Direct topographical effect
Abstract
Before the computation of short-wavelength and long-wavelength components of the geoid undulations from terrestrial data and the two latest satellite missions, i.e. gravity (GRACE mission) and gradiometry (GOCE mission) measurements, the terrain corrections must be determined. Since the corrections enter the first of the three steps of the Remove-Compute-Restore (RCR) procedure for applying Stokes’s integral, this study focuses on determining these corrections. Formulation of the effects introduced and the effects are computed over high elevated topography in Ireland using Helmert’s second condensation method. Finally, the effects of topography on geoid height determinations are presented.
Keywords
- gravity
- gradiometry
- terrain correction
- Remove-Compute-Restore procedure (RCR)
- topographical effects
1. Introduction
The geoid is defined as an equipotential surface along which the Earth’s gravity potential (
The local gravity potential,
1.1 The Remove-Compute-Restore procedure (RCR)
The RCR procedure is a method that fulfils Stokes’ requirements2 for computing the geoid from:
Terrestrial gravity measurements or,
Satellites gradiometry3 measurements (e.g. GRACE, GOCE).
A gravimeter measures magnitude of gravity,
whereas the gradiometer measures the components of the gradient of gravity. We will focus on the
1.1.1 Removing the gravitational effect of the residual topographical masses
Subtracting the gravitational effect of the residual topographical masses,
The gravity attraction of the residual topographical masses is then
at the point of the gravity measurements, and
at point of gradiometric measurements.
To make a potential harmonic in a space above the geoid, these effects have to be calculated and removed from the observations:
1.1.2 Computation of the residual geoid (co-geoid)
The first step of the computation is continuing
Assuming that the data have been continued down to the geoid, the two basic requirements are met and the computation of the geoidal heights can now be carried out. However, the computation gives the height
where
Likewise, the heights of the co-geoid can be determined from gradiometric data as
where the kernel
1.1.3 Adding the contribution of the topography to the solution
Finally, the heights of the geoid will be obtained by adding the difference
To find the expression for
where
Applying Bruns’ formula to the undulation of the co-geoid,
where
and
GRACE and GOCE are the two latest satellite missions for precise, long-wavelength geoid determination from gravity and gradiometry measurements, respectively. Prior processing GRACE and GOCE data, the terrain corrections have to be determined.
The RCR procedure for gravity and gravimetry is summarized in Figure 2.
It shows the motivation for correcting the data by the effects of the topographical masses. These are the direct topographical effects
In the intervening period of the theory of physical geodesy, a wide range of methods have been developed, e.g. Airy-isostatic reduction method [4], Residual Terrain Model (RTM) scheme [5], Helmert’s first and second method of condensation [4, 6, 7, 8]. The choice of each method is area-dependent, and the correlation between the spatial resolution of DEMs, and the elevation of computation points or data surrounding the computation points show the most suitable spatial resolution of DEMs that provide intended geodetic accuracies. This is investigated numerically over high elevated topography in Ireland using Helmert’s second condensation method.
The next section is devoted to express them mathematically and prepare them for numerical realization.
2. Topographical effects
The term
2.1 Topographical masses and the Bouguer plate
2.1.1 Topographical masses
The topographical masses are the masses outside the geoid and below the topographical surface. The gravitational potential
where G is the Newton’s gravitational constant,
The argument notation in
To abbreviate notations, we introduced the symbol
Assuming that the density of the topographical masses does not vary in radial direction, that is
2.1.2 Bouguer plate
The Bouguer plate, used as an approximate model in gravity and gravity anomaly computations accounts for the bulk of topographical effects.
In Cartesian geometry Figure 4a, the topography around the gravity station
In spherical geometry Figure 4b, the Bouguer plate is regarded as a spherical layer of thickness
For evaluating this integral, the geoid is approximated by a sphere of radius
For
2.1.3 Terrain roughness
Since the actual Earth’s surface deviates from the Bouguer sphere, there are deficiencies and abundances of topographical masses with respect to the mass of the Bouguer plate Figure 5. These contribute to the topographical potential
The
2.2 Compensated masses and the Helmert condensation layer
2.2.1 Compensation of the gravitational effects of topographical masses
The equipotential surfaces of
As the masses are compensated in some way [2], we can introduce the
Two extremely idealized
The
The
2.2.2 Helmert condensation layer
In the limiting case, the topographical masses may be compensated by a thin mass layer located on the geoid (somewhat like a glass sphere made over very thin but very robust glass [2]). As shown Figure 7, the topographical masses are condensed as a surface mass layer on the geoid. This kind of compensation is called
where
where
The symbol
that may be evaluated analytically by
The condensation density
and substituting Eqs.(23) and (31) into (32), we find
with
Thus, Eq. (29) will be written
2.3 Indirect topographical effect
2.3.1 Indirect topographical effect on potential
The primary indirect topographical effect is the residual potential
where the Bouguer term and the terrain roughness term are respectively given by
The subtraction of Eqs. (23) and (31) at
and the subtraction of Eqs. (25) and (35) gives
Therefore, substituting Eqs. (39) and (40) in Eq. (36), we obtain the expression of the primary topographical indirect effect on potential:
The unit of this effect is
2.3.2 Indirect topographical effect on the geoid
To correct the geoidal heights
which is the primary indirect effect on the geoid. Where the normal gravity can be taken as the mean value of the gravity of the Earth
2.3.3 The secondary indirect topographical effect (SITE) on gravity
This effect is expressed by means of the Primary Indirect Topographical Effect (PITE) on gravity,
The unit of this effect is mGal.
