GELE Adjustment Computations (Least Squares) — Theory of Errors, Weights and Most Probable ValueStudy Notes

Geodetic measurements are affected by three distinct types of errors. Understanding and managing each type is critical to obtaining reliable survey results. **Blunders (Mistakes or Gross Errors)** These are large, unintended errors arising from carelessness, equipment malfunction, or procedural failure. Examples include: misreading a level staff by one meter, recording a distance as 100 m when it is actually 1000 m, or failing to properly zero an instrument. Blunders must be identified and eliminated before any adjustment computation. Detection methods include: comparing multiple independent measurements, checking against known values, and applying reasonableness tests during field validation. **Systematic Errors** Systematic errors follow a pattern or physical law — they are biased and reproducible under identical conditions. In leveling, curvature and refraction cause systematic bias. In distance measurement, temperature effects cause steel tape to expand or contract. In angle measurement, instrumental errors like collimation error, horizontal circle eccentricity, or vertical circle index error produce systematic bias. These errors must be modeled mathematically and corrected before adjustment. For example, if a steel tape expands 0.001 m per degree Celsius, and the measurement was taken at 5°C higher than the calibration temperature, a +0.005 m systematic correction is applied. **Random Errors (Accidental Errors)** Random errors are small, uncontrollable, and vary unpredictably in magnitude and sign from measurement to measurement. They arise from instrument sensitivity limits, atmospheric fluctuations, personal judgment variations, and countless minor environmental effects. Random errors cannot be eliminated, but they follow a normal (Gaussian) probability distribution. Their characteristics are predictable statistically: they cluster around a true value, small errors are more frequent than large ones, positive and negative errors are equally likely, and the distribution is symmetric. These errors are the focus of adjustment theory and the least squares method.

Random errors conform to the normal probability distribution. Three key statistical measures quantify the magnitude and dispersion of random errors: **Standard Deviation (σ)** The standard deviation measures the spread of observations around their mean. For a sample of n observations, the standard deviation of a single observation is: σ = √[Σv²/(n−1)] where v = x − x̄ is the residual (deviation from the mean) and Σv² is the sum of squared residuals. The denominator is (n−1) because we lose one degree of freedom when computing the mean from the sample (Bessel's correction). Standard deviation has the same units as the measurements. **Variance (σ²)** Variance is the square of standard deviation. It is often used in theoretical work because it has additive properties: the variance of independent sums equals the sum of variances. However, for practical interpretation, standard deviation is more intuitive because it returns values to the original measurement units. **Probable Error (PE)** The probable error is the value such that a random error will exceed it in magnitude exactly 50% of the time: PE = 0.6745 × σ This constant (0.6745) comes from the normal distribution: it is the 75th percentile value (or the z-value corresponding to a cumulative probability of 0.75). While less commonly used in modern practice than standard deviation, it appears in historical literature and some Philippine surveying standards. **Mean Standard Deviation (σx̄)** When we take the arithmetic mean of n equally-reliable observations, the standard deviation of that mean is: σx̄ = σ/√n This critically important formula shows that the uncertainty of the mean decreases as the square root of the number of observations. Thus, to halve the uncertainty, we must take four times as many observations. This principle underlies the design of survey redundancy: taking many measurements improves precision proportionally to √n.

