**Table 1.**

*The statistics of the DTM data in AREA\_1.*

close to the DTM boundaries is not computed according to the standard. However, in the relative comparison between methods, this should not affect the results.

The heights of these 121 prediction points have been assumed coincident with those of the SRTM3 DTM and their statistics are listed in **Table 2**.

Given the geometry of tesseroids and flat tesseroids, SRTM3 orthometric heights were transformed into ellipsoidal heights via the EGM96 geoid undulation [15]. Based on the ellipsoidal coordinates *φellipsoidal*, *λ*, *h* of both DTM and prediction points were converted into spherical coordinates ð Þ *φ*, *λ*,*r* and used in the terrain effect computation with tesseroids and flat tesseroids.

The statistics of the differences among the three methods are given in **Table 3**.

As a first overall comment, it can be stated that the results are in good agreement even in such a rough mountain area with sharp height variations (see **Tables 1** and **2**). By inspecting in more detail the statistics, one can see that values computed by the TC-GRAVSOFT and the UNIPOL approaches are in better agreement than those computed by the TC-GRAVSOFT and the Flat Tesseroid (FT) approaches. The mean of the differences in the first case is nearly 60% of that of the second comparison and the standard deviation is the 86%. This is quite an unexpected result as the geometry of the FT tesseroid is closer to that of the TC prism than to the geometry of the UNIPOL tesseroid. Further investigations will be performed in order to understand this behavior.

On the other hands, the terrain correction values based on the UNIPOL tesseroid procedure and the ones obtained with the FT approach agree very well.

The mean and the standard deviation of the differences between these terrain corrections are of the order of some hundredth of mGal and the maximum difference is of the order of one third of mGal. Although in principle this is quite foreseen, it is important to quantify the differences in view of practical applications.

Larger differences can be seen when comparing these two methods with the one based on prism effect. In these cases, the maximum differences are of the order of 1


**Table 2.**

*The statistics of the heights of the computation points in AREA\_1.*


#### **Table 3.**

*The statistics of the TC computations in AREA\_1.*

*The Gravity Effect of Topography: A Comparison among Three Different Methods DOI: http://dx.doi.org/10.5772/intechopen.97718*

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.

A second test was then devised. Observed gravity data were selected in the area (AREA\_2):

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 **Table 5**.

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 (AREA\_2)

$$4\mathbb{S}^{\circ} \le \mathfrak{q} \le 4\mathbb{S}^{\circ} \qquad \qquad 10^{\circ} \le \lambda \le 13^{\circ}$$

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

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

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


**Table 4.**

close to the DTM boundaries is not computed according to the standard. However, in the relative comparison between methods, this should not affect the results. The heights of these 121 prediction points have been assumed coincident with

**Number of points μ [m] σ [m] Min [m] Max [m]** 1442401 1502.0 644.1 160.0 3460.0

Given the geometry of tesseroids and flat tesseroids, SRTM3 orthometric heights were transformed into ellipsoidal heights via the EGM96 geoid undulation [15].

The statistics of the differences among the three methods are given in **Table 3**. As a first overall comment, it can be stated that the results are in good agreement

On the other hands, the terrain correction values based on the UNIPOL tesseroid

The mean and the standard deviation of the differences between these terrain corrections are of the order of some hundredth of mGal and the maximum difference is of the order of one third of mGal. Although in principle this is quite

foreseen, it is important to quantify the differences in view of practical applications. Larger differences can be seen when comparing these two methods with the one based on prism effect. In these cases, the maximum differences are of the order of 1

**Number of points μ [m] σ [m] Min [m] Max [m]** 121 1471.3 611.7 212.0 2577.0

TC (GRAVSOFT) 13.686 5.194 5.185 35.526 TESSEROID (UNIPOL) 13.553 5.111 5.313 35.065 FLAT TESSEROID (FT) 13.472 5.077 5.275 34.883 TC (GRAVSOFT) - UNIPOL 0.133 0.175 �0.246 0.782 TC (GRAVSOFT) - FT 0.214 0.203 �0.156 1.050 UNIPOL - FT 0.081 0.056 0.006 0.306

**μ [mGal] σ [mGal] Min [mGal] Max [mGal]**

procedure and the ones obtained with the FT approach agree very well.

points were converted into spherical coordinates ð Þ *φ*, *λ*,*r* and used in the terrain

even in such a rough mountain area with sharp height variations (see **Tables 1** and **2**). By inspecting in more detail the statistics, one can see that values computed by the TC-GRAVSOFT and the UNIPOL approaches are in better agreement than those computed by the TC-GRAVSOFT and the Flat Tesseroid (FT) approaches. The mean of the differences in the first case is nearly 60% of that of the second comparison and the standard deviation is the 86%. This is quite an unexpected result as the geometry of the FT tesseroid is closer to that of the TC prism than to the geometry of the UNIPOL tesseroid. Further investigations will be performed in

of both DTM and prediction

those of the SRTM3 DTM and their statistics are listed in **Table 2**.

