Using TPS computations
Release 9.2 Last modified August 23, 2007

About the TPS computations

The Survey Editor in Survey Analyst provides a set of five computations. Four of them are based on the classical survey algorithms for tacheometry, free station, resection, and traverse. The fifth computation is the least squares adjustment.

Once you have mastered the basics of data entry for the General, Setup, Measurement, and Report tabs, you will have an overall familiarity with all the TPS computations.

The following sections cover the general steps required for basic data entry, and also present additional information about specific TPS computations.

Learn more about TPS computations

Tacheometry

The tacheometry computation uses a single instrument setup at a point with known coordinates—known points or reference points. It has a set of horizontal circle readings to other known points and is used to process the horizontal angle and distance observations to define coordinates for previously uncoordinated points—unknown points or measured points.

Free station

To compute coordinates for an instrument setup at an unknown location, distances and observations on the horizontal circle to at least two reference points need to be observed. The free station computation uses these measurements to define a triangle. The solution of the computation is the set of coordinates at the apex of the triangle that defines the setup location.

Traverse

The traverse is a sequence of instrument setups that start at a known location and end at another known location, with the intermediate setups being at points with unknown coordinates. Misclosure (or closure error) is the difference between the computed endpoint coordinate and the known endpoint coordinate. The misclosure is distributed through the intermediate points using one of the following methods:

• Compass rule
  
• Transit rule
  
• Crandall rule
  

The Compass rule, also known as the Bowditch rule, distributes the closure error in the Northings and Eastings in proportion to the distance along all the courses from the first point to each of the unadjusted coordinate locations.

The Transit rule assumes distances have no measurement error, and distributes the error only through the observed angles.

The Crandall method distributes the error in the distances only, assuming observed angles have no measurement error.

You can define the allowable limits for the closure error in the traverse in terms of its lateral misclosure, lengthwise misclosure and angular misclosure.



The formulae used for determining the values of L and Q are:


For a Traverse from P1 to Pn, as shown below, at each point an azimuth a and a distance d are measured to the next point. From these measurements a change in Easting and Northing, DE and DN can be calculated. The locations of a series of points P2' to Pn-1' can be calculated by adding the respective DE and DN values to the coordinates of the previous point, starting with P1. The angular misclosure is given by the difference between the final measured azimuth an-1 and the azimuth an-1' necessary to connect Pn-1' to Pn, the fixed finishing point. (This approach is applicable to both open and closed traverses.)

Resection

It is possible to compute coordinates for a setup location if you have at least three visible reference points and their horizontal circle readings. The advantage of this method is that distances are not required.

It is important when using this computation to be sure you have an approximate coordinate for the point to be computed in order to avoid the resection circle problem. If your known and unknown points all lie approximately on the same circle, then your solution will likely be unreliable. The computation algorithm is unable to determine this without additional measurements to known points that do not lie on the circle.

Least squares adjustment

The techniques and algorithms supported by the least squares adjustment computation make it the best method available for processing multiple instrument setups in a network of measurements.

With a least squares adjustment computation, you strive to remove mistakes in measurements and to improve the quality of your survey points.

Learn more about least squares adjustment

In essence, the least squares adjustment allows many measurements to participate simultaneously in a single computation. This process provides a best fit for survey point locations and allows detection of defective measurements, called blunders or outliers. These outliers are detected based on statistical tests.

Surveyors typically perform extra measurements in the field to improve the quality of the network, and to guard against loss of information and other blunders.

When the number of observed measurements is greater than the number of computed parameters, redundancy exists in the network.

The more redundancy you have in a measurement network, the better your chances are to detect and control problems.

Unlike the other computations, using the least squares adjustment is an iterative process. You remove the measurement or reference point that is detected as the biggest outlier, and recompute. Repeat this until all blunders have been found and the adjustment can be computed successfully. An alternative is to make the testing limits more forgiving by increasing the level of significance.

There are two phases involved when performing a least squares adjustment for your measurement network:

• Free network adjustment
  
• Constrained adjustment
  

These phases separate the testing of measurements from the testing of the reference points. The free network adjustment phase examines the overall geometry of the network by processing the measurements only and using the reference points only for position scale and orientation of the network. In this phase the emphasis is on testing the quality of the measurements rather than computing coordinates. This step is performed to check for outliers in your measurements.

For example, you may have used an incorrect prism offset for a measured distance. If defective measurements are detected, you need to exclude them from the computation to prevent them from affecting the overall best fit of the adjustment. Sometimes, you have enough information to correct the mistake by editing the data. However, it is important to be certain that the correction is valid.

