GeoNetAccuracy

This repository contains Python code for preliminary assessing the accuracy of geodetic (surveying) networks.

The main method for preliminary accuracy assessment is based on the concept of a covariance matrix as an inverse metric tensor in the parameter space, found using the general formula in index notation:

$$\hat{P}_{(ij)}=J_{i}^{l}J_{j}^{k}P_{(lk)}=\frac{\partial x^{l}}{\partial \hat{x}^i}\frac{\partial x^{k}}{\partial \hat{x}^j}P_{(lk)}\quad \text{where}\quad \hat{P}_{(ik)}\hat{K}^{(kj)}=\delta^{j}_{i}$$

where
$\quad x^i$, $\hat{x}^i$ are the components of the vector of measurement and the vector of estimated parameters, respectively,
$\quad J_{i}^{j}$ are the components of the Jacobian matrix, which is the linearized model matrix (design matrix),
$\quad P_{(ij)}$, $\hat{P}_{(ij)}$ are the components of the measurement accuracy matrix (weighing matrix) and the parameter accuracy matrix, respectively,
$\quad \hat{K}^{(ij)}$ are the components of the parameter covariance matrix, if $i=j$, then $\hat{K}^{(ij)}=var(\hat{x}^i)$, otherwise $\hat{K}^{(ij)}=cov(\hat{x}^i,\hat{x}^j)$.

More information about tensors in statistics can be found in monograph by Prof. Peter McCullagh:

McCullagh, P. (1987). Tensor methods in statistics: Monographs on statistics and applied probability (1st ed.). New York: Chapman and Hall/CRC. doi:10.1201/9781351077118.

More information about tensor calculus and index notation can be found in monograph by Prof. Yuri Ivanovich Dimitrienko:

Dimitrienko, Y. I. (2001). Tenzornoe ischislenie [Tensor calculus]. M.: "Vysshaya shkola".

More general information about the theory of measurement errors can be found in:

Grodecki, J. (1997). Estimation of Variance-Covariance Components for Geodetic Observations and Implications on Deformation Trend Analysis. Ph.D. dissertation. Engineering Technical Report No. 186, University of New Brunswick, Department of Geodesy and Geomatics Engineering, Fredericton.

Gordeev, V. A. (2004). Teoriya oshibok izmerenij i uravnitelnye vychisleniya [Measurement error theory and adjustment computations] (2nd ed.). Yekaterinburg: UrSMU. ISBN 5-8019-0054-3

Amiri-Simkooei, A. (2007). Least-squares variance component estimation: theory and GPS applications. PhD thesis. Delft University of Technology, Delft institute of Earth Observation and Space systems (DEOS). Delf: Publications on Geodesy, 64, Netherlands Geodetic Commission. Retrieved from http://resolver.tudelft.nl/uuid:bc7f8919-1baf-4f02-b115-dc926c5ec090.

An example of the program's operation in the form of a plan of the designed geodetic network with ellipses of mean square errors: