SYNTHETIC FIELDS BASED ON SPHERICAL HARMONICS
- EFFECTS MODELS
Some
members have used ultra-high-degree spherical harmonics to construct
synthetic gravity field models. One such model, computed at Curtin
University of Technology, Australia, extends to degree 5400 (to 45
degrees latitude) and to degree 2700 (over the whole Earth). These
limits are set by computer underflow and overflow errors in IEEE
double precision. The ultra-high-degree coefficients have been created
artificially by scaling and recycling EGM96 and GPM98 coefficients (ie.
an 'effects' model). Fully normalised associated Legendre functions
have been modified to avoid numerical instabilities (Holmes and
Featherstone, 2001). The degree-variance of this synthetic model has
been constrained to follow that of the Tscherning-Rapp model beyond
degree-1800.
This
synthetic gravity field has been used to construct self-consistent
geoid heights and gravity anomalies at the geoid, which can then be
used to test geoid-computation algorithms and software as follows. The
synthetic spherical harmonics are used to compute the magnitude of
gravity at the geoid (defined by the synthetic model), then normal
gravity at the ellipsoid subtracted to yield gravity anomalies. These
synthetic gravity anomalies are used as input into geoid computation
algorithms and software, then the geoid output is compared with the
self-consistent, synthetic geoid heights. Any differences between the
computed and synthetic geoid can then be attributed to algorithmic
and/or software errors.
Featherstone
(1999) has used this approach and an earlier version of this synthetic
model over Western Australia to test the theories and software used to
produce AUSGeoid98. This study shows that, when using error-free
gravity data (which, of course, is not the case in practice), the
standard deviation of the error committed due to the algorithms and
computer software is ±0.008m (max=0.035m, min=–0.035m,
mean=0.000m).
A
similar study was conducted by Novak et
al. (2001) to validate the Stokes-Helmert and modified kernel
theories used at the University of New Brunswick, Canada. This shows
that the theory and software, used for some recent Canadian geoid
models, is capable of producing a geoid with centimetre accuracy. The
standard deviation of the error committed when using quadrature-based
numerical integration of the modified Stokes's formula is ±0.010m
(max=0.039m, min=–0.030m, mean=0.003m) and when using the fast
Fourier transform is ±0.011m (max=0.045m, min=–0.036m,
mean=0.003m). Recent work at the University of New Brunswick shows
that extreme care must be used when dealing with terrain corrections
during the construction of mean Helmert anomalies in mountainous
regions.
Experiments,
due to be reported at the 2001 IAG General Assembly, will use a
refinement of the above synthetic gravity field to test the
gravimetric computation of vertical deflections using modifications of
Vening-Meinesz's formulae. Regular geographic grids of self-consistent
geoid heights, gravity anomalies and vertical deflections (at the
geoid) over Greece (18E-30E, 34N-42N) and Australia (108E-160E,
8S-45S) are available at the SSG’s web-site. The Greek data are
given on a 5' by 5' grid (degree 2160), and the Australian synthetic
data are given on a 2' by 2' grid (degree 5400).
SYNTHETIC FIELDS BASED ON MASS-DENSITY -
SOURCE MODELS
Point-mass
models continue to be a useful means of modelling the external gravity
field. Through the numerical (or even analytic; see later) integration
of Newton’s integrals, self-consistent values of the gravitational
potential and acceleration can be generated (ie. a 'source' model).
Claessens
et al. (2001)
use 500 free-positioned point masses beneath the Perth region of
Western Australia to construct a geoid model consistent with gravity
observations. This study led to the identification of some quite
serious errors in the Australian marine gravity database, as well as
confirming the misfit between satellite-altimeter-derived gravity
anomalies in near-coastal regions. As such, this synthetic gravity
field has indirectly found an additional application in detecting
errors in regional gravity data.
