BALTIC QGEOID COMPUTATION AS HEIGHT COMPONENT OF THE GEODETIC INFRASTRUCTURE FOR GNSS POSITIONING

SERVICES IN THE BALTIC STATES

Reiner Jäger 1), Janis Kaminskis 2), Janis Balodis 3), Janis Strauhmanis 2) and Ghadi Younis 1)

1) Faculty of Geomatics, Karlruhe University of Applied Sciences. Moltkestrasse 30, D-76133 Karlsruhe, Germany

2) Faculty of Civil Engineering, Riga Technical University (RTU). Azenes Iela 16/20 – 238A, Riga, LV-1048, Latvia

3) Institute of Geodesy and Geoinformation, University of Latvia, Riga. 19 Raina Blvd., Riga, LV–1586, Latvia

The worldwide ongoing process of the establishment of high precise regional GNSS-positioning services (e.g. LATPOS), which are con-sistent to ITRF-related frames, leads to the new age of GNSS-posi-tioning. The process is further promoted by an increasing number of GNSS orbital segments as well as high precise global GNSS-services. So GNSS becomes an interdisciplinary tool with a broad spectrum, re-lated to precise positioning, precise navigation, robotics, mobile GIS and mobile IT applications. The determination of physical heights H by the transformation H=h-N of the ellipsoidal GNSS heights h to the Geoid or QGeoid height reference surface N as 2nd component of the geodetic infrastructure for GNSS positioning services (GIPS), requires the computation of Geoid- or QGeoid-models N, and belonging RTCM capable databases, as a sustainable geodetic task in the establishment of GIPS. After an overview about the different GIPS components and the realization of concepts and software for all four different GIPS components, which have been developed in international cooperations and projects, the main target of the contribution is to present in detail the mathematical models of the general world-wide valid concept for the computation of height reference surfaces (HRS), namely geoid or quasi geoid (QGeoid) models N as essential part of GIPS. The concept is shown at the computation of the Baltic QGEoid over the territories of Estonia, Latvia an Lithuania. The computation was done in the frame of a research cooperation between the Riga Technical University (RTU), the University of Latvia and Karlsruhe University of Applied Sciences (HSKA), using the DFHRS-software developed at HSKA.

Key Words Geodetic Infrastructures for GNSS-positioning Services (GIPS), Q-Geoid Compu-tation, Spherical Cap Harmonics, EGG97, EGM2008, Vertical Deflections, Gravity Data, RTCM-Transformation Messages

1. Geodetic Infrastructures for GNSS-Positioning Services (GIPS) To enable the full spectrum of GNSS usability, the establishment and maintenance of a respective geodetic infrastructure for GNSS positioning services and applica-

tions (GIPS) in each country is required (Fig.1). Besides the geomonitoring compo-nent (GIPS 4) for GNSS-stations coordinate integrity monitoring as well as early-warning and disaster mitigation, GIPS has two transformation components. Both transformation components concern the GNSS-consistent infrastructure for spatial information in Europe (INSPIRE) and worldwide (cadastre, GIS, urban planning, precise out-/indoor navigation, construction, transportation, meteorology, land ma-nagement, precise agriculture etc.). The horizontal transformation component (GIPS 1) is related to the georeferencing of space-related objects between different frames regional ITRF-related frames (e.g. ETRF89 in Europe) or the global ITRF and the classical national frames, and requires the establishment of respective transformation models and databases. The set-up of RTCM capable transformation parameter databases to enable GIPS-1 and GIPS-2 in online GNSS-application mode is another geodetic task in the establishment of GIPS.

Fig. 1: P Geodetic Infrastructures for GNSS Positioning Services (GIPS)

The determination of physical heights H - by the transformation H = h - N of the ellipsoidal GNSS heights h to the Geoid or QGeoid height reference surface N (fig. 2) as the 2nd transformation GIPS-component (GIPS 2) requires the computation of a Geoid- or QGeoid-model N, and is the main topic is treated in the following chapters 2, 3 and 4. An overview on GIPS at the example of the realization for the GNSS positioning service MOLDPOS in Moldova is given at www.moldpos.eu.

