Direct leveling or geometric leveling is a topographic operation carried out with the aid of a level and a staff, which makes it possible to determine the difference in level (commonly known as geometric height difference) between two points from horizontal sights taken on a staff. The calculation of heights from this operation is based on the knowledge of the level differences and the initial height [2].

This operation is often used in topography to perform a height adjustment, which requires a vertical reference network.

Of course, the level difference between two points does not depend directly on the path followed, unlike measured differences in level ( [3], p 40-p 41). So, is it possible to characterise this variable as a state function? A state function is a physical variable whose variation depends only on the initial and final states and not on the path followed ( [4], p 3).

The back and front readings vary with the path followed and the height of the station, whereas the difference in level along a chosen path depends on the back and front readings. Analysing the elevation potential, it becomes clear that it is not the elevation difference that is a constant but rather the potential difference ( [3], p 41).

On the other hand, considering this element as such, it would be interesting to apply Schwartz’s theorem on the total differential. The application of this formula on the level difference is given by equation number 1 below:

∂ 2 d n ∂ L a r ∂ L a v = ∂ ∂ L a r [ ∂ d n ∂ L a v ] = 1 ? (1)

Applying this formula actually gives a null value to the double differential: this shows that the gradient is not a state function. In a classical way, this difference in level remains a constant between two points. But in the case where one of the points would have undergone a movement then this difference in level becomes a variable function with the environment of measurement that we could estimate. Could a monitoring study within the framework of auscultation estimate in a particular way this temporal variation?

For a path between two points, there is a link between the backward and forward readings. There is a real such that:

l a r = l a v + ε (2)

ε is such that the height difference between these two points remains constant. Therefore, ε does not vary and remains constant.

Thus, the difference in level can be expressed as such:

d n = l a v + ε − l a v = ε = c s t e (3)

∂ ∂ d n ∂ L a r ∂ L a v = 0 (4)

The contradiction between (1) and (4) is an actual limit of direct leveling.

Satellite positioning systems have made a great contribution to the accurate determination of points on the earth’s surface. However, the determination of the altimeter component was one of the limitations of this system, as it could only measure the height relative to the associated ellipsoid. This did not correspond to the physical quantity (altitude) that users were interested in. It was not until the development of geoid models that could be integrated into GNSS receivers or calculation software to obtain heights from these measurements and the undulation provided by the model. Several models have been implemented such as EGM 96 and EGM08.

The functions used to determine the ripples are calculated according to harmonic models [5] [6] [7] [8].

The ripples associated with the EGM08 model are calculated on each of the grid nodes with harmonic functions expandable to degrees n. ( [3], p 44). The following ripple formula can be used:

N = T γ = G M γ ∑ n = 2 ∞ a e n r n + 1 ∑ m = 0 n P n , m ( sin φ ) ( C n , m ∗ cos m λ + S n , m ∗ sin m λ ) (5)

The study area is located in Thies, more precisely in the HLM district of Mbour, just to the right of the road leading to Mbour at the level of the Lat. Dior stadium. The geographical coordinates of the study area vary in longitude between −16.945˚ and −16.950˚ and in latitude between 14.774˚ and 14.776˚.

The study area is illustrated in Figure 1 and Figure 2.

GNSS observation in fast static mode is becoming increasingly easy ( [9], p 462). In addition to the accuracy, these modes require a limited number of operators (one to two people) and can be performed with limited resources. The use of the fast static mode requires more time in some measurement jobs where the number of points to be measured is high or with relatively large baselines. Indeed, resolving ambiguities on each new point requires a considerable amount of time to observe each point and cannot be used for economic reasons on applications such as longitudinal profile measurement or DTM measurement. The use of static and kinematic mode provides speed and accuracy in the realisation of GNSS base points [10]. It is also possible to produce the base map in altimetry as well as in planimetry, both during the day and at night, if sufficient satellites with good geometry are available. The processing of the observations is done with GNSS calculation software and the execution of the calculations is done in a few minutes. However, GNSS methods have some disadvantages: dependence on the measurement environment, dependence on external structures (the constellation), accuracy depending on the length of the baseline, working only externally, etc. For this step, four points were observed by the fast static method and the post-processing was done with the Leica geo office software. The observation time was fifteen minutes on each point and this choice was strongly dependent on the length of the baselines.

Table 1 and Table 2 list the variation of coordinates in meters and degrees respectively. Also, the results of these tables are illustrated in Figure 3 and Figure 4 respectively.

In this section, the graph shows that small (millimeter) deviations were obtained between the RTK and fast static solutions. These small differences in our case are mainly due to the short baselines obtained. These could quickly reach the centimeter (or even a few centimeters) if the baselines exceeded ten kilometers. This could make fast static positioning acceptable in contrast to RTK. It would therefore be better to proceed with the central pivot method, which would consist of creating and calculating a first station centred in the study area by static or fast static methods (depending on the baseline). This station would then serve as a pivot for the different observations in fast static mode or in RTK (if the baselines are weak) as in our case. This would guarantee a certain accuracy and speed.

For this phase, a framed polygon between points K3 and K2 was made. To do this, angle and distance measurements were made on each vertex of the polygon. The result is tabulated in Table 3.

The advantage of this method is that it has the particularity of being the possible technique for making a canvas when the conditions for using GNSS are not met, a possibility for working indoors as well as outdoors. It usually requires a team of three people, including a chief surveyor, an operator, a survey assistant and possibly a driver.

The disadvantage of this method is that it requires a lot of time for execution and precision depending on the length of the sides of the polygonal. Also, it is

only carried out during the day, a restriction due to the total station. It also requires the knowledge of two or more landmarks. Also, it has a low dependency on the measurement environment, some interoperability problems (Prism Constant), independence from external structures. It is generally more easily influenced by certain sources of error.

After the field phase, the raw data has to be compensated when the closure is below the tolerance. This processing can be done with topometric software, by hand calculation or in Excel. This processing can take several tens of minutes compared to the GNSS method. Table 4 lists the validation of the polygonal.

The financial costs for the realization of this method are variable and depend on the number of staff and the execution time as well as on the expected accuracy.

Table 5 gives the results of the compensated polygonal.

The method adopted for connecting the points of the polygon is direct leveling. This method has the following advantages: spontaneous reading of the difference in level, ease of implementation, speed of measurement and millimeter accuracy.

The disadvantages of this method are the limitation of the ranges due to the instrument used, the dependence on the measuring environment, a problem of visibility between two successive measuring points and numerous stations when the points are far apart.

This leveling operation will make it possible to find the altitude of the base points in the study area. It will allow comparison of the variations in undulations deduced by post-processing and by RTK.

A closed path is applied to point TH02. After completing this path, which contains point K1, another closed path is performed around K1 to find the altitude of the post-processed points.

Table 8 gives the elements for calculating the tolerance according to the type of canvas. Once the type of canvas is chosen, the dimension of point K1 is determined from POINT TH02. This path is shown in Table 9.

From point K1, the heights of the other points are found. This path is given in Table 10 and summarized in Table 11.

Table 11 summarizes the different heights of the points.