During
the same study, an issue relevant to generating a synthetic gravity
field from free-positioned point masses was identified. Specifically,
if gravity observations are used to attempt to construct a synthetic
field that is as realistic as possible, large masses may be positioned
in areas devoid of observations. These subsequently cause very large
synthetic gravity and geoid values to be produced in these areas,
which are not necessarily realistic. This study implies that such
synthetic fields should use fixed-positioned masses, which also
reduces the computational burden.
Allasia
(2001) has developed analytic solutions of Newton's integral for a
continuous mass-density distribution. This paper has set a theoretical
framework, but no practical application of this method has yet been
made. It is thus recommended that SSG members begin to collaborate
with Allasia to undertake practical computations to generate a
synthetic gravity field. This method appears to have the potential to
generate a completely error-free (ie. with no approximations)
synthetic gravity field. It may also be possible to generate gravity
field quantities inside the topographic masses, but further work is
probably required.
Lehmann
(2000) has produced a synthetic gravity field model using MATLAB
(version 4.2 or later) script files, principally to test altimetry-gravimetry
problems. However, this synthetic field can also be adapted, or used
directly, to generate other gravity field quantities, such as
spherical harmonics between degrees 11 and 2160. This synthetic field
is based on an axisymmetric model of the Earth that is made as
realistic as possible. Pseudo-random, un-correlated noise can be
introduced into this model. A copy of the 'user manual' and the MATLAB
script files are available via the SSG’s web-site.
Haagmans
(2000) has constructed a global synthetic gravity field model that can
generate gravity field quantities exterior to the Earth's surface at
various spatial resolutions, and at aircraft and satellite altitudes.
The model is based on a spherical harmonic expansion of an
isostatically compensated topography and the EGM96 global geopotential
model. The maximum degree is 2160, which corresponds to a spatial
resolution of 5' by 5'. This synthetic field is available directly
from the author.
Work
on forward gravity field modelling of prism-based mass-density models
continues. Papp and Benedek (2000) have used Newtonian integration of
a three-dimensional topographic mass-density model to determine
curvature of the plumblines. Nagy et
al. (2000) have
published a review-type paper on determining gravity field quantities
from prisms. Papp (2000), Benedek (2000) and Kuhn (2000) have
presented papers related to the use of mass-density data in synthetic
modelling of the gravity field and to geoid computation.
Papp
and his group continue to develop the three-dimensional model of the
lithosphere in Central Europe. The depth-density model of the
sediments has been modified according to the research results of the Eötvös
Loránd Geophysical Institute by separating the sediments into two
groups (Transdanube/West Hungary and Great Hungarian Plane/East
Hungary). The lithospheric model was also extended towards the East
(Romania), where the Transylvanian Basin and the Vrancea region (plate
subduction) are dominant.
The
geophysical community is conducting work relevant to the construction
of a synthetic gravity field for use in geodesy. These studies are
based on forward modelling to generate gravity and magnetic fields due
to reasonably sophisticated geological structures. One of these
software packages, Noddy,
uses Hjelt's dipping prism equations and frequency domain methods to
calculate the potential field response from a three-dimensional
geological model. Other geological forward modelling software, some of
which is in the public domain, is given at http://www.earth.monash.edu.au/~mark/strmodlinks.html.
Work
on the Reference Earth Model (REM), the follow-on from Dziewonski and
Anderson's PREM, also continues, but that group seems to be focussing
more on the seismic properties of the Earth.
SUGGESTED
FUTURE WORK (2001-onwards)
In
order not be too prescriptive over the future activities of SSG 3.177,
the following are suggested directions to the Group. Firstly, it is
important to recognise that different authors are investigating
different, yet complementary, approaches to the construction of the
synthetic gravity field. This in itself is essential so that there is
cross fertilisation of ideas and, moreover, tests on the synthetic
field(s) that may eventually be used as control.
