M. L. Ojigi
Keywords: Precise leveling, network, graphic chart, analysis
The total measurements made along a link (between two Benchmarks) is the sum of the measurements made in both directions; but in case were observations are made in one direction and are assumed to be adequate will surely result in poor control of measurement in the network. This is referred to as the number one advantage of the network analysis by graphic chart.
A single measurement in geodetic leveling is defined as 1 tie, while double measurement is regarded as 2 ties. According to Osazuwa and Ajakaiye , in and ideal network, observation must be complete, homogeneous and symmetrical. Lambert and Beaumont,  defined completeness and homogeneity as “when all possible pairs of station in the network are directly coupled by observation ties” and “when all pairs of stations are coupled with equal number of ties” respectively.
For a network to be symmetrical, the number of the ties in the forward direction, PB1→PB2, must be equal to the number of the ties, PB2→PB1, in the opposite direction between any pair of station. The theories of graphic chart evaluations were applied by Osazuwa and Ajakaiye , to analyse the gravimetric network of Nigeria, and they considered it as a
versatile tool for controlling observations in a network during field work and for subsequent analysis of the quantities of the observation in the network. The application of these theories to geodetic leveling is justified because, it shares common observation and field requirements with gravity network observations.
According to Richardus and Allman,  optimization of a network is generally governed by rules in order to achieve a homogeneous relative precision of the coordinates of the stations; one of such rules is that lines observed at one end or phase only should be avoided. The specifications for field procedures in leveling are usually based on a minimum permissible discrepancy between two independent (Back and fore) leveling of the same time [Blachut, et al. 1979].
The statistical model used for the network evolution allows the inclusion of various network effects (homogeneity, reciprocity, transitivity, cycles, popularity, etc.), of individual covariates (covariates connected to the sender, the receiver, or the similarity between sender and receiver or fore and back as the case may be), and of dyadic covariates [Snijders 2006; Steglich 2006]. The distinct goals and perspectives of network analysis, covers the issues of data collection, validity, completeness, visualization, and mathematical/computer representation (Borgatti & Everett, 2006). The collection and storage of unstructured, natural text data has become fast, cheap, and easy with the advancement in technology [Diesner, 2006].
Geodetic networks, when derived solely from observed GPS baseline vectors, have an inherent datum deficiency of dimension three due to the respective unknown translation parameters; hence estimated coordinates from a (weighted) least-squares adjustment will not be unique, although the adjusted baseline vectors are [Snow
& Schaffrin 2007]. Uniqueness, without affecting the adjustment as such, can be achieved by introducing a minimum number of constraints for the coordinates, or by applying an objective function on the set of least-squares solutions (LESS) that fulfill the so-called normal equations. However, good quality estimate can only be optimized if network completeness analyses are carried out prior the LESS. A network of stable, easily accessible survey control points, referred to as bench marks (BMs), is crucial to a wide range of activities, including coastal zone management, floodplain mapping, stormwater and sewer utility management, large-scale engineering projects, hurricane evacuation and recovery planning, and topographic mapping [Carlson et al, 2009]
Kowalczyk,  observed that the most complete data to determine vertical crustal movements was from 1974–1982 and 1997–2003 leveling campaigns. After identification of common benchmarks from campaigns 2 and 3 (2600 common benchmarks) it was clear that complete data from second campaign can augment vertical movement determination. Comparing all four leveling campaigns it was shown that data from first campaign is not suitable for movement determination mainly because of small number of common benchmarks (78 common benchmarks). It is therefore of great relevance in surveying to have relatively complete and homogeneous network observations as a basis for accurate definition of control points for engineering construction, geodynamic investigations, urban development, land information management and mapping, etc.
Statement of the problem
where is a combination of the n stations taking 2 stations at a time [Osazuwa and Ajakaiye 1986]; which represents the ideal links of a network.
There are situations where due to fatigue on the part of surveyors and field assistants, hazards along geodetic survey routes and some times attempt to reduce financial cost of field surveys,observations made in direction and are assumed to be adequate for both ties (fore and back).This is a technical bias or constraint imposed on the network, which will surely result in a weak, poor and deficient control of measurements in the network. To nip these geometrical and observational consequences in the bud, it is important to carry out a pre-computation analysis of the network using the graphic chart; which is capable of evaluating the completeness status, homogeneity and the symmetry of the network for reliability and accuracy in a survey network. Samaru Campus of ABU and its environs are very important and strategic centre and environment for research, training and development, hence accurate and reliable survey controls must be ensured for its urban development mapping, engineering construction and maintenance; hence the need for the study.
The Zaria community was founded in the 14th century as one of the seven original Hausa city-states (Figure 1). According to oral tradition, Zaria rose to prominence in the early 15th century under the brilliant military leadership of Queen Amina. It became part of the Songhai Empire in the 16th century, fell to the Fulani in the early 19th century, and was captured by the British in 1901 [Stock, 2006]. Zaria (also known as Zazzau) is a road and rail hub in a major agricultural area of the north central Nigeria, the city is a market center for locally produced cotton, peanuts, hides and skins, shea nuts, corn, sorghum, and vegetables. Industries include cotton ginning, peanut and shea-nut milling, tanning, cottonseed-oil production, and the manufacture of cigarettes, bicycles, perfumes,
and soap. Zaria is an important center of education and research in Nigeria with other higher institutions of learning such as Nigerian College of Aviation Technology (NCAT) and The National Leather and Chemical Research Institute, Samaru, etc. The 1995 estimated population of Zaria is 369,800, and projection for 2006 was above 500, 000 people.
