GRAVSOFT - A SYSTEM FOR GEODETIC GRAVITY FIELD MODELLING
C.C.Tscherning
Department of Geophysics
Juliane Maries Vej 30, DK-2100 Copenhagen Oe.
Rene Forsberg and P. Knudsen
Kort og Matrikelstyrelsen
Rentemestervej 8, DK-2400 Copenhagen NV.
4th ed. Jan 1994
1. Introduction.
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Software for regional and local gravity field modelling
has been developed since the early 70's first at the Geode-
tic Institute and later at Kort og Matrikelstyrelsen (National
Survey and Cadastre) and the Geophysical Institute (from 1993
Geophysical Department) University of Copenhagen. In 1988 it
was decided to streamline and collect the software in a
package called GRAVSOFT.
The GRAVSOFT package contains a complete suite of programs for
geoid modelling, conversion of satellite altimetry to gravity data,
prediction of deflections of the vertical etc. by a host of methods:
Least-squares collocation, planar or spherical FFT, Stokes etc.,
and implements supporting system for covariance fit and -approximation,
computation of terrain effects, handling and manipulation of data lists
and grids etc.
The GRAVSOFT package consists of a number of FORTRAN
programs and test data sets. All programs contains a
detailed desription of input parameters and formats.
A sequence of exercises, illustrating the use of the
package are available on request, and the authors are also
prepared to assist at training courses. Improvements for
geoid computation purposes are reported in Tscherning et
al.(1992).
2. Data structures used.
------------------------
Gravity field data are of 4 kinds: potential coeffici-
ents, point or mean value data, grid data and statistics
related to the gravity field (error-degree variances, empirical
covariance values).
2.1. Potential coefficients.
The coefficients are used in the programs GEOCOL and
HARMEXP. They may be on character or binary form. There are
standard formats, which may be used (see loadcs in GEOCOL), but
in general the data consist of records with n, m, Cnm, Snm, where
n, m are the degree and order and Cnm, Snm are the coeffi-
cients. Coefficients on character form may be used of GEOCOL to
produce a corresponding set on binary form. At present the
maximal value of n is 360.
2.2. Point or mean value data.
The general structure is:
No, Lat., Long., Height, data1, data2, data3, ... ,datan,
where No is a station number (integer), Lat. is the
longitude, Long. is the longitude (in degrees, degrees and
minutes or degrees, minutes and seconds or cc-degrees. The
height must generally be in meters, but GEOCOL also accepts km.
If UTM coordinates are used (or another projection), then the
Northgoing coordinate is supposed to be the first. Using
GEOCOL, the longitude may also be first, and no station number
is necessary.
Along track-mean value data (e.g. from airborne gravimetry
must have a separate associated record giving the azimuth of the
line in degrees. Data not observed in the usual geodetic
reference frame must have an associated record giving the
azimuth, tilt and roll in degrees.
The program SELECT (see section 3.1) may be used to
reformat the data.
2.3. Grid data.
Data consist of a label, and then the values are stored
rowwise in East-West rows from North to South. The label
consist of min and max latitude, min and max longitude and the
grid-spacings all in decimal degrees. If an UTM grid is used,
the zone number and an integer identifying the reference
ellipsoid is a part of the label, and latitude and longitude
is substituded by Northing and Easting, respectively. The
grids may also be stored on binary form, see GBIN. Binary storage of
grids is typically used for end-presentation of results with program
GEOID.
In data grids the value 9999 or 9999.99 is used to signal unknown values.
2.4. Statistical data.
Error degree variances are given as a sequence of values
in units of mgal**2 starting by degree 0 (generally equal to
0.0). This kind of data may be produced by GEOCOL, and is used
by GEOCOL and COVFIT.
Covariance values are generally on the form:
spherical distance in decimal degrees, covariance, number
of products and then variance of the covariance. This kind of
data is produced by COVFFT and EMPCOV and used by COVFIT.
3. Programs for data handling and presentation.
-----------------------------------------------
3.1. SELECT.
This program is very useful in converting between different data
structures, e.g. data list -> grid or the opposite. The program
may be used to average data, or thin out data in a pixel mode,
i.e. used to select values located as close as
possible to the nodes of a given grid. If one of the data
elements is an error estimate, values with an estimated error
larger than a given value may be rejected. The program may also
be used to add random noise to data in a file. See the program
for more details.
3.2 FC.
The program fc (File Comparison) is used to produce from
two files a file with the sum or difference of the values in
the files, along with statistics of the various effects.
3.3 GCOMB.
GCOMB (grid combination) is a general purpose grid manipulator.
