Modelling the quasigeoid for the Dobrogea and Seaside area, Romania

PAUL DANIEL DUMITRU, CONSTANTIN COSARCA, ALEXANDRU CALIN

Technical University for Civil Engineering Bucharest

122-124, Bvd. Lacul Tei, Sector 2, Bucharest

ROMANIA

paul.dumitru@gmail.com, constantin_cosarca@yahoo.com, alexcalin1975@yahoo.com

http://geodezie.utcb.ro

Abstract: - The problem of transforming the ellipsoidal height determined with GNSS measurements into the

national altitude reference systems is one of the most important issues in the geodesy field. There are many

solutions for the determination of local or regional transformation parameters depending on the available data.

This paper describes one technique that can be quickly implemented in actual filed works. The results are

showing that the corrections applied to one existing quasigeoid, as EGG97, with values of the altitude anomaly

obtained from GNSS and classic precise leveling is one of the methods. In this project, the geometric modeling

of the quasigeoid was used and the final model is presented as a grid of the conversion surface. The grid file

containing the values of the modeled quasigeoid is formatted so it can be further used with the known GNSS

post processing software applications. The Matlab application was used to develop a program for the

quasigeoid modeling.

Key-Words: -GNSS, precise leveling, quasigeoid, modeling, grid, EGG97, Matlab

1 Introduction Most often the Earth, in terms of geometry,

is approximated by an ellipsoid of revolution

flattened at the poles.

Notwithstanding the irregular terrain, but

including gravity field, Earth's shape is called the

geoid. The geoid surface is perpendicular to every

point on the gravity force (weight).

The geoid (Fig.1), which represents the

theoretical solid Earth produced by the imaginary

extension of the oceans and seas average level

throughout the whole world, has an irregular shape

and therefore cannot be used for geodetic

computations. When speaking of the real shape of

the Earth using a so-called physical surface, that

landforms surface is given by all the measurements

made.

Fig.1 Geoid and Quasigeoid

If HE is the ellipsoidal height of a point, ie

height above reference ellipsoid at the point

considered, measured along the normal to the

ellipsoid at the point considered, for the two

altitudes systems, it can be written:

P

N

P

E

PP

OR

P

E

PHHNHH ζ+=+= ; ( 1 )

where:

- NP - '00PP that is the geoid undulation at

the point considered, ripples geoid being specific

Orthometric altitude system.

- ζP - '''" 00 PP is the anomaly of the

altitudes.

The quasigeoid is the reference surface

having the following definition:

(Quasi)geoid is a surface constructed so that

the normal segment to the ellipsoid will be equal to

the anomaly heights at any point where this value is

unknown.

On large water surfaces the quasigeoid

coincides with the geoid, but under continents there

are differences due to the internal structure of Earth.

A quantity implied in the geoid modeling is

the geoid–ellipsoid separation (sometime named

geoid undulation) which represents the distance,

along the vertical (or plumb line), between the geoid

and the ellipsoid. The separation can be obtained

from gravimetric measurements, from geometric

modeling by comparisons between ellipsoidal

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heights (from satellite techniques) and

orthometric/normal heights (from geodetic leveling)

or from gravity potential coefficients determined by

satellite measurements [6].

2 Problem Formulation In Romania, the reference system used to

determine the altitudes, is called the system of

normal heights - Black Sea 1975 fundamental zero

point. Fundamental zero was considered the

benchmark fundamental point of the Military

Chapel of Constanta, which had its altitude

determined repeatedly by making geometric

leveling work (1962, 1963-1964, 1970, 1972) and

gravimetric measurements.

Studies were performed after this period led

to the idea of creating a new site for the zero key, in

an area "stable" from geologically point of view.

The site was chosen at about 53 km from Constanta,

between the localities Cogealac and Tariverde.

In 2009 it was issued the Order no.

212/4.05. 2009, of the General director of ANCPI

for adoption in Romania of the European Terrestrial

Reference System 1989, which consists of ETRS89

geodetic datum based on GRS80 ellipsoid (Geodetic

Reference System 1980 - Geodetic Reference

System 1980) and ellipsoidal geodetic coordinate

system.