It
is realistic to expect that preliminary synthetic fields will continue
to be customised to accommodate a specific area of study. For example,
comparisons of approaches to Stokesian integration on a regional scale
(eg. Novak et al., 2000)
probably require a synthetic field of different functionality to that
used for, say, detailed investigation of plumbline geometry (eg. Papp
and Benedek, 2000). Of course, a 'complete' synthetic field could be
used for both, but separate synthetic fields are more convenient
initially. With this qualification, the following are offered as a
list of characteristics and features that a complete synthetic gravity
field model could contain.
For
a complete synthetic gravity field model, which is constructed from
geophysical forward modelling, the following should be considered:
·
Realistic
models of the Earth’s topography by the densest available digital
elevation models, which can be extended artificially to higher
resolutions.
·
Realistic
models of the mass-density distribution within the deep Earth, crust
and topography, probably using a
priori geophysical models from other disciplines.
·
Realistic
models of the modes and depths of isostatic compensation and other
boundaries that are characterised by large mass-density contrasts.
·
Realistic
models of noise and systematic errors (correlated and un-correlated),
which can be varied by the user for sensitivity analyses.
Most
importantly, the model should rely on as few assumptions as possible
so that it can be used to test the assumptions currently in use. In
addition, the use of realistic and accepted models of the Earth should
guarantee that the results from the synthetic field can be applied to the real Earth.
It
is envisaged that complete synthetic gravity field models should at
least offer the following features:
·
Generation
of the synthetic field in different formats; these being point, grid
or mean values of geoid, gravity anomalies, gravity gradients and
vertical deflections.
·
A
spherical harmonic series expansion with various spectral error
characteristics.
·
Generation
of point, grid and mean gravity data with various error
characteristics.
·
Generation
of vector gravity data above and within the Earth’s physical
surfaces.
Ideally,
the model will generate gravity to <1microGal and the geoid to
<1mm at all frequencies, though this aim may prove to be
over-optimistic; but let us try!
REFERENCES
Allasia G (2001)
Approximating potential integrals by cardinal basis interpolants on
multivariate scattered data , Computers and Mathematics with Applications (in press)
Benedek (2000) The
effect of point density of gravity data on the accuracy of geoid
undulations investigated by 3D forward modelling. Presented to Gravity Geoid and Geodynamics 2000, Banff, Canada.
Claessens S,
Featherstone WE, Barthelmes F (2001) Experiments with free-positioned
point-mass geoid modelling in the Perth region of Western Australia, Geomatics
Research Australasia, (submitted in 2000)
Featherstone WE
(1999) Tests of two forms of Stokes's integral using a
synthetic gravity field based on spherical harmonics, in: Festschrift Erik Grafarend, University of Stuttgart, Germany.
Haagmans R (2000) A
synthetic Earth model for use in geodesy, Journal
of Geodesy 74: 503-511.
Holmes SA,
Featherstone WE (2001) A generalised approach to the Clenshaw
summation and the recursive computation of very-high degree and order
normalised associated Legendre functions, Journal
of Geodesy (submitted in 1999).
Kuhn M (2000) Density
modelling for geoid determination. Presented to Gravity Geoid and Geodynamics 2000, Banff, Canada.
Lehmann, R. (2000)
Technical description of the axisymmetric experiment for the study of
the altimetry-gravimetry problems of geodesy, (publications details
unknown)
Nagy D, Papp G,
Benedek J (2000) The gravitational potential and its derivatives for
the prism, Journal of Geodesy,
74 (7/8): 552-560.
Novak P, Vanicek P,
Veronneau M, Holmes S, Featherstone WE (2001) On the accuracy of modified Stokes’s integration in
high-frequency gravimetric geoid determination, Journal of Geodesy, 74(11):
644-654.
Papp G (2000) On some
error sources of geoid determination investigated by forward
gravitational modelling. Presented to Gravity
Geoid and Geodynamics 2000, Banff, Canada.
Papp G, Benedek J
(2000) Numerical modelling of gravitational field lines – the effect
of mass attraction on horizontal coordinates, Journal
of Geodesy, 73(12): 648-659.