Adjusting 3D Networks: Keep the Math “Together” for Accuracy’s Sake

By Dr. Robin Steeves

It makes a lot of sense that more and more networks are based on three-dimensional (3D) measurements. Differential leveling measurements are 1D of course, but the other two main groups of measurements, namely GNSS/GPS and total station measurements, provide 3D information for our networks. 

However, it is useful to remember that surveying networks, in the large part, used to be either 2D (horizontal) or 1D (vertical). 

The rationale was that the older measurements themselves were generally either 2D (triangulation, traversing, baselines, and trilateration), or 1D (differential or trigonometric leveling). Furthermore, the software used to perform the least squares adjustments of these networks was either 2D (adjustment on the reference ellipsoid) or 1D (adjustment of leveling measurements).  
 
But now that we are making more and more 3D measurements with higher precision, it is important to use procedures and least squares adjustment models that will result in the most realistic and useful adjusted coordinates and related statistics. Please see my GeoLab Math Models Overview article for more on this. 

We could simply continue to adjust the horizontal (2D) and vertical (1D) measurements separately, but this approach will not produce the best results. Using 3D mathematical models for 3D measurements produces better results and takes less time and effort. The old ways of how "it's always been done", are no longer as useful. 

For example, I just finished a 3D network adjustment, which consisted of two groups of measurements: GPS measurement vectors; and total station distance, direction, and vertical angle measurements. To assess their precision, I first adjusted the GPS measurements by themselves, and then the total station measurements by themselves. In each of these two minimal constraint adjustments, no measurements were flagged as outliers. 
 
When I combined the GPS and total station measurements in the overall adjustment, using GeoLabPX5, the estimated variance factor increased by about 5, caused by the usual small systematic errors in both groups of measurements. Two GPS vectors were flagged for rejection in this case. 
 
The important message here is, if we are interested in estimating the most accurate and realistic adjusted values and their covariance matrix, then it is normally best to combine all types of measurements in one rigorous 3D network adjustment process. 
 
By rigorous, I mean that exact 3D mathematical and statistical models are used for all measurements in the network.  The commercial software package that does this best is GeoLab.

 
Another example is, when GPS measurements are pre-processed, the resulting vector covariance matrices are generally quite optimistic and need to be scaled when the vectors are adjusted together in a network. Requiring measurements to fit together geometrically and statistically within a real network, as well as with other measurement types, gives us the most realistic adjustment results. 
 
I hope you understand what I mean. When you separate measurements, you lose some ability of arriving at the best (most realistic) estimates of the parameters and their statistics. You lose some power of detecting outliers in the measurements, which is a fundamental flaw in the choice to process measurements separately.  
 
The good news is that modern software such as GeoLab can handle your 3D networks rigorously and easily. And you'll get your work done more correctly and more efficiently.