When multiple independent observations of the same quantity are equally reliable — that is, each has the same standard deviation σ — the **most probable value (MPV)** is the arithmetic mean: MPV = x̄ = Σx/n where Σx is the sum of all n observations. **Justification from Least Squares Principle** The least squares principle states that the best estimate minimizes the sum of squared residuals, Σv². For a sample of equally-reliable observations, this objective is achieved precisely by the arithmetic mean. Mathematically, if we propose any value x₀ as an estimate, the sum of squared residuals is: Σv² = Σ(xᵢ − x₀)² Taking the derivative and setting it to zero gives d(Σv²)/dx₀ = −2Σ(xᵢ − x₀) = 0, which simplifies to Σxᵢ = n·x₀, or x₀ = Σxᵢ/n. Thus the arithmetic mean is the value that minimizes squared residuals — the least squares solution. **Computing Residuals and Standard Deviation** Once the mean is established, we compute residuals vᵢ = xᵢ − x̄ for each observation. These residuals have a special property: Σvᵢ = 0 (they always sum to zero). The standard deviation of a single observation is then σ = √[Σvᵢ²/(n−1)], and the standard deviation of the mean is σx̄ = σ/√n. **Practical Application in Geodetic Surveys** In Philippine cadastral surveys (under CA 141 and PD 1529) and control surveys under RA 4374, surveyors often measure the same distance or angle multiple times to reduce random error. The arithmetic mean serves as the most probable value to report. For instance, when measuring a boundary distance in a property survey, taking 4–6 independent measurements and averaging them provides a more reliable result than any single measurement.

In real survey practice, not all observations are equally reliable. Some may be obtained with better instruments, under more favorable conditions, or with greater care. Others may represent accumulated results from multiple measurements. **Weights** provide a mathematical mechanism to reflect this variable reliability and ensure that more reliable observations influence the final result more strongly. **Definition of Weight** The weight of an observation is inversely proportional to its variance: wᵢ = k/σᵢ² where k is an arbitrary constant (often set to 1 for simplicity) and σᵢ is the standard deviation of the i-th observation. Equivalently, weight is inversely proportional to the variance. The key insight is: the smaller the error (smaller σ²), the larger the weight, and vice versa. An observation with σ = 0.01 m has weight w = 1/(0.01)² = 10,000; an observation with σ = 0.02 m has weight w = 1/(0.02)² = 2,500. The first measurement, being twice as precise, receives four times the weight. **Alternative Weight Rules in Geodetic Practice** Beyond the variance-based formula, geodetic practice employs other weight rules suited to specific measurement types: 1. **Leveling (weight inversely proportional to distance):** In differential leveling, error accumulates with distance. A level route of length K km has error proportional to √K. Thus weight w ∝ 1/K. A 1 km leveling route is assigned weight w = 1; a 4 km route has w = 1/4. This reflects the principle that shorter, less error-prone routes are more reliable. 2. **Multiple observations (weight proportional to number of measurements):** If one distance is measured 3 times and another is measured 6 times, the second has twice as many observations and is therefore more reliable. Weight w ∝ n. If the first distance has weight 1, the second has weight 2. 3. **Angle observations:** In theodolite angle measurements, weight may be assigned based on the number of repetitions, quality of instrument, or atmospheric conditions during measurement. **Relative vs. Absolute Weights** In practice, weights are usually expressed as relative values. If one observation has weight 1 and another has weight 4, we can multiply all weights by any constant without changing the result. This simplifies computation: we often work with relative integer weights (e.g., 1, 2, 4, 10) rather than actual probabilities. For example, in a leveling network, if routes have distances 2 km, 1 km, and 4 km, we assign weights 2, 4, and 1 (inverting the distance ratios) or we may normalize to 0.5, 1.0, and 0.25 — either is valid for adjustment purposes.

