### INTRODUCTION

For GNSS-derived ellipsoidal heights to have any physical meaning in a surveying or engineering application, they must be transformed into orthometric or quasi-orthometric heights. The classical method for doing such a transformation is based on the following equation H = h – N, where H is the orthometric height to be determined, h is the ellipsoidal height measured by GNSS, and N is the geoidal height (undulation) (Hofmann-Wallenhof, B., Moritz H. 2005).

The GNSS Leveling method is much more accurate than the classical method since GNSS Leveling substantially reduces the errors associated with the geoidal model. In addition, GNSS Leveling can reduce the time consumed and required to perform classic geometric leveling, this is more evident at distances greater than 10 km.  (Torge W., Muller J. 2012).

GNSS Leveling consists of determining (measuring and calculating) the orthometric height H (or quasi-orthometric) by GNSS measurements in one or more GNSS rover stations, with knowing the orthometric height H (or quasi-orthometric) in the GNSS base station, and measuring the ellipsoidal heights using high-precision GNSS in each station (base and rovers). The geoidal height (undulation) must be calculated or interpolated at both (base and rovers).

The ellipsoidal heights measured by GNSS can be performed under any modality as long their results are highly accurate: Differential / PPP; Real-time/post-processing; Static/Kinematic; simultaneous/non-simultaneous.

This article uses the term “base” to refer to the station where the orthometric height (Hb) is well-known. The term “rover” refers to the remote station where the orthometric height is unknown, and its (Hr) will be calculated.

### CONSIDERATIONS

• GNSS leveling could be understood as a geodesic artifice applied in an operational context to increase geodetic engineering operations’ efficiency.
• GNSS measurements and the geoidal model must refer to the same horizontal and vertical geodetic datum. A zero-height surface is assumed to be the same reference for all the heights involved (Torge W., Muller J. 2012).
• Each geoid model (N) has errors (ɛ) that may vary along its entire surface. Therefore, it is recommended to use GNSS Leveling within a distance range where the error (ɛ) of the geoidal model remains constant or with negligible variations.
• It is recommended to use at least 2 BM (reliable and stable) close to the project area to establish minimum quality control.
• In areas of constant geodynamic activity or affected by vertical deformations (subsidence or uplift), it is recommended to increase the number of BMs and use base stations out of the affected area.

### INPUT DATA AND OBSERVATIONS

• Hb: Known orthometric height at the GNSS base.
• Hb can be a BM or a point with orthometric height known by classical leveling.

• hb: Ellipsoidal height measured by GNSS at the base.
• hr: Ellipsoidal height measured by GNSS in all rovers.
• hb and hr, must be measured by GNSS with the highest possible accuracy.
• Regardless of the GNSS measurement method used, the important thing is to determine the ellipsoidal heights with sufficient precision and accuracy required for the project in question.
• Logically, both the base and the GNSS rover must meet the minimum conditions for high-precision GNSS measurements.

• Nb: Interpolated geoidal height (undulation) at the base.
• Nr: Interpolated geoidal height (undulation) at all rovers.
• Nb and Nr, must be extracted or interpolated from the best geoidal model available for the area.

### MATH MODEL

• ΔH = Δh – ΔN (1) (Torge W., Muller J. 2012);

• where:

ΔH = Hr – Hb (2)

Δh = hr – hb (3)

ΔN = Nr – Nb (4)

• Replacing at (1):

(Hr – Hb) = (hr – hb) – (Nr – Nb) (5) (Hofmann-Wallenhof, B., Moritz H. 2005).

• Calculating Hr:

Hr = (hr – hb) – (Nr – Nb) + Hb (6) (Hofmann-Wallenhof, B., Moritz H. 2005).

### VERIFICATION

• For quality control, it is recommended to calculate the orthometric height (Hr) (described in equation 6) in a second nearby BM to verify the quality of the results.
• If the ellipsoidal heights measured with GNSS are sufficiently precise, the quality control closure at the calculated orthometric heights should be similarly accurate.

