Error ellipse

When adjusting geodetic position networks, the mean errors of the point coordinates in the direction of the respective coordinate axes are obtained in the course of the error calculation . To analyze the error situation, however, knowledge of the maximum errors and their alignment is required.

Using the cofactors from the adjustment, the maximum and minimum errors can be calculated. Since their axes are perpendicular to each other, it is possible to construct an ellipse, which can clearly show the error situation in an overview map.

To compare the designations, the basic equations of an adjustment using the least squares method in the sense of a network adjustment are given below:

  v: Vektor der Verbesserungen der Beobachtungen
  A: Designmatrix; linearisiertes funktionale Modell
  P: Gewichtsmatrix der Beobachtungen
  x: Vektor der gesuchten Koordinatenzuschläge
  l: Vektor der Beobachtungen

Improvement equations:

${\displaystyle v=A\cdot x-l}$

Calculation of the unknown:

${\displaystyle x=(A^{T}\cdot P\cdot A)^{-1}\cdot A^{T}\cdot P\cdot l}$

Calculation of the cofactor matrix:

${\displaystyle Q_{xx}=(A^{T}\cdot P\cdot A)^{-1}}$

Cofactor matrix with its point-related cofactors:

${\displaystyle Q_{xx}={\begin{pmatrix}q_{xx_{1}}&q_{xy_{1}}&\cdots &\cdots &\cdots &\cdots &\cdots \\q_{xy_{1}}&q_{yy_{1}}&\cdots &\cdots &\cdots &\cdots &\cdots \\\cdots &\cdots &q_{xx_{2}}&q_{xy_{2}}&\cdots &\cdots &\cdots \\\cdots &\cdots &q_{xy_{2}}&q_{yy_{2}}&\cdots &\cdots &\cdots \\\cdots &\cdots &\cdots &\cdots &q_{xx_{3}}&q_{xy_{3}}&\cdots \\\cdots &\cdots &\cdots &\cdots &q_{xy_{3}}&q_{yy_{3}}&\cdots \\\cdots &\cdots &\cdots &\cdots &\cdots &\cdots &\cdots \end{pmatrix}}}$

Mean weight unit error of the adjustment (note: standard deviations were previously referred to as mean errors in surveying ):

Representation of the error ellipse derived from an adjustment with the maximum errors.

${\displaystyle m_{0}={\sqrt {\frac {\Sigma vvp}{n-u}}}={\sqrt {\frac {v^{T}Pv}{n-u}}}}$

Mean coordinate error in X direction:

${\displaystyle m_{x}=m_{0}\cdot {\sqrt {q_{xx}}}}$

Mean coordinate error in Y direction:

${\displaystyle m_{y}=m_{0}\cdot {\sqrt {q_{yy}}}}$

Directional angle of the maximum error:

${\displaystyle \tau ={\frac {1}{2}}\cdot \arctan {\frac {2\cdot q_{xy}}{q_{xx}-q_{yy}}}\,}$

Auxiliary variables:

${\displaystyle q_{1}={\frac {q_{xx}+q_{yy}}{2}}}$

${\displaystyle q_{2}={\frac {1}{2}}\cdot {\sqrt {(q_{xx}-q_{yy})^{2}+4\cdot q_{xy}^{2}}}}$

Calculation of the cofactor of the maximum error:

${\displaystyle q_{\mathrm {max} }=q_{1}+q_{2}}$

Calculation of the cofactor of the minimum error:

${\displaystyle q_{\mathrm {min} }=q_{1}-q_{2}}$

Relationship between the cofactors in the direction of the coordinate axes and those of the maximum error:

${\displaystyle q_{\mathrm {max} }+q_{\mathrm {min} }=q_{xx}+q_{yy}}$

Mean maximum error:

${\displaystyle m_{\mathrm {max} }=m_{0}\cdot {\sqrt {q_{\mathrm {max} }}}}$

Mean minimum error:

${\displaystyle m_{\mathrm {min} }=m_{0}\cdot {\sqrt {q_{\mathrm {min} }}}}$

Although the error ellipse is a widely used means of representation, it should be pointed out that the shape of the error ellipse does not allow a correct statement to be made in the case of errors that are neither in the coordinate axes nor in the axes of the maximum errors. For this reason, instead of the error ellipse, it is better to use its base curve, which allows correct statements about the mean error (the standard deviation) regardless of direction, but is much more difficult to calculate. Alternatively, a confidence ellipse can also be calculated and displayed

literature

• Erwin Groten: On the definition of the mean point error. In: Journal of Surveying (ZfV). 11/1969, pp. 455-457.
• E. Gotthardt: Introduction to the adjustment calculation. Verlag Herbert Wichmann, Karlsruhe 1968.