
Midterm
Report of IAG Special Commission 1
MATHEMATICAL
AND PHYSICAL FOUNDATIONS OF GEODESY
for
the period 19992001
by
Petr Holota
Research Institute of Geodesy, Topography and Cartography
250 66 Zdiby 98, Prahavychod, Czech Republic
email: holota@pecny.asu.cas.cz
1. Introduction
The Special Commission on Mathematical and Physical Foundations of
Geodesy (CMPFG) was established by the International Association of
Geodesy on the occasion of the 20th General Assembly of the
International Union of Geodesy and Geophysics in Vienna in 1991. It
expresses the need for a permanent structure working on the
foundations of geodesy. The establishment of the special
commission is essentially associated with the preparatory work done by
K.P. Schwarz (the president of Section IV at that time) and the
Section IV Steering Committee.
The main objectives of the special commission are the following:
 to
encourage and promote research on the foundations of geodesy in
any way possible;

 to
publish, at least once every four years, comprehensive reviews of
specific areas of active research in a form suitable for use in
teaching as well as research reference;

 to
actively promote interaction with other sciences;

to
closely cooperate with the special study groups in Section IV 
General Theory and Methodology. (As an IAG structure the CMPFG belongs
particularly to this section).
This
formulation is short in its form but in reality it represents a
challenging program that may be also found in the 2000 issue of The
Geodesist's Handbook [Journal of Geodesy (2000), Volume 74, No. 1]. In
addition one can read information and details concerning the CMPFG
(including the bibliography) on the website of this special commission
at the address: http://pecny.asu.cas.cz/IAG_SC1/
It is natural that the research program of the CMPFG represents a
continuation of the activities developed already in the period of the
last 8 years when E.W. Grafarend successfully chaired the special
commission. The research program of the CMPFG mainly focuses on statistical
problems in geodesy, numerical and approximation methods, geodetic
boundary value problems, on problems in geometry and differential
geodesy, relativity, cartography, on equilibrium reference models and
also on the theory of orbits and dynamics of systems.
In this field the CMPFG derives important driving impulses especially
from the work of the IAG itself. As a minimum let us mention two
problems that were discussed at a special plenary session held in
Birmingham on the occasion of the 22nd General Assembly of the
International Union of Geodesy and Geophysics in 1999: 1) "Are
our contemporary theoretical and computer models sufficient to handle
the 1:10^{9} accuracy in frame realization, Earth rotation,
positioning etc. consistently?";  2) "Can we be sure that
sensor and/or model deficiencies do not enter into geophysical
interpretation?"
The broad spectrum of research objectives is connected with a
subdivision of the research program into specific tasks. In 1999
immediately upon approval of the CMPFG program by the IAG the
following subcommissions were established:
 Subcommission
1 "Statistic and Optimization"
Chair: P. Xu (Japan)

 Working
Group "Spatial statistics for geodetic science"
Chair: B. Schaffrin (USA)

 Subcommission
2 "Numerical and Approximation Methods"
Chair: W. Freeden (Germany)

 Subcommission
3 "Boundary Value Problems"
Chair: R. Lehmann (Germany)

 Subcommission
4 "Geometry, Relativity, Cartography and GIS"
Chair: V. Schwarze (Germany)

 Subcommission
5 "Hydrostatic/isostatic Earth's Reference Models"
Chair: A.N. Marchenko (Ukraine)

