Least Square Adjustments, Presented by Dane Courville, PLS

Meeting summary for Mentoring Mondays for the Land Surveying Profession (05/13/2024)
Quick recap
The team discussed upcoming presentations, shared personal experiences, and delved into mathematical concepts such as least squares adjustment and the importance of understanding precision and accuracy in various contexts. They also discussed the essential concepts for the upcoming Fs exam, the process of reducing residuals using a perpendicular line, and the methodology of conducting a least squares procedure using a Cartesian graph. Lastly, they discussed strategies for solving equations with two variables, the importance of correctly weighting observations in survey data, and the upcoming mentoring sessions and speaking engagement.
Least Squares, Legacy Projects, and Surveying Journeys
Trent, dane, and Michael discussed their experiences with least squares, a mathematical concept that Richard committed to learning more about. They reflected on their past professional endeavours, with Michael sharing his work on a legacy code conversion project and dane’s journey from land surveying to working as a professional land surveyor in multiple states, including Colorado. dane’s passion for teaching and mentoring was also highlighted, as he continues to share his expertise and presentations on surveying fundamentals.
Dane’s Fs Exam Preparation Insights
dane shared his insights on the essential concepts for the upcoming Fs exam, focusing on right triangles, trigonometric ratios, and the Pythagorean theorem. He also discussed recent revisions to his presentation and section 6 calculation, stressing the need for adjustments when dealing with larger sections. dane emphasized the importance of understanding the concept of least squares adjustment as a ‘best fit line’ to minimize the sum of squares of the residuals. He was in the process of explaining the term ‘residual’ using a drawing.
Understanding and Correcting Rifle Shooting Errors
dane discussed the differences between precision and accuracy in the context of shooting a rifle at a target. He explained the three types of errors: random, systematic, and blunders. Blunders are simple mistakes, systematic errors are due to incorrect equipment or incorrect usage, and random errors are unidentifiable and often environmental. dane emphasized the importance of identifying and correcting these errors to improve the accuracy and precision of the shots. He also noted that while systematic errors can be fixed, random errors are likely to persist and cannot be entirely eliminated.
Residual Concept and Least Squares Objective
dane explained the concept of residual, defining it as the difference between the predicted and observed values of a data point. The objective of least squares, he said, is to minimize this residual for each data point. He gave an example of residual using the calculated position versus the actual position of pins. He further elaborated on the concept using the analogy of a broom hanging from springs attached to a pegboard, which finds an equilibrium position where the least resistance is being exerted. He emphasized that the residual would remain minimal if the broom is not moved at the point of attachment of any spring.
Perpendicular Line Methodology for Residuals Reduction
dane explained the methodology of reducing residuals using a perpendicular line. He illustrated the concept by imagining fence posts and a best-fit line between them, which is typically used in surveying. dane also discussed two different ways to analyze the distances between the points and the best fit line, either perpendicular or vertical, but clarified that they would use the perpendicular method. He walked through the steps and procedures to generate a best fit line, emphasizing that it involves simple arithmetic and staying organized. The objective was to create this line using data points, with the goal of minimizing the area of squares created between the points and the line.
Least Squares Procedure and Cartesian Graphs
dane explained the process of conducting a least squares procedure using a Cartesian graph. dane emphasized the importance of correctly identifying the X and Y values as eastings and northings respectively. The purpose of this process is to create a line that best fits the data points.
Plotting Lines With Differences
He emphasized the importance of examining the area of difference between the green and blue squares. Finally, he explained how to plot the line with multiple points by inserting different values into the equation, thereby visually representing the line.
Solving Equations and Legal Description Strategy
dane explained the method of solving equations with two variables and demonstrated how to find two initial points using the given equation and the intercept on the Y-axis. He emphasized the importance of minimizing the residuals from each data point perpendicular to the line to generate a line that best fits the data points. Additionally, dane proposed a strategy to reduce a legal description from 105 calls from fence corner to fence corner by considering the intent of the deed. The implications of this proposal for the legal process were discussed.
Traverse Adjustment and Survey Data Handling
dane led a discussion on traverse adjustment slide with Michael, who shared his extensive experience in applying least squares adjustment for surveying and triangulation networks. Michael emphasized the importance of correctly weighting observations and dealing with errors in survey data, while dane agreed and stressed the significance of not incorporating obviously erroneous data into the dataset. dane also presented a detailed explanation on how to handle calculations for different sections of a project, cautioning against simple averaging for skewed or irregular sections. The conversation ended with Trent announcing the topics for the upcoming mentoring sessions and dane’s upcoming speaking engagement at a conference in Denver.
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