ic values, this method has a number of disadvantages, among which, • not all stations which are to be used in the datum con ne ct ion problem are necessarily
part of existing levelling networks,
• only for newer networks variance-covariance information will be available.
Therefore, for error the error propagation studies as performed in part Band part C, the precision of the orthometric heights will be based on a general guideline to describe the levelling precision. In general, the standard deviation for levelling measurements can be written as
(47)
with S the levelling distance in km, and a the unit standard deviation. For first order levelling networks, e.g., in Europe, a standard deviation of 0.6 mm/Jkffi, based on the actual levelling distance, is feasible.
Note that according to Vanîcek and Krakiwsky (1986) this so called square root law is only applicable if the individual levelling segments are statistically independent. Due to, a.o., unmodelled systematic effects, for most existing height networks the assumption of statistically independent height segments is not justified. In case of totally depen-dent height differences, a worst case scenario, the standard deviation of levelling can be
estimated from
(48)
Since the actuallevelling distance between the stations and the corresponding fundamen-tal station is hard to recover, Xu and Rummei (1991) assumed a standard deviation of levelling of 1.1 mm/v'kffi, based on the horizont al distance between the stations. Xu and Rummei (1991) furthermore assumed that measurements for the various stations were un-correlated. Although th is assumption of uncorrelated orthometric heights for the stations within one datum zone is not very realistic if these heights stem from the adjustment of the same levelling network, it can be justified by the fact that it is, in principle, possible to determine the height of a station relative to the fundamental station in an individu al levelling process.
The assumption that the variance of the orthometric height measurements is entirely determined by the precision of the levelling measurements can be justified as follows. According to, e.g., Heiskanen and Moritz (1967) the difference in orthometric height between two points A and B is given by
B -
-AH _ A +",9-70.r +9A-70H 9B-70H
u AB - unAB L.., - - u n - - - A - B
A 70 70 70 (49)
in which /:).nAB is the measured height difference, 9A and 9B are resp. the gravity along
the plumb line of A and B, i.e., 9
=
9 - (H~+
27rkp) H. Differentiation of this equation with respect to gravity gives.rH _ /:).nAB.r HA.r H B .r
u - --ug+ - u g - -ug
5 A-priori covariance matrices
Therefore, assuming aU heights in the order of 1000 m wiU for an error óg in the order of 1 mgal give an error t.HAB in de order of only 1 mmo Consequently, the effect of errors
in the gravity measurements as used to determine the orthometric height for the stations ean be neglected.
5.2 A-priori varianee-eovarianee matrix for geometrie heights
In order to determine the a-priori variance-covarianee matrix for the geometric heights of the stations, Xu and Rummei (1991) assumed that aU stations have a variance of 25 em2 and that the measurements for the different stations are uncorrelated.Instead of using a general guideline, varianee-eovarianee matrices based on adjustment of an aetual global referenee frame of SLR, VLBI, and/or GPS stations eould be used. For a large number of SLR and/or VLBI and/or GPS stations, IERS publishes coordinates and
their standard deviations in specific ITRF coordinate systems, e.g., ITRF90 (only VLBI
and/or SLR; see Boucher and Altamimi (1991)), ITRF91 (see Boucher et al. (1992)), or
ITRF92 (see Boucher and Altamimi (1993)).
Assuming that the cartesian coordinates of the stations are uncorrelated, the error
in ellipsoidal height ean be determined through error propagation, i.e., (in a spherieal approximation)
ah (cos2 cp eos2
).) • a~
+
(cos2 cp sin2 ).) . a} + sin2 cp . a}x
2 y2 Z2= . a2
+
.
a2
+
.
a~(51 )
p+~+p x p+~+p y p+~+p
If a speeifie station is not part of agiobal "fundamental referenee network" of VLBI, SLR, or
GPS stations, relative geometrie heights with respect to one or more of these "fundamental spaee stations" can be determined using differential GPS. According to Bock (1996), the varianee of these relative measurements can be estimated as
(52)
in whieh s is the distanee (in km) between the station under consideration and the
"fun-damental spaee station" . I
5.3 A-priori varianee-eovarianee matrix for geoid heights
In the preeeding we have introduced three different sources of gravity information which could be used to determine the geoid height of the stations. How the a-priori covarianee
matrix of the geoid height for the spaee stations can be determined depends on the kind of gravity information used.
In section 5.3.1 the variance-eovarianee matrix for geoid heights solely based on poten-tial eoefficient models wil! be derived. In the next seetion, the variance-covarianee matrix wiU be discussed for geoid heights solely based on terrestrial gravity measurements. In section 5.3.3 we wil! derive how the variance-covariance matrix ean be derived for geoid heights determined from a combination of potential eoeffieients and terrestrial gravity measurements around the stations. FinaUy, in section 5.3.4, the two geopotential models as wil! be used in part Band in part C wil! be described.
22 Part A. Thearetical background ta vertical datum cannectian
5.3.1 Only satellite derived potential coefficients
If geoid heights are solely determined from geopotential coefficients, the geoid height measurements are given by equation 30. From this equation it is clear that two error sources contribute to the a-priori variance-covariance matrix of the geoid heights, i.e,
1. errors in the potential coefficients themselves, referred to as commission error 2. errors caused by the fact that the geopotential models contain information only up
to a maximum degree and order Lo, and information of higher degree is omitted, referred to as omission errors
Assuming these two error sources to be uncorrelated, we can write
(53)
in which 'E,cNpNQ is the variance-covariance matrix of the geoid heights due to the com-mission error, and "ENpNQ the variance-covariance matrix due to the omission error. In the next two paragraphs these variance-covariance matrices will be derived.
Variance-covariance matrix due to commis sion error The variance-covariance matrix of the geoid heights for the stations due to commission error can be derived from the variance-covariance matrix of the potential coefficients as follows. The geoid height measurements, i.e, equation 30, can be written as:
(54)
Denoting the variance-covariance matrix of the potential coefficients by
"Eb,
the variance-covariance matrix for the stations can be found through error propagations as:(55)
where F(P) and F(Q) are coefficient matrices which relate the !::iCnm and !::iSnm to the geoid heights at points Pand Q, respectively. For every station, i, the corresponding row
in the matrix F contains the factors Yim(BpJ. The sequence of these factors depends on the structure of the variance-covariance matrix for the potential coefficients.
Variance-covariance matrix due to the omission error The variance-covariance matrix of the geoid heights for the stations due to the omission errors can be determined, based on a model which estimates anomaly degree variances as:
(56)
In which 'lj;PQ is the spherical distance between station Pand Q, Àl are the eigenvalues relating the anomaly degree variances to degree variances for geoid heights, i.e., Àl
