# 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. Learn more about Least Squares in the Least Squares Overview.

## Least Squares Applications in TPC

In TPC, you can use Least Squares to

## 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.

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.

### Step 1. Analyze

In this step, TPC analyzes your survey data and builds a model that Least Squares can solve. TPC considers protected points, traverses, extra field ties, control points, side shots, and so on. You determine which traverses to consider and whether to do a 2-D or 3-D analysis and TPC does the rest.

Once TPC has built the network, you can edit it. You might want to fix certain points, exclude certain observations or edit precisions. Once you have the network the way you want it, you can solve and evaluate it.

### Step 2. Solve

In this step you tell TPC to solve the network using Least Squares. TPC computes the adjusted coordinates along with all the statistics for coordinates and observations. Your survey is NOT affected during this step. TPC assumes you will want to evaluate the solution before you update your survey.

TPC reports the statistics associated with the network solution. This information allows you to evaluate the strength of the adjustments. If you find weak adjustments, you can edit the data and solve it again. You can repeat the Edit – Solve – Evaluate steps as many times as needed to improve your Least Squares solution.

Once you are ready to accept the Least Squares solution into your survey, you just tell TPC to update the survey points. Your survey doesn’t care if the adjusted coordinates came from a Compass Rule adjustment or Least Squares. The adjusted coordinates are all treated the same. And of course you can always undo any adjustment at any time.

### Step 4. Report

Once TPC has computed a Least Squares solution, you can include information about the solution in a report.

## Chi Squared Test

TPC reports the Chi Squared value of a solution, using the computed standard error of the solution (reference variance) and the degrees of freedom. The test either passes or fails at the 99% significance level.

## Blunder Detection

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.

## Pre-Analysis

When you ‘Pre-Analyze’ a Least Squares solution, you want to see what the adjusted coordinates and observations are without actually applying them to your survey. This step allows you to determine the strength of the Least Squares data and solution.

When TPC solves a Least Squares network or traverse, it is doing the pre-analysis. The survey is not actually updated until you choose to Adjust it with the Least Squares solution.

## Reprocessing

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 analysis file will be JOB1.LSA. This file is stored in the same folder as the survey.