Gravity, Geoid and Geodynamics 2000

GGG2000 lAG International Symposium Banff, Alberta, Canada

July 31 - August 4, 2000

Edited by Professor Michael G. Sideris

Springer

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Professor Michael G. Sideris Department of Geomatics Engineering The U nivesity of Calgary 2500 University Drive N.W. Calgara, Alberta Canada T2N IN4

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Gravity, geoid and geodynamics 2000 : lAG international symposium / GGG 2000, Banff, Alberta, Canada, July 31 - August 4, 2000. Ed. by Michael G.Sideris. - Berlin; Heidelberg; New York; Barcelona; Hongkong; London ;Mailand; Paris; Tokio: Springer, 2001 (International Association of Geodesy symposia; Symposium vol. 123)

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Establishing Global Reference Frames.

Nonlinear, Temporal, Geophysical and

Stochastic Aspects. Athanasios Dermanis

Department of Geodesy and Surveying, The Aristotle University of Thessaloniki

University Box 503, 54006 Thessaloniki, Greece. Email: dermanis@topo.auth.gr

Abstract. The problem of the optimal definition of a Global Reference Frame based on a geodetic network of continuously observing stations is ana­lyzed from various points of view. The non-linear datum definition problem is extended to the time domain and the optimal reference frame for a de­formable network is obtained as a geodesic line on a curved manifold in the vector space of all network coordinates, which is the union of instantaneous shape manifolds. The original ideas of Meissl are extended from the linear to the non-linear case and from the space to the space-time domain. The re­sulting definition of a Meissl Reference Frame is shown to be a geodesic frame as well as a Tisser­and-type frame. The difference between the opera­tional Meissl-Tisserand geodetic network frame and the theoretical geophysical Tisserand earth frame is emphasized and it is shown how a connection be­tween the two can be established by incorporating geophysical hypotheses, such as plate tectonics. Finally the stochastic problem of the optimal com­bination of estimated network frames is examined from both a non-linear and an approximate line­arized solution point of view.

1. Introduction

The undergoing discussions for the restructuring of the International Association of Geodesy indicate that the maintenance of a Global Geodetic Refer­ence Frame emerges as one of the central objectives of future geodetic work. It is therefore appropriate to try to clarify a number of theoretical issues re­lated to this topic, with the hope that applied work will be also benefit from our conclusions.

On a first sight it seems curious that a scientific discipline puts research emphasis on an object that has no physical existence and it is purely a mathe­matical artifice, though of indisputable conven­ience. The introduction of reference frames is asso­ciated with one of the turning points in the history of mathematics: the Cartesian revolution gave rise to analytical methods, which eventually replaced

International AssociatIOn of Geodesy Symposia, Vol. 123 Sideris (ed.), Gravity, Geoid, and Geodynamics 2000 ©Springer - Verlag Berlin Heidelberg 2001

the "more rigorous" geometric ones. If we make an effort to get outside this historical framework we come to the understanding that our "geometric" work is essentially that of the study of shape and its temporal variation. Even motion can be essentially reduced to change of shape. It has been the presence of a "rigid" earth in our immediate environment, which provided the framework (a family of equiva­lent reference frames!) which made indispensable the use of coordinates and made "positioning" a central theme of geodetic work. At the present level of accuracy where the earth is significantly deform­able, the persistence of reference frames and coor­dinates is, nevertheless, not a matter of scientific tradition, but rather one of mathematical conven­ience.

First of all, we need a clear-cut mathematical model for the concept of a global reference frame within the Euclidean-Newtonian physical model. This will give us the tool for choosing an optimal frame, with the understanding that the actual object of our work is the study of the temporal variation of the shape of the earth in relation to the inertial space of surrounding celestial objects. Earth rotation is in fact a way of summarizing the variation of the shape of the earth with respect to the celestial back­ground, where the main (low frequency) character­istics are attributed to earth rotation and the re­maining ones to earth deformation. In fact we study the rotation not of the earth but of a reference frame attached to the earth; one and the same physical phenomenon is artificially separated in two parts for the sake of mathematical convenience.

