SETTOP LEVEL ME: Precision trigonometric leveling

GNSS & Geocom Optics

 

Trigonometric leveling is a technique used in surveying applications to determine elevation differences between two points using angular measurements and distances. Unlike geometric leveling, which determines elevation differences from readings on a leveling staff, trigonometric leveling uses trigonometry to indirectly calculate the height difference.

 

Which one should I use?

The decision of which type of leveling to employ warrants extensive analysis. However, a starting point could be associated with the presence of systematic and random errors (obviously, we exclude gross errors; a 3-sigma rule can help us detect them). So, what errors would be present? According to Ghilani 2017, the systematic errors found in both techniques can be summarized as the effects of Earth curvature, atmospheric refraction, and instrument misadjustment. In all cases, appropriate field procedures (e.g., high-precision geometric leveling criteria), mathematical models (velocity corrections in total stations), and good maintenance can control their effect on observations. Therefore, we should only be left with the random component, which includes errors such as instrument leveling, distance observations, and readings on the staff. An extensive analysis and its corresponding error propagation can be found in Ghilani 2017.

An important aspect that differentiates both methods is the determination of elevation differences. As mentioned before, in geometric leveling, the mathematical operation is approached by addition and subtraction, involving fewer parameters compared to trigonometric leveling, which already considers directions and distances, thus using trigonometry. Sometimes, this difference is associated with better performance and results from geometric leveling. So, is all lost with trigonometric leveling? The answer is NO! and the reason is; LevelMe, but before introducing this solution, let's review some concepts about trigonometric leveling.

Figure 1 shows the schematic of trigonometric leveling.

Figure 1. Trigonometric leveling schematic

From Figure 1, z  corresponds to the observed zenith angle on both faces of the total station (direct and reversed), and Di corresponds to the inclined distance. Considering the errors defined by Ghilani 2017, we could propose the following equation for trigonometric leveling:

In addition to the parameters already defined, we add the instrument height (hi), prism height (hr), Earth's curvature (hcr), and the refraction component (k) (treated as a correction). It is also important to consider elements such as instrument and prism centering, as well as instrumental measurements. 

 

How does Trimble Access solve trigonometric elevation difference?

Initially, let's consider distance. An electromagnetic wave propagates in a medium dependent on pressure, temperature, and humidity, with the latter having the least effect. Thus, a first correction is applied, considering first the reflector constant (prism), and then applying the atmospheric correction. Let's consider that if the wave changes its speed, it affects the determination of distances, therefore, it is key to apply these parameters (theoretically corresponding to the velocity correction). Figure 2 shows the configuration panel from Trimble Access.

 

Figure 2. Atmospheric parameter configuration

 

Once the prism correction and atmospheric parameters are calculated, we can determine the inclined distance Di.

In a second stage, we can analyze curvature and refraction constant corrections. For the first case, Figure 3 schematizes the curvature, which affects the determination of directions.

 

Figure 3. Curvature schematic

 

In the case of atmospheric refraction, its concentric stratification allows for a combined correction with curvature, therefore the model is reduced as follows:

  

From equation 2, we have:

• V2: angle corrected for curvature and refraction
• V1: uncorrected angle
• COnOff: if curvature correction is active, its value is 1; if not active, it is 0.
• ROnOff: if refraction correction is active, its value is 1; if not active, it is 0.
• k: refraction constant (the book Electronic Distance Measurement: An Introduction provides a good explanation for calculating k)
• R: Radius (semi-major axis of the WGS84 ellipsoid 6378137)
• Di: Inclined distance

 

Having presented the configurations for trigonometric leveling, what features do the S-series total stations have for working with trigonometric leveling?

 

Autolock and MagDrive

Autolock can be defined as a technological innovation that automates the process of positioning the collimation axis at the center of the prism, locking onto it, and allowing its tracking. Hand-in-hand with Autolock is the automation of direction determination using servo-controlled motors, which is where MagDrive technology comes in. Figure 4 shows automated prism tracking using Autolock and MagDrive technology.

 

Figure 4. Autolock in operation

 

Having presented the theoretical characteristics and technological capabilities of Trimble S-series total stations, we can delve into LevelMe.

 

LevelMe is an application that allows vertical observation, calculation, and compensation using precision trigonometric leveling with TRIMBLE S Series total stations. So, how is leveling performed with LevelMe? Here, we can identify two key elements: Loop and Leveling. Figure 5 schematizes the sequence of observations performed in LevelMe.

