339

1 INTRODUCTION

The determination of the height in the vertical

reference frame in force in Poland is based both on

national regulations (Council of Ministers Regulation,

2012) and on the resolution adopted by the EUREF

subcommittee (EUREF Symposium, 2000). Currently,

the PL-KRON86-NH vertical reference frame is in use

in Poland – a normal height system referred to a

quasi-geoid of the average level of the Baltic Sea

determined by the zero level of the mareograph in

Kronstadt. According to the Regulation, by the end of

2019, Poland will adopt the PL-EVRF2007-NH vertical

reference frame, i.e. a normal height system referred

to the zero level of the mareograph in Amsterdam.

Necessary calculations can be made with the use of

the official Polish quasi-geoid model, available in

numerical form at the website of Head Office of

Geodesy and Cartography (GUGiK) as well as

numerical data of differences between heights in

vertical reference systems PL-KRON86-NH and PL-

EVRF2007-NH (GUGiK 2015). The above-mentioned

models have a significant drawback – they fail to

cover the entire area of the Polish economic zone as

well as Polish territorial waters. This is why the

authors propose the use of GNSS measurements, the

EGM 2008 geoid model and GUGiK’s quasi-geoid

model, which shows differences between the

reference systems mentioned above. When calculating

the seabed level, we suggest the use of depth

measurement methods used in hydrography and

underwater mining. The EGM 2008 model provides

the heights of the geoid above the ellipsoid. In order

to determine the quasi-geoid height it is necessary to

know the Bouguer anomaly and the vertical gradient

of gravitational acceleration.

Determination of Seabed Heights in the Area of Polish

erritorial Waters in the Official Reference System

.B. Rogowski

Gdynia Maritime University, Gdynia, Poland

R. Galas

Technical University of Berlin,

Berlin, Germany

Kłęk

Maria Sklodowska

-Curie Warsaw Academy, Warsaw, Poland

ABSTRACT: The determination of the height in the vertical reference frame in force in Poland is based both on

national regulations (Council of Ministers Regulation, 2012) and on the resolution adopted by the EUREF

subcommittee in Tromsø (Resolution No. 5, EUREF Symposium, 2000). Currently, the PL-KRON86-NH vertical

reference frame is in use in Poland – a normal height system referred to a quasi-geoid of the average level of the

Baltic Sea determined by the zero level of the mareograph in Kronstadt. According to the Regulation, by the

end of 2019, Poland will adopt the PL-EVRF2007-NH vertical reference frame, i.e. a normal height system

referred to the zero level of the mareograph in Amsterdam. The authors present a method of determining

normal heights of seabed referred to the zero level of the mareograph in Amsterdam for coastal areas of the

Baltic Sea. This method uses GNSS measurements, the EGM 2008 model and depth measuring methods typical

for underwater mining.

http://www.transnav.eu

the

International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 14

Number 2

June 2020

DOI:

10.12716/1001.14.02.09

340

2 DESCRIPTION OF THE METHOD

In the description of the method applied the authors

presented models and formulas necessary to

implement the method, as well as definitions of

height systems used.

2.1 Geodetic height systems

The fundamental concept in the height systems

applied is the so-called geopotential number C. It is

equal to the difference between the potential of the

geoid W

0 and of the point on the surface of the Earth

P. The following relationship can be shown between

the geopotential number C, gravitational acceleration

g and the height h.

C W W gdh=−=

∫

Depending on the gravitational acceleration used

to determine the height system, we may receive:

ort

– orthometric height system

ort

where:

– average value of gravitational acceleration

between the geoid and the physical surface of the

Earth.

nor

– normal height system

nor

where:

– the average value of normal gravitational

acceleration along the vertical line of the normal

gravity field.

dyn

– dynamic height system

dyn

where:

– the normal gravitational acceleration for

latitude 45°.

The normal acceleration of gravity is related to the

concept of equipotential ellipsoid that meets the

following conditions:

− The size and shape of the ellipsoid correspond to

the assumed ellipsoid that best approximates the

geoid.

− The mass of the ellipsoid is equal to the mass of

the Earth.

− The angular velocity of the ellipsoid spin ω

corresponds to the velocity in the rotation of the

Earth.

− The surface of an ellipsoid is by definition an

equipotential surface with potential U

0. This

potential is equal to the potential on the geoid W

This condition is as follows:

= =U W const

Normal acceleration is equal to:

gradU

The applicable global GRS 80 reference system can

be represented by the following relationship:

( )

9.780327 1 0.0053024 0.0000058 2sin sin ms

γ ϕϕ

−

= ⋅+ ⋅ − ⋅ …

where:

φ – latitude.

