TRIMBLE BUSINESS CENTER: Geometric Leveling

GNSS & Optical Geocom

 

Geometric leveling is a technique widely used by engineers and surveyors for determining highly precise elevation differences through vertical observations to a leveling rod (graduated staff). Geometric leveling is performed with an instrument called an automatic level, which produces a level surface intrinsically linked to the direction of the force of gravity, serving as a reference for determining height differences. This technique is fundamental for densifying precise heights with various applications (geodetic control networks, construction, etc.)

Trimble Business Center, on the other hand, allows managing all geometric leveling data using automatic levels, whether optical or digital, accessing automatic calculation and reporting tools with high productivity.

 

Basic theory of geometric leveling

Basically, geometric leveling refers to a difference in vertical distances determined by multiple level surfaces, which are conveniently achieved at each instrument setup.

Due to a simplification made for surveying purposes, the level surface is replaced by the horizontal plane produced tangentially at the intersection of the collimation axis with the vertical axis of the automatic level. In this way, the equations are extremely simple to apply. However, this simplification is not always possible to consider, especially when it is required to densify heights over long distances.

Thus, the elevation difference is calculated using the following equation:



where  and  are the back and fore readings respectively.

The previous equation considers two assumptions: the first is that the automatic level is a perfect instrument (without collimation error) and the second relates to the effect of the earth's curvature and refraction on the elevation difference. Without going into too much detail, the equation that expresses the elevation difference considering all the above is the following:



where  and  are the back and fore readings respectively,  is the collimation error of the automatic level,  and  are the back and fore distances, and  is the radius of curvature of the Earth.

As can be seen in the equation, if the sum of back distances is equal to the sum of fore distances, the two final terms of the equation become zero. For this reason, it is recommended that the distance between the automatic level and the rods be equal for both readings. However, this situation is difficult to reproduce in the field, so, strictly speaking, it is necessary to know both the distances and the collimation error of the level at all times.

 

 Figure 1. Level surface and horizontal surface

 

An interesting fact about this theory is that the name "automatic level" comes precisely from this instrument's ability to produce a level surface on its own, based on the interaction of the compensator (air or magnetic) within a defined range (usually 15').

 

Differences between an optical and a digital automatic level

While a digital automatic level allows for an automatic reading of a rod (through the recognition of a pattern captured by a CCD sensor), with an optical automatic level, it is necessary to directly observe the graduated staff. This is the main difference between the two instruments, which highlights their productivity difference: while a digital level automatically reads a coded rod and stores the data, with an optical level, this process depends on the operator's skill.

With a digital level, it is only required that the pattern recognition be as reliable as possible. This is achieved in optimal light conditions by attempting to take the reading in areas where there is no shadow. Now, an interesting collateral effect is that in the pattern identification process, the digital level observes all the stadia hairs (upper, middle, and lower), which provides height differences between the rods and the distances between the instrument and the rods. With all this information, it is entirely possible to apply the complete mathematical model of the elevation difference (having previously determined the instrument's collimation error).

 

Figure 2. Trimble DiNi (view product)

 

Trimble DiNi and the DAT format

Trimble DiNi is a digital automatic level available in two models with accuracies of 0.3 and 0.7 mm over a 1 km leveling line. Trimble DiNi takes observations on coded rods, which are manufactured in invar or aluminum (single piece), fiberglass (foldable or telescopic), or as adhesive tapes for monitoring applications.

Figure 3. Coded rods for Trimble DiNi

 

The use of each type of rod primarily responds to the required precision of the elevation difference and, secondarily, to productivity-related issues.

Now, Trimble DiNi produces a file containing all the observations made (readings on the rod, horizontal distances, time, temperature, etc.). This file is known as M5 and has a DAT extension.

 

Figure 4. Structure of the Trimble DiNi M5 format

 

This file contains the vertical observation on the rod along with the horizontal distance, as well as other data of interest. Finally, once this data is imported into TBC, a geometric leveling record is automatically created, which can be reviewed and edited by the user:

 

Figure 5. Geometric leveling record in TBC

 

How to import automatic level data into TBC?