In our geodetic work on reference frames we must rely on the pioneering work of Meissl (1965, 1969) and Baarda (1973), which we seek to generalize from its original framework, namely the adjustment of rigid geodetic networks based on linear or line­arized models. Our extension is thus twofold: we must avoid, as much as possible, the linear point of view looking into the nonlinear aspects of reference frame definition and at the same time to extend the original ideas from the spatial to the spatial­temporal domain.

Beyond geodesy we must seek a contact with the concept of reference frame as used in geophysics. for the whole earth. where the objective is to sim­plify the differential equations describing earth ro­tation-deformation and to minimize the deformation part. without any consideration to the actual reali­zation of the frame.

The realization of the reference frame for either a geodetic network or the whole earth relies on the analysis of observations within particular physical models. Uncertainty within both observations and models necessitates the introduction of statistical concepts and methodologies. Thus the deterministic model of a particularly chosen reference frame has to be paired with a stochastic realization. a task which poses interesting problems in both data analysis and probabilistic interpretation of the re­sults. Let us note that the prevalence of linear meth­ods of statistical analysis poses some limitations to our effort for a purely nonlinear approach.

2. A nonlinear spatio-temporal mathe­matical model for a global geodetic reference frame

In order to describe any reference frame for a de­formable geodetic network by a concrete mathe­matical concept we start with the coordinates

Xk=[XkYkZkf. k=1.2 ..... N (1)

of the N network points with respect to the par­ticular orthonormal right-handed cartesian reference frame. The ensemble of all network coordinates

XI

X=

XN

T = [XI YI ZI",xN YN zNl (2)

is a point in the 3N-dimensional Euclidean space

E 3N = E3 x .. · x E3 (n copies). which describes at the same time (and inseparably in some sense) both the shape of the network as well as its position with respect to the reference frame. Or rather. to be more precise. the position of the "fictitious" reference frame with respect to the "real" geodetic network.

In order to separate shape and position we con­sider all the vectors X I, which describe the same network shape but different frame position (fig. 1). Any such vector X I can be derived from the origi-

36

nal vector x by a point-wise defined coordinate transformation

xic = A R(9)xk + b • (3)

z

s

xeS. b. A)

Fig. 1: The shape manifold S in EN, consists of all network coordinates x giving the same network shape. The coordinate transformation, from a fixed reference point z defines a cur­vilinear coordinate set (8, b, A) on S. The related local basis

spans the tangent space TS. at the particular point x.

which (in the most general case) involves 3 rotation

parameters 9 = [81 82 83t ,3 displacement parame-

ters b = [hi h2 h3 f and a scale parameter A. The

induced transformation in E3N. x I = Xa,b,A (x) de­

pends on the 7 transformation parameters 81 , 82 ,

83 , hi' h2• h3• A. i.e .. X = X(9, b, A), or on a par­

ticular subset whenever a particular characteristic of the transformation is defined from the observations. By varying the values of the transformation pa­rameters, we obtain a set of points x I, which for­

mulate a nonlinear manifold S in E 3N , consisting of all points (coordinate sets). which describe net­works with the same shape but different reference frame positions. Each point of the shape manifold S • represents a choice of a reference frame for the geodetic network with shape determined from ob­servations. which is either rigid, or otherwise con­sidered for a single particular time epoch. In fact. the relevant subset of the transformation parameters constitute a set of curvilinear coordinates for S , if a

particular point Z E S is fixed and any other point x = x(9, b. A) E S is uniquely determined by its

curvilinear coordinates

q = [9 T bT At, (4)

through the relations x k = A R(9) Z k + b .

The introduction of a coordinate system allows us the study of the differential geometry of the shape manifold S, i.e., its metric, its connection coeffi­cients (Christoffel symbols) and in particular the local coordinate related frame at any point XES,

which is extrinsically given by

Ox e· =-,

I oqi i =1,2, .. . ,7.