The
theory of orbits and dynamics of systems
is an exception. In general problems that by nature have a tie to this
topic are given a considerable attention in many branches of science.
Here the topic was left within the framework of the special commission
itself. It focuses on the interplay between mathematics (especially
analysis) and applications that together with problems related to
methods of integration, modelling, analysis of perturbations and
qualitative aspects in the evolution of trajectories reach the field
of space geodetic methods and inertial systems. After two years the
original intention is associated with visible achievements. The work
of the CMPFG members resulted in a number of very valuable
contributions. They concern e.g. dynamic satellite geodesy on the
torus; the relation between analytical and numerical integration in
satellite geodesy; energy relations for the motion of satellites
within the gravity field; asymptotic series in mathematics, celestial
mechanics and physical geodesy; satellite geodesy on curved spacetime
manifolds; differential equations in inertial navigation systems etc.
Considerable activities of members develop also in the filed of
dedicated satellite mission and in a contact with IAG Special
Commission 7.
2. Subcommissions
Also the subcommissions are very productive. It can be immediately
seen from the bibliography of the CMPFG that is directly
accessible on the website of the special commission (at the
address given above and in the References). It contains a rich list of
entries that document the research done by the members of the special
commission in the reported period. (On the website the bibliography is
an open material that is under a process of a permanent completion.)
For page limit let us mention some highlights only. Subcommission
1 placed emphasis on areas such as the theory of (geo)inverse
problems, nonconventional models for space applications, spatial
information theory, global optimization methods and also the
advancement of traditional topics. The IAG Executive Committee at its
meeting in Nice, 2000 have decided to give Dr. Peiliang Xu (the chair
of the subcommission) the IAG young authors award for his paper
"Biases and the accuracy of, and an alternative to, discrete
nonlinear filters", published in the Journal of Geodesy, Vol.
73(2000), pp. 3546. Within the general discipline of statistics,
methods which relate each "data point" to a location allow
for an analysis with a spatial resolution that might otherwise be
lost. The data do not need to be point data themselves, but could have
been derived from a certain area by averaging. Key is that we have to
deal with both probabilistic and spatial distributions. This in short
is the field of research of the Working Group that
develops its activities in a close cooperation with Subcommission 1.
Members of the subcommission and the working group brought significant
contributions to the IAG 1st International Symposium on Robust
Statistics and Fuzzy Techniques in Geodesy and GIS in Zurich, March,
2001.
An intensive mathematical research oriented to problems in the
representation and approximation of the Earth's gravitational
potential, to problems in physical geodesy and in the treatment of
modern space geodetic data was in the focus of Subcommission 2.
The season is that one has to think of the geopotential as a
"signal" in which the spectrum evolves over space in
significant way. This spaceevolution of the frequencies is not
reflected in the Fourier transform in terms of nonspace localizing
spherical harmonics. Wavelet transforms are a counterpart. Therefore,
aspects of constructive approximation, decorrelation, data compression
etc. were treated within the wavelet theory. Moreover, an uncertainty
principle was formulated and used as it gives an appropriate bound for
the quantification of space and frequency properties of trial
functions in geodesy. In the focus there were also combined models,
where expansions in terms of spherical harmonics are combined with
local methods, e.g. radial base function techniques as splines,
wavelets, masspoints, finite elements etc. In the limited time span
of the first two years the subcommission also significantly progressed
in the methodology of the treatment of spaceborn observations. In
addition to a number of presentations and entries in the bibliography
an important contribution on "Multiscale modelling of GOCE data
products" was prepared for the ESA International GOCE User
Workshop held in Noordwijk in April, 2001.
Subcommission 3 focused on boundary value problems (BVP)
in physical geodesy. They are essentially connected with the use of
potential theory and the theory of partial differential equations in
the determination of the gravity field and figure of the Earth. In the
reported period the research carried out by the subcommission
concentrated on the refinement of the solution of the standard
problems and new mathematical models, on freedatum and multidatum
BVPs, as they arise from unknown height datums; on mixed BVPs and
especially various types of altimetrygravimetry problems with their
capability to give a mathematical model for a combined use of
different data on the boundary; on stochastic BVPs; overdetermined and
constraint BVPs; BVPs on special surfaces and also on pseudo BVPs. The
research covered also nonclassical methods in the solution of BVPs,
as variational methods with their close tie to the concept of the
socalled weak solution, boundary element techniques, various aspects
in the use of ellipsoidal harmonics and other function bases. Within
traditional concepts the role of the BVPs is rather wellknown in
physical geodesy, but nowadays the work of the subcommission is
strongly influenced by new striking impulses. Among others they
reflect the progress in the data collection, data accuracy, higher
requirements on the accuracy of the solution and also a need for
mathematical modelling associated with the use of modern technologies,
as e.g. airborn gravimetry and dedicated satellite missions (a
spacewise approach, Slepian's problem etc.). The results of the
subcommission were clearly visible at the IAG International Symposium
on Gravity, Geoid and Geodynamics 2000 in Banff, July/August, 2000 and
also at the 26th General Assembly of the EGS in Nice, March, 2001.
Geometry oriented problems, relativity aspects, cartography and GIS
define the field of interest of Subcommission 4. Here
under geometry one understands the MarussiHotine approach to
differential geodesy, foundations of Gaussian differential geodesy,
geometry of plumblines as geodesics in conformal 3manifould, Fermi's
coordinates etc. Nevertheless the main progress was achieved in the
use of the theory of relativity, in particular in the reformulation of
geodetic measurement processes within the framework of general
relativity. Here the metric tensor plays an important role and it was
represented with respect to a set of appropriate charts. Using the
words of the chairman, we knew that almost every quantity of interest
in geodetic and geophysical applications refers to a geocentric,
Earthfixed coordinate system (chart). Therefore, the spacetime
metric with respect to an Earthfixed chart was derived at first post
Newtonian order. The field equations determining the terrestrial
gravitational field were derived and its explicit representation was
outlined. On this basis the impact of the results on the modelling of
geodetic measurement process including spacetime positioning
scenarios as well as the highprecision gravitational filed estimation
was discussed. Finally, results achieved in cartography and GIS were
presented at the IAG 1st International Symposium on Robust Statistics
and Fuzzy Techniques in Geodesy and GIS in Zurich, March, 2001.
Subcommission 5 is a completely new substructure of the
CMPFG. Nevertheless it proved to be very active. In the reported
period it attacked the construction of piecewise radial density
models, stable determination of parameters of radial density models,
variational problems and the interpretation of some reproducing
kernels, it focused on the lowfrequency Earth's gravity field and the
evolution of the Earth's principal axes and moments of inertia
completed with a canonical form of the solution. Some research was
also oriented to incompressible fluid Earth, compressibility and
vicoelestic perturbations. For the density recovery from seismic
velocities the solution was based on three differential equations and
the density function was separated into a hydrostatic (main) part and
an additional small part due to chemical/phase inhomogenieties or
superadiabatic temperatures. Some famous laws (LegendreLaplace,
Roche, Darwin, Gauss) were considered for radial density distribution
in connection with the solution of the famous Clairaut, Poisson and
WilliamsonAdams differential equations. In the interpretation of
reproducing kernels it was shown that the set of all suitable kernel
functions may be interpreted as a finite sum of two point
singularities (pole and dipole) and also straight line singularities.
In addition an optimum point mass model of the global gravitational
filed was compiled.
3. Business Meeting in Banff
The CMPFG is an important discussion forum. This was evident from the
business meeting of the special commission organized in Banff on
the occasion of the IAG International symposium "Gravity, Geoid
and Geodynamics 2000". A circular letter distributed by the
special commission chairman well before the meeting proved to be a
stimulus that met with a good response. "What you think is the
most urgent problem to be solved related to the foundations of
geodesy" this was a key question formulated by C.C.
Tscherning and circulated with the letter. It turned out that what is
natural. The response reflects the impact of the future or upcoming
satellite missions. In particular the following urgent problems
were mentioned (in the formulations by R. Rummel):
 How
to deal in a proper way with the actual Earth boundary when
bringing down the high resolution gravity information from GOCE
from satellite altitude to the Earth's surface? Should one simply
apply some kind of topographic correction or are there better
and/or more correct ways?

