Least Squares Adjustment Method

Horizontal Adjustment

Data input

Once all relevant information has been entered, you can save and exit or run the adjustment.

The observations file is then read, with accompanying messages summarising the input data being displayed on the screen. Any required data blocks not present will cause errors and the program will terminate.

Data validation

Following data input is the validation phase.

This checks :

• All stations used in the observations appear in either the fixed or free stations data blocks.
• No observations occur wholly between stations that are fixed.
• No directions are fixed

Least-squares adjustment

On successful completion of the input and validation stages, the adjustment will begin.

The least-squares process using variation of coordinates may require a number of iterations before the solution converges to a satisfactory result. The number of iterations depends largely on the accuracy of the supplied approximate coordinates for the free stations. The iterations will terminate if one of the following two conditions applies:

• The maximum number of iterations is reached.
• The change in coordinates from one iteration to another is insignificant; that is, less than one millimetre.

In the former case, a message will appear on the screen warning the user that the iterations have been terminated prematurely. The calculation of each iteration is accompanied by a message on the screen.

Warnings during the adjustment

There are four warning messages that may occur during the adjustment phase.

They are :

• C-O limit exceeded on distance.
• C-O limit exceeded on bearing.
• C-O limit exceeded on angle.
• C-O limit exceeded on direction.

These messages are related to the limits that are given in the options data block. See the Options data block under the 'Options' section.

'C-O' is short for 'Calculated minus Observed'. Briefly, if the calculated value of an angle or distance varies from that given in the observations data by more than the limits given in the options data block, then a warning message will appear at run-time during the first iteration. Three asterisks will also placed against the observation in the adjustment report to enable identification of where any warnings have occurred.

Occurrences of this nature imply a discrepancy between the observations and the given approximate coordinates. As such, it COULD be indicative of an error in an observation and thus should be examined closely. On the other hand, it may be that the given approximate coordinates are inaccurate. This latter situation is not a problem unless the adjustment fails to converge to a solution in which case a terminating error will occur.

Errors during the adjustment

There are two irrecoverable errors that may occur during the adjustment phase.

These are:

• Diverging solution.
• Error inverting matrix.

Both these errors generate appropriate run-time error messages and will terminate the program. The causes of these errors fall into the following two categories:

• Highly inaccurate approximate coordinates
• Incomplete observations to resolve the network

In some cases if the approximate coordinates are not accurate enough the solution will diverge, and no result will be forthcoming. This situation should be trapped by the program, possibly after quite a number of iterations, and the program terminated with the appropriate message.

If the network is 'under-observed', that is, cannot be resolved using the observations given, a matrix inversion error will result and the program will terminate.

If ambiguous observations are given, such as a resection on a danger circle, then program termination due to a diverging solution is the most likely result. It must be remembered that another possible result would be a poor adjustment.

Adjustment report

Successful completion of the adjustment generates a report that includes the following:

• A listing of the run-time messages.
• A listing of the input data.
• A listing of the adjusted observations with residuals.
• A listing of the adjusted coordinates along with the standard deviation of the easting and northing and the covariance.
• Additional reports as specified in the options data block.
• If a vertical adjustment is included, its results will also be placed in the report.

The report file will be named identically to the observations file, but with the extension ' .lsa_list '.

Examination of residual sizes, standard deviations of the adjusted coordinates, and, if given, error ellipse parameters, will be necessary to evaluate the success of the adjustment.

Graphics output

If the graphics output option is chosen, then a graphical representation of the network in ground coordinates will be created. If the error ellipses have been generated, then a graphical representation of the ellipses will also be displayed, at an exaggeration based on that defined in the options block.

The name of the Vulcan file will be prompted for in the usual manner; that is, project code and name.

The layer in which this data will be placed is named identically to the survey name in the observations file. It is probably best to have a separate adjustments database away from any other survey data to avoid conflicts with layer names.

All lines observed in the network are created as separate objects in the layer. Each object is colour and line-style coded to reflect the type of observations made on that line. The following criteria are used:

• Lines with both distance and angle observations will have a dotted line type.
• Lines with either only distance observations, or only angle observations, will be solid lines.
• The colours of the three cases mentioned in a) and b) will be different, and examination of the object names will indicate the types of observations of each line.
• Fixed stations will be highlighted by similar coloured crosses.

Examining the graphical output is an easy way of obtaining an overview of the structure of the network, and to see where the network may be strengthened if necessary.