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In Part One I discussed a small part of Ez-Adjust dealing with Four-Parameter Coordinate Transformation and compared it to other programs that offer analysis of the fit of two coordinate systems.
Least-squares programs have two sides. The first is adjustment. The second is blunder detection. Adjustment by least squares can be made by condition equations or by variation of coordinates (sometimes called rigorous). All least-squares adjustments yield essentially the same results provided, of course, that the same weighting is given to the observations or measurements. Therein lays the tale of the adjustment. There are many theories of weighting and algorithms for weighting, but my favorite is weighting by observing. This method, however, is likely beyond the experience of the average surveyor. We depend on the recommendations of the manufacturer or the calculations of various parameters made by the manufacturers’ programs. Rule Number One: if the results of a least-squares adjustment for the same observations or measurements are different, the difference is due entirely to the weighting of the observations or measurements.
From an adjustment point of view, the arguments are purely academic. The Bowditch (compass) rule for the same redundancy is easy to understand, easy to visualize and perfectly adequate for most work. A simple closed traverse with no interconnecting points has only three redundancies (two positional, i.e., sums of latitudes and departures equal to zeroes, and one angular), and a least-squares adjustment offers no advantage over the Bowditch adjustment. When the figure is more complicated than a traverse, Bowditch is extremely difficult to use intelligently and some other method is usually chosen.
The real value of an adjustment is blunder detection; the more redundancy, the finer blunder that can be detected. A traverse with one redundancy is only marginally useful in detecting large, non-specific blunders. The classic example of a fully measured figure is the cross-braced quadrilateral where all of the sides, cross-braces and angles are measured. Look at redundancy in this way. What is the minimum number of measurements necessary to draw the figure on paper? Any additional measurements are redundant. In the quadrilateral, the baseline, the angle between the baseline and the cross-brace, the angle between the cross-brace and the adjoining side are sufficient to draw the figure (all others are merely inverses).
Redundancy is the key to finding the best fit. A least-squares program will calculate the amount by which each observation must be changed to satisfy the basic criteria (that the sum of the squares of these changes must be the minimum). That gives you the best adjustment, but that is not really what it is all about. What you are really looking for are "outliers". That is hard to do since a large change or residual might be the result of a very large observation. If you could use a normalized residual, the problem would be much easier. A normalized residual or change is the result obtained by dividing the change or the residual by the size of the observation. A graph of the normalized residuals is a bell-curve. Small normalized residuals are more frequent than large normalized residuals. When the bell-curve is distorted by "outliers" you can zero-in on possible errors. It is my personal prejudice that an outlier should not be unweighted or removed from the adjustment solely because it is an outlier. It should be removed because inspection shows that it is a blunder (i.e., calculating the wrong height of the antenna, etc.). It has been my experience that some analysts simply remove or unweight the "outliers" until they get a nice smooth fit. Beware of so-called experts who do this. To me, this is like rejecting an original monument because it is 0.20 feet out of position.
Least-squares is very sensitive to blunders (it makes it go crazy). In GPS surveys, it is usual to run loop closures to find the big blunders before even attempting an adjustment. This is exactly the same thing as running a traverse. You should end up where you started. Field crews will set up on the same station and give it different names. They will set up on different stations and give them the same name. They will mistake a nearby mark as a selected station. The list is endless. All of these things have to be discovered and corrected before you can even consider an adjustment. This brings us to the next topic: tools.
Ez-Adjust, with 240 pages of documentation, has all of the tools required to make analysis and adjustment as painless as possible. These tools are divided into three categories: Four-Parameter Coordinate Transformation, Ez-Adjust and EzaPro. (Coordinate Transformation was discussed in Part One [Dec 2005] where it is compared to two other programs sold as boundary analysis programs.
Ez-Adjust is made up of two parts: EzaPro and Eza, which may be purchased separately and automatically provide a license for two seats. EzaPro is a full 3D adjustment package. Eza is really three parts: a 2D adjustment package, a level network adjustment and a coordinate transformation package. There are two other transformation packages being advertised but they are advertised as Survey Retracement Software and the other as Transform. The interesting thing to me is that the coordinate transformation program in Eza is thrown in for free. Eza costs only $225.00 and that is comparable to the other two packages (you also get a 2D adjustment and a level net adjust that will process data from a Leica Digital Level).
Ez-Adjust is the basic adjustment for 2D and 1D (leveling) with things like input, editing and graphics.