Given the foregoing, a quasigeoid

determination model represents a milestone in the

development of geodetic networks. To achieve high

accuracy in determining the altitudes in normal

national reference system for GNSS satellite points

determined by methods of local quasigeoid

modeling is the best solution from a technical

standpoint.

A quasi geoid modeling is the determination

(interpolating) of a conversion between altitudes,

ellipsoidal surfaces and normal altitudes, modeling

abnormalities altitude values in irregular network

points as described by thickening geodetic network.

Geometric methods are known by using multivariate

methods (polynomials of degree higher, Delaunay

triangulation, Ponder the inverse distance, nearest

neighbor or natural neighbor) or by correcting an

existing model by making new precision

measurements [5].

The quasigeoid grid can be defined by a file

containing coordinates of the knots and the values of

the anomaly of the altitude interpolated in each

node. The quasigeoid grid step is chosen according

to accuracy and needs.

In order to interpolate altitude anomalies in

the created quasigeoid model, there can be used

several methods including the most simple and easy

to implement in applications that are specifically

designed bilinear interpolation bicubic spline

interpolation 2D (creates smooth surfaces) [2].

3 Problem Solution The work area is located in the south-east of

Romania, containing 2 counties: Tulcea and

Constanta. In this part of the country we can find the

seaside of the Black Sea and the southern area of

delta of Danube. Thus, the anomaly of the altitude

in this region has low anomaly of the altitude

variations.

To determine the quasigeoid model, there

were used points whose altitudes were determined

in the ETRS89 European reference system and

national altitude system with fundamental zero point

Black Sea 1975. The European EGG97 quasigeoid

model was used and corrected with the values

obtained from the measurements mentioned above

[4].

Normal altitude referenced to the Black Sea

1975 system, 1990 edition, for the points listed in

the table below, which are new points materialized

in the ground through metal pickets, were

determined by double horizon precision geometric

leveling, leaning on leveling marks embedded in

buildings and have not been stationed with GNSS

technology.

Geometric middle leveling was executed,

having equal section of leveling, the length of

section of leveling being between 5 m and 20 m.

Leveling measurements were performed

with Leica electronic level (SPRINTER) and the

digital level TOPCON (DL101C) using invar rods

(bar code) from December 2011 to January 2012.

The use of electronic and digital levels

provides a standard deviation of 1 mm for a level

difference of double km leveling lines, respectively

0.3 - 0.4 mm, using invar rods (bar code).

Precision requirements, field conditions and

- not least - the atmospheric conditions are imposed

taking extra precautions while carrying out

measurements:

− foot plates were used for surveying;

− section of leveling length did not exceed

generally 25 to 30 m;

− were made by two readings on each rod

(double horizon);

If geometric leveling, for integration into

the national altimetric system, the GNSS technology

was used, given that this technology used for

surveying, provides precise relative planimetric

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positioning and precise determination of level

differences. This method was used, in two different

versions:

- the GNSS receiver standing directly on the

known height leveling point (control point), given

that this was possible and ensuring a sufficient

stationary common period with other receivers

installed over the bridging and survey network

points (usually the common session duration was 1

hour);

- the GNSS receiver standing on a "new"

point, materialized near the leveling mark (control

point) - the new point with height which was

transmitted by middle geometric leveling. And in

this case has secured a sufficient stationary common

period with other receivers installed over the

bridging and survey network points (usually the

common session duration was 1 hour).

Following the field measurements

performed by geometric leveling and GNSS

technology that has resulted the heights of the new

points, marked near the leveling mark.

All the measurements realized in the

leveling marks were integrated into the Bridging

and Survey Network (BSN) of the area of interest.

GNSS measurements in BSN were made -

in average - with 10 to 15 receivers type: TOPCON

HIPER Pro, Topcon GR3, GR5 TOPCON, Radian -

SOKIA and Javad, dual frequency, in May 2012.

Achieving The Bridging and Survey

Geodetic Network is a mandatory step in order to

ensure the technical conditions to make all

measurements and subsequent technical operations

necessary to achieve the ultimate goal of this work.

