## 2 Theoretical Basis
### Calculation of Selected Functionals from the GGI Model
#### 2.2.3 The Complete Bouguer Anomalies  
The complete Bouguer anomaly is defined as [20; 21]:  
\[\Delta g_{B}=CBA=\Delta g_{FA}-\delta g_{Top}=g_{o}-\gamma_{e}-\delta g_{Top} \tag{10}\]  
where \(\Delta g_{FA}\) is a free-air anomaly on a geoid, \(g_{o}\) is gravity on the geoid, \(\gamma_{e}\) is normal gravity on the ellipsoid, and \(\delta g_{Top}\) is topographic reduction.  
Used in (10), the value \(g_{o}\) is calculated based on gravity at point \(P\) (\(g_{P}\)) by introducing free-air reduction [4] (Figure 2):  
\[g_{o}=gp-\frac{\partial g}{\partial H}H^{o}=g_{P}+\delta g_{FA} \tag{11}\]  
where \(\delta g_{FA}=-\frac{\partial g}{\partial H}H^{o}\) is free-air reduction, \(H^{o}\) is orthometric height and \(\frac{\partial g}{\partial H}\) is vertical gravity gradient.  
For many practical purposes, vertical gravity gradient is approximated by normal gravity gradient \(\frac{\partial\gamma}{\partial H}\). For areas with small differences between the geoid and the quasigeoid, the orthometric height can be replace with the normal height (\(H^{N}\)). Free-air reduction can thus be defined as:  
\[\delta g_{FA}\approx-\frac{\partial\gamma}{\partial H}H^{N} \tag{12}\]  
Topographic reduction (\(\delta g_{Top}\)) is expressed as:  
\[\delta g_{Top}=G\iint\limits_{\mu}\int\limits_{z=0}^{z=H}\frac{(z-H_{P})}{l^{ 3}}\rho dzd\mu \tag{13}\]  
where \(l\) is the distance between the attracting masses and the attracted point \(P\), \(\rho\) is density, and \(\mu\) is integration area.  
The topographic reduction is classically determined based on DEM grids with different resolutions. This resolution is very high near the gravity point and it decreases with distance. Integration of topographic masses is also carried out to a certain, defined distance around a gravity point, often 167 km [22], introducing corrections for further zones only for large-scale and precise geophysical studies [23].  
When calculating the CBA based on the GGI model, we can use the classical procedure that was presented earlier. By putting Equation (11) into Equation (10) and replacing value \(g_{P}\) with (9) and (8), we get:  
\[CBA=g_{P}+\delta g_{FA}-\gamma_{e}-\delta g_{Top}=-\left(\frac{\partial T_{ \Omega}}{\partial H}+\frac{\partial T_{x}}{\partial H}+\frac{\partial T_{E}}{ \partial H}\right)+\gamma_{P}+\delta g_{FA}-\gamma_{e}-\delta g_{Top} \tag{14}\]  
wherein  
\[\gamma_{P}=\gamma_{e}+\frac{\partial\gamma}{\partial H}h=\gamma_{e}+\frac{ \partial\gamma}{\partial H}\left(H^{N}+\frac{T}{\gamma}\right)=\gamma_{e}+ \frac{\partial\gamma}{\partial H}H^{N}+\frac{\partial\gamma}{\partial H}\frac {T}{\gamma}\approx\gamma_{e}-\delta g_{FA}+\frac{\partial\gamma}{\partial H} \frac{T}{\gamma} \tag{15}\]  
where \(h\) is ellipsoidal height.  
In the GGI model (1), topographic masses are represented by masses \(\Omega\), and topographic reduction (\(\delta g_{Top}\)) is approximated by the vertical derivative of the \(T_{\Omega}\) potential:  
\[\delta g_{Top}\approx-\frac{\partial T_{\Omega}}{\partial H} \tag{16}\]  
By putting Equations (16) and (15) into Equation (14) we get the simple expression that defines the complete Bouguer anomaly:  
\[\Delta g_{B}=CBA\approx-(\frac{\partial T_{x}}{\partial H}+\frac{\partial T_{ E}}{\partial H})+\frac{\partial\gamma}{\partial H}\frac{T}{\gamma} \tag{17}\]  
Note that complete Bouguer anomalies calculated by Equation (17), assume approximation (16), in which the integration takes place in relation to the entire volume \(\Omega\) (not to a certain distance around the gravity point) with densities determined from the GGI model. These parameters are different from those described earlier in the classic approach of determining the topographic reduction.