 A
mission like GRACE drifts (in the course of the entire mission
length) down from, say 500 km to 300 km. While going down the
gravity field is sampled in a changing manner and at the same time
it drifts through several regimes of resonance. At the same time
one tries to recover the temporal variations of the gravity field.
Thus one is faced with a very complicated sampling/aliasing
problem, which to my best knowledge is not sorted out so far.

 A
similar problem in altimetry which certainly affects, for example,
estimates of sea level change. Again the satellite samples time
variable effects such as tides in a very complicated time and
space pattern. What is the aliasing situation there, what could be
done to improve it?

 In
real world satellite sensors such as accelerometers or
gradiometers do not show a normally distributed noise behavior but
drifts and a similar effects cause systematic distortions which
result in e.g. 1/f (f = frequency) behavior. We have taken into
account these things but in an ad hoc manner. Could one work on
stochastic models on the sphere for processes with less favorable
error behavior?

 Currently
several groups try to determine centerofmass changes of the
Earth from the analysis of global tracking data. What is the
proper formulation of datum definition and Stransformation for a
real deformable earth with moving plates, how should a
centerofmass change determination be correctly formulated from
the theoretical point of view?

K.H.
Ilk expressed another view. He pointed out three problem areas
related to the satellite missions CHAMP, GRACE and COCE: 
analysis of the observation system;  modelling and data analysis
aspects;  applications in geosciences, oceanography, climate change
studies and other interdisciplinary research topics.
In addition M. Vermeer suggested, loosely speaking "best
practices" and the use of common sense in connection with
the use of modern techniques in geodesy. Using his words, we know that
in traditional geodesy there were these common sense rules such as
"working from the large to the small" and many many more.
With new techniques, and the availability of fast computers and
complex theories, sometimes it seems that common sense has been a bit
forgotten.
The subsequent discussion at the business meeting concerned some
reflections on the running process towards the new structure of the
IAG. B. Heck, the president of IAG Section IV outlined the key aspects
that motivate this initiative. His information were then amplified by
F. Sanso, the IAG president who first paid a considerable attention to
the work of the CMPFG itself and than focused on a detailed
explanation of the principles and actions that are most frequently
discussed within the IAG executive in preparing the concepts for the
new IAG structure. The business meeting of the CMPFG was well
attended, not only by the members, but also by a number of
participants of the Banff symposium on Gravity, Geoid and Geodynamics
2000. The CMPFG will hold its future business meeting in
Budapest, concurrently with the IAG Scientific Assembly, 28 September
2001. For 2002 the CMPFG prepares an active participation in the
HotineMarussi symposium on mathematical geodesy which by tradition
will be held in Italy under the sponsorship of the IAG.
References: see
please http://pecny.asu.cas.cz/IAG_SC1/
Acknowledgements. Concluding this brief report, I wish
to express my sincere thanks to all my colleagues from IAG Special
Commission SC1 for excellent cooperation and all the results achieved
that often mean months or years of a great endeavor and devoted work.
Much success in your further work!