When observations have different weights, the most probable value is no longer the simple arithmetic mean. Instead, we compute the **weighted mean**: x̄ᵥ = Σ(wᵢ × xᵢ) / Σwᵢ This is the least squares solution when observations have unequal weights. Each observation contributes to the final value in proportion to its weight: highly reliable (high weight) observations pull the result toward their value, while unreliable (low weight) observations have minimal influence. **Derivation from Least Squares** The weighted least squares objective is to minimize the weighted sum of squared residuals: Σwᵢ × vᵢ² = Σwᵢ(xᵢ − x₀)² Taking the derivative with respect to x₀ and setting it to zero: d/dx₀ [Σwᵢ(xᵢ − x₀)²] = −2Σwᵢ(xᵢ − x₀) = 0 Rearranging: Σwᵢ × xᵢ = x₀ × Σwᵢ, which gives x₀ = Σ(wᵢ × xᵢ) / Σwᵢ. This is precisely the weighted mean formula. Thus, the weighted mean is the value that minimizes weighted squared residuals — the least squares solution for unequal-weight data. **Variance of the Weighted Mean** The standard deviation (or standard error) of the weighted mean is: σx̄ᵥ = 1/√(Σwᵢ) = 1/√W where W = Σwᵢ is the sum of all weights. This formula shows that precision improves with the total accumulated weight. If weights are defined as wᵢ = 1/σᵢ², then: σx̄ᵥ = 1/√[Σ(1/σᵢ²)] This general formula applies to any weight definition, as long as weights are internally consistent. **Practical Application: Leveling Network** Consider a height determination using multiple level routes. Route A (2 km) measured: +0.523 m (w = 0.5). Route B (1 km) measured: +0.521 m (w = 1.0). Route C (4 km) measured: +0.525 m (w = 0.25). Weighted mean = [0.5(0.523) + 1.0(0.521) + 0.25(0.525)] / (0.5 + 1.0 + 0.25) = [0.2615 + 0.521 + 0.13125] / 1.75 = 0.91375 / 1.75 = 0.5221 m. The short, high-weight route pulls the result toward its value (0.521 m). Variance of weighted mean: σx̄ᵥ² = 1/(0.5 + 1.0 + 0.25) = 1/1.75 = 0.571; thus σx̄ᵥ = √0.571 = 0.756 (in units of σ for a 1 km standard route).

In geodetic computations, we often combine multiple measured quantities to compute a derived quantity. For example, we measure the base and height of a triangle to compute area; we measure easting and northing to compute distance. The errors in the measured quantities propagate into errors in the computed result. **Error propagation** (also called variance propagation) predicts the uncertainty of derived quantities. **Law of Error Propagation for Independent Observations** If a computed quantity Q depends on measured quantities x, y, z, ... that are independent and have standard deviations σₓ, σᵧ, σᵤ, ..., then the standard deviation of Q is approximated by: σQ ≈ √[(∂Q/∂x)²σₓ² + (∂Q/∂y)²σᵧ² + (∂Q/∂z)²σᵤ² + ...] where ∂Q/∂x, ∂Q/∂y, etc., are partial derivatives of Q with respect to each measured quantity. Each partial derivative is evaluated at the measured values. **Case 1: Sum or Difference** If Q = x + y (or Q = x − y), then ∂Q/∂x = 1 and ∂Q/∂y = ±1, so: σQ = √(σₓ² + σᵧ²) Errors add in quadrature (not linearly). For example, if a traverse section is measured by two distances x = 100 m (σₓ = 0.02 m) and y = 150 m (σᵧ = 0.03 m), the total distance Q = 250 m has error σQ = √(0.02² + 0.03²) = √0.0013 = 0.036 m. **Case 2: Product or Quotient** If Q = x·y, then σQ/Q ≈ √[(σₓ/x)² + (σᵧ/y)²]. The relative error of Q is found by combining the relative errors of x and y in quadrature. If Q = x/y, the relative error is similarly σQ/Q ≈ √[(σₓ/x)² + (σᵧ/y)²]. **Case 3: Power Law** If Q = xⁿ, then (σQ/Q) ≈ n·(σₓ/x). The relative error scales with the exponent. **Practical Application: Distance Correction for Temperature** In Philippine cadastral surveys under CA 141, steel tapes are corrected for temperature. If a tape is calibrated at 20°C and used at 25°C, the expansion is ΔL = α·L·ΔT, where α is the coefficient of thermal expansion (≈ 0.000011/°C for steel), L is the measured length, and ΔT is the temperature difference. The corrected length is L_c = L + ΔL. If the temperature measurement has uncertainty σT = ±0.5°C, then the uncertainty in the correction is σ_correction = α·L·σT. For a 100 m tape at ΔT = 5°C ± 0.5°C: ΔL = 0.000011 × 100 × 5 = 0.0055 m = 5.5 mm σ_correction = 0.000011 × 100 × 0.5 = 0.00055 m = 0.55 mm Thus the corrected distance is 100.00 + 0.0055 = 100.0055 m ± 0.00055 m.