It may add/subtract, expand or overwrite grids in a number of fashions.
The grids need not cover the same area.
3.4 GEOGRID
Fast program for gridding randomly distributed data.
Local collocation or weighted means may be used. Both a geo-
graphical and an UTM grid may be used. Error estimates are
produced. The program may also be used to compute estimates of
individually given points (another randomly distributed data
set). The program may be used to grid very large datasets, and
has been optimized for fast prediction by internal sorting
algorithms. A special version GEOGRIDE may be used for gross-
error detection. Quantities are predicted from closeby points,
and compared.
3.6 GEOIP
GEOIP is a general-purpose interpolation program grids -> point values.
It may add/subtract values interpolated from grids by linear or spline
interpolation. It may also be used to add or subtract two grids of diffe-
rent spacings, e.g. for restoring reference effects after an FFT prediction.
The program may interpolate t w o grids at the same time (e.g. for deflec-
tions) or may be used for "sandwich interpolation" between two different
levels (3-D interpolation). Geoip produces statistics of data reductions
etc., and implements UTM grids fully.
3.7. GEOID and GBIN
GEIOD is a simpler, more user-friendly interpolation routine for
binary geoid grids. It implements also some coordinate transformations.
The shift between binary and ascii grids is carried out by GBIN.
3.8. CRSADJ and ALTSTACK
CRSADJ is used for cross-over adjustment of satellite altimeter data,
with output of a set of adjusted altimeter data.
The adjustment is performed using bias or bias/tilt as parameters.
ALTSTACK stacks co-linear tracks of altimeter observations
and computes variability. See (Knudsen, 1993, 1993a and
Knudsen & Brovelli, 1992).
3.9. POINTMASS.
The program generates a grid or data list of point mass gravitational
effects. This program is most useful for checking various pro-
cedures such as conversion of geoid heights to gravity.
3.10. AZTRACK.
The program will calculate the azimuth of a track segment,
by calculating the azimuth between neighbouring points.
This program is required when using line averages in collocation,
representing e.g. airborne gravity measurements.
3.11. CONVOLVE.
A convolution program for track data, designed e.g. for filtering
predictions corresponding to airborne gravimetry filters.
4. Programs for gravity field modelling.
----------------------------------------
4.1. GEOCOL.
The program has a varity of functions of which the most
important are:
Computation and evaluation of a gravity field model using
least squares collocation. As data may be used spherical harmo-
nic coefficients, point values of the geoid height, and first
and second order derivatives of the anomalous gravity potenti-
al, mean gravity anomalies and coordinate differences (used for
datum shift estimation). GEOCOL may be used for the estimation
of the same kind of point or mean values and their error
estimates. Also parameters such as datum shift or altimetry
bias and tilt values may be estimated. The program may also be
used purely for evaluating spherical harmonic series, cal-
culating error degree-variances etc. It may also be used for
the transformation between different datums.
A separate publication (Tscherning, 1992) exists, which
describes all updates to GEOCOL, suspected errors and planned
improvements. See Tscherning(1974, 1976c, 1976d,1978, 1985) for
further information. The program exists in a UNIX and a
PC-version.
Use of the program and the collocation method for a
varity of applications are described e.g. in Arabelos et
al.(1987), Forsberg and Tscherning (1981), Kearsley et al.
(1985), Knudsen et al.(1988), Tscherning (1975, 1983, 1985,
1990, Tscherning et al. (1990), Tscherning and Forsberg (1986),
Tscherning and Knudsen (1986), and Tscherning and Strykowski
(1987).
The GRAVSOFT package includes 4 test examples, where
different operation modes are illustrated. 3 examples are also
used to verify the correctness of the program. The output from
the examples are also a part of the package.
4.2 EMPCOV
The program empcov is used to produce values of an
emperical covariance function. A table is produced, which sub-
sequently may be used by the program COVFIT to obtain an analy-
tical model for the covariance function.
Two different data sets may be used, in which case both
the auto and the cross covariance functions will be produced.
4.3 COVFIT.
The program has two functions:
(1) evaluation of an analytic expression for the covari-
ance function of the anomalous gravity field and (2) the
fitting of an analytic expression to given empirical values,
see Krarup and Tscherning (1984), Tscherning(1976, 1983, 1993),
Tscherning and Rapp (1974) and Knudsen(1987, 1989), respec-
tively.
The package includes a test input example and an output
example.
