Real-World Applications of COLUMBUS

Orthometric Heights from Ellipsoidal Heights

There are many different approaches to determining orthometric heights from ellipsoidal heights. In some sense, this process is part art and part science, because you generally cannot just plug in numbers to find the best solution for your area. Some factors affecting the determination of orthometric heights from ellipsoidal heights include:

• the accuracy of the geoid model,
• the quality of known orthometric and ellipsoidal heights,
• the quality of the survey (if done),
• and the overall shape of the geoid surface in the project area.

Below we describe a couple methods for determining orthometric heights from ellipsoidal heights. All methods were submitted by our users and have been used on actual projects. If you have a method that we have not included here, please send your approach so that we may add it to this topic.

This article considers two cases:

1. In the first case, you have not performed a 3D survey, but you have published ellipsoidal heights for each station.

2. In the second case, you have performed a 3D survey and intend to use the measured observations to influence your determination of orthometric heights.

Case 1

In this scenario, you have one or more stations with known orthometric and ellipsoidal heights. From this data you want to estimate the orthometric heights of other stations in the area (for which you also have known ellipsoidal heights). Short of developing a complex surface model of your own, here is a common solution:

Using an existing geoidal model (like geoid03 or EGM96), determine the orthometric heights for all unknown stations using their known ellipsoidal heights and any known local orthometric height datum correction in the area (sometimes referred to as a local bias correction).

The local orthometric height datum correction is the difference between the published orthometric height and the height obtained from differencing your adopted ellipsoidal height and the geoid model you are using (geoid03, for example). In small areas, this correction may hold to a single constant. In large areas, you may need a different constant for different portion of the project.

Example:

For a station with an ellipsoidal height of 20.0m, a known orthometric height of 10.0m, and a modelled geoidal height of 9.8m, the correction is -0.2m (10.0 - (20.0-9.8)).

orthometric height (H) = ellipsoidal height (h) - geoidal height (N) + local orthometric height datum correction

To use this approach in COLUMBUS, read Quick Tip Performing geoid modeling at any time within COLUMBUS.

Case 2

In this scenario, you have performed a 3D survey in the project area resulting in latitude, longitude, and ellipsoidal heights for all stations. It does not matter if the survey was performed using GPS, terrestrial, or GPS + terrestrial observations. By adding survey data to the mix, you have the ability to improve on the existing geoid model (or simply the existing geoidal heights), by fitting an improved surface to the existing geoidal height surface.

In COLUMBUS, the source of the geoidal heights is not important, since all geoidal modeling is performed outside the scope of a network adjustment. You could conceivably use geoid03, EGM96 or some localized geoidal surface. The main point is to provide a geoidal height for each station, then try to improve on it, based on the results from the network adjustment.

Each approach described below assumes the following:

1. You have included some stations (preferably scattered throughout the project) with known orthometric heights in the survey.

2. You have already performed one or more minimally-constrained adjustments in order to identify and remove outlier observations. At this point, you are satisfied with your measurements and are ready to perform fully-constrained adjustments, in which you hold fixed additional known control stations.

3. You have performed a constrained adjustment (one or more times) to determine if any of your existing control is adversely affecting your solution. Any control stations degrading the adjustment beyond the desired level should be either removed from the network or simply set to float in subsequent adjustments. Each adjustment should result in a latitude, longitude and ellipsoidal height for each station.

Approach 1:

1. For the stations with known orthometric height, add the modeled geoidal height to the orthometric height (using geoid03, EGM 96 or input known values directly), thus deriving a new biased ellipsoidal height for these stations. These ellipsoidal heights are now biased by the same amount as the bias in the geoidal heights.

2. Perform the constrained adjustment again holding only the stations from Step 1 fixed in 1D or 3D (3D, if it is also horizontal control station). Any other 2D or 3D control stations not altered in Step 1 should be held fixed only in 2D. You will then end up with new biased ellipsoidal heights for all the network stations.

   During the adjustment process, your observation data will be made to fit the new vertical control across the entire network, in effect creating a new "biased" surface. This surface will result in a different built-in bias at each station, rather than assuming one fixed bias for the entire network (as in Case 1 above).

3. From these results, subtract the geoid height values from the adjusted (with bias) ellipsoidal heights (for all stations) to get good orthometric heights. In COLUMBUS, you would simply perform the geoid modeling again based on the new adjusted biased ellpsoidal heights.

Approach 2:

1. Perform the constrained adjustment again, but this time base it on orthometric height instead of ellipsoidal height. Hold the stations with known orthometric height fixed in either 1D or 3D. Do not hold any other stations fixed in 1D or 3D.

2. If you have GPS observations in the network, try the adjustment with and without GPS scaling and rotation enabled.

3. The resulting coordinates will result in orthometric heights (geoid modeling is not required.)

Like all techniques mentioned thus far in this article, it is up to you to compare the results with expectations before deciding on the method to adopt. You may find that different methods fit different projects and that no one method will always be the best choice.