GELE Adjustment Computations (Least Squares) — Least Squares — Observation EquationsSummary

The method of least squares is based on the principle of minimizing the weighted sum of squared residuals: Σ(wᵢvᵢ²) → minimum. A residual v is defined as the difference between a computed value (based on preliminary or estimated unknowns) and the observed value. By finding the unknowns that minimize this sum, the method produces the most probable values when measurement errors are assumed to be random and normally distributed. This approach is optimal in the sense of maximum likelihood for Gaussian errors.

Each observation is expressed as a mathematical function of the unknowns: ℓ + v = f(x̂), where ℓ is the observed value, v is the residual, and x̂ are the unknown parameters. For a linear (or linearized) model, this becomes: v = Ax̂ − l, where A is the design (Jacobian) matrix of partial derivatives, x̂ is the vector of unknown corrections, and l is the vector of (computed-minus-observed) values. The design matrix A is crucial: each row represents one observation, and each column represents one unknown parameter. The element Aᵢⱼ is the partial derivative ∂(observation i)/∂(unknown j).

The design matrix is an n × u matrix where n is the number of observations and u is the number of unknowns. Each element Aᵢⱼ = ∂(observation i)/∂(unknown j). For example, in a level network where the unknown is a point's elevation, each row of A contains the partial derivative of that level observation with respect to elevation changes. In a traverse, rows correspond to angle and distance observations, and columns to coordinates or bearings. The form of A depends entirely on the mathematical model relating observations to unknowns.

The weight matrix P is a diagonal n × n matrix where the diagonal element Pᵢᵢ = 1/σᵢ² (inverse of the variance of observation i). Observations with small variance (high precision) receive large weights. For example, if one level route has a standard deviation of ±1 mm and another of ±2 mm, their weights are in the ratio 4:1. Using proper weights ensures that the adjustment gives appropriate emphasis to each measurement based on its reliability. If weights are not available, equal weights (P = I) can be used, giving the unweighted least-squares solution.

The least-squares solution is found by solving the normal equations: (A^T P A) x̂ = A^T P l. The matrix (A^T P A) is square (u × u) and usually invertible. The solution is: x̂ = (A^T P A)^−1 A^T P l. This is the most probable value of the unknown parameters (in terms of corrections to initial estimates). The matrix (A^T P A)^−1 is the cofactor matrix of the unknowns, from which standard deviations of the adjusted unknowns are derived. The residuals are then computed: v = Ax̂ − l.

Redundancy is the number of observations minus the number of unknowns: redundancy = n − u. It represents the extra observations beyond the minimum needed to solve the problem. For example, three level routes to one point give redundancy = 3 − 1 = 2; a closed traverse with 10 sides has redundancy = 2(10) − 3 = 17 (since 20 measurements—angles and distances—minus 3 coordinate/bearing unknowns). Redundancy is essential: if n = u, there is no adjustment; the solution is unique but cannot be checked. If n < u, the problem is underdetermined. Only when n > u can a least-squares adjustment be performed and the fit assessed.

A residual v is the correction applied to an observation after adjustment. It is computed as v = Ax̂ − l (once x̂ is known). Residuals represent the part of the observation that does not fit the final model; they should be randomly distributed with zero mean and variance proportional to 1/Pᵢᵢ. A large residual for a single observation may flag a blunder or systematic error. The sum of residuals weighted by precision (Σ Pᵢ vᵢ) should be near zero, and the weighted sum of squared residuals (v^T P v) should be commensurate with the expected variance.

The reference variance (also called unit variance or variance of unit weight) is: σ̂₀² = (v^T P v) / (n − u). It is a goodness-of-fit statistic that compares the actual weighted sum of squared residuals to the expected value (which equals n − u if the weighting and model are correct). If σ̂₀² ≈ 1.0 (when v and l are in consistent units), the observations and weights are realistic. If σ̂₀² >> 1.0, the observations have larger errors than expected (either the weights are too optimistic, or there are blunders). If σ̂₀² << 1.0, the observations fit better than expected (possibly over-optimistic weight assignment). Standard deviations of adjusted unknowns are derived by taking square roots of the diagonal of σ̂₀² × (A^T P A)^−1.

When there is only one unknown (u = 1) observed multiple times, the observation equations reduce to n scalar equations: vᵢ = x̂ − lᵢ for each observation i. The normal equation becomes: Σ(Pᵢ) × x̂ = Σ(Pᵢ × lᵢ), giving x̂ = Σ(Pᵢ × lᵢ) / Σ(Pᵢ) = (weighted mean). This is a critical simplification: the least-squares solution is simply the weighted average of the observations. With equal weights, it reduces to the arithmetic mean. This case is the most common on board exams and in practical surveying (e.g., determining a point's elevation from multiple level routes).

If the observation equations are nonlinear (e.g., in angle-distance surveying or GPS positioning), they must be linearized around preliminary estimates x₀: f(x₀ + Δx) ≈ f(x₀) + A Δx, where A is the Jacobian matrix evaluated at x₀. The least-squares solution gives corrections Δx̂, and updated estimates are x₁ = x₀ + Δx̂. If the linearization is poor, the process is repeated with x₁ as the new starting point, yielding x₂, x₃, etc., until convergence. In most geodetic problems, one or two iterations suffice.

