Challenges in defining of Bouguer gravity anomaly

Generally, gravity anomaly is the difference between the observed acceleration of Earth's gravity and a normal value. Topography (all masses above geoid) plays a main role in definition of the gravity anomaly. Based on modeling of the effect of topography, there are different models of gravity anomaly such as free-air and Bouguer anomaly. The main goal of the Bouguer anomaly is removing of gravitational effect of all masses above the geoid (topography and atmosphere). This anomaly is widely used in exploration geophysics. In geodetic applications, in the absence of topography, Bouguer gravity anomaly is smooth and thus more suitable for interpolation and even stable downward continuation.
  In the other hand, gravity anomaly is the difference between real gravity at a point and normal gravity in corresponding point where the real and normal potentials in both points are the same. In geodesy, the gravity disturbance is defined as the difference between the real gravity observed at a point and normal gravity at the same point. In many geophysics literatures, gravity anomaly is replaced by gravity disturbance together a corrective term called geophysical indirect effect. This correction is computed by application of the free-air (and usually the Bouguer) correction over the geoid–ellipsoid separation. This correction must be computed by application of only free air correction to separation of the real equipotential surface and its equivalence in normal gravity field at gravity observation.
The free-air (FA) correction is used to up/downward continuation of normal gravity anomaly. In practice, only linear approximation, 0.3086 mGal/m, is used while a second-order FA correction is more realistic than the linear approximation.  Note that the FA correction is not a reduction formula for downward continuation of gravity anomaly.
One of the most ambiguities in definition of Bouguer effect gravity anomaly arises from formulating the effect of topography. The gravitational of topography can be split into Bouguer term, which is the dominant term, plus minor effect, terrain roughness. In the evaluation of a topographical effect, planar or spherical models of topography can be used. Many studies have shown that planar and spherical model of topography give very different results for Bouguer anomalies. Also, it was shown that the planar topography model (in form of infinite Bouguer plate) yields to a mathematically and physically meaningless quantity. To compute the terrain correction in geophysics, the gravitational effect of only masses up to about distance 167 km (Hayford zone) is considered. In principle the domain of computation of the topographical effect is the whole of the Earth. Despite the fact that the gravitational effect decreases with distance, the effect of beyond Hayford zone is large and should be considered. 
The removal of the topographical masses disturbs the isostasic equilibrium of the crust. As a result, the equipotential surface can be moved up to several hundred meters. The indirect topographic effect is defined as the effect on gravity due to removing the topographical masses. The indirect effect of topography (ITE) in Bouguer gravity anomaly was first introduced by Vanicek, et al (2004). Their computations show that the numerical values of ITE can be reached up to 150 mGal in mountainous area. While, in most studies, ITE does not take into account and only direct topographical effect is considered.
In analogy with topographical effect, in the computation of Bouguer gravity anomaly, the direct and indirect effects of atmospheric masses should be considered. Usually the gravity effect of the atmosphere is evaluated by IAG formula. This formula considers only the direct topographical effect as the correction to gravity anomaly. The indirect atmospherical effect is not discussed in this context. In this study, the method proposed by Sjoberg (2000) is recommended and applied.
In order to investigate differences between classic and new Bouguer gravity anomalies, numerical calculations were performed in a mountainous area bounded by ,  where there are 2385 land gravity observation. The classic planar Bouguer anomalies were computed from
 
where g and  are observed and normal gravity, H is the orthometric height of point and    is the terrain correction computed up to Hayford zone. The new spherical Bouguer anomalies were computed from 
 
where FA is second-order free-air correction, DTE is the direct topographical effect (spherical shell + terrain roughness), ITE is the indirect topographical effect, DAE is the direct atmospherical effect and, IAE is the indirect Atmospherical effect. The results indicate that there are large differences (over 100 mGal) between classical and new Bouguer anomalies. The new Bouguer anomalies are less correlated with terrain heights. Therefore the planar model cannot completely remove the gravitational effect of topography.