**Publication: Scientific Bulletin. Series, Mathematical Modeling in Civil Engineering***Author: Chirila, Constantin*

*Date published: December 1, 2011*

(ProQuest: ... denotes formulae omitted.)

1. Introduction

The concept of "geodetic datum" was recently assimilated by geodetic specialized Romanian literature, and through it, we can operate in the more complex space of coordinate systems along with the reference surface specifications to which they relate [I].

The necessary correspondence between local and global-geocentric data constitutes a current problem which is resolved under he accuracy terms, claimed in general by the terrestrial measurement works and in particular by the cadastral survey works.

As Regards the transition from the local-national datum to the European one it should be said that the solution implementation phase was successfully resolved (creating the permanent GNSS reference station network and coordinates transformations "Transdaf'soft-A.N.C.P.L). This procedure should be extended to other local reference systems such as those created in big cities and municipalities. These local systems served to create some small dimension local geodetic networks, which afterwards formed the base on surveying the terrain details.

In this paper, the study aims to validate the possibility of integrating the existing graphical information, with documentary and cadastre value, from the "Local-Iasi" system, integration in the European geodetic datum, which will be implemented along with the Stereo- 1970 projection adoption of the GRS-80 global geocentric ellipsoid (Stereo-2010) but with the same old characteristic features of the official projections existing in the present.

2. Transformation Coordinate Possibilities Between Geodetic Data

The cartographic projection used in the "Iasi-local "coordinate system is a stereographical tangent projection which is characterized by properties of the conformity, in which linear deformations are null in the central point and then increase, in small value, under a centimeter, for a radius of at least 30 km from the central. This is an important aspect regarding the process of reducing distances via the project plan. From one point of view this phase could be overlooked due to the precision lesser intake in the coordinate transformation. The projection central point is Golia Tower, having Xo = 10 000,000 m ; Yo = 10 000,000 m plane rectangular coordinates. The axes are orientated differently compared to Stereo-70 national projection, with the X on the East and the Y on the north along the place meridian [2].

For the coordinates transformation between the local system and the global one it is necessary first to specify the magnitude order of the trans-calculus precision. Taking in consideration the primordial significant of cadastre information, in the first phase of research we can exclude the vertical Z component of the points, so that a bi-dimensional transformation (2D) should be enough with no need for a tridimensional one (3D). This hypothesis has also the advantage of eliminating inherent errors of not knowing precisely the quasigeoid anomalies , as it is known that the official Romanian height system, namely "Marea Neagra-1975" heights system, is a system of normal heights (Molodensky) which must be related to an ellipsoidal height system.

For further application of 2D coordinate transformation models it is necessary to know the minimum number of points, according to the used model, which have plane rectangular coordinates in both the local and the global reference systems. Data required for the coordinates in the European system (ETRS 89) have been obtained through GPS measurement campaigns conducted in 2005 with the creation of the geospatial network of Iasi and then in 2010 with the network extention to the entire Iasi metropolitan area[3].

The calculation algorithm presents the following steps:

1. Parameters calculation of the WGS-84 coordinates (epoch 2005) 3D transformation in ETRS- 89 coordinates, based on three shared points, by applying the transformation model with 7 parameters, like "Bursa- Wolf ' (table 2.1)[4].

2. WGS-84(epoch 2005) coordinates transformation of 7 shared points from GPS thickening network in ETRS-89 coordinates, expressed in Cartesian ellipsoidal system (table 2.2).

3. Ellipsoidal geodetic coordinates conversion of the 7 shared GPS thickening points in Stereo-70/GRS-80 cartographic projection system (Stereo-2010)(table 2.3).

4. 2D transformation parameters calculation of specified 7 points from the "LocalIasi" into Stereo-70/GRS-80 system, with conformai linear method (table 3.2).

5. 2D transformation parameters calculation of those 7 points from the "Local-Iasi" into Stereo-70/GRS-80 system, with affine method (table 4.1).

6. 2D transformation parameters calculation of those 7 points from the "Local-Iasi" into Stereo-70/GRS-80 system, with polynomial of second degree method (table 5.1).

7. Coordinates transformation of those 7 shared points from "Local-Iasi" in Stereo70/GRS-80 system (Stereo 2010), using trans-calculus parameters previously calculated (table 3.2,table 4.1, table 5.1).

3. Conformai Linear Transformation in 2D Space

This transformation keeps only the topographic conditions and uses the simplified hypothesis of one system translation and rotation in the same plane to the other [5].

The correction equations are written in linearized form:

...

and in matrix form: B^sub 2n4^X^sub 4,1^+L^sub 2n,1^ = V^sub 2n,1^ where:

...

where ? is the number of double points with known coordinates in both systems.

The unknowns of the system are the (Δx, Δy) translation and (a, b) rotation parameters, the unknown coefficients (x^sub i^,y^sub i^) are the coordinates of the double points in the OEEsource" system and the (X^sub i^, Y^sub i^) free terms are the coordinates of the same points in Mhe target" system.

Introducing the minimum condition ([vv] -> min) [6] and canceling the partial derivatives of the function, related to the unknowns of the system, the normal equation system is obtained, with four equations with four unknowns (a, b, Δx, Δy):

...

