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

martina.capponi@polimi.it, daniele.sampietro@g-red.eu

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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