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Step 3 - Reporting
Overview : Step 1- Analyze -----> Step 2 - Solve -----> Step 3 - Report -----> Step 4 - Update
The Least Squares report dialog allows you to select the information you want to see about the solution. TPC creates a separate Least Squares Report view and places the requested information in it. You can edit or select any of the information in this report, print it, copy and paste it to another application or append it to the current Survey Report.
If you need to update your Least Squares adjustment, proceed to Step 4- Update.
Beginning with TPC Desktop 2011, you can create this report even if the solution did not converge. This allows for a post Morten investigation of the non-convergent data.
Least Squares Report
When TPC computes any Least Squares solution it doesn’t actually change any of the current survey points. Instead, you have to explicitly update the survey with the adjusted coordinates. This extra step of updating the survey allows you to review the solution. Once you have reviewed and accepted the solution you can apply it to the survey.
To aid in this review, TPC provides a Least Squares report that can include any of the following information.
Survey Information
Each report begins with the survey and file information for the analysis. The example shown here is a Network adjustment for the Learn Least Squares Network sample survey.
[[ Least Squares Network Adjustment ]]
LEARN LEAST SQUARES NETWORK.trv
Settings
Depending on the type of analysis, the settings sections lists basic information you can use to evaluate the data in the report.
TPC uses the angle units selected in Program Settings, Units in all Least Squares reports.
[[ Settings ]]
Adjustment Type: 2D
Distance Units: Feet (')
Angle: DMS (°'")
[ Datum ]
All observations are reduced to Lambert Conformal Conic (2SP) All listed measurements have been reduced to grid
[ Notes ]
Angle Data Station Order: BS-OP-FS
Precision = estimate of standard error (a Priori)
StdDev = Standard Deviation
* = Redundant Observation
! = Computed value exceeds stochastic model
Stochastic Model
This section includes your aPriori values and the expected Reference Variance of 1.0 which is use for the Chi-Squared test.
[[ Stochastic Model (estimates of standard error) ]]
Reference Variance 1.00
[ Default ]
Distance: Horizontal 0.050' ±5 ppm, Vertical 0.030' ±5 ppm
Angles: Horizontal Point 5.000" Read 3.000", Vertical Point 5.000" Read 3.000"
Direction: Azimuth/Bearing 60.000"
Coordinates: Y 0.00' X 0.00' Z 0.00'
Unadjusted Data
This section lists the initial coordinates and raw observations used in the analysis. If you were to write a Starnet file from a Least Squares analysis, this is the data it would contain.
Notice that TPC includes the Precision (assigned or computed) for each observation. Coordinates are Fixed, Free and weighted. No weighted coordinates are shown here.
Redundant observations are indicated by the * character following the foresight label.
[[ Unadjusted Data ]]
[ Stations Entered = 7 ]
Fixed Y X Desc
A1 10000.000 10000.000
A2 9866.000 7900.000
Free Y X Desc
1 10387.978 11086.004
2 12088.398 11262.443
3 12151.184 9796.049
4 14169.142 9471.772
5 13491.098 8251.257
[ Horiz Angle Observations = 10 ]
BS OP FS Horiz Angle Precision
A2 A1 1 163°59'30" 8.000"
A1 1 2 115°35'00" 8.000"
1 2 3 86°31'40" 8.000"
2 3 A1* 82°07'50" 8.000"
3 A1 1* 75°45'20" 8.000"
1 2 4 133°21'40" 8.000"
2 4 5 101°39'40" 8.000"
4 5 A2* 124°35'20" 8.000"
5 A2 A1* 80°48'50" 8.000"
4 5 3* 69°59'30" 8.000"
[ Horiz Dist Observations = 8 ]
OP FS Horiz Dist Precision
A1 1 1153.23' 0.056'
1 2 1709.55' 0.059'
2 3 1467.74' 0.057'
3 A1* 2160.85' 0.061'
2 4 2745.18' 0.064'
4 5 1396.21' 0.057'
5 A2* 3642.09' 0.068'
5 3* 2044.94' 0.060'
Solution
If you select any of the solution check boxes in the Reports tab, TPC displays the basic information about the solution as follows
[[ Least Squares Solution ]]
Analysis for: Entire Survey
Solve for 5 of 7 points, 10 unknowns
18 Horizontal Observations, 8 redundant (degrees of freedom)
Solution converged in 2 iterations
Statistics
TPC provides the solution’s statistics to help you determine how strong (good) the solution is. For this solution, the Reference Variance of 0.08 is very low, probably a result of not have very many observations.
Notice that TPC then breaks this down into Components for Coordinates and Horizontal (2D solutions don’t give a Vertical component). TPC then breaks the Horizontal into components.These components help you isolate issues to horizontal vs vertical, angles vs distances, etc. A large SumSquares of Residuals or Average StdDev for any of these components indicates where to look for issues with either your stochastic model, your coordinate values or your observations.
[ Statistics ]
Reference Variance ±0.08
Reference Standard Deviation ±0.29
Components Count Sum Squares Average
of Residuals StdDev
Coordinates 10 0.634 0.063
Horizontal 18 0.660 0.287
Horizontal Count Sum Squares Average
of Residuals StdDev
Horiz Angle 10 0.517 0.227
Horiz Dist 8 0.143 0.134
Chi-Square
This example solution did not pass the Chi-Square test at the 95% level. So,what does that mean?