Notice that the radial derivative of the Newton surface and volume integrals, which is required for computing
and using the derivation of
where substituting the expression (45) in (44), the analytical expression is given by,
2.4 Direct topographical effect (DTE)
2.4.1 Direct topographical effect on surface gravity
Differentiating the residual topographical potential
Substituting for
where
are the radial components of the gravitational attraction induced by the topographical and compensated masses a the point on the Earth’s surface, respectively. Considering Eq. (24) for
where
Let us start with the determination of the Bouguer term
where
Subsituting the radial derivative of Eq. (23) at
similarly, the radial derivative of Eq. (31) at
Using Eqs. (33)–(34) for
where
Substituting the radial derivations of Eq. (25) for
2.4.2 Direct topographical effect on satellite gravity
Differentiating
Analogically to the direct topographical effect on surface gravity Eq. (57), where the radius of the computation point is
where
and
2.4.3 Direct topographical effect on gradiometry
Differentiating
Substituting for the residual topographical potential
where
are the
where
Let us start with the determination of the Bouguer term
where
Taking the radial second derivative of Eq. (23) at
the radial second derivative of Eq. (31) at
and using Eqs. (33)–(34) for
where
Therefore, differentiating
Here
2.5 Computations of the topographical effects
2.5.1 The integral Newton kernels for numerical computation
The distance
Thus, we can write the Newton kernel
To solve this integral, we use the following equations Gradshteyn and Ryzhik [12]
for
Replacing Eq. (75) and “our” notations in Eq. (74), we obtain the analytical expression:
Hence, the analytical expression
where
2.5.2 First radial derivative of the Newton kernel
The radial derivative of the Newton kernel gives
By Eq. (18), since
we readily get
By substituting the expression (79) in (78), and integrating the equation, we obtain the analytical expression
The computation of
2.5.3 Second radial derivative of the Newton kernel
Expression of
where
To obtain the second derivative of
We finally obtain:
Including Eq. (84) in Eq. (81), the second derivative of
After deriving a second time the reciprocal distance
where
The computation of
3. Numerical studies
The determination of topographical effects from DEMs is a very time-consuming process, particularly when computations are required for large areas, such as a country or a continent. With a fine grid resolution, for instance, 50 m Quadratic Grid Resolutions (QGR) a unified spatial data structure, computations are beyond what a multi-processor computer can accomplish within a reasonable time-frame for an area such as Ireland. Numerical investigations have resulted that increasing the spatial resolution of DEM by a factor of two increases CPU computational time by a factor of fourteen.
Although the computational time is a factor to be taken into account, it is less important since it is the spatial resolution of DEM which is critical for improving accuracy required when a precise geoid is determined.
The Fast Fourier Transform (FFT), which relies on linearization and series expansions of the non-linear terrain effect integrals, provides a reduction in computational time by several orders of magnitude, compared to space domain integration methods [13]. In the FFT, the higher-order terms of the radially integrated Newton kernel expressed by the Taylor series expansion are neglected. In addition, the reciprocal distance 1/
3.1 The study area
Topographical effects in high elevated topography in Ireland are computed from 50 m QGR, Ireland with a relatively flat terrain surrounded by the ocean. The study area is
3.2 The bound of integration area
The gravitational potential of Topographic Masses of finite thicknesses behaves like the potential of a thin layer when it is observed from a larger distance. This is explained by the behavior of integration kernels generating the potential of the gravitational potential
generating the potential of Helmert’s condensation layer
generating the gravitational potential
generating the potential of Helmert’s condensation layer
generating the gravitational potential
The choice of varying angular distance and fixing the elevation of integration point to 1039 m and computation point to 1 m or vice versa, enables us to determine the most attainable differences (maximum or minimum) between the kernels in question.