After eliminating or correcting the blunders in the measurements, the reference points are used in a constrained adjustment. In this second phase, the emphasis is on testing the reference points as well as computing final coordinates.

Including reference points imposes additional constraints on the solution of the adjustment.

There are two possibilities for performing a constrained adjustment:

• Absolutely constrained
  
• Weighted constrained
  

In the absolutely constrained adjustment, the coordinates of the reference points keep their original value. The standard deviations of their coordinates are held at 0 and, therefore, do not shift in response to the adjustment. You would use this method when, for instance, the reference points are coordinates published by a government authority and should, therefore, remain unchanged in the survey dataset.

In the weighted constrained adjustment, the reference point coordinates are treated as observed values (measurements), and their standard deviations are applied in the adjustment. Unlike the absolutely constrained adjustment, both the reference point coordinates and the measurements receive corrections during the adjustment. This method provides an optimal “best fit” solution and can be used when you are free to define and publish coordinates for your survey control network.

Statistical testing

The aim of statistical testing is to detect possible outliers (blunders) in the measurements. This provides an important component in the process of quality control. The statistical testing is part of the least squares adjustment computation.

It is used to verify statistical hypotheses that define a set of assumptions. A special set of assumptions is referred to as the null-hypothesis (H0). This hypothesis implies that:

• There are no blunders (mistakes) in the measurements.
  
• The relations between the measurements and the coordinates for survey points are correctly defined.
  
• The chosen a priori standard deviations for the measurements are appropriate.
  

Level of significance

There are two possible outcomes for the testing of a hypothesis: acceptance or rejection. A specific cutoff point, or critical value, determines the acceptance or rejection of a hypothesis.

Critical values are determined by the choice of a level of significance (α). The level of significance is the probability of an incorrect rejection. The level of confidence 1-α is the complement of the level of significance and is a measure of the confidence one can have in the decision.

Two types of statistical tests are available: the F-test and the W-test.

The F-test

The F-test is a commonly used test that assesses the model in general. The information provided by the F-test, namely acceptance or rejection of the null-hypothesis (H0), is not very specific. This means that if H0 is rejected, it is necessary to find the cause of the rejection by finding problems in measurements. If you suspect that the H0 is rejected due to a gross error present in one of the measurements, you can use the W-test to identify those measurements.

The W-test and datasnooping

A rejection of the F-test does not directly lead to the source of the rejection. In case the null-hypothesis is rejected, other hypotheses must be formulated that describe possible errors. A simple, but effective hypothesis is the conventional alternative hypothesis, based on the assumption that there is an outlier present in a single measurement in the network. The test associated with this hypothesis is called the W-test.

The process of testing each measurement using the W-test is called datasnooping

Setting Quality Beta

When used together, the F-test and W-test are referred to as the B-method of testing, for which a power can be defined. The power, or Quality Beta (β) of this method is defined as follows: the probability that H0 is accepted while in fact it is false is equal to 1-β.

The least squares adjustment computation allows you to set the β. The Quality Beta defines a tolerance for a level of accuracy. Setting a lower β will, for example, result in the F-Test and W-Test being more tolerant of systematic error.

How to use the TPS computations

Processing a single TPS setup using Tacheometry

1. Click the Project dropdown arrow in the Survey Editor toolbar, and click the survey project that should own the new computation.
   
2. Click the Editor menu on the Editor toolbar and click Snapping.
   
3. Check Survey Points.
   
4. Click the Computation tool palette dropdown arrow, move the mouse to the last row on the palette, and click the Tacheometry button .
   
5. Type a name for the computation and press Tab.
   
6. Type a comment for the tacheometry computation and press Tab.
   
7. Type the limits you will allow for the standard deviation of the orientation.
   
8. Click the Setup tab.
   
9. Snap to and click the setup point on the map.
   
10. If the point is not visible on the map, type its name in the text box and press Tab.
    


11. Click the Setup Name dropdown arrow and click the name of the setup you want to process.
    
12. Click the Value field for the Setup Name attribute if you want to change the name of the setup and type a new name.
    
13. Repeat step 12 if you want to change the stored attributes for Instrument Height, Date, or Comment.
    
14. Click the Measurements tab and resize the Survey Explorer if needed.
    
15. Check the reference points you want to use for orientation.
    
16. Click the Compute button .
    
17. Verify that the computation was processed successfully by ensuring the state is valid.
    
18. Click the Overview tab to see the orientation and its standard deviation.
    
19. Click the Details tab to see the orientation and residuals for each of the reference points. The Orientation field is the direction of the zero reading on the horizontal circle of the TPS instrument.
    