4.4. GEOFOUR.
The program implements the flat-earth Fourier method of modelling
the gravity field. Gridded data are used and produced. Geoid,
gravity and Tzz values may be used as input and a varity of
other quantities may be produced. The relevant theory and many
results are found in Schwarz et al. (1990) and Forsberg and
Solheim (1988). The advantage of the FFT methods is that large
grids may be transformed very fast. In the program Wiener
filtering is implemented, which is essential for converting
noisy satellite altimeter data into gravity values.
4.5 SPFOUR.
The program calculates geoid, upward continued gravity, isostatic
effects and other quantities by the multi-band spherical FFT method.
This allows FFT's to be carried out virtually exact on the spehere,
see Forsberg & Sideris(1992).
4.6. STOKES.
The program computes geoid heights using Stokes formula by integration
from gridded data. A spline densification scheme is used to
handle inner zone effects efficiently.
4.7. HARMEXP.
The program exaluates a spherical harmonic expansion,
computes geoid, gravity or deflections using gpotdr, the same
subroutine used bt GEOCOL. Harmexp has been programmed by D. Arabelos, Greece.
4.8. COVFFT.
The program calculates 1-D and 2-D covariance functions
and power spectra using gridded data using Fourier analysis of
gridded data files. See Forsberg (1986) for result examples.
4.9 TC.
The program calculates terrain, bathymetric and/or
isostatic effects using several overlapping grids with terrain
data with varying spacing.
The space domain prism method is used.
Gridded topographic data from New Mexico, USA, and
gravity and deflections of the vertical from the same area are
included in the GRAVSOFT package for test purposes. Correspon-
ding input and output examples for tc are also included. The
program is described in detail in Forsberg (1984a).
4.10. TCGRID.
A support program to average grids for use with TC. The same functions
may also be obtained with SELECT, but TCGRID may in addition also filter
height data for RTM height reference surfaces.
4.10. TCFOUR.
Computation of terrain effects, terrain corrections etc. by Fourier
methods, as described in Forsberg (1985, 1987). From one or two height
grids the quantities are computed in similar grids using a transformed
space-domain kernels.
5. Auxillary programs.
----------------------
5.1. GRREDU
LaCoste and Romberg gravity observation preprocessing program.
The program computes gravity differences using observed
gravimeter readings, and calculates the earth-tide effect.
Instrument calibration tables are or must be available as a
part of the BLOCK DATA module.
5.2. SORTADR.
The program sortadr may be used to extract references or
adresses of a file containing records with the structure
-1
text ......
ending with an empty record.
5.3. GEOPLOT.
A program for displaying and contouring data based on
UNIRAS subroutines. Color plots may be produced. A correspon-
ding Pascal program is available at KMS, which is not UNIRAS
based, but generates HPGL code.
5.4. TRANS13.
The program is used to transform point files or grid
files from or to geographical coordinates to plane projection
coordinates (UTM, Stereographic, Mercator, Lambert Conform
Conical, Danish S34). It is only realeased after permission
from KMS.
5.5. ORBIT8.
The program generates a file of points along a satellite
orbit computed using the C20 term. May be used when simulating
a satellite mission.
5.6. JULIA.
The program computes the Julian date, the modified Julian
date and the ERS-1 time or the reverse (find date and time).
6. Data files.
--------------
Generally the package will include one set of potential
coefficients, namely the GPM2 set. We recommend the use of the
OSU91A field, but we will only distribute this if the permis-
sion has been obtained from R.H.Rapp, OSU.
The terrain and gravity files used to test TC are dis-
tributed with the permission of NGS for scientific purposes.
Adress and litterature list files are only distributed
after special request.
7. References.
--------------
Arabelos, D., P.Knudsen and C.C.Tscherning: Covariance and bias
treatment when combining gravimetry, altimeter and gradiometer
data by collocation. Proceedings of the IAG Symposia, pp.
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Forsberg, R.: Local Covariance Functions and Density Distribu-
tions. Reports of the Department of Geodetic Science and
Surveying No. 356, The Ohio State University, Columbus, 1984.
Forsberg, R.: A study of terrain reductions, density anomalies
and geophysical inversion methods in gravity field modelling.
Rep. of the Dep. of Geodetic Science and Surveying, the Ohio
State Univ., Columbus, 1984a.
Forsberg, R.: Gravity Field Terrain Effect Computations by
FFT.Bulletin Geodesique, Vol. 59, pp. 342-360, 1985.
Forsberg, R.: Spectral Properties of the Gravity Field in the
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Forsberg, R.: A new covariance model for inertial gravimetry
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Forsberg, R.: NKG Nordic Standard Geoid 1989. Proc. 11th
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of Geosat ERM and Seasat Altimeter Data in the Mediterranean Sea.
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