Let us note that the matrix

E = [e 81 ' e 82 ' e 8, ' e d, ' e d2 ' ed, ' e A 1 =

= Ox (x) = E(x) , oq

(5)

(6)

is no other than the coefficient matrix Exo = E(xo)

of the Meissl inner constraints E~o (x - xo) = 0 ,

involving the approximate coordinate values X O

used in the network adjustment linearization. The columns of E(x) span the tangent space TS x

to the manifold S at the point x (fig. 1), while the

columns of the inner constraint matrix span TS~o,

tangent to the approximate shape manifold Sa at

the point xO. The extension from a single epoch t to a time in-

terval [to,t F J, is now straightforward: To every

epoch t corresponds the manifold St of the in­

stantaneous network shape, and the union of such shape manifolds constitutes the time-shape mani­fold

M =USt (7)

of the deformable geodetic network. The submani­folds St constitute a fibering (Dermanis, 1998) of

M . The choice of a reference frame for a deform­able network is equivalent to the choice of a refer­ence frame x(t) E St for every single epoch

t E [to, t F] in a smooth way (no jumps, x(t) a

smooth function of t). It can be therefore modeled by a section of the above fibering (fig. 2), i.e., by a

37

smooth curve x(t) on M , which extends from So

to SF and crosses every St in a single point x(t).

x~

SF

10 ~.,

Fig. 2: Alternative reference frame deftnitions RF and RF' on the time-shape manifold M of a deformable network.

In order to obtain an analytical description of any such curve, we need only to pick up a fixed refer­ence curve, i.e. a particular reference frame z(t), in

which case any other reference frame

x(t) = XS(t) ,d(t),A(t) ~(t)) ~

Xk (t) = A(t)R(9(t»zk (t) + bet) (8)

is uniquely determined by the 7 functions 9(t),

b(t), A(t). The problem of choosing an optimal

reference frame x(t) is thus reduced to the deter­

mination of the corresponding optimal time func­

tions 9(t), b(t), A(t), which will result from a

particular criterion of frame optimality. Such a cri­terion may be related to the "metrization" of the

"host" Euclidean space E3N , the simplest choice being the metric defined by the Euclidean norm

p(x,x') =11 x -x'il = ~(X _X)T (x -x). (9)

This metric induces a metric on M and defines its intrinsic differential geometry in relation to the co­

ordinate set p = [qT tf .

3. The geodesic-Meissl-Tisserand optimal reference frame.

Among the various choices of reference frame curves those with smaller length are preferable,

since they involve less coordinate variation for the description of the same variation of shape. Thus an optimal choice is a geodesic curve and in particular a minimal geodesic curve from So to SF defined

by the conditions of being orthogonal to both So

and SF (fig. 3). A different idea is to extend the

concept of the discrete Meissl inner constraints, which applied to 2 discrete epochs have the form

E(X(t))T (x(t + t) -x(t) = 0, to a continuous otho­

gonality condition involving the tangent vector to the reference frame curve (velocity vector) vet) = lim I~ 0 [(x(t + t) - x(t)) / t] = (dx / dt)(t) .

The resulting condition

T dx E(x(t)) -(t) = 0 .,.

dt

dx v =-l.TS x ' "It (10)

dt

is equivalent to the orthogonality of the reference frame curve at each epoch to the instantaneous shape-manifold SI (fig. 4).

I .'~ " " ,

10 SI

X/I So

Fig. 3: Minimal geodesics as optimal reference frames.

Yet another idea is to view the network as a set of point masses and to introduce a Tisserant frame with respect to which the relative angular momen­tum vanishes

dx · h ==" m·[x·x]-' = 0 xL' I dt

I

(11)

It has been shown (Dermanis, 1995) that the con­cepts of minimal geodesic and Meissl frames coin­cide. The same is true for the concept of the net­work Tisserand frame with fixed scale and equal point masses mi (Dermanis, 1999). We may

therefore refer to the geodesic-Meissl-Tisserand

38

frame, determined from the solution of a set of first order differential equations (see Box 1) for the functions 9(t) , b(t) , which define the transforma-

tion xk(t) = R(9(t))zk(t)+b(t) to the desired op­

timal reference frame x(t) from an initial one z(t) ,

which is only subject to the easily implemented condition z == l:iZi = O.

M IF ,'-'

"

"

fO~

So

Fig. 4: The concept of a Meissl reference frame

4. Relating a geodetic network frame to a geophysical earth frame.

In theoretical studies of earth rotation and deforma­tion the introduction of a geocentric reference frame is aiming at the simplification of the equa­tions of motion and involves the whole earth. The principal or figure axes, for which the matrix of inertia becomes diagonal (Munk & MacDonald, 1960), demonstrate a large diurnal variation (Moritz and Mueller, 1987); instead the Tisserand axes are used for which the relative to the frame angular momentum of the earth vanishes and at the same time the relative kinetic energy is minimized.

