GELE Adjustment Computations (Least Squares) — Error Propagation, Variance-Covariance and Error EllipsesSummary

When a computed quantity y = f(x₁, x₂, …, xₙ) depends on measured quantities with independent errors, the variance of y is given by: σₑ² = Σᵢ(∂f/∂xᵢ)² σᵢ². Each term (∂f/∂xᵢ)² is a partial derivative squared, weighted by the variance of that observation. This fundamental law describes how measurement uncertainties combine and amplify (or reduce) in calculations. The key insight is that errors are added in quadrature (squares sum, not absolute values), and each partial derivative acts as a sensitivity weighting factor. For independent errors, correlations vanish; for correlated errors, cross-product covariance terms appear in the full matrix expansion.

Several practical cases simplify the general law: (1) Sum or difference (y = x₁ ± x₂): σₑ = √(σ₁² + σ₂²)—errors always add in quadrature regardless of sign. (2) Scaled quantity (y = kx, constant k): σₑ = |k| σₓ—the error scales proportionally with the coefficient. (3) Sum of n equal independent observations: σ_sum = σ√n (not σ/√n; that applies to the mean). These shortcuts avoid repeated partial-derivative algebra and appear frequently in field computations and exam problems.

The matrix formulation generalizes propagation to multi-variable functions. If y = f(x) is a vector function and J is the Jacobian matrix (∂f/∂x), then the covariance matrix of y is Σₑ = J Σₓ Jᵀ, where Σₓ is the covariance matrix of the input measurements. This elegant form automatically handles correlations: the off-diagonal elements of Σₓ (covariances between x₁ and x₂, etc.) propagate into Σₑ through matrix multiplication. Diagonal elements of Σₓ are variances; off-diagonal elements are covariances σᵢⱼ. The beauty of this notation is that it scales from 2 variables to hundreds and naturally incorporates geometric strength from least-squares adjustments.

After a least-squares adjustment of survey observations, the estimated parameter covariance matrix is Σ_hat_x = σ₀² (AᵀPA)⁻¹, where σ₀² is the reference variance (unit weight variance), A is the design matrix (partial derivatives), P is the weight matrix (inverse of measurement covariance), and the term (AᵀPA)⁻¹ is the matrix inverse of the normal equations. The diagonal of Σ_hat_x holds parameter variances (σ²ₓ, σ²ᵧ, σ²ᵤ for coordinates); the off-diagonal elements are covariances (e.g., σₓᵧ between E and N coordinates). A large diagonal element indicates high uncertainty in that parameter; a large off-diagonal covariance indicates strong correlation (if one coordinate is over-estimated, the other tends to be as well).

A 2-D position (E, N or X, Y coordinates) has uncertainty represented geometrically as an error ellipse. This ellipse is centered on the computed position and oriented according to the covariance between E and N. The semi-axes of the ellipse are derived from the eigenvalues λ₁ and λ₂ of the 2×2 covariance submatrix Σ_2D = [[σ²ₓ, σₓᵧ], [σₓᵧ, σ²ᵧ]]. The semi-major axis a = √λ_max and semi-minor axis b = √λ_min. The orientation angle α is found from the eigenvector of λ_max. A circular ellipse (a ≈ b) indicates well-balanced uncertainty in all directions—strong geometric control. An elongated ellipse (a >> b) indicates weakness in one direction: poor geometry, weak intersections, or correlated systematic errors. The standard error ellipse corresponds to approximately 39% probability; scaling by 2.45 gives the 95% confidence ellipse (following the χ² distribution with 2 degrees of freedom).

The standard error ellipse from the eigenvalues of the covariance matrix Σ corresponds to one standard deviation in the principal directions. Statistically, this contour encloses approximately 39% of the probability distribution for a bivariate normal position. To achieve a 95% confidence region (the confidence level most commonly used in engineering and legal surveys), the semi-axes must be scaled by the factor k = √(χ²₂, 0.95) ≈ 2.45. Thus, the 95% confidence ellipse has semi-axes a₉₅ = 2.45 √λ_max and b₉₅ = 2.45 √λ_min. Some older references use a factor of 2 or 2.2 (corresponding to ~86% confidence), so always clarify which confidence level applies in a project specification. Philippine professional standards and international geodetic practice favor the 95% level.