20. Click the Computed Points tab to view the coordinates computed.
    

Processing a single TPS setup using a Free Station Computation

1. Click the Computation tool palette dropdown arrow, move the cursor to the last row on the palette, and click the Free Station button .
   
2. Type a name for the computation and press Tab.
   
3. Type a comment for the computation and press Tab.
   
4. Type the limits you will allow for the standard deviation of the orientation.
   
5. Click the Setup tab.
   
6. Snap to and click the setup point on the map.
   
7. If the point is not visible on the map, type its name and press Tab.
   
8. Click the Setup Name dropdown arrow and click the name of the setup you want to process.
   
9. Click the Measurements tab and resize the Survey Explorer if needed.
   
10. Check the reference points you want to use for orientation.
    
11. Click the Compute button .
    
12. Verify that the computation was processed successfully by ensuring the state is valid.
    
13. Click the Overview tab in the results section to see the computed coordinates and their standard deviations. The orientation and its standard deviation is also reported.
    
14. Click the Details tab to see the orientation and residuals for each of the reference points. The Orientation field is the direction of the zero reading on the horizontal circle of the TPS instrument.
    
15. Click the Computed Points tab to view the coordinates computed.
    

Processing multiple TPS setups using a Traverse Computation

1. Click the Computation tool palette dropdown arrow, move the mouse to the last row on the palette, and click the Traverse button .
   
2. Type a name for the computation and press Tab.
   
3. Type a comment for the traverse and press Tab.
   
4. Type the limits you will allow for lateral and lengthwise misclosure.
   
5. Type the limits you will allow for the standard deviation in the horizontal position, elevation, and orientation.
   
6. Check Compute Tacheometry for every Traverse Station if you want to compute all the side-shots. This option will automatically process each instrument setup as a tacheometry computation after the initial traverse has been computed.
   
7. Click the Setup tab.
   
8. Click the Point identity field in the Setup Point field.
   
9. Snap to and click the setup point on the map. If the point is not visible on the map, type its name and press Tab.
   
10. Double-click the Setup field dropdown arrow in the text box and click the name of the setup you want to use for the traverse. Press Tab.
    
11. Check the box in the fixed field, if available, if you want to use the setup point as a reference point.
    
12. Press Enter.
    
13. Repeat steps 9 through 12 for each instrument setup that you want to process in the traverse.
    
14. Click the Measurements tab.
    
15. Click the Setup dropdown arrow and click the name of the setup with which you want to work.
    
16. Click the Results/Method dropdown arrow and select the adjustment method.
    
17. Click the Compute button .
    
18. Verify that the computation was processed successfully by ensuring the state is valid.
    
19. Click the Traverse Overview tab to see the coordinate misclosures. In the overview you can also see the angular, linear, latitude, and departure misclosures.
    
20. Click the Traverse Details tab to see the coordinated locations defined by the traverse. The changes from the initially computed coordinates and the coordinates after the adjustment are shown in the Delta East and Delta North fields.
    
21. Click the Tacheometry Overview tab and the Tacheometry Details tab to view the results of the tacheometry computation for this setup.
    

Typing measurements from a field book: an example based on the Resection

1. Click the Computation tool palette dropdown arrow, move the mouse to the last row on the palette, and click the Resection button .
   
2. Type a name for the computation and press Tab.
   
3. Type a comment for the traverse and press Tab.
   
4. Type the limits you will allow for the standard deviation in the horizontal position, elevation, and orientation.
   
5. Click the Setup tab.
   
6. Click the Point identity field for the Setup Point, type a name for a new survey point, and press Tab.
   
7. Type a name for the setup and press Tab.
   
8. Complete the properties for the Setup Details.
   
9. Click the Measurements tab.
   
10. Snap to and click the observed point name on the map. If the point is not visible on the map, type its name in the text box and press Tab.
    
11. Check the Orientation box if the observed point is a reference point. Press Tab.
    
12. Type an angle value in the HzAngle field. Hold down Ctrl and press Enter. When the focus goes to the last field, press Enter.
    
13. Repeat steps 10 through 12 to add new rows for each entry from the field book.
    
14. Click the Compute button .
    

Processing multiple TPS setups using a least squares adjustment

1. Click the Computation tool palette dropdown arrow, move the mouse to the last row on the palette, and click the Adjustment button .
   
