STANDARD DEVIATION OF THE GEODETIC QUADRILATERAL POINT COORDINATES DETERMINED BY THE APPLICATION OF HANSEN'S METHOD

The Danish astronomer Peter Andreas Hansen, who also worked on theoretical geodesy, provided an original solution for solving the geodetic quadrilateral in which the four angles and the length of one side are known. This procedure, identified in scientific literature as Hansen’s problem, has been applied in geodesy but significantly more so when solving various problems related to mining measurements in underground exploitation. Nonetheless, the literature does not cover the determining of the standard deviation of the coordinates of unknown points which are established using Hansen’s method. In this paper then, equations are derived to calculate the standard deviation of the coordinates determined by the above mentioned procedure.


INTRODUCTION
Peter Andreas Hansen (1795-1874) was a Danish astronomer whose most famous work is in the field of celestial mechanics: theories on the motion of comets, small planets, the Moon, as well as lunar tables (Ephemerides).In addition to astronomy, Hansen worked on optics, probability theory and theoretical geodesy.In the field of Geodesy, he is well-known for his method of solving the geodetic quadrilateral in which four angles and the length of one side are known.Based on these elements, the unknown angles φ and ψ (Figure 1) are calculated, followed by all the other elements of the quadrilateral.
Such a way of solving the geodetic quadrilateral, known as Hansen's problem can be applied in: -Determining the elements of eccentricity (Mihailović, 1987); -Determining the base for terrestrial photogrammetric surveying (Schofield and Breach, 2007); -Determining the points for vertical shaft surveys (Patarić, 1990); -Connecting pits to the base on the surface of the terrain (Baturić, 1959), etc.On the whole, Hansen's problem is applied in the instances where two points of the quadrilateral are inaccessible or are, for reasons of safety, inconvenient for stations at which it is necessary to focus the surveying instrument.The four angles required in the quadrilateral are measured only from the remaining two points of the quadrilateral.
When it is necessary to evaluate the coordinates of the points in the geodetic quadrilateral, then it is, as in every free trigonometrical network, essential to know the coordinates of the two points.In that case, the length of the side between these two points is calculated on the basis of the coordinates of the given points, while only angles are measured on the terrain.It follows that two situations are then feasible: - The angles are measured from the given points, and - The angles are measured from unknown points.The first situation, where the angles are measured from the given points is more straightforward and does not involve calculating the elements of the geodetic quadrilateral.Based on the directional angle of the side between the given points and the measured angles, oriented directions are calculated in terms of unknown points or their coordinates (Chandra, 2005).
The second situation, when the angles are measured from unknown points, is more complex and requires that the geodetic quadrilateral be previously solved in accordance with Hansen's method.
The five known elements (four angles and one side) represent the mathematical minimum hence providing a unique solution.All additional measurements of an element of the geodetic quadrilateral (redundant measurements) offer an ambiguous solution necessitating the leveling of all measured values.In this case the calculations for the coordinates of unknown angles will be carried out by applying the methods of indirect leveling, thus obtaining the standard deviation of the coordinates of the unknown points.

CALCULATING ELEMENTS OF THE GEODETIC QUADRILATERAL USING HANSEN'S METHOD
The geodetic quadrilateral problem which consists of two given but inaccessible points and two unknown but accessible points can be solved using Hansen's method (www2.washjeff).
Points A and B are given points with known coordinates Y A and X A , or Y B and X B , but they are inaccessible and angles cannot be measured from them.The horizontal length d between them is also known and is calculated on the basis of the coordinates of these points.
In order to solve the quadrilateral or to determine the coordinates of the unknown but accessible points 1 and 2 it is sufficient to measure, on the terrain, the horizontal angles from these points: The unknown angles γ and δ are derived from triangles: from A12: 2) The unknown angles φ and ψ are calculated on the basis of their semi-sums and semi-differences, or: and ∆A12), the semi-sum of the unknown angles is: In accordance with the Sine rule: Or: 1 2 1 2 sin sin sin sin tan sin sin sin sin As: The semi-difference of the unknown angles is: The unknown sides of the quadrilateral d 1 and d 4 , as well as its diagonal d 2 and d 3 , are calculated using the Sine rule based on the known side d: While side d 5 is calculated in terms of the previously evaluated sides (11): The coordinates of the unknown points 1 and 2 are calculated on the basis of the evaluated elements of the quadrilateral, in terms of the known points A and B, using the equations: As:

STANDARD DEVIATION OF THE COORDINATES OF UNKNOWN POINTS
The standard deviation of the coordinates of unknown points 1 and 2, due to extensive deriving, will be evaluated only on the basis of equations ( 13) and ( 15).
Previously, in these equations, all evaluated values will be expressed only in Where shifts are introduced: The standard deviation of the coordinates of unknown points 1 and 2 will be calculated only on the basis of standard deviations of measured angles  1 ,  2 ,  1 and  2 , while the coordinates of the given points A and B, as well as the value of the grid bearing B A  and the length d will be considered true values.
In accordance with that, the standard deviation of point coordinates will be calculated using the equations: Where: 1 2 1 2 ; ; ;     -standard deviations of measured angles; and partial derivatives of functions are: s 2 1 tan 2cos 180 cos sin sin sin sin 2 sin 2 tan sec 2 cos 2 1 tan Where shifts are:

Example
An example of the analysis pertaining to the standard deviation of unknown points has been performed on the geodetic quadrilateral in the shape of a regular square whose elements are shown in Figure 2. The coordinates of unknown points 1 and 2 have been evaluated using equations ( 13) and ( 15): The values of the partial derivatives are: Based on these values and assuming various standard deviations of the measured angles using the equations (21-24), the standard deviations of the coordinates of the points 1 and 2 have been evaluated, where: As Table 1 shows, considering the regular shape of the geodetic quadrilateral, the standard deviations of the unknown points are mutually equal, and their value increases proportional to the decrease in the accuracy of the angle measurements in the quadrilateral.

CONCLUSION
The standard deviation of the coordinates of unknown points which is determined using Hansen's method is not presented in scientific literature.One reason for this, among others, must lie in the fact that it involves extensive and complex calculations, primarily of the coefficients which represent partial derivatives with respect to variable or measured quantities.Although contemporary measuring techniques enable the determining of the coordinates of any point providing there is mutual visibility, there is a whole range of different situations on the terrain which require the application of Hansen's method.This is, above all, the case when solving specific engineering problems in regard to mines with underground exploitation.
For this reason, it is vital to perceive the optimal shape of the geodetic quadrilateral, having in mind that errors in point coordinates, besides being influenced by errors in the measured angles, are also impacted by the shape of the quadrilateral.This paper serves as the first step in the analysis of the impact of the geodetic quadrilateral shape on the standard deviation of the coordinates of the unknown points where standard deviation equations have been derived.

Figure 1 -
Figure 1 -Elements of the geodetic quadrilateral which is solved using Hansen's method

Figure 2 -
Figure 2 -Geodetic quadrilateral in the shape of a regular square

Table 1 -
Standard deviation of the coordinates of unknown points