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

The least-squares method rests on a simple but powerful idea: given more observations than unknowns, find the unknown parameters that make the observed data 'fit' as well as possible. The measure of fit is the weighted sum of squared residuals, denoted Σwᵢvᵢ², where vᵢ is the residual (the correction needed to each observation) and wᵢ is its weight (reciprocal of variance). Why squared residuals? Because: 1. Squaring penalizes large errors more heavily than small ones, forcing the solution to avoid outliers. 2. The method produces unbiased estimates and, under normal error distribution, is statistically optimal (maximum likelihood). 3. It is mathematically tractable—the solution is linear in the observations. Why weights? In geodetic work, observations have different precisions. A distance measured with a high-precision EDM should carry more weight than one measured with a tape. Weights are inversely proportional to variance: wᵢ = 1/σᵢ². A weight matrix P (usually diagonal) incorporates all weights formally. The least-squares solution gives the most probable values under the assumption that random errors follow a normal (Gaussian) distribution with mean zero. This assumption is standard in geodetic practice and is validated by the reference variance check after adjustment.

In the observation-equation (parametric) model, each observation is related to unknowns through a functional equation. For the ith observation: lᵢ + vᵢ = fᵢ(x₁, x₂, ..., xᵤ) where lᵢ is the observed value, vᵢ is the residual (unknown correction), and xⱼ are the unknown parameters we wish to estimate. For a nonlinear function, we linearize about approximate values x₀ⱼ: fᵢ(x) ≈ fᵢ(x₀) + (∂fᵢ/∂x₁)·Δx₁ + (∂fᵢ/∂x₂)·Δx₂ + ... + (∂fᵢ/∂xᵤ)·Δxᵤ Denoting the partial derivatives as aᵢⱼ = ∂fᵢ/∂xⱼ and collecting into matrices: **v = A·Δx − l** where: - **v** is the n × 1 vector of residuals - **A** is the n × u design matrix (or Jacobian), with entry aᵢⱼ - **Δx** is the u × 1 vector of parameter corrections (what we solve for) - **l** is the n × 1 vector of "observed minus computed" (given the approximate values x₀) - n is the number of observations, u is the number of unknowns The weight matrix **P** is an n × n diagonal matrix: **P** = diag(w₁, w₂, ..., wₙ) where wᵢ = 1/σᵢ². For observations with equal precision, all weights are equal (or can be set to 1). The normal equations are derived by minimizing **v**ᵀ**P****v**: (**A**ᵀ**P****A**) · **Δx** = **A**ᵀ**P****l** Solving for the corrections: **Δx** = (**A**ᵀ**P****A**)⁻¹ **A**ᵀ**P****l** The adjusted parameter values are: **x̂** = **x₀** + **Δx**

A key concept in least-squares adjustment is redundancy (or degrees of freedom): **r = n − u** where n is the number of observations and u is the number of unknowns. • If r = 0 (n = u): The system has a unique solution, but no redundancy. There is no way to check for errors or assess the quality of the solution. This is called a critically determined system. • If r > 0 (n > u): The system is overdetermined. Least-squares adjustment is necessary and meaningful. You can assess internal consistency and detect blunders. • If r < 0 (n < u): The system is underdetermined. There are infinite solutions, and adjustment is impossible. In geodetic practice, redundancy is essential. It allows: 1. Detection of blunders or outliers in observations. 2. Statistical testing (reference variance) to check whether the assumed weights and errors are realistic. 3. Rigorous uncertainty estimates (variance-covariance matrices) for adjusted parameters. Example: A leveling network connecting four points requires three independent measurements (three heights relative to a datum) to determine all four elevations—thus u = 3. If you have 8 measurements, then r = 8 − 3 = 5 redundant observations. These five extras allow you to 'vote' on the best estimate and flag inconsistent data.

After solving for adjustments **Δx**, you compute the adjusted residuals: **v̂** = **A**·**Δx** − **l** The reference variance (or unit variance) is: **σ̂₀² = (v̂ᵀ·P·v̂) / r** where r = n − u is the redundancy. Interpretation: - If σ̂₀² ≈ 1 (or close to a baseline value if weights have been scaled), the observations fit the model well. The assumed standard deviations and weights are realistic. - If σ̂₀² >> 1, residuals are larger than expected. This may indicate blunders, model deficiencies, or weights that are too optimistic (σᵢ assumed too small). - If σ̂₀² << 1, residuals are smaller than expected. The weights may be too conservative (σᵢ assumed too large). A rough guideline: σ̂₀² should lie in the range 0.5 to 2 for a well-conditioned adjustment. Outside this range, investigate your observations and model. Uncertainties of adjusted parameters can then be estimated: **Cov(x̂)** = σ̂₀² · (**AᵀPA**)⁻¹ The standard deviation of the ith adjusted parameter is σ(x̂ᵢ) = √[Cov(x̂)]ᵢᵢ. This propagates the observational uncertainties through to the final estimates—a requirement under PRC standards and Philippine surveying regulations (RA 4374).