Adjustment computations exist along a spectrum of complexity, each suited to different survey scenarios: **Direct Observation Adjustment (Most Probable Value Method)** When we have n direct observations of the same quantity with only random errors, the most probable value is the arithmetic mean (for equal weights) or weighted mean (for unequal weights). This is the simplest adjustment — one calculation that produces the best single value. Its strength lies in simplicity and intuitive interpretation; its limitation is that it handles only redundant direct observations, not systems of indirect observations or conditional equations. **Indirect Observation Adjustment (Least Squares for Indirect Observations)** Often we cannot measure the desired quantity directly. Instead, we measure related quantities and compute the desired result. For example, in a rectangular parcel survey, we measure the four sides and two diagonals (6 observations). The four sides alone would determine the rectangle (4 unknowns); the two diagonals are redundant. Least squares processes all 6 observations simultaneously to find the best estimates of the 4 side lengths. This method is more general than the direct observation case and handles cases where observations outnumber unknowns. **Conditional Adjustment (Least Squares for Conditional Equations)** In network problems (leveling networks, traverse closures, triangulation), observations must satisfy closure conditions. In a closed level circuit, the sum of height differences around the loop must be zero (up to measurement error). Conditional adjustment enforces these constraints while minimizing the sum of squared weighted residuals. This is typical in geodetic networks where redundant observations must be adjusted to satisfy geometric or physical constraints. **Parametric Adjustment (Least Squares for Indirect Observations with Parameters)** The most powerful and general approach combines indirect observation adjustment with parameter estimation. Observations depend on a set of unknown parameters (coordinates, heights, calibration constants, etc.). Least squares solves for the best parameter estimates. This method encompasses all others as special cases and is the standard approach in modern surveying software. **Role of Most Probable Value in Least Squares Framework** The most probable value concept underlies all least squares adjustment. In every case — direct, indirect, conditional — we seek the estimate that minimizes the weighted sum of squared residuals. The MPV for a single quantity is the simplest manifestation of this principle. Understanding MPV and weights is foundational: it provides the intuition and mathematical basis for appreciating how least squares works in complex network adjustments. Professional geodetic engineers in the Philippines (under RA 4374, RA 8560, PD 1529) must command this progression from simple MPV to full least squares analysis.

This section collects the essential formulas from error theory and most probable value for rapid reference during problem-solving: **Arithmetic Mean (Equal Weights)** x̄ = Σx/n **Residuals** vᵢ = xᵢ − x̄ (sum to zero: Σvᵢ = 0) **Standard Deviation of Single Observation** σ = √[Σvᵢ²/(n−1)] Alternatively: σ = √[Σxᵢ² − (Σxᵢ)²/n] / (n−1) (more numerically stable) **Standard Deviation of the Mean** σx̄ = σ/√n **Variance** σ² = [Σvᵢ²/(n−1)] **Probable Error** PE = 0.6745·σ **Weight (Inverse Variance)** wᵢ = k/σᵢ² (typically k = 1) Alternative: wᵢ = 1/σᵢ² or wᵢ ∝ 1/distance (leveling) or wᵢ ∝ n (repetitions) **Weighted Mean** x̄w = Σ(wᵢ·xᵢ) / Σwᵢ **Standard Deviation of Weighted Mean** σx̄w = 1/√(Σwᵢ) = 1/√W **Error Propagation (General)** σQ = √[Σ(∂Q/∂xᵢ)²·σᵢ²] **Error Propagation — Sum/Difference** If Q = x ± y, then σQ = √(σₓ² + σᵧ²) **Error Propagation — Product/Quotient** If Q = x·y or Q = x/y, then σQ/Q = √[(σₓ/x)² + (σᵧ/y)²] (relative error) **Most Probable Error (95% Confidence)** MPE ≈ 1.96·σ (approximately 2σ rule: 95% of errors lie within ±2σ) **Confidence Interval at 68% (1-sigma)** Range: x̄ ± σ (contains ~68% of possible single observations)

