In the DFHRS concept at its first stage of development, a continuous polynomial surface over of a grid of finite element meshes (FEM) with polynomial parameters p (fig. 2a, thin blue lines) is used as a carrier function for the HRS. The FEM surface of the HRS is therefore called NFEM(pB,L,h), (fig. 2b). For some old height systems H a scaledifference factor Δm has to be considered in addition, so that the DFHRSmodel of N (fig. 2a) consists of two parts. The principle of a GNSSbased height determination H (fig. 2) requires submitting the GNSSheight h to the DFHRS(B,L,h)correction N, reading:
H=hN=hDFHRS(\mathbf{p},\Delta mB,L,h)=h(NFEM(\mathbf{p}B,L)+\Delta m\cdot h)

(1) 
The DFHRS height N = DFHRS(B,L,h) is provided by means of a database (DFHRS_DB), which contains the HRS parameters (p, Δm) together with the meshdesign (fig. 2a) information. DFHRS_DB have become an industrial and user standard for all GNSSreceiver types worldwide, and can also be used for the setup of new RTCMtransformation messages. In the 1st development stage of the DFHRS approach, geoid or geopotential model (GPM)based nonfitted Geoid or QGeoid/geoid heights N, observed astronomical or GPMbased deflections of the vertical (x,h) in any number of groups, and fitting points (B,L,hH), fig. 2a) were exclusively used as observation groups in a common least squares computation for evaluation of the DFHRS_DB parameters p and Δm. The mathematical model for these observations is given by formulas (2a–f) below. In the case of an adequate stochastic model, the use of gravitybased geoid or quasigeoidmodels N (2b) computed from Stokesbased models or from GPMs, together with the covariance information, is equivalent to the use of the original observed gravity values g. The mathematical model for computation of the DFHRS_DB parameters (p, Δm) using the above socalled geometrical part of observation components reads:
Functional Model  Observation Types and Stochastic Models  
\begin{array}{l}
h+v=H\text{ } +\text{ } h\cdot \Delta m\text{ } +\text{ NFEM(} \mathbf{p}\text{x,y),} \\
\text{ with NFEM(} \mathbf{p}\text{x,y) } =:\mathbf{f}(x,y)^{} T\cdot \mathbf{p}\end{array}

Ellipsoidal height h observations. Covariance matrix
\mathbf{C}_{h} =diag(\sigma _{h_{i} }^{2} ).

(2a) 
N_{G} (B,L)^{j}+v=\mathbf{f}(x,y)^{T} \cdot \mathbf{p}+\partial N_{G} (\mathbf{d}^{j} )\text{ }

Geoid height observations. With real covariance matrix CNG or evaluated from a synthetic covariance function.  (2b) 
\begin{array}{l}
\xi ^{} j+v=\mathbf{f}_{B} ^{T} /(M(B)+h)\cdot \mathbf{p}+\partial \xi (\mathbf{d}^{} j_{\xi ,\eta } )\text{ } \\
\eta ^{} j+v=\mathbf{f}_{L} ^{T} /((N(B)+h)\cdot \cos (B))\cdot \mathbf{p}+\partial \eta (\mathbf{d}^{} j_{\xi ,\eta } )\end{array}

Deflections from the vertical (η,ξ) observed with a zenith camera or derived from a gravity potential model. 
(2c)
(2d) 
H+v=H

Physical height H observations with covariance matrix
\mathbf{C}_{H} =diag(\sigma _{H_{i} }^{2} ).

(2e) 
C+v=C(\mathbf{p})

Continuity conditions as pseudo observations with small variances and high weights.  (2f) 
By B and L the geodetic latitude and longitude are described. The (x,y) are the projected coordinates.
With f_{B} and f_{L} we introduce the partial derivatives of f(x(B,L),
y(B,L)) (2c) with respect to the
geographical coordinates B and L, while M(B) and N(B) mean the radius of meridian and normal curvature,
respectively, at a latitude B. The continuity of the resulting HRS representation
NFEM(px,y) = f(x,y)^{T}p over the meshes (fig. 2a, thin blue lines) is automatically provided by the
continuity equations C(p) (2f).
A number of identical fittingpoints (B,L,hH) shown in fig. 2 by green triangles are introduced by the
observation (Eqs. (2a) and (2e)). In the practice of DFHRS_DB evaluation, one geoidmodel a number of
different such models  e.g. the EGM 2008 or, in Europe, the EGG97  are used in the least squares
estimation related to the mathematical model (2a–f), which is implemented in the DFHRSsoftware 4.3.
To reduce the effect of medium or longwave systematic shape deflections, namely, the natural and
stochastic “weakshapes”, in the observations N and (x,h) from geoidmodel (GPM) these observations are
subdivided into a number of patches (fig. 2, thick blue lines). These patches are related to a set of
individual parameters introduced by the datum parameterizations
δN_{G}(d^{j}) (2b) and
δη(d^{j}_{ξ,η})
(2c,d). In this way,
it is obviously possible to introduce geoid height observations and vertical deflecttions from any number
of different geoidmodels (e.g. the EGM 2008) in the same area, or vertical deflections
(ξ,η)_{P} observed with modern zenith cameras (fig. 3, right).