Introduction

In geodesy, the length element known as distance must be positioned on a reference surface to be quantified. The main problem is that these surfaces change depending on the effect that needs to be established. This is how the problem is simplified by using horizontal distance as the main element for the development of engineering and construction activities. However, there are other reference surfaces, such as the ellipsoid and the cartographic projections themselves, which, although they deform the horizontal distance, still represent the same positions.

With the use of GNSS, this problem is completely overcome since, due to its nature, it determines geocentric coordinates which are converted to geodetic coordinates and then to projected coordinates, achieving a transition without loss of representation. However, total stations measure inclined distances that must be reduced to one of the three reference surfaces described above: horizontal, ellipsoid, or cartographic projection. Therefore, this effect must be considered in the daily use of total stations.

 

Some definitions

  • Horizontal distance: Distance between two points measured on a horizontal plane (Dh)
  • Sloping distance: Distance measured along an inclined plane (Di)
  • Projected distance: Distance referred to a cartographic projection (Dc)
  • Geodetic distance: Distance between two points on a geodetic reference surface (ellipsoid) (De)

Fig 1. Distance reduction

When evaluating distances, it is identified that the distance between two points is different depending on the surface used as a reference. This is how the compatibility of distances is addressed by means of a scale factor, which can mathematically relate the magnitudes obtained.

Initially, the following stages can be considered:

For a distance observed in the field using a total station, the following can be considered:

  • Electronic distance: first distance provided by the distantiometer of a total station.
  • Sloping distance: electronic distance corrected by atmospheric parameters and prism constant, plus curvature corrections.
  • Horizontal distance: obtained from the sloping distance using a classical approach associated with topographic methodologies.
  • Distance reduced to the geodetic chord
  • Geodetic distance
  • Projected distance: use of a cartographic projection such as UTM, LTM or PTL.

 

Relationship between horizontal and projected distances

A widely used scale factor in engineering is that which relates projected and horizontal distances; this factor, called the combined scale factor, works with the UTM projection (0.9996) and the height scale factor. Its approach is presented below (variables defined previously):

Fig 2. Simplification of distance reduction

 

Case 1: use of a unit scale factor

In most cases when using a total station, a unit scale is used. This means that distances will not be modified by the scale module:

Fig 3. Configuring a job in Trimble Access with a unit scale factor

     For example, an observation is made as indicated in the following figure:

    Fig 4. Observation carried out with a total station

     

    This results in the following:

     

    Fig 5. Calculations performed by the total station after the observation

     

    If the observation is accepted, the point will be established through the determined coordinates.

    If the inverse calculation is applied, the horizontal distance and the elevation determined previously can be verified:

    Fig 6. Inverse problem applied on Trimble Access

     

    Conclusion Case 1

    The grid components (cartographic projection), ellipsoid (geodetic distance) and terrain (horizontal distance) do not vary due to the definition of the coordinate system used.

    Fig 7. Choice of reference surface for distance reduction

     

    Case 2: use of cartographic projection

    A coordinate system will be configured according to the UTM projection with an average height of 900 m:

    Fig 8. UTM cartographic projection configuration under SIRGAS-Chile 2021.00

     

    In the creation of the job, it will look like this:

    Fig 9. Configuring a job in Trimble Access with a scale factor associated with the cartographic projection

     

    A new observation is made:

    Fig 10. Observation carried out with a total station

    Fig 11. Calculations performed by the total station after the observation

     

    In the same way as before, the inverse calculation of the Trimble Access COGO functions is applied:

    Fig 12. Inverse problem applied on Trimble Access

     

    In this case, it is possible to change the reference surface:

    Fig 13. Distances calculated with different reference surface definitions

     

    It is possible to appreciate the change between distances by modifying the reference surface.

    Now, it is important to clarify that the coordinate calculation is based on the grid distance, having configured the cartographic projection initially.

     Fig 14. Inverse problem applied on Trimble Business Center

    Case 2: use of cartographic projection

    The grid components (cartographic projection), ellipsoid (geodetic distance) and terrain (horizontal distance) experience changes due to the influence of the cartographic scale factor and the elevation scale factor, which originate the combined scale factor. However, the coordinates are calculated on the plane of the cartographic projection, that is, using the grid (or projected) distance, which is fully compatible with the use of GNSS.

     

    Final conclusion

    The grid components (cartographic projection), ellipsoid (geodetic distance) and terrain (horizontal distance) experience changes due to the influence of the cartographic scale factor and the elevation scale factor, which originate the combined scale factor. However, the coordinates are calculated on the plane of the cartographic projection, that is, using the grid (or projected) distance, which is fully compatible with the use of GNSS.