Least Squares Network Adjustment

This Getting Started article, covers the Least Squares Network Adjustment option, which is supplied with all editions of the ⁿ4ce software.  This is based on the variation of coordinates principle when the coordinates of unknowns are approximated and then corrected.  Weights can be applied to measurements using standard errors, and a full statistical analysis of the results can be presented including error ellipses.  Reports and Error Ellipse plots are provided post-adjustment and can be printed.

Whilst it may appear to be a black box solution, you have some control over the process, especially when dealing with weights.  You can apply a “what if scenario” changing parameters and re-running the adjustment.

Least Squares Reports and Plots

ⁿ4ce Least Squares Network

Least Squares Option

The Least Squares is a standalone option, meaning when you enter it, you effectively leave the ⁿ4ce environment.  Data is prepared in ⁿ4ce and then passed onto the application.  The reverse happens when leaving the application with adjusted coordinates.

Data is prepared as per any other adjustment, with control measurements being recorded with a -99 point number.  Control measurements with -88 point number (bearings) and no distance will be ignored.

Meaned readings of sightings using FL and FR measurements to the same stations are used.  See later for warnings.

At the end of the adjustment process, you will run into Reduction.  This is where the computed station coordinates are used to calculate the coordinates of all survey observations.

Data Extraction

The principle of 'Least Squares Variation of Coordinates' used here depends upon supplying both control survey observations and fixed stations.  You MUST have at least two fixed stations or one fixed station and a fixed bearing. In addition, there MUST be a redundancy of observations.

Fixed control can be provided in the Stations folder on the ⁿ4ce Project Tree, or entered manually in the Least Squares dialog that appears during the adjustment process.  Any manually entered, or imported stations with a Known status (Type) will appear as a Fixed station in the Least Squares option.  All other station coordinates identified in the Stations folder will be treated as being known.  The Status of all stations can be altered prior to the adjustment.

During the extraction process, multiple observations to the same station will be meaned.  A spread about the mean will be calculated as part of this process, and a spread for Horizontal angles, Vertical angles and Slope distances will be calculated.  If this spread exceeds the Standard Error allocated to this measurement, an error warning will be displayed as seen below.

Control Data Warning Message

When you have passed through this initial check, you will enter the Least Squares Adjustment Dialog Box.  This is different to the traverse adjustment, which is a wizard allowing you to move backwards and forwards.  Here you step between stages using Tabs at the bottom of the Dialog.

Least Squares Adjustment Dialog Showing Poor Data (Red) and Standard Error as a tooltip

If you select the Readings Tab, this will display the extracted (mean) control readings for each station.  Hovering over any readings highlighted in red will display the standard error as a tooltip.  This exceeds the default weight for this reading.  In the above, it’s a Vt Angle weight which is set to 3 sec, and the tooltip is showing Std Err = 7.1 sec.  The white/cyan banners are used to distinguish different instrument setups.

 

Note: There are NO tools for editing the control data that appears in the Readings Tab. You will have to return to the Project to make changes.

Pre-Calc

When you first enter the Least Squares Adjustment, the screen below will be displayed. This contains the stations used in your survey.  If the coordinates are found in the Stations folder, these will appear as either Fixed or Approx.  You can enter new values or simply let the adjustment process recalculate the approximate station coordinates for you.  Those shown below are pre-calculated and therefore blank.

Pre Calculated Coordinates

Note: Any or your network which does not tie in (hanging leg), will be ignored in the LS Network Adjustment and treated as normal survey detail.  Their status becomes Derive during reduction.

In the above example there is one fixed station, C1.  Since only one station is fixed, a bearing between C1 and C2 is set with a value of zero. The standard errors (weights) are shown with corrections for OS UTM Scale, Curvature & Correction and Mean Sea Level turned off.  3D adjustment is enabled.

By changing Standard Errors and repeating the Adjust its possible to get closer to the a Total Error Factor of unity.

Note: Warning Tolerances shown below, used when importing data and analysing the final adjustment.

Changing Settings

Bear in mind that the whole process is a statistical estimate for the best value, within a 95 percentile range, plus or minus RMSE.

The calculation process is iterative and in situations where it’s not possible to converge on a result, the Number of Iterations and a Convergence Limit can be set.

Adjust

Pressing the Adjust button completes the “Black Box” calculation, based on the data supplied and settings, which can be changed and the process re-run.

The first tab, called Pre-Calc, displays all the stations identified, that will be used in the adjustment.  Any side shots, not part of the adjustment will be stripped out.  You will have to set at least one as being Fixed, and an Orientation or two or more Fixed with multiple Orientations.