Based on the ellipsoidal coordinates *φellipsoidal*, *λ*, *h*

*The statistics of the heights of the computation points in AREA\_1.*

*The statistics of the TC computations in AREA\_1.*

order to understand this behavior.

*The statistics of the DTM data in AREA\_1.*

**Table 1.**

**Table 2.**

**Table 3.**

**52**

effect computation with tesseroids and flat tesseroids.

*Geodetic Sciences - Theory, Applications and Recent Developments*

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


#### **Table 5.**

*The statistics of the observed gravity values.*


#### **Table 6.**

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

**Figure 7.** *The DTM (AREA\_2) and the points for TC computation.*

the EGM96 geoid undulations when computing the terrain correction with the TC software of the GRAVSOFT package.

Given the three different terrain corrections, by applying Eq. (4), three different sets of Bouguer anomalies have been derived.

In Eq. (4), as previously stated, the orthometric height *H* is derived via the EGM96 geoid undulation and the Bouguer plate is accounted for by using Eq. (1). We further assumed that

$$\frac{\partial \chi}{\partial h} = -0.30877 \, m \text{Gal}/m \tag{14}$$

the maximum values (around 2.6 mGal) is not so significant if compared with other

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

Thus, the statistics of **Tables 3** and **7** prove the substantial equivalence of the

Three different methods for terrain correction have been compared in two areas over the Alps. The standard computation given by the TC-GRAVSOFT program has been compared with the terrain corrections evaluated via spherical tesseroid and flat tesseroid formulas. In the first test, the SRTM DTM was clipped in a 1° 1° window and *TC* effect was computed in a set of gridded points in the same area. In the second test, observed gravity values in a 1° 1° area have been used in the computation of Bouguer anomalies considering the 3° 3° SRTM DTM values centered on the area containing the gravity data. Despite the fact that the topography in the two selected DTM windows is quite rough, no significant differences among the methods have been revealed. The statistics of the values obtained by modeling in different ways the shape of the discretized topography elements are practically equivalent. Differences among *TC* effects and Bouguer anomalies computed with parallelepiped, spherical tesseroid and flat tesseroid amount to maximum values that are around 1 and 3 mGal respectively. As a matter of fact, there are other error sources (e.g., density heterogeneities, DTM and gravity point heights mismatch) that can have impacts on the terrain correction computation larger than 3 mGal. However, if in the second test on Bouguer computation we consider the values *per se*, spherical tesseroid and flat tesseroid models perform slightly better when RMS values are compared, i.e. the spherical tesseroid and flat tesseroid based

Finally, we remark that the concept applied in the flat tesseroid modeling can be adapted to the terrain effect computation when shaping the topography according to the Triangulated Irregular Network model [17]. In this way, a more detailed terrain effect evaluation will be possible, particularly in the neighbor of the com-

We thank DTU Space for providing us with the TC program of the GRAVSOFT

biases occurring in the Bouguer reduction.

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

Bouguer anomalies are smoother.

**Acknowledgements**

**Conflict of interest**

package.

**55**

putational points, by better modeling the terrain slopes.

The authors declare no conflict of interest.

**4. Conclusions**

three approaches used for the *TC* computation.

and the normal gravity in a point *Q* of latitude *φ<sup>Q</sup>* on the ellipsoid is given by the GRS80 normal gravity formula [16]:

$$\gamma(Q) = 978032.7 \left( 1 + 0.0053024 \sin^2 \varphi\_Q - 0.0000058 \sin^2 2\varphi\_Q \right) mGal \tag{15}$$

Although this formula has an accuracy of 0.1 mGal (see [16]), it can be used in the context of this relative comparison among different terrain correction computation methods.

**Table 7** summarizes the statistics of the Bouguer anomalies obtained with the three terrain correction methods.

Comments similar to those given on **Table 3** hold for the Bouguer values in **Table 7**. The Bouguer anomalies obtained by applying the three methods have quite similar statistics. Those computed via TC-GRAVSOFT software have the smallest standard deviation and the highest mean while those obtained with the other two methods have smaller mean and higher standard deviations. If the RMSs are considered, one can see that the Bouguer anomalies based on the flat tesseroid have the smallest value. However, as pointed out before, even the largest difference among


**Table 7.** *The statistics of the Bouguer anomalies.*