Furthermore, a numerical examination of
Eqs. (36) and (57) assume the compensation is strictly local, which means
3.3 Direct topographical effect on gravity
3.3.1 On the Earth’s surface - residual effect δ A surf
The computation of the residual direct topographical effect on gravity on the Earth’s surface
Min | Mean | Max | rms | |
---|---|---|---|---|
Topographic Heights (m) | 0 | 58.600 | 920.000 | |
−17.4 | 0.006 | 5.7 | ||
−27.5 | −0.6 | 2.6 |
Due to the low elevation of topography in the test area (maximum 920 m),
Since the residual effect
where
3.3.2 At satellite altitudes—residual effect δ A sat
The computation of the residual direct topographical effect on satellite gravity
The residual effect on gravity
Min | Mean | Max | rms | |
---|---|---|---|---|
−3.1 | −0.002 | 138.9 | ||
−0.87 | −0.74 | −0.51 |
3.4 Direct topographical effect on gradiometry—residual effect δE
The computation for the direct topographical effect on gradiometry
Min | Mean | Max | rms | |
---|---|---|---|---|
−2.13 | −0.00001 | 0.7 | ||
0.01 | 0.7 | 0.11 |
The total direct topographical effect
where
3.5 Topographical effect on geoid heights
3.5.1 The effect of PITE on geoid heights
The primary indirect topographical effects on the geoid height determination are computed by Eq. (42),
Figure 12b illustrates the effect of PITE in a high elevated topography at 50 m grid resolution.
3.5.2 The effect of DTE on geoid heights
The direct topographical effects on geoid heights,
where
4. Conclusions
The RCR procedure for gravity and gravimetry is summarized in Figure 2. It shows the motivation for correcting the data by the effects of the topographical masses. These are the direct or indirect topographical effects computed at surface or satellite altitude. This study summarized the field corrections for the determinations of geoids by terrestrial data or the latest satellite missions. The integral of newton presented in Sec:2.1.1 in the form of the gravitational potential of topography Eq. (16), proves necessary to the expression of topographical effects. It made it possible to understand the observations made either
Presentation of the
In order to express the effects, we need to know the topographic masses defined by their height and density. We have seen that Newton’s integral for determining the geoid allows expressing the effects. The topography effects on the satellite measurements are calculated in a way similar to the effects on ground measurements.
The effects of topographic masses have much less impact on gravity measurements at the satellite level than on ground measurements.
Computing topographical effects for large areas is a very time-consuming process. Increasing the resolution of sampled DEM by a factor of 2 (e.g. from 100 m to 50 m quadrangle) increases the number of data by a factor of 4, and it increases the computational time by a factor of approximately 14. Thus, it is suggested to restrict the integration area to a small area of radius
A sparse grid size, particularly in rugged areas, is not sufficient to express the irregularities of the terrain and thus does not reveal properly the contribution to geoidal height due to terrain height variations.
With a tiny grid step size, the magnitude of the Bouguer component becomes comparable with that of the terrain roughness component, which reduces the correlation between DTE and PITE. Since DTE does not contain a Bouguer component, the correlation between DTE and DEM is generally smaller than that for PITE.
Numerical investigation shows that the Bouguer components of PITE have a larger contribution to topographical effects than the terrain roughness components.
Numerical examination of Kernel’s, controlled by varying topographic height at a fixed angular distance, in the immediate neighborhood of the computation point, shows that the larger the height of the integration point, the more significant the difference between these kernels, and therefore, the topographical effects are more substantial.
Comparing the results with different grid sizes shows (not shown here) an improvement in computation accuracy. Contrary to our expectations, it is not the case for calculations at satellite altitudes, so griding can be reduced, and a more refined grid does not change a long-wavelength feature of
Acknowledgments
The authors would like to thank Dr. Pat Gill (TUS) and Dr. Nic Wilson (UCC) for their comments that have helped to improve the manuscript.
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Notes
- Gravimetry is the method of measuring gravity and the instrument used is called a gravimeter. In the past, gravity data were exclusively provided by terrestrial surveys. Later, transportable relative gravimeters were designed for the use on ship and airborne, however, the data accuracy was very variable and geographically unevenly distributed. Recently, satellites gravimetry has emerged providing the global coverage of repeated measurements. The unit of gravity is the Galileo, 1 mGal=10−5m.s−2.
- No masses outside the geoid, and the measurements are referred to the geoid.
- Gravity gradiometry is the study of variations in the acceleration due to gravity. It is the measurement of the rate of change of gravitational acceleration called gravity gradient is the spatial. The unit of gradient is the Eötvös, 1E°=10−9m/s2/m.
- Note that the same formula applies to air-borne gravity measurements, replacing the radius of computation by r=R+Hplane where Hplane is the flying altitude of the plane performing gravity measurement.
- Unit of gravity: 1 mGal=10−5m.s−2.
- unit of gradiometry: 1E∘=10−9m/s2/m.