My colleague sent an E-mail to a person associated with the project, asking, "Could you provide the conditions or cases in which one can have such an "over-constrained' adjustment?"
The response came back: "Over-constrained adjustments are used whenever one wants to make sure that pre-existing control points really do stay fixed at their published values. However, such adjustments do not have the minimum variance property, and one can no longer use the common equation that says that the covariance matrix of the results is proportional to the inverse of the normal equation coefficient matrix."
If this information is over your head, don't worry-you are not alone. Those who adjust GPS networks on a regular basis should understand that statement. For those of you who don't, let me try to explain.
In every network of stations observed by GPS or surveyed using conventional methods, there are seven parameters that must be defined:
- An origin that can be X, Y, Z; φ, λ, h; or a local system (three parameters)
- The rotation with respect to the origin (three parameters). For conventional surveying this would be the backsight to the azimuth mark, the horizontal angle to the foresight and the zenith angle to the foresight. With GPS, this comes from the orbit; you are observing the satellites as they are moving so it's inherent in the observations.
- The scale (one parameter). For conventional surveying this would be the slope distance to the foresight. With GPS the scale is automatically determined by the speed of light; it's inherent in the observations.
Performing a Minimally Constrained AdjustmentPlainly stated, with GPS you need seven parameters to define a network, and this can be done by holding only one station fixed. To illustrate this situation, look at Figure 1. This is a sketch of a small network observed by GPS; every line in this figure represents an observed vector.
This network should be adjusted by a minimally constrained adjustment, and this adjustment should always be performed first. To do so, assume Station 1 is a station with published coordinates in the National Spatial Reference System (NSRS). In your GPS network adjustment program, fix the coordinates of Station 1 by inserting the published coordinates and giving each component (for example, φ, λ, h) a small standard deviation (σ) like 0.00001 mm.
Now perform a least squares adjustment. In most cases, the math model is based on observation equations where the observed vectors are the observations and the other stations in the network are the parameters. The observations are adjusted until the sum of the squares of the residuals, for all observations, are a minimum. The results of the adjustment are the coordinates of each station, with standard deviations, and the standard deviation of unit weight, σ o. σ o tells you the quality of the adjustment. The expected value is 1; if σ o is close to 1, this means the observations are consistent with few if any blunders. If σ o is a larger number (and different people have different ideas on what is large), you need to look at the standard deviation of the adjusted parameters to see if you can detect bad observations. An analyst who has extensive network adjustment experience is extremely valuable in this situation.
Look at Figure 1 again. Let's assume that Station 2 is also a station with published coordinates in the NSRS. Let's say we want to also hold this station fixed. As before, enter the published coordinates of this station and a small standard deviation. Then perform a least squares adjustment.
What will happen? Holding both Stations 1 and 2 fixed destroys the orientation and scale as determined by GPS. Also, the observed vector between the two stations is ignored. The network will adjust, but the error estimates will not be representative of a true least squares adjustment.
Applying Weighted ConstraintsUnforunately, some surveying firms today are fixing all stations that have published coordinates. The least squares model was not designed to handle this situation. This is especially a problem where some of the known stations are part of the Continuously Operating Reference Station (CORS) system and others are part of the state High Accuracy Reference Network (HARN). The two networks are not consistent and errors are introduced by fixing both CORS and HARN stations in the same network.
What should one do in a case where both CORS and HARN stations are observing stations in the same network? My opinion, and the opinion of many of my geodetic colleagues, is that a "weighted constraint" should be applied to all stations with published coordinates except the one used as an origin. A weighted constraint is where a standard deviation is given to the stations' components through the weight matrix W based on your knowledge of the relative accuracy. As an example, let's say Stations 1 and 2 in Figure 1 are both Order B HARN stations. Let's say the distance from 1 to 2 is 36 km.
36 km = 36,000 m = 3,600,000 cm = 36,000,000 mm
An Order B station has a relative accuracy to other Order B stations of 1 part per million. This means that the distance from Station 2 to Station 1 can be off by Â±36 mm and still meet Order B requirements. If I were adjusting this network, I might try giving the latitude and longitude of Station 2 a standard deviation of 2 cm. Doing this allows Station 2 to move within the allowable tolerance and not destroy the adjusted statistics.
From many years of experience adjusting geodetic networks, I can say there is nothing wrong in applying weighted constraints to parameters when you have knowledge of relative accuracies. Applying weights to the weight matrix is acceptable. Some people who read this article will say I'm wrong because the National Geodetic Survey fixes all CORS. Yes, NGS does, but it uses a different math model for absolute constraints-one that you probably don't have in your least squares adjustment package.
The model for absolute constraints is explained in detail in "The Effects on Unestimated Parameters," an article by Dr. Charles R. Schwarz that was published in the June 2005 issue of Surveying and Land Information Science (SALIS). Dr. Schwarz developed a math model for a least squares adjustment that solves this problem. Rather than give his mathematical development, I'll quote his opening abstract and conclusion:
Constrained adjustments are the means by which new points are brought into an existing geodetic datum. In most of these adjustments the coordinates of existing control points are held fixed. Some fixed control points are necessary to define the coordinate system, but usually more than the minimum number of existing control points are held fixed. When this happens, the coordinates of the extra control points are called "unestimated" parameters, emphasizing that they could have been determined from the observations but were not. The coordinates determined in the least squares adjustment are valid, but the conventional equations for linear error propagation must be extended to take account of the uncertainties of the estimated parameters. The equations for doing this are known for the case of absolute constraints. Since most adjustment programs use weighted constraints to hold fixed the coordinates of existing control points, the equations for estimating the effects of unestimated parameters are extended to the case of weighted constraints. ...
The effect of the variance of the a priori values of the control points used in an adjustment can be computed. There are separate formulations for programs using absolute constraints and those using weighted constraints. These formulations allow us to hold existing control points fixed, yet still perform a correct error propagation.1
The NSRS Readjustment of NAD 83The same issue of SALIS that featured Dr. Schwarz' article also contained a timely report on the NSRS readjustment of NAD 83. Chris Pearson, the NGS geodetic advisor for Illinois, reported on the readjustment and wrote that the adjustment has been developed for two major reasons. The first of these is a requirement to develop individual local and network accuracy estimates for each point in the network. The second purpose is to resolve inconsistencies between the existing statewide HARN adjustments and the nationwide CORS system, as well as between states. Pearson listed nine key points of the adjustment; I will cite the first two:
1. Only GPS will be adjusted. Classical geodetic observations will not be included.
2. CORS stations will serve as control, i.e. CORS positional coordinates will be held fixed. 2
This is outstanding news for the geodetic community, as we will have a national coordinate system for all stations observed by GPS. But until that adjustment is completed in 2007, you must remember that the CORS and HARN networks are from different adjustments. Be careful what constraints you use when stations from both systems are in the same network.