2. Height Reference Surface (HRS) Computation by the DFHRS-Con-cept and –Software used for Baltic QGeoid

2.1 Continuous Polynomial-based Representation of HRS

The DFHRS (Digital-Finite-Element-Height-Reference-Surface) research and de-velopment project (www.dfhbf.de), in German DFHBF, is aiming at the compu-tation of height reference surfaces (HRS). Depending on the height system type, the physical heights H are called orthometric heights, normal height (European height system), or spheroidal normal heights (NN-heights). The respective HRS are the geoid, the QGeoid or the NN-surface, and are represented by the height N of the HRS above the reference ellipsoid (fig. 2, N = ”Geoid”).

Fig. 2: Principle of GNSS-based height determination: H = h – N

The main pratical target of the DFHRS project, and the 2nd GIPS component, is to enable the conversion of the ellipsoidal GNSS height h determined at the earth topography, into the physical earth gravity field based standard “sea-level” height H by H = h – N (fig. 2).

2.2 Geometrical Observation Components and Parametrization – 1st Stage of the DFHRS concept

In the DFHRS concept in its first stage of the developments, a continuous polyno-mial surface over of a grid of finite element meshes (FEM) with polynomial para-meters p (fig. 3, thin blue lines) is used as a carrier function for the HRS ([1], [2]) The FEM surface of the HRS is therefore called NFEM(p|B,L). For some old height systems H a scale-difference factor ∆m has to be considered in addition, so that the DFHRS-model of N (fig. 2) consists of two parts. The principle of a GNSS-based height determination H (fig. 2), requires to submit the GNSS-height h to the DFHRS(B,L,h)-correction N reading:

)hm)L,B|(NFEM(h)h,L,B|m,(DFHRShNhH ⋅∆+−=∆−=−= pp (1)

The DFHRS height N=DFHRS(B,L,h) is provided by means of a DFHRS database (DFHRS_DB), which contains the HRS parameters (p, ∆m) together with the mesh-design (fig. 3) information. DFHRS_DB have become an industrial and user standard for all GNSS-receiver types worldwide ([3]), and can also be used for the set-up of the new RTCM-transformation messages ([4], [6]).

In the 1st stage of the DFHRS approach development, geoid- or geopotential model (GPM) heights N, observed astronomical or geoid/GPM-model based deflections of the vertical (ξ,η) in any number of groups, and fitting points (B,L,h|H), fig. 3 , were exclusively used as observation groups in a common least squares compu-tation for the evaluation of the DFHRS_DB parameters p and ∆m. The mathemati-cal model for these observations is given by formulas (2a-f). In case of an adequate stochastical model, the use of gravity-based geoid-/GPM model as together with the covariance information is equivalent to the use of the original observed gravity values g, as shown in [3]. The mathematical model for the computation of the DFHRS_DB parameters (p, ∆m) using the above so-called geometrical part of ob-servation components reads:

Functional Model Observation Types and Stochastic Models

pfp

p

⋅=+∆⋅+=+

)y,x(: y)x,|NFEM( with

y),x,|NFEM( mh Hvh

Ellipsoidal height h observa-tions. Covariance matrix

)(diag 2ihh σ=C .

(2a)

)(N)y,x(v)L,B(N jG

TjG dpf ∂+⋅=+ Geoid height observations. With

real covariance matrix CNG or

evaluated from a synthetic cova-riance function.

(2b)

)(

))Bcos()h)B(N/(v

)()h)B(M/(v

,j

TL

j

,jT

Bj

ηξ

ηξ

η∂+

⋅⋅+−=+η

ξ∂+⋅+−=+ξ

d

pf

dpf

Deflections from the vertical (η,ξ) observed with a zenith camera or derived from a gra-vity potential model.

(2c)

(2d)

HvH =+ Physical height H observations with covariance matrix

)(diag 2HH i

σ=C .