The program will automatically calculate the approximate coordinates of other stations used in the adjustment, if not found in the Stations folder.

The standard errors used in the adjustment are shown to the right of the screen.  Use the Settings button to make changes, if necessary.

The Readings Tab will show you all the meaned readings being used in the adjustment process.

Also note the Corrections for MSL, C&R and Scale.  Adjustments can be either 2D or 3D.

Pre-Calc Coordinates

Additional tabs will appear at the bottom of the dialog box after pressing the Adjust button. You can step between these tabs at any time.

The station coordinates shown in Pre-Calc above, will be the “latest” values up to that point in the calculation process.

Selecting the Adjust Tab will take you to a summary of the adjustment process.  The upper window displays the Adjusted Station Values and highlights potential problem stations in red.  This is where the calculated RMSE exceeds tolerance checks.

The Max Residuals will indicate max corrections to readings in Hz Ang, Vt Ang and Slope Distance to and from given stations.  

The Max Adjusted Station RMSE values, give you the worst case adjustment in X,Y & Z.

Adjust Summary

The Total Error Factor (Sigma Zero) and Err Factors/SdErr are used to give you an overall quality of adjustment.  The TEF should approach unity for well-balanced adjustments.

Selecting the Map Tab takes you to a graphical representation of the network, with error ellipses.  These are created from a Covariance Matrix, and give you a graphical representation of the potential errors in the adjustment.  Clearly, the larger the ellipse the worse the predication in the direction of  the major axis.

Various tools are available to view and scale this image, including displaying a grid and creating a plot. The roller ball on your mouse can be used to zoom in/out and left mouse button to pan, if held down.

Map of Error Ellipses

Vt Angles, Hz Angles and Pl Distances – Final Adjusted values

The last three Tabs allow you to step through the adjusted readings.  The Residual column identified the correction made to the original measurements, whilst the Weight column is the weights used in the adjustment.  Clearly, if the correction exceeds the weight by a large amount, this throws some doubt on the reading.  These will be show in red.

Report

The Following is a copy of the Report created after the Adjustment has been carried out.  It is suggested that you study this carefully!  See boxed notes.

Reduction

Following exit from the LS Network Adjustment, ⁿ4ce will automatically go into reduction of the detailed observations.  As you can see from the results below, this survey closes very well but this is not always the case!

Note: that the status of the Stations involved in the adjustment are now shown as Least Squares.

Survey Observation Reduction

Tolerance checks are made between the adjusted coordinates to the backsight and  those calculated from the raw survey data.  The difference is called the RO Review and should there be a discrepancy, this will be identified if it exceeds the Tolerances, shown top left.  A small number between 2 – 4 will appear alongside the station name in the upper window, indicating the number of tolerance failures.

Theory and Looking for Problems

The Least Squares Network Adjustment feature provides a general 2D or 3D adjustment but has limitations.  Data will come from survey measurements in terms of Hz Ang, Vt Ang and Sl Dist to stations which are identified as measurements with a -99 point number.  If multiple sightings are found, these will be meaned to provide single values.  There is currently no preview of the meaned readings and residuals.  

This is not based on a floating system and at least one station should remain fixed, with a fixed bearing or a second fixed station.  Readings to side shots that are not part of a traverse or network will be ignored.

Individual weights are not allowed, so all Distances, Hz Ang and Vt Ang are allocated with the same weight in each category.

The -99 readings will automatically be extracted from your survey, meaned (including FL & FR readings) and presented to the LS engine.

The Standard Error of these meaned readings will be compared to the Warning Tolerances and if found to exceed these will appear in red on the readings page.

If you hover over these red readings, a tooltip will appear showing you what the standard error is.

Displaying Errors in Readings

Note: Elsewhere in the software you will see items highlighted in red.  This indicates where these items exceed checking tolerances.

Readings are banded white/cyan per setup.

The standard error is found by calculating the difference from a mean (offset), using the following equation.

Std Err =  √[∑ (offset²)/(n-1)]

where n = number of sightings

The standard error is used to assign a suitable Weight to this type of reading.  As a guide, you may wish to use the quoted precision of your total station as the standard error.  But remember, the more times you take a reading using both faces, the better the prediction of the standard error becomes.  It’s important to remove collimation errors by measuring on both FL and FR!

The weights have a massive influence on the outcome of the adjustment process and should be changed to achieve a Total Error Factor of Unity.  In this regard, also check the Error Factors of Distances, Hz Angles and Vt Angles.

