Zilkoski
In the first article of this series, the basic concepts of GPS-derived heights were discussed. The article discussed the three types of heights involved in determining GPS-derived orthometric heights: ellipsoid, geoid and orthometric. It was also mentioned that each of these heights has its own error sources that need to be detected, reduced and/or eliminated by following specific procedures or applying special models. The second article discussed guidelines for detecting, reducing and/or eliminating error sources in ellipsoid heights. This article will discuss a few basic procedures for analyzing GPS project results to ensure the desired ellipsoid height accuracy standard has been met.

GPS results can be evaluated by analyzing repeat base line differences, network loop misclosures and residuals from a minimum constraint least squares adjustment. It was noted in the second article that if GPS users follow the NGS guidelines, they will reduce and/or eliminate errors in ellipsoid height and, at a minimum, detect problems or errors in data. It was also mentioned that the basic concepts are very simple, but they all need to be followed exactly as prescribed. For example, “the observing scheme for all stations requires that all adjacent stations (base lines) be observed at least twice on two different days and at two different times of the day.”

GPS can provide “absolute” and relative positioning information much easier, faster and more precisely than some classical techniques. However, the wrong station can still be occupied, the height of the antenna can be measured wrong or incorrectly entered during the base line reduction processing phase, the receiver can malfunction, an abnormal atmospheric condition can cause large errors in the height component, or some “unknown Gremlin” can cause an error source.

Classical techniques of establishing horizontal and vertical control used networks, which consisted of many loops, triangles and braced quadrilaterals. This design provided enough redundant observations to detect data outliers. NGS guidelines for establishing GPS-derived heights were designed with this same concept in mind. Since all base lines must be repeated and adjacent station observed, analyzing repeat base line differences, loop misclosures and residuals from minimum constraint least squares adjustments are very effective analysis tools for detecting data outliers.

Figure 1. Repeat base line differences by distance.

Comparing Repeat Base Lines

The procedure is very simple: subtract one ellipsoid height from the other, i.e., the ellipsoid height from base line A to B on day 1 minus the ellipsoid height from base line A to B on day 2. If this difference is greater than 2 cm, one of the base lines must be observed again. (See Figure 1.) This is a very simple procedure, but also one of the most important. Many users complain about having to repeat base lines, but requiring an extra half-hour occupation session in the field can often save many days of analysis in the office.

Analyzing Loop Misclosures

Loop misclosures can be used to detect “bad” observations. If two loops with a common base line have large misclosures, this may be an indication that the common base line is an outlier. Since users must repeat base lines on different days and at different times of the day, there are several different loops that can be generated from the individual base lines. If a repeat base line difference is greater than 2 cm then comparing the loop misclosures involved with the base line may help determine which base line is the outlier. According to NGS guidelines, if a repeat base line difference exceeds 2 cm then one of the base lines must be observed again, and base lines must be observed at least twice on two different days and at two different times of the day. If it can be determined which base line is the potential outlier then the user will know which time of the day to reobserve the base line. Loop misclosures can be very helpful in isolating errors.

Figure 2. Ellipsoid height residuals from minimum-constraint least squares adjustment.

Plotting Ellipsoid Height Residuals from Least Squares Adjustment

If users follow NGS guidelines and evaluate all repeat base lines the adjustment results should confirm what has already been determined, i.e., if a repeat base line indicates a large difference between two vectors then typically one of the residuals should be larger than the other. Following NGS guidelines provides enough redundancy for the adjustment process to detect outliers and usually apply the residual to the appropriate observation, i.e., the bad vector.

Like comparing repeat base lines, analyzing ellipsoid height residuals is also important. During this procedure the user performs a 3D minimum-constraint least squares adjustment of the GPS survey project, i.e., constrain one latitude, one longitude and one ellipsoid height; plots the ellipsoid height residuals; and investigates all residuals greater than 2 cm.

Notice that the plot of repeat base line differences indicates that base line A to B exceeds 2 cm and the plot of ellipsoid height residuals show that one of the base line’s residuals is 2.2 cm while the other is 0.0. That’s all there is to it—when the user follows NGS guidelines exactly as prescribed.

A few more items should be noted. First, the reader may have noticed that some large residuals on the residual plot, i.e., 5 cm around 10 km, did not show up as large differences on the repeat base line plot. There are several reasons why this could occur. For example, the stations involved in the base line are not adjacent stations, so the base line wasn’t repeated; the repeat base line misclosure was large, but not greater than 2 cm; and the pair of stations are involved with many vectors and the other vectors are consistent with each other. Regardless of the reason, if there is enough redundant observations to the station and the repeat base lines don’t indicate a problem then the adjustment is doing what it is suppose to do, that is, detecting outliers and reducing their influence on the final adjusted height. Second, the reader may have wondered why the largest residual on the repeat base line plot, i.e., 3.1 cm around 2 km, didn’t show up as a large residual on the height residual plot. This is because the least squares adjustment could not identify one of the base lines as an outlier. The two residuals on the repeat base line were 1.5 cm and –1.6 cm. The NGS guidelines require all adjacent stations to be repeated. But due to the design of the network, the adjustment still may not have enough redundant data to determine which observation is the outlier. Repeating the base line is the recommended solution. c

Note: The NGS guidelines have been documented in a publication titled “Guidelines for Establishing GPS-derived Ellipsoid Heights (Standards: 2 cm and 5 cm), Version 4.3” and can be downloaded from NGS’ website at http://www.ngs.noaa.gov/PUBS_LIB/

NGS-58.html.

NGS has also developed a few routines that read a set of vectors and compute and list repeat base line components. For more information on the guidelines and routines, please contact Edward Carlson, Spatial Reference System Division at 301/713-3191 or by E-mail at Ed.Carlson@noaa.gov.

This series of articles has discussed basic concepts of GPS-derived heights, NGS guidelines for establishing GPS-derived heights and basic procedures for detecting GPS-derived height data outliers. The next article (September 2001), the last of this series, will briefly discuss basic procedures that need to be followed for establishing North American Vertical Datum of 1988 (NAVD 88) GPS-derived orthometric heights.