(2e)

)(CvC p=+ Continuity conditions as pseudo observations with small varian-ces and high weights.

(2f)

With fB and fL we introduce the partial derivatives of f(x(B,L),y(B,L)) (2c) with re-spect to the geographical coordinates B and L. M(B) and N(B) mean the radius of meridian and normal curvature at a latitude B. The continuity of the resulting HRS representation NFEM(p|x,y)=f(x,y)T⋅p over the meshes (fig. 3, thin blue lines) is automatically provided by the continuity equations C(p) (2f).

A number of identical fitting-points (B,L,h | H) are introduced by the observation equations (2a) and (2e) (fig. 3, green triangles). In the practice of DFHRS_DB eva-luation, one or a number of different geoid-/GPM such as the EGM 2008 or in Europe the EGG97 are used in a least squares estimation related to the mathema-tical model (2a-f), which is implemented in the DFHRS-software 4.0 (fig. 3, 5). To reduce the effect of medium- or long-wave systematic shape deflections, namely

the natural and stochastic “weak-shapes” [4], in the observations N and (ξ,η) from geoid- or GPM models, these observations are subdivided into a number of patches (fig. 3, thick blue lines). These patches are related to a set of individual parameters,

which are introduced by the datum parametrizations ∂NG(dj) (2b) and )( ,j

ηξξ∂ d

)( ,j

ηξη∂ d (2c,d). In this way, it is of course possible to introduce geoid height ob-servations and vertical deflections from any number of different geoid- or GPM models (such as e.g. the EGM 2008) in the same area, or vertical deflections (ξ,η)P observed with modern zenith cameras (fig. 4, right).

Fig. 3: DFHRS-computation design for the DFHRS_DB of Baltics with a 10 km mesh-size. FEM-Meshes (thin blue lines), patches (thick blue lines) and fitting

points (B,L,h | H) (green triangles)

3. Physical Observation Components and New Parametrization - 2nd stage of the DFHRS concept

The extension of the DFHRS-concept and -software to physical observation types - such as e.g. terrestrial, air- or space-borne gravity measurements (terrestrial gravity meter, see fig. 4), or physical observation types taken from geopotential models

(GPM) such as EGM 2008 - is based on a regional spherical cap harmonic parame-trization (SCH) of the earth’s gravitational potential V ([4],[5],[7]). The benefit of a SCH-paremetrizatiob (3) with a local cap pole and a limited cap size area, instead of an ordinary global spherical harmonic (SH), is that the same resolution of V can be achieved with SCH by a much less number of parameters than by SH. E.g. for a 2 mm resolution for the HRS, a degree of 7200 for the SH parametrization by (Cn,m; Sn,m) is required, while for a cap size area of 100 km a degree of only

80k max = is enough in case for a SCH parametric model [7]. So SCH is the key for enabling the computation of high resolution HRS in the 2nd stage of the DFHRS research and development, meaning an integrated over-determined HRS-compu-tation for all types of geometrical and physical observations.

Fig. 4, left: Gravity meter for the terrestrial observation of the gravity vector T

PLAV ]g,0,0[ −=g . Right: Zenith camera for vertical deflections determination

The representation of the gravitational potential V of the earth in terms of SCH with parameters )'S,'C( m),k(nm),k(n reads ([4], [6], [7]):

∑ ∑ θ⋅λ⋅+λ⋅

=θλ

= =

+maxk

0k

k

0mm),k(n'm),k(nm),k(n

1)k(n

)'(cos'P)'msin'S'mcos'C(r

a

)',',r(V

.