If you have a DAT file from Trimble DiNi, simply drag it to TBC to start working. Otherwise, if you have a manually written record, it must be transcribed into TBC using the level editor following the steps (also applies to editing digital level data):

Creating a new run: First of all, you must indicate how many setups were performed with the level to prepare the digital record. Also, you must specify the method of leveling: back and fore observation, fore and back observation, or double instrumental position.

 

Figure 6. New run in the level editor

 

Data entry: A digital record is enabled where the point name, back observation, fore observation, horizontal distance, and point type (reference or calculated) must be indicated. Of course, in this section, the back and fore staff readings are entered, but it is also necessary to indicate the distance (even if approximate). The latter is important for the least squares adjustment process. Another important point has to do with the difference between a set point and a turning point: this is activated to create a point (left side of the window) or deactivated when it is a turning point.

 

Figure 7. Level editor

 

Graphical representation of elevation differences in TBC

To graphically represent the elevation differences in TBC, it is necessary to indicate the horizontal position of the points for which heights are being determined. While this is not strictly necessary, the graphical visualization of elevation differences is a great help in verifying the vertical network or the set of elevation differences.

In addition, the graphical visualization of elevation differences allows for direct selection of them, making it possible to review the sum of readings, applied corrections, number of setups, and precision.

 

Figure 8. Elevation difference in TBC

 

Preparation for least squares adjustment

It is essential to define the precision with which a geodetic observation has been made in order to evaluate, a priori, a scalar called reference variance. This is extremely relevant for evaluating the precision of the determined heights.

TBC provides two possibilities for determining the reference variance:

• Error for 1 km of leveling: This is the precision specification with which all automatic levels operate. It is evaluated for a particular elevation difference using , where  is the level's precision and  is the distance in kilometers of the leveling route. For example, for the data displayed in Figure 8, there is a 1.44 km route with a precision of 1mm/km, which results in a precision of 1.2 mm at 68% (multiplied by 2 to reach 95%).

 

• Error per setup: calculated exactly the same as in the previous case, but taking into account that all distances between the level and the rod are similar for all setups. In this way, the precision of the elevation difference is given by the propagation of errors in each setup.

 

Figure 9. Precisions for geometric leveling

 

Calculation example

There are 140 km of geometric leveling from more than 1200 setups, in which 32 elevation differences were observed for 17 points, 1 of which is fixed (known height).

 

Figure 10. Vertical network in TBC

 

Regarding reporting, by selecting an elevation difference, it is possible to obtain a report in the style of a geometric leveling record:

 

Figure 11. Geometric leveling report in TBC

 

In this case, the elevation differences between points were observed without the possibility of checking the closing error. To do this, both elevation difference determinations are combined:

 

Figure 12. Determination of the closing error for a leveling loop

 

Finally, the least squares adjustment of the network is performed:

 

Figure 13. Least squares adjustment

 

Figure 14. Least squares adjustment report

 

From Figures 13 and 14, the post-adjustment results statistically validate the elevations obtained. Firstly, the reference factor and the chi-square test provide an overview of the adjustment process, particularly the adoption of suitable weights and the chi-square hypothesis test do not indicate any errors in the observations (elevation differences). In turn, the identification of outliers using the Tau test validates the obtained elevation differences regarding possible errors in them.

 

Conclusions

Geometric leveling is traditionally associated with a high-precision, costly process that involves extensive logistics for its execution. While some of these aspects are intrinsic to this technique, it is observed that optimizing resources in the field, by reducing gross or systematic errors, contributes to improving the quality of the results. This is evident through the automation of elevation difference capture in the field (Figure 4) and the automatic observation methodology in the case of digital levels.

Another important aspect is data processing, whether in leveling lines or leveling networks. Traditional methods for "distributing" errors and subsequently providing elevations are presented under different approaches, which can make their application somewhat arbitrary. In contrast, the use of least squares provides a single approach extensible to networks of different dimensions, i.e., not only geometric leveling.

Finally, an automated process, such as the one presented, optimizes field operation aspects such as data capture. In turn, the processing performed in TBC allows for providing elevations with precision associated with a certain confidence level.