One of the most important and frequently tested scenarios in geodetic adjustment is the single-unknown case. When you have one unknown (such as an elevation, a distance, or a single parameter) observed multiple times with different precisions, the least-squares solution reduces elegantly to the **weighted mean**. Setup: - One unknown: x (e.g., elevation H, distance d) - n observations: l₁, l₂, ..., lₙ - Weights: w₁, w₂, ..., wₙ (proportional to precision, or 1/σᵢ²) - Functional form: lᵢ + vᵢ = x (each observation estimates x with error vᵢ) The observation-equation matrix form is: - **A** = [1, 1, ..., 1]ᵀ (n × 1 column of ones) - **l** = [l₁, l₂, ..., lₙ]ᵀ - **P** = diag(w₁, w₂, ..., wₙ) The normal equation becomes: (**AᵀPA**) · x̂ = **AᵀP****l** (Σwᵢ) · x̂ = Σwᵢlᵢ Solving: **x̂ = Σwᵢlᵢ / Σwᵢ** This is the **weighted mean**. Special cases: - If all wᵢ = 1 (equal weights), x̂ = (Σlᵢ) / n, the simple arithmetic mean. - If weights are inversely proportional to distance (common in leveling), longer routes have less weight. - If weights are inversely proportional to variance, more precise observations are favored. For the variance of the weighted mean: σ(x̂)² = 1 / Σwᵢ The uncertainty decreases as you add more (or more precise) observations—a fundamental principle of error propagation. This reduction to the weighted mean is more than a computational trick; it validates the least-squares method. Examiners often ask, "What does least squares give you when there's only one unknown?" The answer: the weighted mean—simple, intuitive, and correct.

A systematic approach ensures accuracy and compliance with professional standards (PRC-governed under RA 4374, RA 8560): **Step 1: Define Unknowns and Observations** - Clearly identify all unknowns (u), e.g., coordinates, elevations, scale factors. - List all observations (n) and their measured values. - Confirm redundancy r = n − u > 0. **Step 2: Choose Approximate Values** - For each unknown, provide a reasonable initial estimate x₀ⱼ (from a quick calculation, previous survey, or external reference). - Linearization about these values is essential for nonlinear models. **Step 3: Compute the Design Matrix A** - For each observation i and unknown j, compute aᵢⱼ = ∂fᵢ/∂xⱼ evaluated at x₀. - Record the n × u matrix A with all partials. **Step 4: Compute Observed Minus Computed** - Using approximate values x₀, compute the 'computed' value of each observation: fᵢ(x₀). - Compute the residual vector: l = observation − computed = lᵢ − fᵢ(x₀). - Arrange into the n × 1 vector **l**. **Step 5: Assign Weights and Form P** - Determine or estimate the standard deviation σᵢ of each observation. - Compute weights: wᵢ = 1/σᵢ² (or normalize as needed). - Form the weight matrix: **P** = diag(w₁, w₂, ..., wₙ). **Step 6: Form and Solve Normal Equations** - Compute **N** = **AᵀPA** (u × u symmetric matrix). - Compute **b** = **AᵀPl** (u × 1 vector). - Solve **N·Δx** = **b** using Gaussian elimination, LU decomposition, or matrix inversion: **Δx** = **N⁻¹·b**. **Step 7: Compute Adjusted Parameters** - **x̂** = **x₀** + **Δx**. **Step 8: Compute Residuals and Reference Variance** - **v̂** = **A·Δx** − **l**. - σ̂₀² = (**v̂ᵀ·P·v̂**) / (n − u). - Check if σ̂₀² is close to 1. If not, investigate. **Step 9: Compute Variance-Covariance Matrix** - **Cov(x̂)** = σ̂₀² · **N⁻¹**. - Extract standard deviations: σ(x̂ᵢ) = √[Cov(x̂)]ᵢᵢ. **Step 10: Document and Report** - Report adjusted parameters x̂ with their standard deviations. - Report the reference variance σ̂₀². - Summarize the redundancy, residuals, and any notes on data quality. - Compliance: ensure documentation meets RA 4374 (Professional Regulation) and RA 8560 standards for accreditation and legal defensibility.