Realization of The Bridging and Survey

Geodetic Network aims to meet the administrative,

economic and legal requirements and represents the

basis of the topographic measurements performed

for various purposes:

− realization of the overhead photos flights,

resulting orthophotomap;

− realization of the preparation of the control,

in order to produce orthophotomap;

− making measurements with LIDAR

technology resulting the digital terrain

model (DTM) and its derivatives;

− realization of the model of quasigeoid used

to correct all altitudes values (heights)

determined with modern positioning

technologies;

− realization of transverse and longitudinal

profiles of the river water bodies in space

Dobrogea - Litoral, both by topographic and

bathymetric measurements;

− making topographic measurements

(topographic surveys) in order to draw up

plans of special character;

− realization of the informatic system, specific

for the domain and its maintenance.

Given the purpose of this network and

respecting rules and regulations in force

(no.534/2001 Order of the Minister of Public

Administration, no.634/2006 Order of the General

Manager of National Agency for Cadaster and Land

Registration, Order 108/2010 of the General

Manager of National Agency for Cadaster and Land

Registration, Decision no.1/2008 of the Director of

the Geodesy and Cartography Direction ANCPI) the

Bridging and Survey Geodetic Networks were

designed in a preliminary form.

At the preliminary design was used

information taken from the National Agency of

Cadaster and Land Registration and from the

Authority of Hydrographic Basin Dobrogea -

Seaside, consisting of coordinates points inventories

of Spatial National Geodetic Network of Class A

(GNSS permanent stations), Class B (ground

points), class C (ground points belonging to national

geodetic triangulation network) and information

about Romanian Geodetic Leveling Network for the

interest area and information about CSA landmarks

from the works for water courses cadastral axis

determination, located in the ABA database.

When designing the bridging and survey

networks aimed to provide the corresponding

density in the network, thus providing the possibility

of using the entire network for development in the

next step polygonal traverses necessary to the

detailed topographic survey, in order to obtain the

above-mentioned products.

To this purpose, two points groups - the

visibility between them has been assured

(materialized at distances of about 200-300 m),

located outside the influence of future construction,

which could be the starting base and - respectively -

closing of subsequent measurements made with

GNSS technology or polygonal traverses.

This network design is a guarantee for

having the accuracy details for topographic survey.

At the preliminary design, there were

identified potential positions for a total of

approximately 150 points (75 pairs of points,

denoted by sign P001..... P075) that will be part of

the bridging network Dobrogea – Seaside of the

basin area.

Given the purpose and usefulness of the

bridging network points, the following criteria were

observed to the design:

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− points to be as close to geodetic leveling

network points of Romania to achieve and

transmit - the geometric precision leveling -

altitude in the national reference system;

− points to be located at the junction of two or

more streams river located in the Dobrogea

- Seaside area;

− sites should be easily accessible for

measurement and possible preserving the

points in time;

− a total of two benchmarks (benchmark

pairs) will be used for every possible

designed position in order to provide

infrastructure for achieving both

measurements with GNSS modern

technology and traditional technology;

− to be located near lakes and major hydro

construction;

− to be evenly distributed over the entire area

of interest.

The processing of the measurements was

made using GNSS technology, performed using

specialized software (software component of the

company TOPCON), mainly aiming to obtain

solutions of "fixed" type (fixing ambiguities) for

each vector (base, distance between points, length)

and their compensation as a constrained network

using measurements and coordinates of the National

Space Geodetic Network (A Class) - provided by

Romanian ROMPOS service of determination of the

position (GEO ROMPOS service)

The Bridging and Survey Geodetic Network

processing determined by GNSS technology has

followed the following steps:

− verification of data at their disposal;

− importing data into a new project;

− their visualization;

− establishment of GNSS data processing

parameters (elevation, type solution,

ionosphere and troposphere models used);

− preliminary processing of GNSS data;

− analysis results (Loop Closure, statistical

tests);

− processing the determined vector;

− generate reports on determined vectors

(relative coordinates and accuracy);

− establishment of the network compensation

parameters;

− defining constraint points by entering their

coordinates in ETRS89 system and

declaring that fixed;

− bridging and survey geodetic network

compensation, as a constrained network;

− analysis results after compensation;

− generate reports on network compensation

(compensated vectors and ellipsoidal

coordinates B, L, h / XYZ in reference

system ETRS89).