The vertical gradient of gravitational acceleration

can be calculated using the formula:

(

)

f q fsin B

dh a

γγ

=− + +−

where:

a – semi-major axis of ellipsoid,

f – flattening of the ellipsoid,

B – ellipsoidal width,

b – semi-minor axis of ellipsoid,

– rotational velocity of the Earth,

GM – geocentric gravitational constant.

According to the national regulation (Council of

Ministers Regulation, 2012) and to the resolution

adopted by the EUREF subcommittee – Resolution

No. 5 of the EUREF Symposium in Tromsø (EUREF

Symposium, 2000). By the end of

2019 current

vertical reference frame will be replaced with

PL-EVRF2007-NH reference system. It is a

normal height system in which the quasi-

geoid refers to the average level of the North

Sea defined by the zero level of the

mareograph in Amsterdam.

For hydrotechnical works the system of dynamic

heights is of key importance.

2.2 Determining the normal height of a point at sea level

The principle of determining the normal height of a

point at sea level, e.g. a drilling platform, research

vessel, etc. is shown in Figure 1.

341

Figure 1 The principle of determining the normal height

nor

= −

where:

h – ellipsoidal height determined through GNSS

observations,

N – distance between the geoid and the ellipsoid

calculated from the model,

– height anomaly.

The ellipsoidal height of a point on the sea surface,

for example on a drilling platform, can be determined

by GNSS observations with the use of RTK network

technology implemented on the ASG-EUPOS

network. The evaluation of the potential use of this

technique in the offshore area was the subject of prior

research (Rogowski et al., 2015). The cited work

involved a number of measurement experiments

conducted to evaluate the usability of the ASG-

EUPOS system with the use of RTK/VRS technology.

Data for correction was obtained via the Internet from

Orange mobile network. In the offshore area,

discrepancies in the horizontal component reach up to

3 cm because the RTK/VRS correction is extrapolated.

Since the height component is determined with an

error twice as large, one can estimate that it will be

determined with an error of up to 5 centimetres.

Another problem is the range of mobile network

which covers almost the whole Gulf of Gdańsk but in

the coastal zone does not exceed 10 Mm. These are

only estimates because, unfortunately, network

administrators do not want to share any data. On the

area beyond 10 Mm it will be necessary to apply the

method which in turn will result in the deterioration

of accuracy to approx. 25 cm (Przestrzelski, 2017),

(Rogowski et al., 2015).

As follows from the principle presented in Fig. 1,

the knowledge of the geoid model as well as the

difference between the ellipsoidal height of the quasi-

geoid ζ and the height of the geoid N are necessary in

order to establish the normal height. Values of this

correction in numerical form are available at the

GUGiK’s website. We propose that the EGM 2008

model is used to calculate the value N. It is an official

Earth Gravitational Model 2008 that has been publicly

released by the U.S. National Geospatial-Intelligence

Agency (NGA) EGM Development Team. This

gravitational model consists of spherical harmonic

coefficients complete to degree and order 2159, and

contains additional coefficients extending it to degree

2190 and order 2159. For this model, there are online

calculators available on the following websites:

http://earth-

info.nga.mil/GandG/wgs84/gravitymod/egm2008/,

http://icgem.gfz-potsdam.de/ICGEM/,

http://www.softpedia.com/get/Science-CAD/AllTrans-

EGM2008-Calculator.shtml.

These calculators allow you to determine the

height of the geoid with a relatively high accuracy (3 –

5 cm) and grid resolution (1' x 1'). For the area we are

interested in, we can create a map that can further be

used to determine the value N corresponding to the

object position. An example of such a map for Gulf of

Gdańsk is shown in Fig. 2 (Pałczyńska, 2017). The fact

that all of the above calculators use the WGS 84

ellipsoid proves to be problematic. A slight difference

in the parameters of the WGS 84 and GRS 80

ellipsoids (see Table 1) can be found in the data

obtained from the abovementioned calculators. For

accuracies we are interested in, this difference can be

omitted.

Table 1

_______________________________________________

Parameters WGS 84 GRS 80

_______________________________________________

a 6 378 137 6 378 137

1/f 298.257 223 563 298.257 222 101

_______________________________________________

Figure 2. Map of EGM 2008 quasi-geoid for Gulf of Gdańsk

(Pałczyńska, 2017)

The distances between the geoid and the quasi-

geoid can be calculated with the following

relationship (Barlik, 2000):

where the second element represents the influence of

Bouguer gravimetric anomaly. This anomaly is

defined in the following formula:

where k is the Newton's constant, and s is the density

of subsurface material. The formula includes also the

vertical gradient of gravitational acceleration

Normal heights for the PL-EVRF2007-NH system

can be determined using Δζ height anomaly

correction for data for the closest points from the

official GUGiK model. The transition from the normal

heights referred to the EGM 2008 quasi-geoid to the

342

quasi-geoid passing through the zero level of the

mareograph in Amsterdam is shown in Figure 3.

where:

∆

– height anomaly correction referred to the

Amsterdam system

Figure 3 Normal height correction referred to the

EGM 2008 quasi-geoid to the Amsterdam system.