If , then there is no tie between stations i and j. We must be consistent in determining the elements of matrix Â; hence is regarded as the number of measurement in forward direction if i < j, and the backward direction if i>j.
Figure 2: Sketch of the precise leveling network in part of ABU, Samaru Campus, Zaria.
Evaluation of completeness in the network observation
The sum total of leveling ties, Ni from station i to other stations and number of ties, Nj from other stations to station i in the network is given by Osazuwa and Ajakaiye,  as:
where N is the total number of ties in the network. In order to determine the number of links in the network, the matrix is transformed to such that the elements of are given by
Therefore, considering the upper right triangle of the leveling graphic chart only, the number of links between Benchmarks is
where k = i+1. Completeness (C) may be defined as the ratio of number of the observed links in a real network to the total number of observable links in the ideal network; hence from (1) and (6), we have:
Using the graphic chart and equation (7), an estimate of the completeness of the network in this study was determined or calculated.
Estimation of homogeneity in the network observation
The following limits were set in order to assess the degree of homogeneity.
From the chart and equation (8), the total inhomogeneity was found.
Evaluation of symmetry in the network observation
For large geodetic leveling networks, equations (8) and (9) are not all true due to operational difficulties. The degree of symmetry was evaluated using equation (10). Let T be the sum of the product of the forward and backward observations in the entire network. From equation (10) T was calculated as 13. It should be noted that T is identical to folding the graphic chart along its zero element diagonal and finding the sum of the products of coinciding elements; that is a form of convolution of the backward observation with the forward observation.
Next, the larger of each pair of coinciding elements is sought, and squared and summed up. In this application, the elements in matrix are uniform; which is 1. Taking the square of 1 in thirteen placed, then summing the result will yield R.
Symmetry (S) is given by dividing the results of equations (10) by (11) and finding the square root using the equation (12)
Results and discussion
The observed leveling network over part Samaru Campus of ABU Zaria by Ojigi  consists of 8 stations, out of which two (CSZP79 and CSZP78) were the reference Benchmarks, while the other 6 located within the Campus were the object stations. Only 7 stations, made up of the 6 object points and CSZP78 were occupied with 26
leveling ties (fore and back) or loop observations.
Table 1: The graphic chart (Matrix) of the precise leveling network in parts of ABU,
Table 1 shows that in the precise leveling network, there are 26 ties and 13 links. The calculated completeness for the network was 0.619, and expressed in percentage to yield approximately 62% completeness. For ideal network the completeness is 1 (100%); which is practically impossible except in a very small network with little or no problems of time constraints, operational difficulties and project expenses.
The estimated degree of homogeneity for the network observation is zero; hence there is an absolute homogeneity of the network (H=0). This implies that all pairs of benchmarks were tied with equal number of ties (2 ties per pair of benchmarks).
The symmetry S of the observation was estimated from the graphic chart (Table 1) to be 1, which is the optimum value for and ideal network. This result is therefore considered as absolute symmetry of the precise leveling network observations in part of ABU, Samaru Zaria.
It is therefore of great relevance in surveying to have adequately measured and homogeneous network as a basis for accurate definition of control points for engineering construction, geodynamic investigations, urban development, land information management and mapping, etc This technique could be used for dual network signal propagation-delay in Electromagnetic Distance Measurement (EDM) system and other surveying applications. This study is a proof that graphic chart is a veritable and valid tool for leveling network quality control and analysis.
Carlson, E, Doyle, D; and Smith, D. (2009). Development of comprehensive geodetic vertical datums for the United States Pacific Territories of American Samoa, Guam, and the Northern Marianas. The National Society of Professional Surveyors, The American Association for Geodetic Surveying, and The Geographic and Land Information Society. Surveying and Land Information Science, Volume 69, Number 1, March 2009 , pp. 5-17(13).
Diesner, J. (2006). Dynamic network analysis (DNA) and automap. Carnegie Mellon University.
Kiviniemi, A. (1974). High precision measurements for studying the secular variation in
Lambert, A. and Beaumont, C., (1977). Nano variations in gravity due to seasonal groundwater movements: implications for the gravitational detection of tectonic movements. Journal of Geophysical Research, 82, pp. 297-306.
Osazuwa, B. I and Ajakaiye, E. D. (1986). The use of the graphic chart for the analysis of
Richardus, P., and Allman, S. J., (1977). Project surveying- general adjustment and
Snijders, T., (2006). The analysis of longitudinal social network data using SIENA program: Introduction to models for network dynamics and working with the SIENA program. Oxford: University of Oxford and University of Groningen
Surveying. Engineering. ASCE, Volume 133, Issue 3, pp. 114-122 (August)