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M. L. Ojigi
Department of Surveying and Geoinformatics, Federal University of Technology, Minna, NigeriaE-E-mail:

Relative measurements made between any pair of Benchmarks (PB1 and PB2) in a precise leveling network often involves at least one forward and one backward observation in a PB1→PB2, PB2→PB1 sequence respectively. However, due to fatigue on the part of the surveyors and field assistants, hazards along geodetic survey routes and some times attempt to reduce financial cost of field surveys, observations made in one 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 ensure completeness and geometrical strength and reliability in a survey network, analysis by graphic chart is relevant. This paper used graphic chart technique to analyse and evaluate the completeness, homogeneity and symmetry of a precise leveling network over part of Samaru Campus of Ahmadu Bello University, Zaria. The results show about 62% completeness, absolute homogeneity (H=0) and absolute symmetry (100%). It is recommended that network analysis for all precise leveling networks be carried out prior the rigorous adjustment computations in order to detect any missing links and ties in the network and to improve the network reliability and accuracy. Engineering and applied science problems that require a two-way measurement or feedback system could benefit tremendously from this technique.

Keywords: Precise leveling, network, graphic chart, analysis

In precise surveying engineering applications, observation networks are designed and established in order to have a framework for accomplishing the purpose(s) of such projects. The established network has to satisfy certain geometrical and observational requirements for the purpose of reliability and accuracy.
Some features in a network such as observational completeness, homogeneity and symmetry of observations (including geodetic or precise leveling) can be evaluated with graphic chart [Osazuwa & Ajakaiye, 1986]. In precise leveling, relative measurements made between any pair of Benchmarks (PB1 and PB2) in a precise leveling network often involves at least one forward and one backward observation in a PB1→PB2, PB2→PB1 sequence respectively.
Precise leveling operates the principles similar to gravity measurement, and Kiviniemi, [1974)] observed that, the PB1→PB2→PB1 observations in one drift control, carried out on the same day constitute a double measurement, while PB1→PB2 or PB2→PB1 observation, without drift control constitutes a single measurement.

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 [1986], in and ideal network, observation must be complete, homogeneous and symmetrical. Lambert and Beaumont, [1977] 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 [1986], 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, [1977] 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, [2008] 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
For a complete observation of a network of n stations, the total number of links (U), is given by 

            dfdff                                                                    (1),

where adfd 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 study has the following objectives:

  1. To show the concept and design of graphic chart in the analysis of precise leveling network observation
  2. To evaluate the observational completeness of the leveling ties and links of the network
  3. To estimate the homogeneity of the network observation
  4. To evaluate the symmetry of the network links.

Study area
The study area is the campus of the Ahmadu Bello University (ABU), Zaria, located at Samaru along Zaria-Funtua Road, Kaduna State of Nigeria. The Samaru Campus of ABU (the study area) is made of ten (10) faculties, which include: Faculties of Medicine, Engineering, Environmental Design, Art, Social Sciences, Pharmaceutical Sciences, Vertinary Medicine, Science, Agriculture and Education.

fadfd                fadaafd
Figure 1: Kaduna State and Zaria Local Government Area in Nigeria

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.

The design of the graphic chart
The graphic chart of the leveling network was designed into an n x n matrixadfd, with the stations arranged in sequence. Other elements,adfadf, which may be zero or greater than zero, represent the number of ties between the two stations i and j, which  are defined by (2)

              adfdf                                                                                       (2)

adfdfdIf adafadfasdf, then there is no tie between stations i and j. We must be consistent in determining the elements of matrix Â; hence adfdf  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.
               [Source: Modified after Ojigi, 1993]

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, [1986] as:

                          afafd                                                                             (3)
For an ideal network and Ni = Nj, but for a real network Ni   may or may not be equal to Nj, but
               adafd                                                                              (4)

where N is the total number of ties in the network. In order to determine the number of links in the network, the matrix fa is transformed to adfada such that the elements adf of adfadfa are given by

adad            a for  dfdf  or adfdaff
            afadfaf for adada

Therefore, considering the upper right triangle of the leveling graphic chart only, the number of links adfad between Benchmarks is

          fadfaf                                                                                   (6)

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:

       adfadf                                                                            (7)

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 value of homogeneity was estimated from the set of observation of the leveling network. The homogeneity can only be related to the actual links in the network as against the case of completeness; which was determined in relation to the observable links in the ideal network (equation 7). Let H be the total inhomogeneity in the network, the H can be defined as

   adad                                                        (8)

The following limits were set in order to assess the degree of homogeneity.

  1. For absolute homogeneous observations, H=0
  2. For sufficiently homogeneous observations,  afadfa
  1. For poorly homogeneous observations, adfada

From the chart and equation (8), the total inhomogeneity was found.

Evaluation of symmetry in the network observation
The condition for symmetry in network observations is that   adfadfa. Therefore modifying equation (3) into the form of (9) solves the problem.


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 chartada 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.

            adfadfa                                                                             (10)

Next, the larger of each pair of coinciding elements is sought, and squared and summed up. In this application, the elements in matrix  d are uniform; which is 1. Taking the square of 1 in thirteen placed, then summing the result will yield R.

              adfdfa                                                                                (11)

Symmetry (S) is given by dividing the results of equations (10) by (11) and finding the square root using the equation (12)
                 adada                                                                                                  (12)


Results and discussion

The observed leveling network over part Samaru Campus of ABU Zaria by Ojigi [1993] 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.
Therefore, the forward measurements constitute the upper right triangle in matrix  ada , while leveling measurements in backward direct constitute the lower left triangle in matrix adadfa  (see Table 1). The diagonal elements of adfadf  are all zero because a benchmark cannot be tied to itself.

Table 1: The graphic chart (Matrixadfdffa) of the precise leveling network in parts of ABU,
                Samaru , Zaria.


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. 

The network completeness, homogeneity and symmetry estimation and evaluation carried out in this study by the use of graphic chart are three essential factors that determine how well a leveling survey network is controlled. However, the 62% completeness obtained in this study is not absolute, but could be regarded as sufficiently good result. This result could only be enhanced by establishing more links between the benchmarks in the leveling network. The absolute estimates for homogeneity and symmetry of the leveling network shows that the network benchmarks were sufficiently observed with enough redundant links or leveling lines; which will eventually strengthen the determination of the network controls during the adjustment process.

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.
The following recommendations are hereby made for the furtherance of purposeful application of graphic chart in network analysis.

  1. This technique should be used as a priori quality control and assurance test for all precise leveling projects, and engineering and applied science problems that requires a two-way measurement or feedback system, in order to detect any missing links and ties in the network and to improve the network reliability and accuracy.
  2. Redundant measurement should be ensured in a precise leveling network in order to secure absolute homogeneity and symmetry in the network observation.
  3. Step-down controls between major control points and benchmark should be considered a necessity in order to achieve adequate completeness in the network before the commencement of rigorous adjustment computation.
  4. Any link in a precise leveling network not observed in a fore and back mode should be counted as an incomplete observation; hence should not be used in the network adjustment.

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