Table 2 presents differences between the GUGiK

geoid and the EGM 2008 geoid for selected points on

the Baltic shore.

Table 2.

_______________________________________________

B L EGM2008 PL-geoid-2011 Differences

_______________________________________________

54,72 18,46 29,574 29,136 0,438

54,7 18,46 29,595 29,161 0,434

54,69 18,47 29,586 29,153 0,433

54,66 18,46 29,636 29,207 0,429

54,63 18,48 29,632 29,206 0,426

54,64 18,52 29,546 29,119 0,428

54,55 18,56 29,615 29,192 0,423

54,5 18,56 29,710 29,288 0,422

54,44 18,58 29,788 29,371 0,417

54,41 18,66 29,698 29,285 0,413

54,37 18,75 29,626 29,211 0,415

54,35 18,92 29,484 29,057 0,427

54,35 19,18 29,319 28,884 0,435

54,4 19,5 29,045 28,619 0,426

54,34 19,55 29,154 28,731 0,423

54,79 18,42 29,583 29,134 0,449

54,78 18,45 29,533 29,085 0,447

54,75 18,4 29,661 29,216 0,445

54,72 18,41 29,673 29,235 0,439

_______________________________________________

Calculating the height of the sea floor requires the

knowledge of the height difference between the

position of the GNSS antenna and the seabed. It can

be determined with an echo sounder or with other

methods used in underwater mining.

3 SUMMARY AND CONCLUSIONS

The data necessary for the transition between

reference systems is available on the website of the

Head Office of Geodesy and Cartography at the

following address: http://www.gugik.gov.pl/bip/

prawo/modele-danych. The data to determine the

correction to the EGM 2008 geoid in order to calculate

heights in the PL-EVRF2007-NH system can be

extrapolated from the GUGiK model available at

http://www.gugik.gov.pl/__data/assets/text_file/0016/

1843/gugik-evrf2007.txt. The authors estimate that

lthe seabed height in the area of Polish territorial

waters in the PL-EVRF2007-NH frame can be

determined with the accuracy of approximately 10

cm.

LITERATURE

Barlik M., (2000), On the contribution of vertical gravity

gradient anomalies to the separation between the geoid

and Molodensky’s quasigeoid, Reports on Geodesy, no. 2

(50).

Czarnecki K., (2010), Geodezja współczesna w zarysie.

Katowice: Wydawnictwo Gall.

Rogowski J., Specht C., Weintrit A., Leszczyński W., (2015),

Evaluation of Positioning Functionality in ASG EUPOS

for Hydrography and Off-Shore Navigation, TransNav,

the International Journal on Marine Navigation and Safety of

Sea Transportation, Vol. 9 No. 2, pp. 221-227.

Pałczyńska I., (2017), Opracowanie mapy geoidy EGM2008 dla

Zatoki Gdańskiej. (Master’s thesis), Faculty of Navigation,

Gdynia Maritime Academy, Unpublished manuscript.

Przestrzelski P., (2017), Sieciowe pozycjonowanie różnicowe z

wykorzystaniem obserwacji GPS i GLONASS, (Doctoral

dissertation), University of Warmia and Mazury.

Regulation of the Council of Ministers of 15 October 2012 on

the state spatial reference system, Journal of Laws of the

Republic of Poland, Warsaw, 14 November 2012, Item

1247 (in Polish).

Resolution No. 5 of the EUREF Symposium in Tromsø, 22 –

24 June 2000. Available from:

http://www.euref.eu/symposia/book2000/P_340_341.pdf.

Internet sources

http://www.euref.eu/symposia/book2000/P_340_341.pdf.

http://earth-info.nga.mil/GandG/wgs84/gravitymod/

egm2008/.

http://icgem.gfz-potsdam.de/ICGEM/.

http://www.softpedia.com/get/Science-CAD/AllTrans-

EGM2008-Calculator.shtml.

http://www.gugik.gov.pl/bip/prawo/modele-danych.

http://www.gugik.gov.pl/data/assets/text_file/0017/1844/gu

gik-geoid2011.txt

http://www.gugik.gov.pl/data/assets/text_file/0016/1843/gu

gik-evrf2007.txt