### RESULTS IN THE UNITED STATES OF AMERICA (USA)

• From the National Geodetic Survey (NGS) data base, 643 BMs (1st order leveling network) were carefully selected. Those BMs have also already been measured by GNSS, class A (NGS. 2022).
• The selected BMs are distributed throughout the conterminous US (without Alaska state). In addition, all are in good condition and stable.
• All these BMs are referenced to NAVD 88 (Zilkoski, D. B., et al. 1992), and its horizontal datum is NAD 83(2011) (NGS. 2012).
• For all BMs, the geoid undulation (Ni) was calculated from the geoid model GEOID 18, referenced to NAVD 88 in the conterminous US.
• The geoidal heights (Ni) were calculated for each BM using the most recent US geoid model, GEOID 18 (NGS. 2020).
• Now, the three types of heights are known for each BM: orthometric height by classical leveling (Ho), ellipsoidal height by GNSS (h), and geoidal height (Ni).
• The 643 BMs generated 206.403 unique combinations of GNSS Leveling lines, regardless of their geographic location. In other words, for each BM, its orthometric height (Hr) was calculated 642 times from the rest of the BMs by using equation 6.
• The distances between BMs were: Min: 9 km., Max: 4.549 km., Average: 1.834 km., Mode: 1.027 km.
• The slopes between BMs were: Min: 0.002 m., Max: 3.519 m., Average: 792 m., Mode: 127 m.
• The orthometric heights (Hr) derived from GNSS Leveling were compared against their respective classical orthometric heights (Ho) values to estimate the errors and method reliability.
• The orthometric heights differences between (Hr) and (Ho) were: Min: -0,384 m., Max: 0,291 m., Mean: -0,003 m., Mode: 0 m., RMSEz (95%): ±0,067 m.
• It is essential to highlight that the maximum observed errors (± 0,4 m) could be insignificant, considering some leveling lines across the country, from the east(E) to the west(W) and reverse. However, in this study, there was no clear correlation between the observed errors, distances, and levels.

### APPLICATIONS

• The establishment of vertical control networks has been faster thanks to trigonometric leveling. Today that has taken a drastic turn with the advent of GNSS Leveling in connection with high-resolution geoids or quasi-geoidal models (Torge W., Muller J. 2012).
• Large-scale GNSS leveling is an efficient tool to improve or renew entirely national or continental height systems (Torge W., Muller J. 2012).
• GNSS Leveling is ideal in hard-to-reach areas such as transferring heights to islands or high mountains such as Mount Everest (Chen et al. 2010).
• Vertical geodetic control in topography, hydrography, cadastre, roads, remote sensors, and geophysical methods.
• Vertical geodetic control for Photogrammetric or LIDAR projects.
• Vertical geodetic control for terrestrial seismic projects (2D, 3D, 4D).
• Vertical positioning for coastal and/or offshore operations.
• Construction of power lines and telecommunications.
• Construction of oil or gas pipelines.
• Determination of the height for towers, high-rise buildings, etc.

### BIBLIOGRAPHY

• Chen, J., Zhang, Y., Yuan, J. et al. (2010). Height Determination of Qomolangma Feng (Mount Everest in 2005. Survey Review. Volume 42, 122 – 131.
• Hofmann-Wallenhof, B., Moritz H. (2005). Physical Geodesy. New York: SpringerWien-New York. (Pág. 171-772).
• National Geodetic Survey (NGS). (2012). The National Adjustment of 2011 Project. Web site: https://beta.ngs.noaa.gov/web/surveys/NA2011/.
• National Geodetic Survey (NGS). (2020). NOAA Technical Report NOS NGS 72. GEOID 18. Web site: https://geodesy.noaa.gov/GEOID/GEOID18/.
• National Geodetic Survey (NGS). (2022). Survey Marks and Datasheets. Web site: https://geodesy.noaa.gov/datasheets/.
• Torge W., Muller J. (2012). Geodesy. 4th Edition. Walter de Gruyter. Berlin/Boston. (Pág. 256-257).
• Zilkoski, D. B., J.H. Richards, and G.M. Young. (1992). “Results of the general adjustment of the North American Vertical Datum of 1988.” Surveying and Land Information Systems, 52(3), 133–149.

By Hermógenes David Suárez (Eng).