 

Figure 5. LevelMe workflow schematic

 

From Figure 5, we can observe the LevelMe observation methodology. Here, we find vertical direction readings (z1 and z2) as well as inclined distance measurements (Di1, Di2), the presence of the hj and hi readings (back and front), and finally, it is possible to determine the elevation difference Δh.

 

Loop

A loop is considered a back-and-forth leveling line that closes at the same point. It is also possible to close at a different point with a known elevation, considering it an open leveling line with control. Figure 6 shows the general configuration options.

 

Figure 6. New Loop

 

Once a Loop is created, it is possible to start a Leveling. Here, the observation methodology is summarized in measurements towards "Back" and "Front," that is, back-sight and fore-sight readings. When taking the measurement, LevelMe automatically provides the observation direction, reducing errors and optimizing field time (Figure 7 and Figure 8).

 

Figure 7. Measurement direction, reference point name, elevation

 

Figure 8. Measurement direction

 

As preliminary results, we can "review" a Loop, which presents information related to the survey (Figure 9).

 

Figure 9. Observations made using a Loop in LevelMe

 

From Figure 9, we find information related to the setup point, target height (prism), vertical angle, inclined distance, and the corresponding elevation. Here we can find information related to instrument and prism heights, as well as prism constants or starting elevation.

So, how do we get results with LevelMe? The answer lies in the Close Line function (Figure 10).

 

Figure 10. Close loop

As a result, we find a summary table with the results of the leveling line (Figure 11).

 

Figure 11. Line closure in LevelMe

 

From Figure 11, we find information related to leveling, tolerances, and adjusted observations. The next step is to export the results. Here, we can choose a classic view, meaning the results provided by LevelMe, or integrate these observations into a least squares adjustment in Trimble Business Center. Figure 12 presents the export options.

 

Figure 12. Export options in LevelMe

 

As a result for the "Full format", we have a summary of compensated observations and their corresponding elevations (Figure 13).

 

Figure 13. "Full format" export

 

If we select the Dini M5 option in the options of Figure 12, we already have a file that can be incorporated into TBC for analysis and processing using least squares. Figure 14 shows the import.

 

Figure 14. Dini M5 file from trigonometric leveling with LevelMe

 

The procedure for processing leveling lines using Trimble Business Center can be found in our course TBC terrestrial network processing. Thus, as results, we could obtain a report of the leveling line adjustment (Figure 15).

 

Figure 15. Elevations adjusted by least squares

 

Experience of leveling line adjustment in TBC

In order to evaluate the characteristics of LevelMe, an experiment was conducted with geometric and trigonometric leveling points. Having the horizontal coordinates, it was possible to graphically represent the adjusted leveling line (Figure 16).

 

Figure 16. Leveling line – plan view

 

Regarding this example, a comparison was made with a geometric leveling line. The results are presented in Figure 17.

 

Figure 17. Comparison of trigonometric and geometric elevation differences

 

From the table in Figure 16, we find millimeter-level differences. In that sense, how could we estimate the performance of LevelMe? Figure 18 presents the error per kilometer leveled trigonometrically, based on the nominal precisions of an S-series total station (0.5° and 1 mm + 1 ppm) and the distances between sight lines.

 

Figure 18. Error per kilometer leveled trigonometrically based on the nominal precisions of an S-series total station (0.5° and 1 mm + 1 ppm) and the distances between sight lines

 

Analysis and conclusions

In Figure 17, we have an inverse relationship between the distance between sightings and the error per kilometer (similar to the fundamentals of precision geometric leveling), which can be seen in the graph in Figure 18. There is also a direct relationship between the nominal directional accuracy of the instrument and the error per kilometer. This is one of LevelMe's advantages, in addition to the data comparison in Figure 15, where, on average, the differences between trigonometric and geometric elevation differences are millimetric. There is an optimization in field procedures while keeping the error per kilometer controlled, as shown in the table in Figure 18, where a 1" accuracy station in directions provides only two tenths of a millimeter difference in the error per kilometer if 100 m sightings are used instead of 25 m sightings, which implies a 4-fold optimization of the distance between sightings.

While LevelMe does not seek to replace geometric leveling as such, it presents itself as an alternative for engineering projects that seek to optimize their process without losing precision, all through an automated workflow in the field and with multiple possibilities for interoperability with geodetic data analysis software such as TBC.

 

References

Ghilani 2017, Adjustment Computations and Spatial Analysis

Rüeger 1980, Electronic Distance Measurement: An Introduction

Trimble Access Calculations, 2023