During the GNSS measurements campaigns

there were measured about 1800 GPS bases

(vectors) for determination of 196 new points,

across the entire area of Dobrogea – Seaside area,

thus ensuring a high redundancy in the network.

In the figure below the sketch of the

network could be found.

Fig. 2 Network sketch

Some of the leveling landmarks were

stationed directly with GNSS technology. In this

case the precision of the execution of the leveling

measurements was not necessary.

There were chosen and designed a total of

48 points. Distributed points were evenly over the

entire work area (Hydrographic Space Dobrogea –

Seaside, Tulcea and Constanta counties).

The steps followed to develop a model for

quasigeoid geometric method are:

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− Geodetic network design for thickening;

− Materialization points by thickening

network ground terminal;

− Making determinations by geometric

leveling measurements GNSS for a

sufficient number of points to ensure a high

precision its shape;

− Perform additional measurements used to

verify the quasigeoid model;

− Modeling using geometric altitude anomaly

values obtained by difference ellipsoid and

the normal altitudes. Geometric modeling

involves quasi geoid surface interpolation

by obtaining a surface modeling conversion

between ellipsoidal altitudes determined in

the system ETRS89 (GRS80 ellipsoid) and

normal altitudes determined in the national

reference system;

− Generating the grid (rectangular grid) of the

quasigeoid model;

− Perform action to verify and confirm the

quasigeoid model developed and its

accuracy;

− The final product of the quasigeoid model

as a grid;

Determination of the altitudes in ETRS89

reference system was achieved with GNSS

technology, in points of Romanian Leveling

Network that were stationed directly with GNSS

equipment and where it was not possible

(benchmark embedded in bridges, churches and

other buildings) was made by geometric leveling

from new point located nearby.

From processing the bridging network

consisting of 48 points determined by GNSS

technology (static method), the determination of

geometric leveling for leveling control points which

were not GNSS stationed and for which new points

were materialized and which underlying the

quasigeoid surface model generation, these values

of altitude anomaly were determined. These will

form the basis for the quasigeoid generation in the

interest area.

For the determination of the surface model,

there were used the quasigeoid altitudes anomaly

values for the considered points. Because only a

geometrical modeling is not sufficient to generate a

quasigeoid surface with adequate precision, it was

used and the European gravity geoid model EGG97.

The European model correction EGG97 anomaly

with values determined from GNSS measurements

altitude and leveling will be generating a surface

model for the quasi gravimetric work.

The designed points to be measured by

GNSS technology and geometric precision leveling

have the following position as shown in Fig.3:

Fig.3 Position of the points used measured

for quasigeoid modeling

The solution was to correct the quasigeoid

EGG97 model and interpolate the surface anomaly

differences between the values determined from

GNSS measurements and precise leveling and

interpolated values in the model EGG97 anomaly.

For the inside perimeter points determined by the

underlying approach to creating model generation is

the bicubic spline interpolation using triangulation

with finite element method. For areas outside the

perimeter the nearest neighbor method was used.

Table 1. Differences between the anomaly

of the altitude of EGG97 model and the GNSS and

leveling measurements

Point Ellipsoidal

altitude

ETRS89

h (m)

Normal

altitude

Black

Sea

1975

H (m)

Anomaly

of the

altitude

GNSS –

precise

levelling

(m)

Anomaly

of the

altitude

ζ

EGG97 (m)

Differences

ζ (m)

A050 43,194 10,548 32,646 32,934 -0,288

Biserica

Adamclisi 148,216 112,987 35,229 35,322 -0,093

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Point Ellipsoidal

altitude

ETRS89

h (m)

Normal

altitude

Black

Sea

1975

H (m)

Anomaly

of the

altitude

GNSS –

precise

levelling

(m)

Anomaly

of the

altitude

ζ

EGG97 (m)

Differences

ζ (m)

Biserica

Cerbu 276,484 242,908 33,576 33,671 -0,095

Biserica

Cerchezu 175,923 140,198 35,725 35,555 0,170

Biserica

Cotu Vaii 95,212 60,412 34,800 34,833 -0,033

Biserica

Crucea 114,843 80,688 34,155 34,277 -0,122

Biserica

Cumpana 90,597 56,830 33,767 34 -0,233

To model the quasigeoid were used 34

points of the 48 evenly distributed over the entire

work area with ellipsoidal altitudes determined in

ETRS89 reference system and in normal national

reference system with fundamental zero point Black

Sea 1975, as well as altitude anomaly values

interpolated in European geoid model EGG97[7][8].