The unknowns matrix in calculated with the matrix inverse method:

...

The determination precision of each trans-calculus parameter from the unknown matrix is expressed with a mean square error:

...

In the end, based on trans-calculus parameters, these formulas will determine the new point coordinates:

...

In table 3.1 there are the coordinates of the seven GPS thickening points, in "Local-Iasi-" and Stereo-70/GRS-80 coordinate systems. The coordinates from the local system have been obtained with the specific method of thickening the local geodetic network (backward multiple intersection method), rigorously compensated, applying the least square principle, the indirect measurement process.

4. Affine transformation in 2D space

In the bidimensional affine transformation, different corrections are introduced for each of the two coordinates of the axis directions [5]:

The correction equations are written in linearized form :

X^sub i^ * a + y^sub i^ * b + Δx - X^sub i^ = ν^sub xi^,

X^sub i^ * c + y^sub i^ * d + Δy - Y^sub i^ = ν^sub yi^,

and in matrix form: B^sub 2n,6^X^sub 6,1^+L^sub 2n,1^ = V^sub 2n,1^, where:

...

The system has (2n) equations with (6 + 2n) unknowns (a, b, c, d, Δx, Δy, v^sub x1^, v^sub y1^ ...v^sub x^sub n^^ ,v^sub y^sub n^^) so that a minimum of 3 shared points with known coordinates are necessary in both systems.

The normal equation system becomes:

...

The matrix of the unknowns calculated with the matrix inverse method:

...

The determination precision of each trans-calculus parameter from the unknown matrix is expressed with the mean square error:

...

In the end, based on trans-calculus parameters, these formulas will determine the new point coordinates:

...

For the affine transformations, seven GPS thickening point coordinates are considered known in both systems, "Local-Iasi" and Stereo-70/GRS-80, from table 3.1.

The transformation results are in table 4.1.

5. Polynomial of Second Degree Transformation

The olynomial of second degree transformation, in order to obtain a proper coordinate transcalculus precision, is using second degree polynomials[5].

The polynomial coefficient determinations are made with the least square method. For this type of transformation it is necessary to know 6 shared points in order to find all the 12 (a^sub ij^, b^sub ij^) coefficients.

The general transformation formulas for the second degree polynom are:

...

The correction equation is written in matrix form:

...

The equation system unknowns are a^sub ij^ and b^sub ij^ polynomial coefficients which are calculated with the matrix inverse method:

...

The mean square error for trans-calculus of a point is:

...

For the second degree polynomial transformation, the same seven GPS thickening point coordinates are considered to be known in both system, "Local-Iasi" and Stereo-70/GRS-80, from 5.1.

The transformation results are in table 5.1.

6. Conclusions

The existing graphical information in the OEMLocal-Iasi" system can be spatially integrated in the new European geodetic datum, through the coordinate transformation with a precision appropriate to the real estate cadastre rules.

In order to determine the 2D trans-calculus parameters, were applied three coordinate transformation models (linear, affine and second degree polynomial), with similar results falling within the tolerance of 5 cm for the position of a point in the plan. The best result of ± 3,6 cm was obtained for the linear transformation but it is possible that an extension of the shared points increase in number the precision on the other two transformation methods, which are related to a complex distribution of the relative position error of those two work systems.

In conclusion, it can be acknowledged, that on this level, the 2D space coordinates transformations can be applied successfully, between local and European datum, following that in a later stage, when a very precise local model for the geoid is available, a 3D coordinates transformation can be applied with an acceptable precision.

References

[1] Hies Alexandria s.a., Retele Geodezice de Sprijin, vol. I- Sisteme de referintä si de coordonate, Ed. Compress, Bucureçti, 2004

[2] *** - The file nr. 2067/68 - "Triangulatia orasului Iasi", D.S.A.P.C. Iasi.

[3] Chirilä C, Mänutä ?.- The realization of the gps geodesic network necessary for the implementation of the real - urban building cadastre and the database formation on the administrative territory of the Iasi municipality, Simpozionul çtiintific international GeoCad'06 - RevCAD nr. 6, 2006, Alba Iulia, pp.35-42

[4] *** - "Reteaua Geodezicä a Municipiului Ia§i", S.C. Geotop 2001 S.R.L.

[5] Bofu C, Chirilä C- Sisteme Informationale Geografice. Cartografierea ci editarea hàrtilor, Ed. Tehnopress, Iasi,2007

[6] Nistor Gh., Teoria prelucrarii masuratorilor geodezice, Universitatea Termica "Gheorghe Asachi", Iasi, 1996

[7] Chirilä C, Dumitraçcu A. - Transformäri de datum în Sistémele Informationale Geografice, Simpozionul International "Sisteme Informationale Geografice", a XIV -a editie, 2006Cluj Napoca, pp49-54, 2006

Author affiliation:

CONSTANTIN CHIRILA- Lecturer, PhD Eng "Gheorghe Asachi" Technical University, Department of Terrestrial Measurements and Cadastre, email: tynelro@yahoo.com

MIHALACHE (cas. FICIUC) RALUCA MARIA- PhD Candidate "Gheorghe Asachi" Technical University, Department of Terrestrial Measurements and Cadastre, email: rmihalacheficiuc@yahoo.com