EzaPro contains the sophisticated tools for analysis and adjustment of 3D networks. We mentioned loop closures. Be forewarned that every loop closure based on one setup (no matter how many receivers) will be perfect! To find a valid loop closure the loop must contain a minimum of two setups. (There has been a running battle over this for decades. There is a minority of one that insists that a single setup has all independent legs. To me and others this is ridiculous. If you have three receivers in a single setup you get two independent legs. If no other logic could convince you, the fact that you will always get a perfect closure on such a setup should at least make you suspicious. This is the same thing as inversing the last leg of a traverse and always getting a perfect closure. Yet, there are a few who cannot see this logic.) EzaPro has all of the tools you need to perform, edit, print, save and evaluate loop closures, plus, it has a wonderful geodetic calculator.
Ez-Adjust (or Eza) is like other programs out there that tell you they do a 3D adjustment. What they really do is 2D adjustment and a 1D or level adjustment.
Figure 1 shows a typical 2D adjustment problem using a total station or a transit and tape. You will give your initial point 5000/5000 and you will use some point for a fixed azimuth which you can use for verifying that the gun has not slipped or been bumped or whatever. You will need to set the linear units and the angular units. If you are using an EDM or total station you can define it with the manufacturer’s specifications or your own specifications if you have the facilities to determine them. All these tools are provided in Eza. Input all of the measurements. Hit the adjustment button and see Figure 2.
Figure 2 is really the first of two very long pages of information which includes the angles, distances azimuths, and circular probable error. But the first page displayed in Figure 2 shows the meat of the adjustment for looking for errors and goodness of fit. Figure 2 shows the residuals, the normalized residuals, and the scale factor that results from the fit. It shows the first few angle observations as well (this is a different survey but of the same kind). You can mix different kinds of observations–from chaining to GPS.
Figure 3 just shows the way in which a level net is drawn to avoid confusing it with a 2D or 3D net (to show that the lines were not measured exactly).
Figure 4 shows the first page of the results of the adjustment. The next page shows the adjusted elevations and 95% circle error. Here the normalized residuals are in order of the most to the least.
Eza reads data directly in native format from the following Leica digital levels: NA2000, NA3000, DNA3 and DNA10. This will make life simpler for those who have these levels. I would like mention something about First Order leveling. A surveyor (who shall remain nameless) made a contract with a large research facility to create a First Order Level Net of some 30-50 stations. He had his crew perform the three-wire observations with an N2 level on uncalibrated (actually, ordinary Philadelphia) rods and since he had no adjustment package, he had his office person with no network adjustment experience adjust it from the outside-in. I did not know whether the engineer who accepted the work or the surveyor who signed the work ought to have been approached by the Board of Registration. This surveyor obviously had not a clue as to what was a First Order Level, calibrated rods or the extensive protocol for making the observations. The USGS has a great protocol. Follow it.
EzaPro is a full 3-D adjustment package which contains every tool you will need to analyze any survey. It is just an ordinary complete package for doing least squares adjustments by variation of coordinates with very versatile weighting tools.
I would like to mention one tool in particular. Figure 5 shows the screen for the geodetic calculator that comes with EzaPro. With geodetic coordinates every point in the world has a unique set of coordinates. The unique coordinates can be expressed in different ways, geocentric (x,y and z), latitude and longitude and ellipsoidal height, State Plane Coordinates, UTM, and many other systems. If you know the coordinates in one system, you know them in all systems because they are unique. The differences are straight mathematical calculations. That is what a geodetic calculator does. Put the coordinates into the calculator and out comes all the other standard coordinates. EzaPro also tells you how and where to download the latest geoid model so that not only can you get the ellipsoidal height, you can get a close approximation of the orthometric height. You can define a system or enter the definition of any model you have to work with. Not only do you get all the bells and whistles, but you can batch process files as well. This is really a good geodetic calculator.
EzaPro has all of the standard tools for preprocessing, such as loop closures, so that you do not have to keep re-entering data. EzaPro automatically calculates the loops but if you want to add or change the loops you can do so.
EzaPro also has a neat set of graphics tools that allow you to print, save to clipboard (for inclusion in reports), zoom, pan, inquire, or scale the error ellipses. Figure 6 shows a mixed survey. Notice that it is color coded.
The geodetic calculator and the coordinate transformation programs are terrific. Add to that the fact that you also get a full-blown professional adjustment package thrown in! For $495.00, this is an unbelievable bargain.
Joe Bell is licensed in California and New Mexico. He has been reviewing software for surveyors since 1982.
A 1.194Mb PDF of this article as it appeared in the magazine complete with images is available by clicking HERE