**Pitfall 1: Incorrect Residual Sign** Definition matters. If you define v = computed − observed, you'll get opposite signs to v = observed − computed. Be consistent and state your convention clearly. The least-squares principle minimizes Σwᵢvᵢ² regardless of sign, but the matrix equation changes: - If v = observed − computed (as in these notes): **v** = **A·Δx** − **l**. - If v = computed − observed: **v** = **l** − **A·Δx** (and normal equations reverse sign). Both are correct if used consistently. On exams, show your definition. **Pitfall 2: Forgetting to Linearize** For nonlinear functional relationships (e.g., intersection of two circles, distance-angle relations), you must linearize using Taylor series. Using the original nonlinear equation without linearization will give wrong answers. Always expand ∂f/∂x around approximate values. **Pitfall 3: Weight Matrix Confusion** Don't confuse weights and variances: - Weight wᵢ = 1/σᵢ² (inverse of variance). - Variance σᵢ² (square of standard deviation). - Standard deviation σᵢ (uncertainty of observation i). If told an observation has σᵢ = 2 mm, then wᵢ = 1/4 mm⁻², not 2 or 1/2. Incorrect weighting leads to biased solutions. **Pitfall 4: Insufficient Redundancy** If r ≤ 0, the system is underdetermined or just-determined. You cannot use least squares; you cannot check for blunders. Always verify r > 0. **Pitfall 5: Ignoring the Reference Variance** A reference variance σ̂₀² = 10 is a red flag. Don't ignore it; investigate. Common causes: gross errors in observations, incorrect model (missing variables), or unrealistic weight assumptions. Professional practice (and the PRC) expect you to diagnose and resolve these issues. **Pitfall 6: Single-Unknown Confusion** When there is one unknown, least squares simplifies to the weighted mean. Don't over-formalize: if asked "adjust a single measurement with three observations," the answer is just x̂ = Σwᵢlᵢ / Σwᵢ. A full matrix solution is correct but unnecessary and time-wasting on an exam. **Pitfall 7: Rounding and Numerical Precision** Keep sufficient decimal places (at least 0.1 mm for coordinates, 1 mm for elevations) during intermediate steps. Premature rounding propagates errors. For normal matrix inversion, use double precision and be aware of ill-conditioning if AᵀPA is nearly singular. **Pitfall 8: Coordinate Systems and Datums** When adjusting coordinates under WGS84 or PRS92, ensure all observations and unknowns are in the same reference frame. Mixing datums or confusing PPCS/UTM zones is a common error on exams and in practice. Reference RA 4374 and PD 1529 for datum requirements. **Exam Tip 1: Board-Style Problems** Exam questions often give simple numerical cases (e.g., three observations of one unknown with specified weights). Draw up the design matrix, compute normal equations by hand, and solve. Show all steps. Examiners look for method, not just the final answer. **Exam Tip 2: Conceptual Understanding** Questions like "Explain why redundancy is necessary" or "What does the reference variance tell you?" test understanding. Be ready to explain not just 'how' but 'why.' A good answer explains the statistical reasoning and practical implications. **Exam Tip 3: Mixed Cases** Don't assume all problems are single-unknown. Some may involve two unknowns (e.g., X and Y coordinate shifts) or require you to set up (but not fully solve) a larger system. Know how to formulate A and interpret the normal equations even if exact numerical solution is beyond the scope. **Exam Tip 4: Practical Context** Problems may mention leveling networks, traverses, or intersection surveys. Recognize the problem type, identify the unknowns and observations, and apply the standard 10-step procedure. Context helps, but the mathematical approach is universal.

For rapid recall during exams, here is a condensed summary of essential formulas: **Observation-Equation Model:** - Observation i: lᵢ + vᵢ = fᵢ(x₁, ..., xᵤ) - Linearized form: **v** = **A·Δx** − **l** **Matrices:** - **A** (n × u): Design matrix; entry aᵢⱼ = ∂fᵢ/∂xⱼ - **l** (n × 1): Observed minus computed at x₀ - **P** (n × n): Weight matrix, P = diag(w₁, ..., wₙ), wᵢ = 1/σᵢ² - **v** (n × 1): Residuals (computed after solving) **Normal Equations:** - **N·Δx** = **b**, where **N** = **AᵀPA**, **b** = **AᵀPl** **Solution:** - **Δx** = **N⁻¹·b** = (**AᵀPA**)⁻¹·**AᵀPl** - **x̂** = **x₀** + **Δx** (adjusted parameters) **Residuals and Quality:** - **v̂** = **A·Δx** − **l** - σ̂₀² = (**v̂ᵀ·P·v̂**) / (n − u) (reference variance) - **Cov(x̂)** = σ̂₀² · **N⁻¹** (variance-covariance matrix) - σ(x̂ⱼ) = √[Cov(x̂)]ⱼⱼ (standard deviation of jth parameter) **Special Case: Single Unknown** - x̂ = Σwᵢlᵢ / Σwᵢ (weighted mean) - σ(x̂)² = 1 / Σwᵢ - σ(x̂) = 1 / √(Σwᵢ) **Redundancy:** - r = n − u (degrees of freedom) - Require r > 0 for least-squares adjustment **Key Relationships:** - Higher precision (smaller σᵢ) → higher weight wᵢ → larger influence on solution. - More observations → larger **A**, more equations, better overdetermination, smaller uncertainty on x̂. - Better weights (more realistic σᵢ) → σ̂₀² closer to 1. This summary is a reference checklist for problem-solving and self-grading.