Experienced instructors have identified recurring mistakes that cost points on the PRC Geodetic Engineer Licensure Examination. Awareness and careful practice prevent these costly errors: **Pitfall 1: Using n instead of (n−1) in Standard Deviation** Incorrect: σ = √[Σv²/n] Correct: σ = √[Σv²/(n−1)] The (n−1) factor (Bessel's correction) accounts for the constraint that Σvᵢ = 0. When computing σ from a sample, we "lose" one degree of freedom. Using n instead gives a biased underestimate. On board exams, explicitly state "n−1 degrees of freedom" to demonstrate understanding. **Pitfall 2: Confusing Weight w = 1/σ² with w = 1/σ** Incorrect: w = 1/σ Correct: w = 1/σ² Weight is inverse variance, not inverse standard deviation. This mistake compounds through calculations. If σ₁ = 0.01 and σ₂ = 0.02, then w₁ = 10,000 and w₂ = 2,500 (ratio 4:1), not w₁ = 100 and w₂ = 50 (ratio 2:1). Double-check by dimensional analysis: if σ is in meters, then σ² is in m², and 1/σ² has units m⁻². Weights are dimensionally inverse squared units. **Pitfall 3: Forgetting √n in Standard Deviation of the Mean** Incorrect: σx̄ = σ/n Correct: σx̄ = σ/√n This is a critical relationship. Taking four times as many observations reduces the mean error by a factor of two (√4 = 2), not four. Mistakes here underestimate the benefit of redundant observations. **Pitfall 4: Adding Errors Linearly Instead of in Quadrature** Incorrect (for independent errors): σQ = σₓ + σᵧ Correct: σQ = √(σₓ² + σᵧ²) Errors combine geometrically (vector addition) when independent. Linear addition (σₓ + σᵧ) overestimates error and is conservative but incorrect. For two equal errors σ: correct result is σ√2 ≈ 1.41σ, not 2σ. **Pitfall 5: Mixing up Weighted Mean and Arithmetic Mean** For unequal weights, the arithmetic mean is wrong: x̄ = Σx/n is incorrect. Correct for unequal weights: x̄w = Σ(wᵢ·xᵢ)/Σwᵢ On exams, always check whether weights are given or implied. If all weights are equal (wᵢ = w for all i), then x̄w = Σ(w·xᵢ)/(n·w) = Σxᵢ/n, recovering the arithmetic mean as a special case. **Pitfall 6: Forgetting Units or Confusing Unit Conversions** Example: σ = 5 mm in some places and 0.005 m in others; mixing these causes a 1000× error. Always convert to consistent units (SI) before calculations. **Pitfall 7: Failing to Check that Σvᵢ = 0** Once residuals are computed, verify Σvᵢ ≈ 0 (within rounding). This is a quick sanity check. If Σvᵢ ≠ 0, an arithmetic error was made. **Pitfall 8: Reporting Excessive Significant Figures** The uncertainty itself determines the appropriate precision. If σ = 0.0347 m, report as σ ≈ 0.035 m (2–3 significant figures). A mean of 145.23143 m with σ = 0.002 m should be reported as 145.231 ± 0.002 m, not 145.23143 ± 0.002 m. **Pitfall 9: Misapplying Leveling Weight Rules** For leveling, weight w ∝ 1/K (inversely proportional to distance), not w ∝ K or w ∝ √K. A 4 km route has 1/4 the weight of a 1 km route, not 4 times the weight. Errors accumulate with √K in leveling, but weights are defined as 1/K to achieve the least squares solution.