The geodetic global reference frame based on a set of discrete points limited on the surface of the earth cannot be identified with the reference frame used in theories of earth rotation. In order to com­pare observed rotation with that of any theoretical model, it is necessary to provide a modified geo­detic frame, which will implement geophysical hy­potheses, which, if true, will bring it close to the Tisserand axes of the whole earth. A first step in that direction is the use of network Tisserand frame, with unequal point masses mi , reflecting the part of

the earth surface closer to each point, or even the estimated lithospheric mass of this area based on terrain models and isostatic hypotheses.

db =0 dt '* bet) = bet 0) = const.

dO = n-1RC;lhz dt

N

where: L Zi =0 i=1

8R [O>k x]=-RT

88k ' n = [0>10>20>3]

N

C z = L [(z; zi)I -ZiZ;] (matrix of inertia)

i=1

N dz h z = L [z i x]--' (relative angular momentum)

i=l dt

Box 1: The differential equations defining the optimal geo­desic-Meissl-Tisserand frame x(t), by means of the trans­formation functions Oct), bet) from an initial arbitrary

reference frame z(t).

A more sufficient model could be based on the implementation of tectonic plate models. Departing from a common reference frame x(t) an optimal

reference frame x I'; (t) can be determined for the

subnetwork of each plate P;, by solving the differ­

ential equations of Box l. Point masses may be also implemented by including the factor mi in each

summand of the initial moment of inertia matrix and vector of angular momentum.

The transformation parameters e I'; (1), b I'; (t) for

each plate provide the means for the calculation for each plate of the contribution C I'; (t) to the total

matrix of inertia and hI'; (t) to the angular momen­

tum vector with respect to the initial frame x(t)

r T T CI'; = .R, [(x x) I - xx ]dm(x) (12)

J dx h p = [xx]-dm(x)

, ; dt (13)

39

The "revised" relative angular momentum of the geodetic network, viewed as an ensemble of plates,

hx,p (t) = L hI'; (t) + h other (1) (14)

can be transformed into a vanishing one hx(t) = 0,

by transforming the original geodetic network x(t)

into an "optimal" one x(t) , closer to the geophysi­

cal Tisserant frame, by means of the appropriate

transformation parameters 0(1), b(t).

Essential in such an approach are the adopted mod­els for the extensions of the plates, the terrain heights (depths), Moho depths and mean lithos­pheric densities. In addition, the total contribution h other (t) to the variation of the angular momentum,

from any other known (or included in a particular theory) sources, must be taken into account. Since any change of frame implies a corresponding revi­sion of the earth rotation parameters, the ones rele­vant to the modified optimal frame x(t) are appro­

priate for comparison of observed and theoretical earth rotation, as well as the remaining deforma­tions. In addition we obtain revised transformation

parameters 0 I'; (t), b I'; (1) for each plate.

5. Statistical aspects of frame definition

The above discussions are confined to the deter­ministic point of view, which is in a certain sense unrealistic. The geodesist (unlike the geophysicist) must provide not only a definition but also a reali­zation of the reference frame, based on uncertain observations of a stochastic character.

In contrast to the global and nonlinear approach taken above any practical statistical data analysis is both local and linear, i.e. valid in linear approxima­tion within a small neighborhood around a Taylor

point Xo of approximate coordinates used in model linearization. The resulting coordinate estimates x contain both an estimate of the shape of the network as well of the position of a chosen reference frame, in a way that it does not allow the straightforward separation of the two, as in the deterministic case of perfectly known coordinates. Indeed for a set of parameters y describing a concrete physical object,

a set of estimates with known distribution, e.g. the normal one y - N (y, C y ), allows us to perform a

statistical inference on the object (confidence re­gions, hypotheses testing). To do the same for coor­dinate estimates, e.g. x - N(x,C x)' we must first

separate the physical part (shape) from the arbitrary mathematical artifice (reference frame placement with respect to the shape). When no frame informa­tion is contained or forced into the data analysis, the distribution of x is flat along the tangent space TS x ' a fact that is reflected in the rank deficiency of

the covariance matrix Cx . Confidence regions ex­

hibit zero dimension along TSx (space of frame

transformation parameters) thus demonstrating the luck of any information concerning the frame posi­tion. In order to combine various data sets we need to perform a type of "shape statistics" by first sepa­rating shape from frame position by passing from coordinates to another more appropriate set of pa­rameters, where the two components are clearly separated. Unfortunately an obvious way to do that suffers from the localization and linearization, in­herent in data adjustment.

