GELE Adjustment Computations (Least Squares) — Least Squares — Observation EquationsRevision Notes

Formulas

Exam Tips

• Memorize: for leveling, weight = 1/(route length in km). This appears in almost every board exam leveling problem.
• Check: always verify that r = n − u > 0 before proceeding with adjustment.
• When the problem says 'equal precision' or 'equal weight,' use P = I (identity matrix); the solution becomes the simple arithmetic mean.
• The phrase 'most probable value' in board problems almost always signals a least-squares (weighted mean) computation.

Key Points

• Least squares is applied whenever the number of observations n exceeds the number of unknowns u, creating redundancy r = n − u > 0.
• Each observation contains a small random error; least squares finds the most probable values of the unknowns given those errors.
• The criterion: minimize the weighted sum of squared residuals, Σ wᵢvᵢ².
• A residual vᵢ = (computed value from adjusted parameters) − (observed value); it is the 'leftover' after adjustment.
• When all observations have equal precision, weights are all equal (w = 1) and the method reduces to minimizing Σvᵢ², equivalent to finding the arithmetic mean for a single unknown.
• Weights are inversely proportional to variance: wᵢ = σ₀² / σᵢ², where σ₀ is the reference standard deviation. For leveling, wᵢ = 1/Kᵢ (inverse of route length in km).
• The least-squares solution is unbiased and has minimum variance among all linear unbiased estimators (Gauss-Markov theorem).

Definitions

Section Title

The Least-Squares Principle

Common Mistakes

• Using equal weights for observations of clearly different precision (e.g., short and long leveling routes).
• Forgetting that r = n − u must be positive; trying to 'adjust' a system with n = u (no redundancy).
• Confusing 'error' (true, unknown) with 'residual' (computed, observable after adjustment).
• Applying wᵢ = Kᵢ (proportional to length) instead of wᵢ = 1/Kᵢ for leveling — a classic sign error.

Formulas

Exam Tips

• For board exam problems, A is usually given or easily constructed — focus on correctly forming N = AᵀPA and AᵀPl.
• A quick check: N must be square (u×u) and symmetric. If it is not symmetric, recheck your matrix multiplication.
• For a single unknown with aᵢ = 1 for all rows, AᵀPA = Σwᵢ and AᵀPl = Σwᵢlᵢ, giving x̂ = weighted mean directly.
• In Philippine board exams, 'set up the normal equations' typically means write out (AᵀPA)x̂ = AᵀPl in scalar form — show the arithmetic.

Key Points

• Each observation ℓᵢ is modeled as a function of the unknown parameters: ℓᵢ + vᵢ = f(x̂).
• For nonlinear functions, linearize by Taylor expansion about initial approximate values x₀: the result is a linear system.
• The design matrix A (also called the coefficient matrix) contains the partial derivatives ∂f/∂xⱼ evaluated at x₀.
• The misclosure vector l contains (observed value − value computed from x₀); it is NOT the raw observation.
• The system of observation equations in matrix form: v = Ax̂ − l.
• The normal equations are derived by minimizing vᵀPv with respect to x̂: (AᵀPA)x̂ = AᵀPl.
• The solution is: x̂ = (AᵀPA)⁻¹ AᵀPl. The matrix N = AᵀPA is called the normal matrix.
• For linear problems (e.g., leveling, traverse angle adjustment), no iteration is needed — A is exact.
• For nonlinear problems (e.g., geodetic positioning), iterate: update x₀ ← x₀ + x̂ and recompute until x̂ converges to zero.

Definitions

Section Title

Observation Equations — Matrix Formulation

Common Mistakes

• Confusing l (misclosure = observed − computed from x₀) with the raw observation vector ℓ.
• Setting up A with incorrect signs — always differentiate carefully; a sign error propagates through the entire solution.
• Forgetting to include P in the normal equations when observations have unequal weights (using AᵀA instead of AᵀPA).
• Attempting to invert a singular or near-singular normal matrix without checking the network datum (free networks need constraints).
• For nonlinear problems, stopping after one iteration without checking convergence of x̂.

Formulas

Exam Tips

• Set up a neat tabular solution: columns for ℓᵢ, wᵢ, wᵢℓᵢ, vᵢ, wᵢvᵢ, wᵢvᵢ² — this organized format earns full marks.
• For leveling: immediately write weights as w = 1/K before doing anything else.
• Board exams often ask for the 'most probable value' or 'adjusted elevation' — this is always the weighted mean for single-unknown problems.
• Carry sufficient decimal places (at least 4 significant figures) in wᵢℓᵢ to avoid rounding errors in the final answer.

