9.4 Linear Algebra, Adjustments, and Least-Squares Foundations

Observation Equations in Matrix Form

A survey network produces many equations: each angle, distance, or leveling observation relates the measured value to unknown coordinates or elevations. Linear algebra collects them compactly as AX = L, where A is the design (coefficient) matrix, X is the column of unknowns, and L is the column of observations. When the number of independent observations equals the number of unknowns, the system is determined and you solve it directly. When observations exceed unknowns, the system is over-determined (redundant), and because real measurements contain random error, no single X satisfies every equation exactly.

The number of extra observations is the redundancy or degrees of freedom: r = (number of observations) − (number of unknowns). Redundancy is desirable — it lets you detect blunders and estimate measurement quality. A leveling loop that returns to its starting benchmark, or a traverse that closes back on a known point, provides redundancy through a closure condition.

Solving a determined 2×2 system. Suppose two distance/coordinate relations give:

1.0·ΔE + 0.0·ΔN = 250.0
0.0·ΔE + 1.0·ΔN = −175.0

Here A is the identity matrix, so ΔE = 250.0 and ΔN = −175.0 immediately. Most real systems are not this clean, but the structure — coefficients on the left, observed values on the right — is identical.

The Least-Squares Principle

When the system is over-determined, least squares finds the X that minimizes the sum of the squares of the residuals (the leftover misfits). A residual is v = (computed observation) − (measured observation). Among all possible adjustments, the least-squares solution is the one for which Σ w·v² is a minimum, where w is each observation's weight. This is the same principle that defines the most probable value: for repeated measurements of one quantity, the value that minimizes Σv² is simply the arithmetic mean.

The solution comes from the normal equations:

(AᵀWA) X = AᵀWL

so X = (AᵀWA)⁻¹ AᵀWL, where W is a diagonal matrix of weights. With equal weights, W is the identity and this reduces to the ordinary least-squares form X = (AᵀA)⁻¹AᵀL.

Why squaring matters. Squaring residuals (rather than summing them directly) prevents positive and negative misfits from canceling and penalizes large discrepancies disproportionately, which is exactly the behavior wanted when a single bad observation should be pulled toward the consensus. Least squares is the standard adjustment method precisely because, under the assumption of normally distributed random errors, it yields the statistically most probable answer.

Weights and Residual Inspection

Weights express relative confidence. By convention, weight is inversely proportional to the variance (the square of the standard error) of an observation: w ∝ 1/σ². A distance measured with σ = 0.01 ft has four times the weight of one measured with σ = 0.02 ft, because (0.02/0.01)² = 4. For repeated measurements averaged into a mean, weight is also proportional to the number of repetitions n.

Worked weighted mean (most probable value). An angle is measured three times with these values and weights:

Convert to seconds past 42°10': v₁ = 20", v₂ = 26", v₃ = 14". Weighted mean of the seconds = Σwv/Σw = (2·20 + 3·26 + 1·14)/(2+3+1) = (40 + 78 + 14)/6 = 132/6 = 22.0". The most probable value is 42°10'22". The higher-weight second observation pulls the result toward 26", as expected.

Finally, inspect residuals rather than discarding them. After adjustment, residuals that are all small and randomly signed indicate good data; a single large residual flags a probable blunder; a consistent one-directional pattern suggests an uncorrected systematic error.

Adjustment in Practice: From Closure to Corrections

Most FS adjustment questions do not require inverting a matrix by hand; they test whether you understand what an adjustment is doing. A traverse closure is the simplest case. After computing departures and latitudes around a loop, the sums should be zero for a closed traverse. They rarely are, and the leftover misclosures are the closure in departure (CΔE) and closure in latitude (CΔN). The linear misclosure is √(CΔE² + CΔN²), and the precision (relative accuracy) is that misclosure divided by the total traverse length, expressed as 1:N.

Worked closure. A 2,400 ft traverse closes with CΔE = +0.08 ft and CΔN = −0.06 ft. Linear misclosure = √(0.08² + 0.06²) = √(0.0064 + 0.0036) = √0.01 = 0.10 ft. Relative precision = 0.10/2,400 = 1/24,000. Whether this passes depends on the specification (an urban boundary survey may demand 1/10,000 or better).

Two correction methods distribute the misclosure. The compass (Bowditch) rule distributes corrections in proportion to each leg's length — appropriate when angular and linear measurements are of comparable quality. The transit rule distributes in proportion to the magnitude of each departure or latitude. Rigorous least-squares adjustment generalizes both: it weights observations by their variances and produces corrections that minimize Σwv² across the whole network simultaneously, rather than treating departures and latitudes separately.

The practical takeaway for the FS exam: a misclosure that is small relative to traverse length signals good work, and the chosen correction rule should match the relative quality of the angle and distance measurements.