# A New Model of Quasigeoid for the Baltic Sea Area
## 2 Materials and Methods  
In the classical approximation, the geoidal undulation can be estimated by [PERSON] formula  
\[N=\frac{R}{4\pi\gamma}\iint_{\sigma}\Delta gS(\psi)d\sigma \tag{1}\]  
where \(R\) is the Earth's radius, \(\Delta g\) is the gravity anomaly, \(\gamma\); is normal gravity, and \(S(\psi)\) is Stokes' function, provided the topography has in some way been removed or shifted inside the geoid. Gravity is usually measured on the Earth's surface. To calculate the geoid using [PERSON] formula, the gravity (or gravity anomaly) must refer to the geoid and not to the Earth's surface. The gravity must be reduced to the geoid. This may be done using one of several available methods. In [16], reduction methods such as the Bouguer, isostatic, Rudzki, and Helmert methods are mentioned. Additionally, the gravity may be reduced by the residual terrain model (RTM) method [17]. Thus, the gravity anomaly in Equation (1) has to be reduced to the geoid by applying free-air reduction, assuming there are no topographic masses  
\[\Delta g=g+\delta g_{F}-\gamma_{0} \tag{2}\]  
These masses are then placed on the surface of the geoid in the form of a layer with a density \(\kappa=\rho\times\mathrm{H}\). This method of placing topographic masses on a geoid is called the Helmert condensation method. The displacement of the topographic masses changes the gravitational field of the Earth, including the potential of the geoid (Figure 1). The level surface that possesses the geoid potential after displacement is called the cogeoid: therefore, the equipotential surface computed from Stoke's integral is the cogeoid instead of the geoid. The vertical distance between the geoid and the cogeoid \(N_{ind}\) can be computed according to [18] from  
\[N_{ind}\approx-\frac{\pi G\rho}{\gamma_{m}}H_{P}^{2} \tag{3}\]  
which converts the cogeoid into a geoid. To compute the cogeoid, additional reduction of the gravity value from the geoid to cogeoid is necessary (secondary indirect effect) and here, the free-air reduction is sufficient  
\[\Delta g_{ind}\approx\frac{2\pi G\rho}{R}H^{2} \tag{4}\]  
This term is usually much smaller than 1 mGal and is often neglected. To obtain the quasigeoid the geoid undulation, \(N\) minus height anomaly \(\zeta\) is needed. This separation is given in [16] as  
\[N-\zeta\approx=\frac{\Delta g_{B}}{\gamma_{m}}H \tag{5}\]  
where \(\Delta g_{B}\) is the Bouguer anomaly, and \(\gamma_{m}\) is the mean normal gravity (9.81 ms\({}^{-2}\)). To get the true value of the geoidal undulation, the integration should be extended to the whole sphere. For local or regional geoid computation, usually, a global geopotential model (GM) is combined with discrete local gravity data as well as height data. The geopotential model provides the low-frequency or long-wavelength part of the geoid. Local gravity anomaly data and height data provide the medium- and high-frequency components of the geoid spectrum, respectively [19]. The formula for the practical computation of the local geoid by the remove-restore technique is  
\[N=N_{GM}+N_{\Delta g_{m}}+N_{ind} \tag{6}\]  
where \(N_{GM}\) is the geoidal undulation implied by the geopotential model, \(N_{\Delta g_{m}}\) is the contribution of the terrain-corrected mean free-air gravity anomalies reduced to the reference field, and \(N_{ind}\) is the indirect effect of the terrain reduction.