Key Points

• When a single unknown x is measured directly n times, every row of A equals [1], so A is an (n×1) column vector of ones.
• The normal equation reduces to: (Σwᵢ)x̂ = Σwᵢlᵢ.
• The least-squares solution is simply the weighted mean: x̂ = Σwᵢℓᵢ / Σwᵢ.
• If all weights are equal (wᵢ = 1), the weighted mean becomes the arithmetic mean: x̂ = Σℓᵢ / n.
• This is the most common board-exam scenario for least squares in leveling, angle measurement, and EDM distance measurement.
• After solving, compute residuals: vᵢ = x̂ − ℓᵢ (note: here the 'computed' value is x̂ for each observation).
• The sum of weighted residuals always equals zero: Σwᵢvᵢ = 0 — use this as a numerical check.

Definitions

Section Title

Special Case: Single Unknown — Weighted Mean

Common Mistakes

• Computing the simple arithmetic mean instead of the weighted mean when weights differ.
• Using wᵢ = Kᵢ (length) instead of wᵢ = 1/Kᵢ for leveling route weights.
• Forgetting to check Σwᵢvᵢ = 0 as a numerical verification.
• Rounding intermediate Σwᵢℓᵢ and Σwᵢ values before dividing, causing accumulated rounding error.

Formulas

Exam Tips

• Memorize: σ̂₀² = vᵀPv / (n − u). This formula appears directly in board exam questions.
• When asked to 'check the adjustment,' compute σ̂₀² and state whether it is close to 1.0.
• r = n − u is also tested directly as a standalone calculation — a 2-point question in many previous board exams.
• For the single-unknown case, always verify Σwᵢvᵢ = 0 first, then compute σ̂₀² = vᵀPv / (n − 1).

Key Points

• After solving for x̂, compute the adjusted residuals v = Ax̂ − l and the weighted sum of squared residuals vᵀPv.
• The reference variance (a posteriori variance factor) is: σ̂₀² = vᵀPv / (n − u) = vᵀPv / r.
• For a properly weighted network, σ̂₀² should be close to 1.0 (if weights were set using σ₀ = 1 as reference).
• σ̂₀² >> 1 indicates: blunders, incorrect weights, or systematic errors in the observations.
• σ̂₀² << 1 indicates: over-pessimistic weights (observations are better than assumed) or insufficient redundancy.
• The reference standard deviation is σ̂₀ = √(vᵀPv / r).
• The covariance matrix of the adjusted parameters is Σx̂ = σ̂₀² (AᵀPA)⁻¹ = σ̂₀² N⁻¹.
• Individual observation precision after adjustment: σ̂ᵢ = σ̂₀ / √wᵢ.
• A formal chi-squared test can be applied: the statistic r·σ̂₀²/σ₀² follows a χ²(r) distribution.

Definitions

Section Title

Post-Adjustment: Reference Variance and Quality Check

Common Mistakes

• Using n (total observations) instead of r = n − u as the denominator in the variance formula.
• Forgetting to compute residuals after adjustment — vᵀPv cannot be calculated without them.
• Interpreting σ̂₀² > 1 as always indicating blunders — it may simply mean the a priori weights were too optimistic.
• Reporting σ̂₀² without specifying the units or the weighting convention, making the value meaningless to others.

Formulas

Exam Tips

• Practice writing A from word problems without referring to notes — this matrix setup step is where most time is lost in exams.
• For a three-route leveling problem converging on one point: A = [1; 1; 1] (3×1 column of ones) — instantly recognizable as the weighted mean case.
• Label the A matrix explicitly with observation names on rows and unknown names on columns during the exam to prevent errors.
• The design matrix A for a linear model never contains the numerical values of observations — only coefficients.

Key Points

• For leveling: if each observation directly gives the elevation of unknown Hⱼ, then Aᵢⱼ = +1 if Hⱼ is the 'forward' BM, −1 if 'backward', and 0 otherwise.
• For a traverse: angle observations link adjacent azimuths; distance observations involve coordinate differences.
• For the single-unknown case with all direct observations: A is simply a column vector of 1s.
• For two unknowns (x₁, x₂) observed in combinations: A has two columns; row i contains the coefficients of x₁ and x₂ in the i-th observation equation.
• Always label rows (observations) and columns (unknowns) clearly before writing any entry in A.
• The number of rows of A equals n (observations); the number of columns equals u (unknowns).
• For GPS baseline components in a Philippine CORS network, A contains direction cosines linking baseline vectors to station coordinate corrections.

Definitions

Section Title

Setting Up the Design Matrix A — Practical Examples

Common Mistakes

• Writing A with the wrong number of columns (e.g., u+1 columns due to miscounting unknowns).
• Putting raw observations in l instead of the misclosure (observed − computed from x₀).
• For leveling networks: using elevation differences as entries in A instead of ±1 coefficients.
• Transposing rows and columns — always: rows = observations, columns = unknowns.