Best-Fit Computing - COLUMBUS Network Adjustment Software

Adjustment Combinations Tool

COLUMBUS supports an advanced technique to automate the search for poor quality control stations in your fully-constrained network adjustments. This technique, available for 1D, 2D and 3D networks, is accessed through the Adjustment Combinations Tool. It is based on combinational theory, which is applicable to many real-world problems.

Combinational Theory

Suppose you own a small surveying firm and have just received a GPS control survey contract. To complete this survey on time, you have estimated at least three experienced GPS surveyors must be dedicated to the project. Within your small company, there are five people qualified from which to choose (Tom, Mary, Dave, Alice and Andy). At the moment, all five are free to work on this project. The question, then, is which people should be put on the project? How many combinations of solutions are possible?

You could decide to put only three on the project. Or you could add a fourth or fifth to finish the project sooner. Some of the possible combinations are listed below:

  • Tom, Mary, Dave
  • Tom, Mary, Alice
  • Tom, Mary, Andy
  • Tom, Mary, Dave, Alice
  • Tom, Mary, Dave, Andy
  • Tom, Mary, Dave, Alice, Andy
As you might expect, the total number of possible combinations can grow very large. In fact, the actual number of combinations is given by the formula:



	K! * (N - K)!

    ! = factorial
    N = total number of qualified people available (in this case, five)
    K = number of qualified people for any given solution
For this example, the formula needs to be applied three times: once for K = 3, once for K = 4 and once for K = 5. The results from each are then added together.

The answer to this example is: 10 + 5 + 1 = 16

5! = 5 * 4 * 3 * 2 * 1

4! = 4 * 3 * 2 * 1

3! = 3 * 2 * 1

How Does This Apply To Network Adjustment?

For a network adjustment the same type of process can be applied, but instead of different people, we have several different control stations. Each control station is uniquely identified by its known coordinates.

The Adjustment Combinations Tool can be used when you are ready to perform a constrained adjustment. Before using this tool, you should have performed repeated free adjustments and removed any possible bad observations. If you have several control stations tied to your network, several possible combinations of control station scenarios exist. You could use all the control you have, or you might decide to use a subset of your total control.

If you had several control stations, in normal practice you might perform several constrained adjustments, altering which stations are set up to be controlling for each adjustment. After finding the most favorable results, you would perform your final constrained adjustment and be done. This process could require days or weeks worth of work if you had to manually change the configuration of control for each combination. Plus, you might leave one of the possible combinations out, never knowing if it would have been the best solution. With the Adjustment Combinations Tool, COLUMBUS performs these iterative steps for you quickly, easily, and without leaving any possible solution untested!

Example: You have a 3D network consisting of 50 stations and hundreds of observations. You have completed the free adjustment process and have removed all poor quality observations. Of the 50 stations in your network, you have known 3D coordinates for four of the stations and known 1D height for three additional stations.

Within COLUMBUS you would select these stations to be fixed in 3D and 1D, respectively. From within the Set Up Adjustment Combinations dialog box, you would then specify which combinations of control configurations are desirable in any final solution. This is accomplished by setting up the minimum and maximum number of control station types you want to have in each acceptable configuration.

For any constrained 3D network, you need a minimum of three fixed combination parameters (two fixed parameters are required for 1D or 2D networks). Stations fixed in 1D or 2D count as one parameter. Stations fixed in 3D count as two parameters - since they have both a 1D and 2D component. A constrained 3D network would require (at a minimum) any of the following:

    One 1D fixed station + one 3D fixed station
    One 2D fixed station + one 3D fixed station
    Two 1D fixed stations + one 2D fixed station
    Two 2D fixed stations + one 1D fixed station
    Two 3D fixed stations
Note: Three or more stations fixed in only 2D or only 1D will not work, since you need at least 3D worth of control for a 3D constrained adjustment.

By selecting the minimum and maximum number of control stations you are willing to accept in any possible solution, the range of possible solutions is decreased or increased, respectively.

For this example, we have decided that each solution must have at least one station fixed in 1D and one station fixed in 3D. You would declare this by setting the Minimum boxes for 1D and 3D stations to the number '1'. Additionally, we have decided we want a maximum of two 1D stations in any given solution. You would declare this by setting the Maximum box for 1D stations to the number '2'. Finally, for any given solution, we want no more than two stations to be fixed in 2D (a 3D stations can be used as a 1D, as a 2D, or as a 3D control station). You would declare this by setting the Maximum box for 2D stations to the number '2'.

The edit boxes would then look like this:
Station Type Minimum Maximum
1D 1 2
2D 0 2
3D 1 4
For the settings above, each combination solved by COLUMBUS will have:

    At least one station fixed in 1D
    At least one station fixed in 3D
    No more than two stations fixed in 1D
    No more than two stations fixed in 2D
    No more than four stations fixed in 3D
Given these parameters, when the Compute Combinations button is selected, the total number of combinations to solve would be computed to be 700 (based on Combinational Theory). WOW! Each one of these combinations would satisfy the constraints defined above.

As you can see, there are many possible solutions even with as few control stations as described above. To perform this task manually would be mind-boggling and time-consuming. The risk of not attempting each solution could be the difference between an acceptable survey and one which requires more work.

Why Not Just Use All Known Control?

The answer depends on the quality of the control in your area. Some control points may be more reliable than others, based on how they were established. Some control points may have been disturbed since they were originally surveyed. This could be a result of geological movement, vandalism or even wildlife. Simple instrument centering/height measurement errors could also influence the observed quality of any given control station. You may never know if the control point has these problems, because your first inclination may be to suspect your own observations.

By solving all possible combinations, you have the best opportunity to eliminate control which may be introducing significant errors into your project.

How Solutions Are Ranked

As COLUMBUS processes each new combination, it keeps track of the top 500 solutions for two independent tests: the A Posterior Variance Factor test and the Residual Distribution Test. When processing is complete, these solutions can be displayed and printed. You can then pick which of these solutions to use as your final constrained adjustment. Or you might take a closer look at the top 10 solutions by solving them individually and looking at several other statistical indicators presented by COLUMBUS after the traditional network adjustment process.

The A Posteriori Variance Factor ranking is always performed. The closer the A Posteriori Variance Factor is to 1.0, the better the ranking for a given solution. This test does not require a final inverse on each solution, which results in faster processing times for larger networks.

The Residual Distribution Test is optional. It will give you a numerical indicator as to how evenly the Standardized Residuals (for all observations) are distributed about their mean. Since this test requires additional statistical information found within the final inverse for each solution, it will lengthen processing times for large networks when enabled.


While the process automated by the Adjustment Combinations Tool can be done manually for each combination, the task would be difficult and error-prone for projects containing several control stations. In the medium-sized network described above, the number of valid constrained network combinations is 700 - that's a lot of combinations to solve!

The Adjustment Combinations Tool can save you days of work spent processing different constrained network scenarios looking for the Best-Fit Solution!

Network Adjustment and Coordinate Transformation
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