Setting x = XO + Ox, we need to separate the

space E3N localized at xO, into two subspaces

T = TS 0 and a complementary one T C so that x

E 3N = T e T C . The separation can be numerically implemented by passing from the natural basis {i l , ... ,i 3N } (columns of the 3Nx3N identity

matrix) to a basis {el , ... ,ed' gl , ... , g3N-d}, where

d is the relevant rank deficiency. The components of the coordinate vector Ox in the new basis

d 3N-d

Ox= L qkek + L skgk = k=1 k=1

ql Sl

=[el···ed] +[gl···gd] ==

qd Sd

q ==Eq+Gs=[EG] , (15)

s

are separated into shape parameters s and frame position parameters q. This allows us to (locally

and linearly) combine various shape estimates s in order to obtain a combined estimate, ignoring posi­tion estimates q, and finally introduce a unique

40

desired frame to return to coordinate estimates OX. The matrix E is of course the local matrix Exo of

the Meissl inner constraints and our task is to for­mulate a matrix G such that rank[EG] = 3N . The

particular choice G = E.J. where E.J. is a matrix

satisfying also ETE.J. = 0, corresponds to the or-.J.

thogonal decomposition E 3N = T e T C .

In geodesy we have also used a concept of refer­ence frame which has a concrete physical meaning and it is not merely a mathematical artifice. These are frames introduced by d trivial minimal con-

straints C T Ox = 0, where d coordinates are as­signed constant, say zero, values. For example the frame with its origin at a particular point p, its x­axis along the line defined by two points P and Q while its xy-plane is the plane of 3 points P, Q, R

(constraintsxp =yp =zp =YQ =zQ =ZR =0)

is a well-defined natural object, which furthermore maintains its deterministic character in any statisti­cal analysis! The trivial minimal constraints on

Ox = [Oxi Ox~ f have the form CT Ox = Ox 2 = 0

and the remaining coordinates Oxl give a set of

r = 3N - d shape parameters s. This allows the combination of various such shape estimates OX I in

a statistically optimal way. Trivial minimal con­straints have been considered undesired in the past because they contribute coordinate variances and covariances in a most uneven way and for this rea­son have been replaced by the Meissl inner con-

straints ET Ox = 0, which provide a most even dis­tributions. However, from our point of view they are more advantageous because they assign all co­variance to the shape parameters, leaving zero vari­ance to the (deterministic!) frame position parame­ters.

Minimal constraints are also an indispensable tool for studies of the estimability of partial frame posi­tion information in a particular data set (Marana and Sanso 1999), such as the position of the geocenter and the direction of the rotation axis. Only with respect to a trivially minimally constrained frame do such parameters obtain a physical meaning and the problem of their estimability-determinability can be addressed. From a similar point of view, information remaining in estimates of frame posi-

tion parameters q, should be used for determining,

e.g. the position of the geocenter with respect to an adopted frame, rather than forcing the frame itself to be geocentric. The latter usual approach has the disadvantage that uncertainty in the estimated posi­tion of the geocenter with respect to the frame itself. Instead a fixed deterministic translation could be used to bring the network center in coincidence with the geocenter position estimate.

Another choice of basis for the frame position pa-

rameters q is to choose G, i.e. T e , in a way that

they become uncorrelated with the shape parame­ters S (Sillard and Boucher, 1999).

