Triangles, squares and rings: computation of terrain correction
close to ground stations
Capponi, M.(1-2); Sampietro, D.(3)
(1) Politecnico di Milano, Italy; (2) Università di Roma La Sapienza, Italy; (3) Geomatics Research & Development srl (GReD), Italy
[email protected], [email protected]
Introduction
The computation of the vertical attraction due to the topographic
masses (terrain correction) is still a matter of study both in
geodetic as well as in geophysical applications.
The increasing resolution of recently developed digital terrain
models, the increasing number of observation points and the
increasing accuracy of gravity data represents nowadays major
issues for the terrain correction computation. Classical methods
such as prism or point masses approximations are indeed too
slow while Fourier based techniques are usually too approximate
for the required accuracy.
In this work we improved the GTE algorithm, an innovative
solution based on a combined FFT-prisms approach expressively
developed for airborne gravimetry, to compute TC also on the
DTM surface, close to the ground stations.
This requires, a part developing a proper adjustment of the FFT
algorithm of GTE software, also to face the problem of the
computation of the gravitational effect due to the actual slope of
the terrain close to the station. Here the latter problem is
discussed by testing different solutions like concentric
cylindrical rings, triangulated polyhedrons or ultra high
resolution squared prisms.
Some tests to prove the performances of the final software to
compute high accurate terrain corrections on ground stations in a
very short time are also shown.
GTE software main features
GTE for ground data TC close to ground stations
Conclusions
In the present work, the GTE software, developed in order to compute the gravitational terrain effect at airborne level in a fast and accurate way, has been improved thus allowing the computation of the TC also at
ground level. The improvement consists in two main issues: the development of the new FFT kernels and the computation of the effect of the DTM slope close to the ground station.
SW Time [s] Std [mGal]
Segmented Rings 0.02 0.1
Squared prisms 0.05 0.1
Triangular polyhedrons 0.001 0.1
• Easy to compute with a close formula
• It requires to manage the combination of right
rectangular prisms with pieces of rings
• Very simple managements of the geometries
• It requires computational power due to the
complexity of the prism equation
• Allow to compute less elementary elements
• Very complex managements of the geometries. It
requires an elaborated processing of the initial
DTM
Segmented rings
Squared prisms
Triangulated polyhedrons
Results in terms of accuracies with respect
to a pure prisms solution to compute the
gravitational effect of a single ground
station (with std 1.5 mGal).
A region with radius 3000 m centered on the
computation point has been modelled by
means of the three algorithms
Numerical tests
The numerical test performed to analyze the software in terms of computational time and accuracy,
consists in computing the TC on the DTM surface of a complex terrain model by means of the
improved GTE software and by simple prisms computation
Statistics and computational time on a grid
Algorithm Time
[s]
Mean
[mGal]
Std
[mGal]
Min
[mGal]
Max
[mGal]
Prisms 12950 53.67 46.56 -7.84 213.14
FFT 222 4x10-4 0.002 -0.004 0.02
Another test performed that consists in computing the TC on a set of
1000 random distributed points shows that the accuracy degrades to
0.2 mGal in terms of standard deviation.
About the computational time the GTE software uses a proper
threshold to discriminate if the direct prisms computation is faster or
slower than the FFT solution
Starting from the Newton integral in planar approximation, the gravitational effect of the DTM is
I2 computed with an exact close formula (Nagy, 1966)
I1 can be split into two parts
IIn
IOut
is the region of points “close” (i.e. ~1 km) to the observation point
is the region of points far away from the observation point with FFT techniques
with prisms
for distances < ~ 500 m the DTM slope is considered too with segmented rings
References
Sampietro, D., Capponi, M., Triglione, D., Mansi, A. H., Marchetti, P., &
Sansò, F. (2016). GTE: a new software for gravitational terrain effect
computation: theory and performances. Pure and Applied Geophysics, 1-19.
Capponi, M., Mansi, A. H., & Sampietro, D. GTE software improvement to
compute terrain correction close to ground stations. Submitted to Geophysical
Prospecting
About the first improvements, the performed test shows that the algorithm is able to compute the TC from a DTM 1001 x 1001
cells on the same grid in less than 5 minutes (the corresponding prism solution takes more than 3.5 hours) with accuracies of the
order of 0.002 mGal (standard deviation). If the correction is required on a set of stations (not coinciding with the DTM cells) the
accuracy degrades to 0.2 mGal in terms of standard deviation for the current test. However it should be remarked that this is
partially due to the extreme DTM used in the test area, in fact in the southern part of region (far away from the mountainous region
of the Italian Alps) the differences with prism solution decrease to less than 0.1 mGal. About the computational time the software
uses a proper threshold to discriminate if the direct prisms computation is faster or slower than the FFT solution.
As for the second problem, i.e. the one related to the effect of the DTM slope, a solution based on the computation of the
gravitational effects of a set of rings sectors has been developed. This solution has been chosen after the analysis of the accuracies,
the computational times and the complexity of the algorithms related to the implementation of three different methods (squared
prisms, triangulated polyhedrons and segmented rings).
Slicing GTE performances
GTE theoretical aspects Multiresolution
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