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Using Least Squares





TPC can reduce random error inherent in survey measurements through a Least Squares adjustment. A Least Squares adjustment can be applied to something as simple as a resection and as complicated as a 3-dimensional network. The result is adjusted coordinates that are statistically more accurate than those derived from simple intersection and traverse adjustments.

Types of Least Squares Solutions

TPC uses Least Squares for number of solutions.

Adjusting a Network

A Least Squares network is not limited to the sequence of points defined by a traverse. Although Least Squares can use the points defined by a traverse, it can also combine traverses to form a network then solve the network simultaneously.

Solving a Least Squares Network allows you to reconcile all your data at once providing the strongest solution possible.

You can find the Least Squares Network Adjustment tool in Tools | Least Squares Network Adjustment.

Adjusting a Traverses

TPC can adjust traverses of Closed Loop or Close Point-To-Point types using Least Squares. By definition, traverses have just one degree of freedom unless additional, redundant observations are taken.

Solving a Resection

TPC automatically uses Least Squares to solve resection points. This method allows the greatest flexibility in the data you collect to generate the resection position.

Fitting Lines & Curves

TPC uses Least Squares to fit lines and curves to points.  See Horizontal Line Fitting and Horizontal Curve Fitting.

These routines are sometimes called best fit routines, since they determine the line or curve that best fits the selected points.  In terms of Least Squares, these are the fitted lines and curves that have the smallest sum of the residuals when fitted to the points.


When TPC computes a calibration, it uses Least Squares to compute a best fit for that calibration.  As a result, you will generally want 3 or more point pairs to compute a calibration, with more pairs producing better calibration parameters.  See Calibrations.

Learning TPC's Least Squares

A step-by-step guide on using Least Squares in TPC is included in the Least Squares Learning Guide.

You can also learn more about Least Squares in the Least Squares Overview.

General Steps to a Least Squares Network Solution

A Least Squares solution requires you to follow certain steps on your way to a solution. These steps insure that you give proper consideration to each phase of the Least Squares solution. These steps are outlined in detail in the Least Squared Dialog.

The resection and traverse solutions go through these same steps but do it behind the scenes without your intervention since the solution is based on such a well defined set of data.

Stochastic Model

A Least Squares solution starts with estimates of standard error for the coordinates and observations used in the solution.  These are part of what Least Squares refers to as the Stochastic Model.

Finding and Resolving Blunders

A blunder is an observation that is in error by more than the typical systematic or random error normally encountered. It's a BIG error that happened because two numbers were transposed or the wrong back sight point was sighted or there was a glitch in the data transfer or something. Blunders are generally 'one time' errors that don't appear elsewhere in the survey.

TPC can report observation and coordinate blunders it identifies during a Least Squares solution. Once identified, you can exclude the offending observations or coordinates and compute a new solution or you can take whatever steps are necessary to correct the observation or coordinate then compute a new solution.

Positional Tolerance

Positional tolerances define the radius of a circle about a theoretical point based on its distance to the nearest controlling station. Positional tolerances are derived from a Least Squares solution that gives consideration to stations that control the positions of dependent stations. Generally, these controlling stations are monuments of fixed legal position and the dependent stations are new monuments that are being set.

Think of this as a standard of accuracy for survey points. A computed position either meets the standard or it does not. The standard defines the radius of a circle about a theoretical point based on its distance to the nearest controlling station. As long as a computed position lies within the positional tolerance of the theoretical point, it meets the standard. The higher the standard the smaller the radius about the theoretical point and thus the smaller the positional tolerance.

Evaluating a Solution Before Accepting It

As a rule, you will want to evaluate a Least Squares solution before you accept it, allowing you to determine the strength of your stochastic model, observations, controlling coordinates and the reported solution.

Solving the Least Squares network or traverse does not update the survey. The survey is not actually updated until you choose to update it with the Least Squares solution.


What do you do when you have more points to add to a survey that has been adjusted by Least Squares? Or what if you have an updated position for one of your fixed control points?

These situations and others require you to re-run or re-process the Least Squares solution.

Least Squares File

When you do a network adjustment, the Least Squares analysis, including all data and adjustments is stored in a file with the same name as the survey but with a .LSA extension. If your survey is JOB1.TRV, the Least Squares file will be JOB1.LSA. This file is stored in the same folder as the survey.

Related Topics

Least Squares
Least Squares Overview
Least Squares Resection
Least Squares Traverse Adjustment
Least Squares Network Adjustment
Least Squares Blunder Detection
Least Squares Positional Tolerance
Least Squares Files
Redundant Data
Miscellaneous Settings


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