(3)

Here the space position refers to the triple of spherical cap coordinates ( ',',r θλ ). In the frame of that article, the observation equation for terrestrial and air- or space-borne gravity observations Pg is briefly worked out. The gravity observation Pg at the earth surface, taken with a gravity meter (fig. 4), is referring to the local astronomical vertical system (LAV), and so we have for the respective observed three-dimensional gravity vector in total:

TP

LAV ]g,0,0[ −=g - Original gravity observation and vector (4a)

The astronomical vertical ( ξ+=Φ B , )Bcos(/L η+=Λ ) is set up by the ellipsoi-dal vertical (B,L) and the deflections from the vertical ),( ηξ . The original vector

gLAV (4a) is first rotated to the earth-centred earth-fixed system (ECEF) using the astronomical direction ),( ΛΦ from a geopotential model. In that coordinate frame, the centrifugal part of gLAV (4a) is removed, and so the original observation (4a) is strictly reduced with respect to the vertical deflections (“topography”) and the cen-trifugal acceleration, and we arrive at:

TrEN

LGVred ]g,g,g[=g - Reduced gravity observation vector. (4b)

The observation vector LGVredg (4b) is the further rotated as SCH

gravg to the SCH-repre-

sentation frame (3) with

- TSCHgrav ]

r

V,

V

'sinr

1,

'

V

r

1[

∂∂

λ∂∂⋅

θ⋅θ∂∂⋅=g - SCH-parametrization of (4a) . (4c)

The ”vertical” and principal component of the reduced observed gravity observa-

tion LAVredg (4b) is related to the third parametric component in (4c), and the accura-

cy of the vertical component is nearly unaffected by the above reductions. So we have the following observation equation for a gravity observation in the so-called integrated DFHRS approach, reading:

∑ ∑ θ⋅λ⋅+λ⋅+

=+

∞

= =

+

0k

k

0mm),k(nm),k(nm)),k(n

1)k(n

grSCHgrav

)'(cosP)'msin'S'mcos'C(r

)1)k(n(

r

a

vg

.

(4d)

With gv we describe the observation correction in an adjustment approach.

By the disturbance potential Prefm),k(nm)),k(nP )V)'S,'C(V(T −= applied to

the theorem of Bruns in theory of Molodenski (5a), fitting-points (h-H) as well as quasi-geoid heights (5a) and vertical deflections P),( ηξ (5b,5c) observed at a point P can be introduced as follows:

Q

P

Q

PrefQGNormal

T)VV(NNh

γ=

γ−

==− ,

PQ

pQ

QG

N

QG)

B

T(

)hM(

1T

B

1

)hM(

1

B

N

s

B

s

B

B

N

∂∂⋅

+⋅γ−=

∂∂⋅

γ⋅

+−=

∂∂

⋅∂∂−=

∂∂⋅

∂∂

−=ξ ,

PQ

PQ

QG )L

T(

Bcos)hN(

1T

L

1

Bcos)hN(

1

L

N

s

L

∂∂⋅

⋅+⋅γ−=

∂∂⋅

γ⋅

⋅+−=

∂∂

⋅∂∂−=η .

(5a)

(5b)

(5c)

In [5] a new adjustment-based approach has been published, which enables to estimate the coefficients ( m),k(n'C , m),k(n'S ) for regional SCH modelx V (3)

from the coefficients ( m,nC , m,nS ) of a global geopotential model, e.g. EGM

2008. The estimated coefficients ( m),k(n'C , m),k(n'S ) can be introduced as

direct observations in the integrated approach

m),k(nm),k(n 'Cv'C =+ and m),k(nm),k(n 'Sv'S =+ . (6)

The integrated DFHRS approach represented by the equations (3) to (6) is a far rea-ching alternative to the model (1) to (2f) , because it allows both “geometrical” and “physical” observations. The integrated approach is presently further investigated and implemented in the DFHRS-software version 5.0. Formula (5a) is used to derive the QGeoid NQG from the estimated SCH-parameters. A QGeoid model NQG or grid-respectively – having been computed by the DFHRS approaches (1 – 2f) or (3 – 6) - can be transformed to a Geoid-model by

. Hg

NN QGG ⋅γ

γ−+= , (7)

The mean true gravity g along the plumbline (fig. 2) can be taken from a density model or with from a geopotential model, the mean reference gravity value γ from the closed formulas related to the GRS80.

4. Computation Results The computation of a QGeoid for the Baltics (Estonia, Latvia, Lithuania) was car-ried out with the DFHRS-software version 4 in the approach (1) to (2f) (fig. 5).