Changes to standard errors can be made after each run, and then re-adjusted.  As part of the adjustment procedure, approximate coordinates are calculated for the unknown values.  If repeating the adjustment, you will be invited to use the last adjusted values.

Note: Corrections for Mean Sea Level, Curvature and Refraction and Scale Factor (OS Transfer Mercator Projections) can be applied as part of the adjustment.  These corrections will be carried through to the Reduction, if applied.

Isolating and Eliminating Observations

If you are aware of possible errors or inconsistencies in your observation, you can isolate these using the ,I comma code.  This will leave the observation in the database, displaying them in orange, but will ignore them during adjustment and reduction calculations.  You may wish to use this after observing the least squares report.

Using Comma ,I to Ignore Readings

When entering the LS option, if meaned readings have standard errors exceeding a tolerance check, a flag will appear as shown below.

Go to the Readings tab to see the standard error as a tool tip.  You will have to leave the LS option to go back to the control grid, if you wish to eliminate of check readings.

Checking the Report for Errors

Once the adjust process has been complete you should look at the Report.  There is a section which identifies the corrections to distances and angles, as shown below.  The process tries to adjust the readings to make them match the final coordinates.  The adjustment to these readings is identified as the Residual.  This is then compared to the standard error, and if this falls with 1.5xStd. Err, this is considered to be acceptable.  Anything greater would indicate a possible blunder, or that the nominated standard error for that reading is too small.  Problem readings are identified with asterisks, with *** being unacceptable.  See results below.

Conditioning networks

There are ways in which you can help yourself making our network more stable.  This involves taking bracing measurements, especially across the network, and fixing control points at extremes of the network.  If you start off with just one fixed point and a bearing, the further you go from the start,the worse the RMSE for each station will appear.  This is clearly seen in the error ellipses, as shown below.

The fixed station is at C1 with a bearing of 90deg to C2.  The further away from the C1, the larger the error ellipses.  But note the bracing.  Joining between C1B to H2 would improve the conditioning.

Adjusting Weights

You will have appreciated that the standard errors of readings has a massive effect on the quality of your results, as they are used to define the weights used in calculations.  The smaller the standard error, the larger the weight and hence its influence in calculations.

But this needs to be controlled and indications of suitable standard errors will be provided in reporting.  Key to a suitable solution and appropriate weights is the Total Error Factor (TEF), which should approach unity.  These are provided on the Adjustment tab.

If the TEF is unity, as above, then the standard errors for measurements are considered satisfactory and station coordinates are with 95% confidence limits +/- RMSE values.  Clearly, we also need to reduce the RMSE values as well!

The above report also shows worst case scenarios for corrections to observations and RMSE values for X, Y and Z computed coordinates at stations.

Let’s see what happens when we adjust the standard errors for individual readings.  The current Error Factors for Dist, Hz and Vt angles are 0.61, 1.5 and 0.82.  These should fall within the bounds 0.5 to 1.5.  Current standard errors are 1.0, 5.0 and 10.0 respectively.

Return to the Pre-Calc tab and select Settings.  Change Hz angle SE to 2, Vt angle to 5 and Distance to 2.  This will alter the weights accordingly.

Now exit and select the Adjust button.  You will be asked to confirm using the last computed coordinates.

The process just completed is iterative and what we are trying to achieve is consistency in the selection of standard errors of measurements.

With the current set of standard errors, the TEF has gone up to 1.9 and the individual Error Factors have changed to 0.71, 3.45 and 1.50.

There is an argument now to increase the standard error for Hz angle to 5, or more to reduce the error factor from 3.45.

Once a suitable set of standard errors have been found these should be appropriate for the surveyor and instrument being used, next time.

Once you are happy with the results, then Exit and the adjusted coordinates will be saved in the Stations folder, on the Project Tree.

The Type will indicate Lst Squares adjustment and Remarks will show the RMSE values in mm.

Exercise 11 – Sample Network Adjustment

From the ..\Training\Intermediate folder open the Ex11 – LS Network.sdb project.  This is similar to the data that appears at the head of  this Getting Started Manual and has a number of cross linked traverses.  The following is a summary of what you see when exploring this Project.

Exercise 11 – Selecting Least Squares Option

 Exercise 11 - Station Control

Control Readings – Open Legs are Ignored

1. One fixed station, called C1, has been entered into the Stations folder.  This could have been entered after selecting the LS Adjustment option.  At least one station must be fixed with a bearing to another station, or two fixed stations.