The position of the points is represented in

the following figure (Fig. 4):

Fig.4 Position of the points used in

quasigeoid modelling

The surface obtained after correction of the

interpolation of the anomaly of the altitude

differences is presented in the following figure

(Fig.5):

Fig. 5 Correction surface resulted after

interpolation (3D view)

After modeling the quasigeoid surface the

interpolated surface result as a rectangular grid with

nodes containing its altitude anomaly values.

Correction and generation of quasigeoid

model was performed using a own application,

accuracy of modeling the quasigeoid surface being

+ / - 0.010m plus the European model EGG97

accuracy of + / - 1 ... 5 cm distance up to 10 km

(according to official sources of the European model

developers - Erdmessung Institute, University

Hannover - Institut für Erdmessung – IFE [4]).

To determine the altitude anomalies in quasi

geoid model is determined using bilinear

interpolation on regular grid.

Some features of the following grid (grid

generation format is consistent with the national

scheme has reference standards and regulations in

force and is compatible with the application you

wish TransDatRO its use interpolation method

implemented in the application):

− Minimum latitude: 43.60 (d.ddd)

− Maximum latitude: 45.60 (d.ddd)

− Minimum longitude: 27.15 (d.ddd)

− Maximum longitude: 29.40 (d.ddd)

− Step on latitude : 0.010 (d.ddd) –

aprox.1080m

− Step on longitude: 0.015 (d.ddd) – aprox.

1080m

− The sense in grid: colum1: min Lat -> max

Lat, min Long; colum2 min Lat - > max

Lat, min Long +dLong ; colum3 ……

− No. of grid nodes: 30351 (201 lines * 151

columns)

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Example of the grid:

Table 2. Example of the quasigeoid grid

No. Node Latitude Longitude Anomaly of

the altitude.

1 43.600 27.150 38.233

2 43.610 27.150 38.203

3 43.620 27.150 38.174

4 43.630 27.150 38.144

5 43.640 27.150 38.112

6 43.650 27.150 38.081

7 43.660 27.150 38.051

8 43.670 27.150 38.020

9 43.680 27.150 37.988

11 43.690 27.150 37.938

12 43.700 27.150 37.946

13 43.710 27.150 37.934

14 43.720 27.150 37.925

………… …………….. ……………… …………..

Representation of quasigeoid model determined for

the area of interest is shown in the following figure:

Fig.6 3D representation of the quasigeoid

model

Checking the model was based on points

whose coordinates in the system of reference

ETRS89 (new determinations GNSS) and the

national reference system normally Black Sea 1975,

ed.1990 (new geometric leveling measurements) are

known and not used in the creation of the model

mentioned above (10 points). After checking the

model were obtained the following results:

Table 3. Checking the quasigeoid model Control

Point

Ellipsoidal

altitude

Normal

altitude

Normal

altitude

from

quasigeoid

model

Differences

Biserica

Adamclisi 148,216 112,987 113,007 -0,02

Biserica

Ostrov 80,255 47,239 47,236 0,003

Biserica

Peceneaga 52,093 19,314 19,38 -0,066

Biserica

Rachelu 59,289 27,532 27,467 0,065

Gara Ciobanita

125,077 90,354 90,382 -0,028

Gara Zebil 39,795 7,555 7,54 0,015

Piatra km

183 144,562 108,804 108,803 0,001

Pod DN39

km 47 57,561 23,385 23,388 -0,003

Podet km

152 76,033 44,16 44,173 -0,013

Zid fantana

CAP Olteni 134,058 98,416 98,375 0,041

After checking, the average difference is

approximately + / - 0.034 m with a mean square

error of + / - 2mm.

The final quasigeoid grid data file in ASCII

format has the following view (Fig. 7):

Fig.7 The ASCII file of the quasigeoid grid

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4 Conclusion As mentioned, the determination of the

quasigeoid model is one of the most important

activities in geodesy. Once determined a quasigeoid

model using the results of high precision GNSS

measurements made with modern technology can

lead to accurate determination of altitudes in the

national altimetric reference system.