Fig. 5: DFHRS-computation of Baltic the Q-Geoid as a screen-shot of the DFH-

RS-software with the computation results showing the maximum and mean residuals for the observations N, H and h

The result of a closed continuous QGeoid has an accuracy of (1-3) cm. A number of 213 identical points (B,L,h|H) were used (fig. 3). The levelled normal heights H (2e) were introduced with an accuracy of 5 mm, and the GNSS heights h (2a) with an accuracy of 7 mm. The QGeoid-observations N(2b) have been taken from the EGG97 and were introduced with 7 cm, and the vertical deflections from EGG97 also with 7“. Alternatively the EGM2008 could have been introduced here, which ha the same accuracy in the area.

Tab .1 shows an extract of the list of the observations H after the adjustment, where v-H are the observation residuals. The test-statistics NV is the standardized residual with a critical value of 3.3. The values “Repro” indicates how the respective final height H and QGeoid N would change, if the respective observation H would not take part in the adjustment. The “Repro” values give the external accuracy of the computation result N of the Baltic QGeoid N at the respective location.

Nr H v-H [m] Red NV t-Test Repro [m]

:: :: :: :: :: :: ::

107 9.483 0.00262 6.24 2.1 25.0 -0.042** 108 16.791 0.00217 6.34 1.7 20.5 -0.034** 109 112.277 0.00078 7.85 0.6 6.6 -0.010 110 84.981 0.00078 13.83 0.4 5.0 -0.006 111 72.252 -0.00106 12.33 0.6 7.2 0.009 112 82.675 -0.00032 14.14 0.2 2.0 0.002 113 83.174 -0.00202 13.24 1.1 13.2 0.015 114 116.477 0.00189 12.21 1.1 12.9 -0.015 115 94.802 0.00072 11.96 0.4 5.0 -0.006 116 123.068 -0.00207 11.59 1.2 14.5 0.018 117 117.445 -0.00351 14.87 1.8 21.7 0.024 118 131.976 -0.00380 12.08 2.2 26.1 0.031** 119 144.385 0.00084 11.86 0.5 5.8 -0.007 120 163.986 -0.00137 10.77 0.8 9.9 0.013 121 116.893 0.00429 12.59 2.4 28.9 -0.034** :: :: :: :: :: :: ::

Tab .1: List of height observations H after the DFHRS adjustment (2a-f)

From the Repro-values (Tab. 1) a locally varying external accuracy of (1-3) cm is estimated for the QGeoid over the Baltics and the carrying DFHRS database, re-spectively. The (1-3) cm QGeoid for the Baltic states Estonia, Latvia and Lithuania is ready for use in GNSS heighting. It can be applied for the determination of physical normal heights H, both on GNSS-controllers, and or for the set up RTCM transformation messages by GNSS-services (see www.moldpos.eu).

5. Conclusions The research cooperation between RTU, the university of Latvia and HSKA re-cently led to the successful computation of a closed QGeoid for the Baltics (Estonia, Latvia, Lithuania) with a surface accuracy of (1-3) cm and the DFHRS-database (224 KB) is ready for use on GNSS-controllers and for the set up the

RTCM height transformation message (www.geozilla.de, [6]) for GNSS positio-ning service providers in the Baltic area. The further research will be dealing with the improvement of the Baltic QGeooid the target of a 1cm solution, based comparatively on both approaches above, whereas only the integrated parame-trization by Sperical Cap Harmonics (Cnm’, Snm’) alos allows the introduction of gravity values. Here the EGM 2008, mapped to the regional Spherical Cap Harmonics Coefficients (Cnm’, Snm’) as prior information, identical fitting points points (B,L,h | H), gravity observations g(B,L,h) and vertical defections (ξ,η) from zenith cameras (fig. 4 right) will be used as observation input. A further topic of the research will be the optimum design concerning the use of gravity data and ver-tical deflections, as well as the question, whether vertical deflections may replace by parts or completely gravity data.

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