**3. The numerical tests on the three proposed approaches**

The three different mathematical models presented in Section 2 have been applied in two *TC* computation tests. Both tests have been carried out in the Alpine area.

In the first test, the SRTM3 DTM (see [14]) have been selected in the area (called AREA\_1):

$$4\mathfrak{G}^{\circ} \le \mathfrak{q} \le 4\mathcal{T}^{\circ} \qquad \mathbf{11}^{\circ} \le \lambda \le \mathbf{12}^{\circ}$$

The statistics of the height data in this area (see **Figure 6**) are given in **Table 1**. Points for the *TC* computation have been selected on a 3<sup>0</sup> � 3<sup>0</sup> regular grid in the inner area:

$$4\text{€}.25^{\circ} \le \mathfrak{q} \le 4\text{€}.75^{\circ} \qquad \text{11.25}^{\circ} \le \lambda \le \text{11.75}^{\circ}$$

The computation points are thus in an inner area which have 0.25° from the extents of the outer one containing the DTM. So, the terrain correction in points

**Figure 6.** *The DTM (AREA\_1) and the points for TC computation.*

*H φ <sup>j</sup>* þ *Δφ <sup>j</sup>*, *λ <sup>j</sup>*,*rj* 

DTM point *Q <sup>j</sup>*

built.

**Figure 5.**

*point* Pi *.*

**50**

. Note that the two planar surfaces identified by the closed poly-

.

. **Figure 5** illustrates in graphical form how the single polyhedron is

gons *ABCD* and *EFGH* have their vertices whose radiuses depend in the first case on the radius of the computation point *Pi* while in the second case on the radius of the

*Geodetic Sciences - Theory, Applications and Recent Developments*

This procedure defines the spherical coordinates of the vertices of the polyhedron that are at the height of the terrain and at the height of the computation point, i.e. the level at which the Bouguer plate is computed. The computation of the firstorder derivative along the direction of *ri* of the gravitational effect of such polyhedron corresponds to the terrain correction to be applied at *Pi* as contribute of *Q <sup>j</sup>*

Such value is obtained by running the code *polyhedron.f* made available by the author [4] implementing the linear integral approach. As input, the relative cartesian coordinates of the polyhedron vertices with respect to the computation point and their topological relationships are required. These are obtained applying a change of reference system to the polyhedron vertices. In particular, their coordinates were roto-translated into a local reference system having the origin at the computation point *Pi*, the *z* axis pointing up along the direction of *ri* and the *x* and *y* axes parallel to the local East and North directions, respectively. Regarding the topological relationships defining the outer normal direction of the six planes of the polyhedron, they are defined by a topology matrix containing the counterclockwise sequence of the vertices as seen from outside. As output, the absolute value of the computed *Vr* is taken. This procedure contemplates two nested loops over all the *m* DTM points *Q <sup>j</sup>* and the *n* computation points *Pi*. Within the loop over the computation points, different local reference systems are defined. This leads to slight changes in the directions of the *x* and *y* axes but not on the *z* axis, always normal and pointing outside the reference sphere defined on *Pi*, then maintaining

*Sketch of the polyhedron vertices building procedure on the basis of the DTM point* Qj *and the computation*


mGal. Even though this value is high if compared with the precision of the gravity observations (which can reach few μGals), one has to consider that other error sources in the topography reduction process can have a larger impact. As an example, the discrepancy between the heights of the point associated with the gravity observations as compared with those obtained by the DTM in the same points can amount to ten meters (or even more) in mountain areas. Given that the absolute value of the free-air gradient is 0*:*30877 mGal*=*m, this implies 3 mGal in 10 m due to this mismatch. Also, biases can occur due to the assumption of constant density. In view of that, even the maximum difference between the GRAVSOFT terrain correction and the spherical tesseroid/flat tesseroid values are not so significant.

*The Gravity Effect of Topography: A Comparison among Three Different Methods*

*DOI: http://dx.doi.org/10.5772/intechopen.97718*

A second test was then devised. Observed gravity data were selected in the area

46° ≤φ≤47° 11° ≤λ≤12°

Gravity point coordinates were surveyed with GNSS and framed to ITRF94. Statistics of the ellipsoidal heights of these gravity points are listed in **Table 4**. Gravity values have been measured with a Lacoste&Romberg G-367 relative gravimeter. The standard deviation of the observed values is of the order of 0.02 mGal. Gravity data are referred to IGSN71 and their statistics are summarized in

For the computation of the terrain component, the SRTM3 DTM have been selected in the 3° 3° area centered on the one containing the gravity data area

45° ≤φ≤48° 10°≤λ≤ 13°

**Figure 7** shows the DTM features of AREA\_2 and the position of the gravity

**Number of points μ [m] σ [m] Min [m] Max [m]** 116 1161.74 384.22 312.67 2217.08

**Number of points μ [mGal] σ [mGal] Min [mGal] Max [mGal]** 116 980404.459 71.605 980208.060 980545.806

**Number of points μ [m] σ [m] Min [m] Max [m]** 12967201 1054.4 853.2 35.0 3865.0

Similarly to what has been done in the first test, SRTM3 and gravity point coordinates were transformed into spherical coordinates for the computation of the terrain correction with the UNIPOL and FT approaches. On the other hands, ellipsoidal heights of gravity points have been converted into orthometric heights via

The statistics of the SRTM3 in AREA\_2 are described in **Table 6**.

*The statistics of the heights of the computation points in AREA\_2.*

*The statistics of the observed gravity values.*

*The statistics of the DTM data in AREA\_2.*

(AREA\_2):

**Table 5**.

(AREA\_2)

points.

**Table 4.**

**Table 5.**

**Table 6.**

**53**
