Introduction

This document introduces your "Adjustment Theory" course, and points out some of the more important sections and ideas in the textbook's Introduction.
 
​Please study this document carefully, while following along in the Introduction. 
 
Before we get started with the work of this lesson, ensure you have download the “Theory Course Resources” package from the main page of this course.

Basics
​
Definition of a measurement 
- The application of a device or apparatus for the purpose of ascertaining an unknown quantity.
- An observation made to determine an unknown quantity.​​
​

Characteristics of Measurements 
​
- No measurements are exact.
- No matter how hard we try, there is always some error in an observation - e.g. angles have errors due to 
  - pointing on target,
  - reading the circles,
  - miscentering of instrument,
  - miscentering of target,
  - ​misleveling of instrument.
​​
- No measurement is exact.
- There is always error. Electronically measured distances have errors due to
  - Atmospheric conditions
  - Instrument-reflector constant
  - Scaling errors
  - Miscentering of instrument and target. 
​
All measurements contain errors.
​

​Types of Measurements

- Direct
  - Direct measurements are the application of a device or apparatus to determine an unknown quantity.
  - Direct measurements always contain errors, e.g. 
    - Measuring the length of a table.
    - Measuring an angle with a total station.
    - Measuring an elevation difference with a level.

- Indirect
  - Application of some mathematical formula to determine some unknown quantity.
  - Errors from direct measurements are propagated into the formula resulting in errors in the indirect measurements, e.g. 
    - Latitudes and departures from angle and distance observations
    - Errors in angles and distances are propagated into latitudes and departures
    - Computation of area/distance/directions from coordinates
    - ​Errors in coordinates are propagated into areas/distances,directions
​

Definition of Error
An error is a measured quantity minus the true value. 
​

Sources of Errors

- Instrumental errors
  - Caused by imperfections in instrument construction or adjustment
    - Length of tape to short or long
    - Instrument-reflector offset of an EDM
    - Sensitivity of level vials

- Natural errors
  - Caused by changing conditions in the environment
    – Changing atmospheric conditions along the sight line of an
    – Electronically observed distance
    – ​Leveling observation

- Personal errors
  - Caused by limitations in human senses and manual dexterity
    - Ability to read graduations on a rod, tape, etc.
    - Ability to center optics on a target
    - Ability to center a leveling vial 

- Mistakes
  - Not really an error, but caused by confusion or carelessness of the observer.
  - Also called blunders and outliers
    - Technically, outliers are either blunders or large random errors

Examples
- Forgetting or failing to set the ppm-correction when measuring a distance electronically.
- Recording errors
- Failing to properly correct observations .

- Systematic errors
  - Errors that follows a physical law
    - Can be corrected by applying proper field methods or making mathematical corrections
  - Also called biases since they bias the results of the observation
  - Examples
    - Tape corrections
    - ​Vertical axis of total station not perpendicular to horizontal axis.
​
- Random errors
  - All the errors that remain in an observation after systematic errors and mistakes are removed
  - ​Also called compensating errors
    - ​Results of imperfections in instruments and operators

Random Error Theory

- Random errors are
  - Generally small in magnitude
  - ​Large random errors seldom occur
  - ​Follow the laws of probability
  - ​Are as likely to be negative as positive in sign
  - ​Impossible to avoid

Quality of Observations

- Discrepancy
- The algebraic difference between two observations of the same quantity
- Example
  - Two distances are measured as 39.163 m and 39.175 m
  - Discrepancy = 39.175 – 39.163 = 0.012 m
- The smaller the discrepancies the more precise the observations.
​

- Precision
- Refers to the degree of consistency between observations
  - Based on the size of the discrepancies in a data set

- Accuracy
- Refers to the absolute nearness to the true value of an observation
  - Never known since true value is never known.
  - Can be estimated by statistical laws.

Discrepancies
- The largest discrepancy in 
  - Pacing: 31
  - Taping: 0.37
  - EDM: 0.051

- Most precise observations made by an EDM
  - Not necessarily the most accurate, BUT highly probable that they are if proper field procedures followed and instrument/reflector are calibrated.

How do we know there are errors?

- Sum of angles in a polygon traverse do not equal (n – 2)180°
- Sum of latitudes or departures in closed traverse do not equal 0.
- Sum of elevation differences in leveling loop do not equal 0.
- Horizon closure does not equal 360°

Redundant Observations

- Observations in excess of those absolutely needed to determine the unknown or unknowns.
- Also called degrees of freedom
  - ​They enable discrepancies to be computed
  - ​Perform adjustments
  - ​​Make checks on observations

Importance of Adjustments
​
- Sizes of errors can be assessed enabling valid decisions to be made regarding acceptance or rejection of observations
- All quantities in a survey are mathematically consistent
- Precisions of final quantities are increased

Facts 
- Compass Rule:
  - Developed to respond to a contest in the Mathematical Analyst in Philadelphia in 1810.
  - ​Bowditch won the contest!
  - ​Adjustment performed assuming that angles and distances are of equal quality.
    - ​So why are angles adjusted twice?

- Least squares
  - Method developed between 1774 and 1800.
  - ​Developed by Carl F. Gauss to fit astronomical observations to Kepler’s Laws of Planetary Motion
  - ​Difficult method for hand computations.
  - ​Easily performed with a computer.