6. Discussion

We conclude with a discussion of the basic issues associated with the establishment and maintenance of a global reference frame by the geodetic commu­nity. A geodetic frame, such as the ITRF, based on a set of fundamental observing stations with high accu­racy and varying collocated observations, should be

defined by the coordinate functions x(t) according

to a well defined theoretical principle, such as the geodesic-Meissl optimality criterion discussed here. Specific computational procedures must be adopted for (a) treating the data in a way that allows the optimal combination of estimated shapes of subnetworks (hidden within coordinates in varying frame reali­zations) into a single network shape as well as the estimation of the time varying position of the geo­center and the direction of the rotation axis with respect to the network, (b) applying the time-continuous principle in a dis­crete way, for assigning the optimally defined frame to the estimated shape of the global network and producing a time sequence of station coordinates.

The purpose of such a reference frame is to pro­vide a basis for positioning and for this reason low frequency shape variations must be removed from its definition. This is currently realized by removing modeled tidal variations from observed station co­ordinates as well as by fitting a piecewise linear model for the time coordinate functions (e.g. the coordinate-velocity model for ITRF). By the way the current ITRF approach does not preserve conti­nuity since coordinates predicted by subsequent ITRFs do not coincide for intermediate epochs. A

41

continuous model has of course the disadvantage that all past coordinates must be revised when new future data are incorporated in the analysis. More elaborate models are needed which maintain conti­nuity also of first or higher derivatives, such as spline models or time series models, which are ap­propriate for spectral analysis of shape variation.

In addition to the geodetic-positioning reference frame we need to establish geophysically relevant global frames which incorporate plate motion as well as any other known source of variation of an­gular momentum. The purpose of such a frame is to provide the link between geodetic observations and geophysical theories concerning the total deforma­tion of the earth. This deformation is conveniently separated into (a) the rotation of the adopted terres­trial Tisserant frame with respect to an adopted ce­lestial frame (ICRF) and (b) the remaining defor­mation of the earth with respect to the Tisserant frame. The establishment of such a frame requires an extensive number of stations, well distributed over the whole earth, which, more than absolute position with respect to a global frame, provide information on local deformations. The density of the network will allow the identification of "outlier" stations, which reflect motion of localized character (e.g. landslides) and not deformation of the earth as a whole. In addition spectral components present in the relevant theory should be included in the frame definition and they should not be removed as in the previous case. The techniques for establishing such a frame have only been outlined in the limited space provided here and the relevant mathematical and computational details will be published elsewhere.

Since any type of frame discussed above is not unique but depends on initial conditions, the geo­detic community should decide and provide the necessary conventions which implement the choice from a set of theoretically and practically equivalent frames.

References

Baarda, W. (1973): S-Transformations and crite­rion matrices. Netherlands Geodetic Commission, Pub!. in Geodesy, New Series, v. 5, no. l.

Dermanis, A. (1995): The Nonlinear and Space­time Geodetic Datum Problem. Presented at the Inter. Meeting "Mathematische Methoden der Geodasie ", Mathematisches Forschunginstitut Oberwolfach, 1-7 Oct. 1995.

Dermanis, 1998): Generalized inverses of nonlinear mappings and the nonlinear geodetic datum problem. Journal of Geodesy, 72,71-100.

Dermanis, A. (1999): On the maintenance of a proper reference frame for VLBI and GPS global networks. In: "Quo Vadis Geodesia?", Festschrift for Erik W. Grafarend's 60th Birthday, Dept. of Geodesy and Geoinformatics, Univ. of Stuttgart, Rep. Nr. 1999.6-1, 1,79-89.

Marana, B. and F. Sanso (1999): Why do we need a worldwide network of GPS stations? Politecnico di Milano - DIIAR.

Meissl, P. (1965): Uber die Innere Genauigkeit drei­dimensionaler Punkthaufen. Zeitschrift fUr Verme­ssungswesen, 90, 4,109-118.

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Meissl, P. (1969): Zusammengfassung und Ausbau der inneren Fehlertheorie eines Punkthaufens. Deutsche Geodiitische Kommission, Reihe A, Nr. 61,8-21.

Moritz, H. and I.I. Mueller (1987): Earth Rotation. Theory and Observation. Ungar, New York.

Munk, W.H. and GJ.F. MacDonald (1960): The Rotation of the Earth. A Geophysical Discussion. Cambridge University Press.

Sillard, P. and C. Boucher (1999): A review of al­gebraic constraints in terrestrial frames datum definition. Submitted to Journal of Geodesy.