Since the measurements of precisise

leveling for any point on Earth's surface is long term

activities, the only option for normal altitudes

determinations (national reference system) with

GNSS measurements is to use an interpolation into a

quasigeoid model.

After interpolation into the model of

quasigeoid the anomalies of the altitude are

obtained, which, as it looks, may decrease the

values determined by technology GNSS ellipsoidal

altitudes to altitudes with high precision values

obtained in the national reference system.

All GNSS measurements performed for the

determination of the coordinates of the network

points (the thickening network) will be made using

the altitudes in the national system. Also, any

leveling of detail through modern technology using

GNSS DGNSS / RTK using differential corrections

in real time or post processing, from a fixed base

that transmits corrections via radio (UHF radio

modem) or using differential corrections from the

Romanian ROMPOS position determination system

can lead to high precision results from applying the

interpolated solutions of quasigeoid model created

to determine normal altitudes.

Determination of the quasigeoid model is a

crucial stage in the network regardless of geodetic

surveying work performed and any leveling

methods used.

Since hand equipment and other new

technologies in the topographic measurements

cannot determine altitudes in the national reference

system for the determinations of the ground

checking points for photogrammetric or LIDAR

projects or the generation of altimetric Digital

Terrain Model without using a precise quasigeoid

model based on a sufficient number of GNSS and

leveling points will ensure high accuracy for

determining altitudes in the national reference

system.

The code developed in the Matlab

application could be easily used further in case of

the determination of a new conversion surface for

the (quasi)geoid model [9]. The completion of the

initial ASCII file used at the quasigeoid modeling

with new coordinates and values of the anomaly of

the altitude for new known points is the only

operation that should be made for a new quasigeoid

generation.

The actual algorithm for interpolation

permit the using of the horizontal coordinates in

national reference system S-42, Stereographic 1970

projection. This flexible way of development

permits with small efforts the further possible

modifications for direct interpolation of the

ellipsoidal coordinates.

The generated model could be improved

through the utilization of other methods or by their

combination and using the regional (European

EGG97 or EGG08) model or global (EGM2008

model) [1],[3].

References:

[1] Denker, H., W. Torge, G. Wenzel, J. Ihde, U.

Schirmer: Investigation of Different Methods

for the Combination of Gravity and

GPS/Leveling Data. In: K.P. Schwarz (Ed.):

Geodesy Beyond 2000 - The Challenges of the

First Decade. IAG Symposia, Vol. 121, 137-

142, Springer Verlag, Berlin, Heidelberg, 2000

[2] Franke, R.: Scattered Data Interpolation: Test of

Some Methods, Mathematics of Computations,

1982, 33(157):181

[3] Denker, H.: Evaluation and Improvement of the

EGG97 Quasigeoid Model for Europe by GPS

and Leveling Data. Continental Workshop on

the Geoid in Europe, Budapest, Hungary, March

10-14, 1998, Reports of the Finnish Geodetic

Institute 98:4, 53-61, Masala, 1998

[4] Denker, H.: Evaluation and Improvement of the

EGG97 Quasigeoid Model for Europe by GPS

and Leveling Data. Continental Workshop on

the Geoid in Europe, Budapest, Hungary, March

10-14, 1998, Reports of the Finnish Geodetic

Institute 98:4, 53-61, Masala, 1998.

[5] Ch. Yang, S. Kao, F. Lee, P. Hung, Twelve

Different Interpolation methods: A Case Study

of Surfer 8.0, Geo-Imagery Bridging

Continents, XXth ISPRS Congress, 12-23 July

2004 Istanbul, 2006

[6] P. Dumitru, M. Plopeanu, A. Jocea, A. Calin, O.

Badescu, Approaches on geoid modelling, 13th

International Multidisciplinary Scientific

GeoConference, SGEM2013 Conference

Proceedings/ June 2013, Albena, Bulgaria (In

press)

[7] http://www.ife.uni-

hannover.de/forschung/egg97_e.html

[8] http://www.gfy.ku.dk/~iag/egg97.html

[9] www.mathworks.com

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