The Method Of Simulated Annealing For The Optimal Adjustment Of The Nigerian Horizontal Geodetic Network

ABSTRACT

The Horizontal Geodetic Network of Nigeria is made up of terrestrially arranged chains of

triangles augmented by precise traverses. The work on the network began early 19th century

but by 1930 the work was discarded and a re-observation of the network was carried out to

the highest possible accuracy then, enhanced with high order geodimeter traverses which

linked to other neighboring African networks.

The full network consists of 515 stations, with 2411 observations which comprise 2197 angular

observations, 40 Laplace azimuths and 174 measured distances, part of which substituted for

the sparse triangulation observations especially in the southern part of the country. The added

observations contributed to strengthening of the network in the 1977 adjustment which

however was not a holistic optimized adjustment, but rather, a phase adjustment. Based on the

1977 state of adjustment of the network, no meaningful distortion monitoring exercise can take

place until the network is adjusted by an optimized simultaneous technique in order to

ascertain the state and consistency of the network.

The use of the simulated annealing method, which has been successfully applied in other fields,

is presented for the classical geodetic problem of simultaneous adjustment of the entire

triangulation net using the least squares observation equation method. This method is an

iterative heuristic technique (a method of solving problems by learning from past experience

and investigating practical ways of finding a solution) in operations research. It uses a thermo

dynamic analogy (Cooling theory) to adjust a network of unstable stations (changes to gaseous

state) through fairly stable station coordinates (liquid state) to a stable station coordinates

(solid state) so as to offer a solution that converges in a probabilistic sense (statistically based)

to the global optimum. The simulated annealing method of optimization serves to help

determine the position of all triangulation stations by means of minimizing the volume of the

error hyper ellipsoid inherent in the solution to give an optimal configuration of the geodetic

network. Computer programs were developed using Matlab Software and run on an adequately

configured Pentium IV computer. Creation of an intelligent database was achieved through the interactive network of the data storage, processing, manipulation, analysis and retrieval of

results of the adjustment.

The result of the new adjustment produced a generally consistent trend of changes in the

distances and azimuths compared to the previous adjustments. Error analysis of all lines were

carried out and the respective standard errors in distances and azimuths were determined.

Relative and absolute error ellipses of all stations were determined and plotted. Statistical plots

and analysis of the error ellipses of the network stations were also determined. The absolute

and relative weakness/strength of the network stations coordinates after adjustment were

shown and confirmed by the error plots to have the following geometry error distributions.

That is, 90.5% of the 515 Network Stations fell within Network Standard deviation of 1- Sigma,

94.2% within 2-Sigma, while 98.3% fell within 3- Sigma

The distributions confirmed the high reliability of the Nigerian Horizontal Geodetic Network and

its data quality. Re-strengthening exercise would be necessary using either the 1-Sigma or 2-

Sigma region of network standard deviation

A data structure for the entire network was developed and necessary conclusions and

recommendations are made for further action to update/upgrade the precision of the Nigerian

horizontal geodetic network for future study.