The off-diagonal covariance σₓᵧ (or σ_EN) in the variance-covariance matrix quantifies linear correlation between two parameters. If σₓᵧ = 0, the parameters are uncorrelated (independent errors). If σₓᵧ ≠ 0, they are correlated: positive covariance means if x is over-estimated, y tends also to be over-estimated; negative means they tend to compensate. The correlation coefficient ρₓᵧ = σₓᵧ / (σₓ σᵧ) ranges from -1 to +1 and measures correlation strength normalized by standard deviations. High positive correlation (ρ ≈ 0.9) creates an elongated ellipse tilted ±45°; zero correlation creates axes aligned with the coordinate system. Understanding covariance structure reveals geometric deficiencies: for example, in a poorly-formed traverse, all stations tend to shift together (high positive covariance), whereas in a well-braced network, errors are more independent.

Error propagation applies to nearly all survey calculations: (1) Traverse distance: if each taped segment d₁, d₂, … has error σ, total distance D = Σdᵢ has σ_D = σ√n. (2) Area: for a rectangle with length L (σ_L) and width W (σ_W), area A = LW, so σ_A = √((W σ_L)² + (L σ_W)²). (3) Coordinate transformation: converting from raw measurements to UTM or PRS92 datum introduces Jacobian-based errors. (4) Height network: closing misclosure δ ≈ √(Σ σᵢ²) for segments with variances σᵢ². (5) Angle sum: if three angles each have σ = 5″, their sum has σ_sum = √3 × 5″ ≈ 8.66″. These recurring patterns appear on every professional exam and in daily fieldwork.

A project requires X, Y coordinates staked to ±0.05 m at 95% confidence. Using error ellipse theory, surveyors compute the required standard deviation (divide 0.05 m by 2.45 ≈ 0.02 m standard error). From this, the design specifies which measurement methods (GNSS, total station, tape) and network configuration achieve this precision. The variance-covariance matrix from the proposed adjustment is checked: if the semi-major axis of the 95% error ellipse exceeds 0.05 m, the design is revised (additional tie-ins, higher-accuracy instruments, etc.).

A precision leveling network closing a loop has segments with variances σᵢ². The closing misclosure δ is expected to be approximately δ ≈ √(Σσᵢ²). If the observed misclosure exceeds, say, 3√(Σσᵢ²), the fieldwork contains a gross error or systematic bias and must be re-surveyed. Error propagation quantifies the tolerance; post-adjustment, the variance-covariance matrix confirms that each height's uncertainty meets the PRS92 datum requirements.

After adjusting a triangulation network, the error ellipse at each station is plotted on a map. An ellipse is nearly circular at well-tied stations but elongated perpendicular to a poorly-constrained direction. For example, an island chain measured by trilateration from shore may show ellipses stretched in the offshore direction if the shore tie-in is weak. This visual feedback guides re-measurement of critical angles or distances, or repositioning of control points to strengthen weak geometry.

Modern GNSS processing software (e.g., Trimble Business Center, Leica LGO) outputs a variance-covariance matrix for each solved position in WGS84. When transforming to PRS92 using the Helmert or affine transformation, the Jacobian matrix method (Σ_PRS92 = J Σ_WGS84 Jᵀ) propagates errors through the transformation. The resulting 95% error ellipse defines the confidence region in the Philippine datum, essential for compliance with RA 4374 and international standards.

A closed traverse with n legs, each measured to an accuracy σ_d for distance and σ_θ for angle, accumulates error roughly as σ_closure ∝ √n. The Law of Propagation allows surveyors to compute the expected closure misclosure and set acceptable tolerances (typically ±0.05 m/√n in Philippines). If observed misclosure exceeds this, systematic error (e.g., compass variation unaccounted, tripod settling) is suspected and corrected before coordinate adjustment.

A parcel boundary is measured with coordinates having covariance matrix Σ. The area A is computed from the coordinates (shoelace formula or cross-product). Using the Jacobian ∂A/∂(x_i, y_i) for each vertex, error propagation yields σ_A, the uncertainty in the computed area. This is critical for real estate transactions where valuation depends on area; clients must be informed: 'The lot is 500 m² ± 2 m² at 95% confidence,' demonstrating professional rigor under Philippine property law (PD 1529).

Repeated surveys of an infrastructure (bridge, dam, building foundation) measure coordinates over time. Error ellipses from each epoch are compared. If a structure's position shifts beyond the 95% error ellipse of the previous epoch, movement is statistically significant; otherwise, the shift may be just measurement noise. Error propagation correctly sets the decision threshold, enabling early warning of structural distress.

A survey manager must justify whether a high-accuracy total station (σ ± 2 mm) or a standard instrument (σ ± 5 mm) is required. Error propagation to the final deliverable (e.g., building floor coordinates ±20 mm at 95%) answers this: if σ ± 2 mm propagates to ±18 mm final but σ ± 5 mm to ±25 mm, the high-accuracy instrument is justified to stay within tolerance. This cost-benefit analysis uses variance-covariance mathematics to make equipment purchasing decisions.