Remember, the Chi-Square test failed because the computed Reference Variance of 0.08 is statistically different from the assumed stochastic Reference Variance of 1.0 at the 95% confidence interval.
In statistical terms, it means that we reject our null hypothesis that the computed reference variance equals 1.0. In very general terms, this test simply asks how well a sample data set represents a population. In the case of a survey, if we took a lot more observations(the population), how well would this sample data set represent the survey. Given that we have such a small data set, we could still accept this solution if the residuals and standard deviations are small.
Now is a good time to look at the standard deviations reported below for all coordinates and observations. You will see that they are very small. Notice also that because the Chi-Squared test failed, TPC multiplies the standard deviations computed for the solution by the computed Reference Standard Deviation of 0.29. If the Chi- Least Squares Network Adjustment22Squared test had passed, TPC only multiplies them by the computed Reference Standard Deviation if you tell it to in the Miscellaneous tab.
You can choose the confidence interval you want in the Solve tab, where 90%, 95% and 99% are available from the pull down list.
[ Chi-Square ]
Failed Chi-Square test at 95% significance level!
Chi-Square = 0.66
Lower/Upper Bounds (2.180/17.530)
The computed reference variance 0.08 is statistically different from the aPriori value of 1.
Standard deviations of computed coordinates and observations shown below are multiplied by 0.29.
Consider using these to improve your stochastic (weighting) model.
Adjusted Coordinates
The adjusted coordinates section lists the values that will be used to update the survey if you choose to do so.
[ Adjusted Coordinates ]
Station Y X
1 10387.977 11086.001
2 12088.392 11262.453
3 12151.204 9796.061
4 14169.157 9471.790
5 13491.117 8251.277
Standard Deviation of Coordinates
The standard deviations are listed for each coordinate that was computed in the solution along with their standard deviations in Y, X and Z (3D only).
You can think of this Sy and Sx as how much each coordinate was moved from their unadjusted position as a result of the Least Squares solution. In most cases, smaller movement is better than larger movement. However, you can have a situation where the Least Squares solution improves an unadjusted position that wasn't where it should have been.
[ Standard Deviation of Coordinates ]
Station ±Sy ±Sx
1 0.011 0.013
2 0.017 0.019
3 0.013 0.020
4 0.015 0.033
5 0.016 0.026
Coordinate Error Ellipse Data
Error ellipse data provides information about the size and direction of possible error in the computation of the adjusted position of each adjusted coordinate.
Su is the size of the semi-major axis of the ellipse and Sv is the size of the semi-minor axis, while t is the angle of the ellipse.
[ Station Coordinate Error Ellipse ]
F-Statistic Multiplier (95% Confidence) 2.864 is applied to Su, Sv below.
Because the ChiSquared test failed, the Reference Standard Deviation 0.287 will also be applied.
Su=Semi-Major, Sv=Semi-Minor, t=Theta=angle from north (+CW, -CCW)
Station ±Su ±Sv t
1 0.035 0.033 25°31'53"
2 0.063 0.037 -51°39'08"
3 0.058 0.038 -83°16'39"
4 0.093 0.043 -89°46'27"
5 0.068 0.054 -73°39'14"
Residuals / Corrections
This section lists the residuals of each observation used in the solution along with the corrected observation and it's standard deviation. Remember, standard deviations are multiplied by the computed standard deviation for the solution if the Chi-Square test fails. If it passes, you have the option to do this in the Miscellaneous tab.
If TPC encounters a computed standard deviation that exceeds it's aPriori value, TPC will display ! next to it.
[ Horiz Angle Corrected Observations = 10 ]
BS OP FS Residual Correct ±StdDev
A2 A1 1 -0°00'00" 163°59'30" 1.793"
A1 1 2 0°00'02" 115°35'02" 1.630"
1 2 3 0°00'02" 86°31'42" 1.577"
2 3 A1* 0°00'04" 82°07'54" 1.617"
3 A1 1* 0°00'02" 75°45'22" 1.520"
1 2 4 -0°00'00" 133°21'40" 1.701"
2 4 5 -0°00'01" 101°39'39" 1.543"
4 5 A2* -0°00'01" 124°35'19" 1.525"
5 A2 A1* 0°00'01" 80°48'51" 1.438"
4 5 3* -0°00'01" 69°59'29" 1.667"
[ Horiz Dist Corrected Observations = 8 ]
OP FS Residual Correct ±StdDev
A1 1 -0.00' 1153.22' 0.013'
1 2 -0.00' 1709.55' 0.013'
2 3 -0.00' 1467.74' 0.013'
3 A1* -0.00' 2160.85' 0.013'
2 4 0.01' 2745.19' 0.015'
4 5 -0.00' 1396.21' 0.015'
5 A2* 0.01' 3642.10' 0.017'
5 3* -0.02' 2044.93' 0.013'
Blunders
The blunder factor can be changed in the Miscellaneous page dialog.
[ Horiz Angle Blunders ]
At From To Residual Correct
CP215 CP214 CP107 0°00'24.89" 76°58'54.89"
CP216 CP215 CP107 -0°00'21.70" 29°58'38.30"
2 Horiz Angle Blunders Detected (blunder > 5 X StdError
[ Horiz Dist Blunders ]
From To Residual Correct
CP211 CP212 -0.1811 856.9149
CP215 CP216 -0.1885 756.0185
2 Horiz Dist Blunders Detected (blunder > 5 X StdError
Related Topics
Least Squares
Editions
Premium, Professional
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