8.6 Least Squares, Residuals, and Error Propagation

What Least Squares Does

Least squares is named in the FS Survey Computations area, usually tested at the concept and small-computation level rather than as full matrix algebra. Given more observations than the minimum needed, a network has no single exact solution; least squares finds the most probable values by minimizing the weighted sum of squared residuals:

minimize Σ p·v²

where v is a residual (the correction applied to an observation: v = adjusted value − observed value) and p is the observation's weight. Squaring prevents positive and negative residuals from cancelling and penalizes large discrepancies more heavily; weighting lets reliable observations dominate.

Key vocabulary the exam expects you to use precisely:

Residuals are not blunders or excuses; they are the small, expected corrections that distribute closure error rigorously across all observations in proportion to their precision.

Weights, Redundancy, and Standard Error

Weight is inversely proportional to variance: p = 1/σ². An observation with σ = 0.01 ft has weight 1/0.0001 = 10,000, while one with σ = 0.03 ft has weight 1/0.0009 ≈ 1,111. The first is therefore about nine times as influential — weights scale as the inverse square of the standard deviations (the ratio (0.03/0.01)² = 9). Distances measured many times, or angles turned with more repetitions, earn higher weight.

Redundancy (degrees of freedom) is r = m − n, the number of observations m minus the number of unknowns n. A leveling loop with 6 observed height differences solving for 3 unknown station elevations has r = 6 − 3 = 3. Redundancy is what makes error estimation possible; with r = 0 there is a unique solution and nothing to check.

The standard error (or standard deviation) of unit weight, sometimes the reference standard deviation, summarizes the adjustment's overall fit:

S₀ = √(Σ p·v² / r)

If the observations' stated precisions were realistic, S₀ should come out near 1.0 (for properly scaled weights). An S₀ much larger than 1 suggests an undetected blunder or overly optimistic precision estimates; much smaller than 1 suggests the precisions were overstated. This single number is a powerful blunder screen the exam may reference.

Error Propagation

Error propagation estimates how uncertainty in measured quantities transfers to a computed result. For independent random errors, errors add in quadrature (root-sum-square), not linearly:

• Sum or difference: σ = √(σ1² + σ2² + … + σn²). For a traverse of n equal legs each with σ, the total is σ_total = σ·√n.
• Product y = a·b: relative error (σ_y/y)² = (σ_a/a)² + (σ_b/b)².
• General function: σ_y² = Σ (∂y/∂xᵢ)² σ_xᵢ² (the general law of propagation of variances).

Worked example

A traverse has 9 angle stations, each turned with a standard deviation of 5". The expected angular misclosure (standard error of the sum) is σ = 5"·√9 = 5"×3 = 15", not 9×5" = 45". Random errors partly cancel, so they grow with the square root of the count, while a systematic error would accumulate linearly — distinguishing the two is a recurring exam theme.

A distance example

A line is taped in three independent segments with standard deviations 0.02, 0.03, and 0.01 ft. The standard deviation of the total length is √(0.02² + 0.03² + 0.01²) = √(0.0004 + 0.0009 + 0.0001) = √0.0014 = 0.037 ft. Adding the errors arithmetically (0.06 ft) would overstate the uncertainty — quadrature is the correct combination for independent random errors.

Error Types and Why Least Squares Beats Simple Distribution

Before any adjustment, classify the errors present, because least squares only handles random error:

A blunder — a transposed digit, a wrong target, a miscounted tape length — must be located and removed, not adjusted; this is why an oversized residual or an S₀ far above 1 is a warning, not something to average away. Systematic errors (tape too long, instrument index error, atmospheric refraction) have a consistent sign and must be corrected by a physical model before adjustment. Only the residual random error — the inevitable ± scatter of careful observations — is properly distributed by least squares.

Why not just divide the misclosure equally?

Simpler rules (the compass rule, equal distribution of angular misclosure) distribute closure error by length or count. Least squares instead weights every observation by its precision and minimizes Σpv² over the whole network simultaneously, so a one-time GPS vector and a short, precise total-station distance each receive corrections matched to their reliability rather than an arbitrary share.

What the FS exam actually asks

FS items rarely demand a full matrix solution. They test whether you can: define a residual as adjusted-minus-observed; state that weight = 1/σ²; compute redundancy r = m − n; recognize S₀ = √(Σpv²/r) and that it should be near 1; and combine independent random errors in quadrature. Mastering these five ideas — not memorizing matrix algebra — is the efficient path to the points in this topic.