Surveying
Least Squares Adjustment for Surveyors
Summary
Least squares adjustment finds the coordinates that minimize the weighted sum of squared observation residuals. Unlike the Bowditch or Transit rules, it uses every redundant observation, weights each by its precision, and produces statistics such as standard deviations, error ellipses and blunder tests.
Definition
Least squares adjustment: an estimation method that minimizes Σ w·v², where v are observation residuals and w their weights (inverse variances).
- Minimizes Σ w·v²
- Weights = 1/σ²
- Redundancy = observations − unknowns
- Gives error ellipses
The model
Each observation (distance, angle, GNSS vector, height difference) is written as a function of unknown coordinates. The equations are linearized around approximate coordinates and solved iteratively:
A is the design matrix, W the weight matrix, ℓ the misclosure vector and r the redundancy.
Reading the results
- Reference variance σ₀² near 1 means the weights matched reality; a chi-square test checks this.
- Standardized residuals larger than about 3 flag possible blunders.
- Error ellipses show the precision of each point at a chosen confidence level.
When to use it
Use least squares for networks, traverses with extra cross-ties, GNSS baseline networks and leveling networks. For a single simple traverse, the Bowditch rule gives practically similar coordinates.
FAQ
Is least squares always better than Bowditch?
It is statistically rigorous and handles redundancy and weights, but for a single well-balanced traverse the coordinate differences are usually small.
What does a reference variance of 4 mean?
Observations disagree about twice as much (√4) as the a-priori standard deviations predicted. Check for blunders or re-evaluate the weights.
Does GeoCAD Bridge offer least squares?
The Traverse tool currently offers Bowditch and Transit with weighted angle balancing. Network least squares is on the roadmap.
Related tools and articles
Tools
Articles
- Bowditch Rule (Compass Rule) Traverse AdjustmentThe Bowditch rule, also called the compass rule, adjusts a closed or link traverse by distributing the misclos…
- Transit Rule Traverse AdjustmentThe Transit rule adjusts a traverse by distributing the easting misclosure in proportion to the absolute depar…
- Traverse Adjustment: Complete Guide for SurveyorsA traverse adjustment turns measured angles and distances between control points into a consistent set of coor…
- Differential Leveling: Height of Instrument MethodDifferential leveling measures height differences by reading a staff on a known point (backsight) and on the n…
References
- Ghilani, C. D. Adjustment Computations: Spatial Data Analysis, 6th ed., Wiley, 2017.
- Mikhail, E. M. & Gracie, G. Analysis and Adjustment of Survey Measurements, Van Nostrand Reinhold, 1981.
Cite this page
GeoCAD Bridge. “Least Squares Adjustment for Surveyors.” GeoCAD Bridge Knowledge Center, updated 2026-10-05. https://www.geocadbridge.com/knowledge/survey-least-squares/