2. Type a name for the computation and press Tab.
   
3. Type a comment for the adjustment and press Tab.
   
4. Type the a priori standard deviation for the horizontal angle observations and press Enter.
   
5. Repeat step 4 for the vertical angle observation.
   
6. Type the standard deviation for the distance, press Tab, and enter the parts per million ratio. Press Enter.
   
7. Type the a priori standard deviation for centering the setup. Press Enter.
   
8. Repeat step 7 for the standard deviations of Target Centering, Instrument Height, and Target Height.
   
9. Click the Level Of Significance Alpha dropdown arrow and click the alpha percentage you want to use.
   
10. Click the Quality Beta dropdown arrow and click the beta value you want to use.
    
11. Click the Setup tab.
    
12. Snap to and click the setup point on the map. If the point is not visible on the map, type its name and press Enter.
    
13. Press Enter.
    
14. Skip to step 16 if the correct setup name appears in the Setup field.
    
15. Double-click the Setup field dropdown arrow and click the name of the setup you want to use for the traverse. Press Enter.
    
16. Repeat steps 12 through 15 for each instrument setup that you want to process in the adjustment.
    
17. Make corrections to the Setup Name, Instrument Height, and Comment, if required.
    
18. Click the Measurements tab.
    
19. Click the Setup dropdown arrow and click the instrument setup name to view the measurements belonging to the different instrument setups.
    
20. Make corrections to the measurements, if necessary, by editing their values. Only do this if you know the existing values are mistakes and if you are absolutely certain of the correct value.
    
21. Click the Reference Points tab.
    
22. Click the Free network adjustment option.
    
23. Highlight the Point Name field in the first row by clicking in the field.
    
24. Snap to and click the reference point on the map. If the point is not visible on the map, type its name in the text box and press Enter.
    
25. Skip to step 28 if the reference point should be used as control in three dimensions.
    
26. Double-click the Fixed Type field to get the Fixed Type dropdown arrow.
    
27. Click the dropdown arrow and click the dimension to be used for the reference point.
    
28. Press Enter.
    
29. Repeat steps 24 through 28 for each reference point that should be added to the least squares adjustment.
    
30. Click the Compute button .
    
31. Assess the state of the computation by looking at the State icon.
    
32. Follow the steps outlined in the next procedure, 'Detecting and disabling outlier measurements using datasnooping', if the icon state indicates that the computation is incorrect. Continue with step 33 once you have completed the datasnooping process.
    
33. Click the Results tab.
    
34. Click the Adjusted Objects dropdown arrow and click Points to view the computed coordinates and their quality information.
    
35. Click the Adjusted Objects dropdown arrow and click Measurements to view the adjusted measurement values. You can also view the deltas, which provide the amount of change from the original unadjusted values.
    

Tips

In the least squares adjustment computation, the setup point can be added multiple times, once for each instrument setup. Avoid making edits to measurements from raw data collected in the field unless you are completely convinced that they are incorrect and you are certain of the correct value. You can choose the dimension of the reference point in the Reference Points tab of the least squares adjustment by selecting the dimension field and pressing the 1, 2, or 3 keys. Original measurement values do not change the results of the least squares adjustment, nor are they altered in the survey dataset in which they are stored. You can view the final adjusted measurement values and their deltas–this information is stored with the computation.

Detecting and disabling outlier measurements using datasnooping

1. Complete steps 1 through 29 defined in the preceding procedure, 'Processing multiple TPS setups using a least squares adjustment'.
   
2. Click the Compute button .
   
3. Assess the state of the computation by looking at the State icon.
   
4. If the icon state indicates that the computation is incorrect, continue with the following steps. Otherwise, you do not need to disable outlier measurements.
   
5. Click the Test-Statistics tab to start the data snooping process.
   
6. Click the Test Type dropdown arrow and click Suspected outliers in measurements.
   
7. Click the Setup Name dropdown arrow and click the different setup names to view their outlier measurements.
   
8. Identify the measurements that have the largest estimated error for angles and distances and note their setup point and to point names for the next few steps.
   
9. Click the Test Type dropdown arrow and click W-Test Measurements.
   
10. Click the Setup Name dropdown arrow and click the setup points you noted in step 8.
    
11. Find the outlier measurements highlighted in red, and identify the largest W-Test value.
    
12. Click the leftmost column of the row with the largest W-Test value. This selects the measurement's row.
    
13. Right-click the selected row's leftmost column and click Disable Measurement. Do not disable more than one measurement at a time.
    
14. Return to step 2.
    

Tip

The W-test assumes an outlier in a single measurement. Therefore, outliers should be disabled one at a time and the computation recomputed and tested after each change.

Please visit the Feedback page to comment or give suggestions on ArcGIS Desktop Help.
Copyright © Environmental Systems Research Institute, Inc.