2. Each of the above instrument setups has control readings.  Readings from C1H are shown below.  If multiple sightings have been made to station these will be meaned and the mean value sent to the LS Adjustment option.  Spreads about this mean are used to calculate the Standard Error of measurements.

Control Readings – C1H

Note: Control readings will be identified in bright yellow with a -99(STN) point number.  It is NOT necessary to separate the Control from Detail observations, as shown here.

Only the Reduction Correction defaults are carried across to the LS Option.  The others defaults used with in the application are controlled and saved from the option itself.  This includes the tolerance checking and setting weights.

3. Having selected the Least Squares Option a small delay may appear as checks are made on observational data.  Here we’re looking at the offset from mean values to calculate a standard error for  that reading.  This is checked against tolerance values and if exceeded warnings will be given and the reading highlighted in red, as previously discussed.  The following will then be displayed. 

Note: Ensure that the Standard Errors and Corrections are as displayed below otherwise your results will not compare to  those in this exercise. 

Pre-Calc Tab Giving Access to Readings, Settings and Adjustment

This dialog shows all the stations that are part of  the network.  There has to be at least one fixed station, as shown here, which is supplemented with a known or fixed bearing, o two fixed stations.  The current Standard Errors and Corrections (if applied) will be shown on the right.  If necessary got to Settings and change them accordingly.  Also note that the 3D check button is selected.

Note:  This is where you can return to change the standard errors in Settings and rerun the Adjustment.  It’s not possible to alter the observations at this stage.  You must exit and return to the detail observations in the main Project to do  this.

4. Set the Orientation Bearing from C1 to C2Z to 0.0. and then press  the Adjust button.

5.

 

Adjust Tab with a Summary of  the Results

This screen provides a summary of the adjustment using the parameters used with additional Tabs, and highlights worst case scenarios.  It’s important to look at the Error Factors first.  These should approach Unity.  Sigma Zero is 1.3 but the separate error actors for angles are 1.38 and 1.64, indicating the standard errors for these are too low.  Likewise the distance error factor is 0.29 indicating the standard errors used here are too high.

Change the standard errors for Hz, Vt angles and distances to as follows and repeat the Adjustment.  Hz Ang = 5”, Vt Ang = 8” and Dist= 1mm and 1mm ppm.

Interaction of the Adjustments with New Weights

Provided the individual error factors fall within 0.5 and 1.5 this is acceptable.  But note the Total Error Factor closer to unity, which is what we are trying to achieve.

Also look at the Max Adjusted Station RMSE vales.  These are smaller in X & Y but slightly larger in Z.  The worst case maximum corrections to Observations has also increased.  The final coordinates are given a 95% approval within the specified RMSE shown in the Report, discussed below.

6. If you select the Plot Tab the following Error Ellipse diagram will appear.

The fixed station C1 id shown with a small triangle with a bearing of zero to station C2Z.  The 95% prediction of the point coordinates fall within  the error ellipse (2D) and clearly the further they are away from the fixed points the greater the potential for error.  Fixing a leg between C1 and H1 will strengthen the network as will fixing coordinates for S2, if known.

Note: the open ended legs are not included in the adjustment.

7. It’s important to look at the Report which can be found in the Adjust Tab.

The adjustment process uses approximate coordinates for the unknowns.  These are shown at the top of the report, together with the original observations.  Half way down the report, an Adjustment Summary will appear, together with the Sigma Zero value, which should be close to unity. 

This is then followed by corrections to the approximate coordinates, noting that the fixed vales have zero corrections.

The next part shows corrections to the original observations to achieve the results that were calculated.  Look out for any stars (***) and large corrections (residuals).  This is where possible measuring errors might have been made!  Distances look fine, but note Hz angles between C1C, C1B and C1D are throwing up 20” corrections (**).  Since we set the standard error for Vt angles high no (***) are appearing but look at the corrections.  There are seven corrections greater than 10”.

And finally, the Adjusted Coordinates are showing large RMSE’s for stations away from the fixed control.

Overall, this survey could have been improved by strengthening the network using more redundancy and improvements to the measured data.  FL and FR will remove collimation errors and more rounds with give improved accuracy.  It’s essential to use traversing kit and if available ATR for locking onto targets, to achieve the best results.

Summary of Exercise 11

Exercise 11 illustrates a further useful tool in ⁿ4ce, least squares network adjustment, but you really need to understand the procedure to get the most out of it.  In this exercise the readings were particularly poor, with RMSE’s exceeding 20mm in plan and 5mm in height.  These can’t be corrected in the software and may require re-measurements!  See earlier example.