The matrix Q = (A^T P A)^−1 is the cofactor matrix of the unknowns. When multiplied by the reference variance σ̂₀², it gives the covariance matrix Cov(x̂) = σ̂₀² × Q. The diagonal elements Cov(x̂ᵢᵢ) = σ̂₀² × Qᵢᵢ are the variances of the adjusted unknowns; their square roots are the standard deviations. Off-diagonal elements represent correlations between unknowns. For example, if adjusting a traverse's coordinates, the cofactor matrix will show that x and y coordinates of the same point are correlated. This information is crucial for assessing the precision of computed derived quantities (e.g., areas, distances).

The design matrix A is derived from the partial derivatives of the observation equations. Each element Aᵢⱼ = ∂(observation i)/∂(unknown j). Without correctly formulating A, the observation equations cannot be set up properly, and the least-squares solution will be meaningless.

The observation equations v = Ax̂ − l are the starting point. Applying the least-squares condition (∂/∂x̂)[v^T P v] = 0 yields the normal equations (A^T P A) x̂ = A^T P l. Solving this system gives x̂.

The weight matrix P (determined from observation precisions 1/σᵢ²) influences both the solution x̂ and the standard deviations of the adjusted unknowns. Better-weighted observations lead to better-conditioned normal equations and more reliable results.

After solving, residuals v = Ax̂ − l are computed. The weighted sum of squared residuals (v^T P v) is then used to calculate σ̂₀² = (v^T P v)/(n−u). This statistic assesses whether the model and weighting are appropriate. A reference variance far from 1 signals issues.

Redundancy determines whether adjustment is possible and how many degrees of freedom exist. Only when n > u can a least-squares adjustment be performed and statistical tests applied. Redundancy = 0 means no adjustment; redundancy < 0 means the problem is unsolvable.

When u = 1 (one unknown), the observation equations and normal equation simplify dramatically. The solution becomes the weighted mean of all observations. This is both a practical calculation and a sanity check for more complex problems.

The cofactor matrix Q = (A^T P A)^−1 contains the variance-covariance structure of the adjusted unknowns. Multiplying Q by σ̂₀² gives full covariance information. Diagonal elements provide variances; square roots give standard deviations.

For nonlinear observation equations, an initial linearization around approximate values is necessary. The correction Δx̂ is found via least-squares. If convergence is poor, the process repeats with updated estimates, forming an iterative loop.

Input observation precisions determine the weights. Weights influence how the solution is computed and what precision the adjusted unknowns will have. A single poor-precision observation, if not down-weighted appropriately, can degrade the entire adjustment.

A point's elevation is determined by three level routes with observed values ℓ₁ = 152.34 m (std. dev. σ₁ = 2 mm), ℓ₂ = 152.30 m (σ₂ = 1 mm), ℓ₃ = 152.36 m (σ₃ = 4 mm). Weights are w₁ = 1/(2²) = 0.25, w₂ = 1/(1²) = 1.0, w₃ = 1/(4²) = 0.0625. The least-squares (weighted-mean) elevation is x̂ = [0.25(152.34) + 1.0(152.30) + 0.0625(152.36)] / (0.25 + 1.0 + 0.0625) = 152.312 m (approximately). The most precise route (σ = 1 mm) has the most influence. This is a direct application of the single-unknown weighted-mean formula.

A closed traverse with n = 10 sides has 20 observations (10 distances + 10 angles) and u = 3 unknowns (two coordinate differences Δx and Δy of the closing error, plus bearing correction). Redundancy = 20 − 3 = 17. The design matrix A relates each observation to these three unknowns (via partial derivatives). After least-squares adjustment, the traverse closes exactly, residuals are distributed, and a reference variance is computed to check the survey's quality against PD 1529 standards.

A network of n = 50 GPS baseline measurements (e.g., from five receiver pairs, multiple epochs) determines u = 6 coordinates (x, y, z of two stations; the third is fixed). Redundancy = 50 − 6 = 44. Observations are ranges; the design matrix contains partial derivatives of range with respect to coordinates. Weights are based on dilution of precision (DOP) and receiver noise. The least-squares solution gives precise, adjusted coordinates and quality metrics (standard deviations, dilution of precision). This is standard practice in Philippine geodetic surveys (WGS84 to PRS92 transformation).

A cadastral survey of a parcel uses n = 15 angle and distance measurements (tape and theodolite) to determine u = 4 corner coordinates. Redundancy = 15 − 4 = 11. The observation equations relate measured angles/distances to corner coordinates. After adjustment, residuals are inspected; the reference variance is checked against DENR limits (typically σ̂₀² should be near 1.0 to 1.5 for class A surveys). If σ̂₀² is too large, blunders may be present and must be identified and eliminated.

In photogrammetry, a block of n = 1000 image point measurements (2D pixel coordinates on multiple overlapping photos) is used to solve for u = 300 unknowns (3D object space coordinates of ground points, plus camera exterior/interior parameters). Redundancy = 1000 − 300 = 700. The observation equations relate image coordinates to object coordinates via collinearity equations (nonlinear). Least-squares iterative adjustment produces precise 3D coordinates and an orthophoto. The reference variance confirms the quality of the solution.

A secondary control network, observed in WGS84, is adjusted to connect to existing primary control points (fixed in the Philippines' PPCS UTM system). n = 20 observations, u = 8 unknowns (coordinates of 4 new points). Redundancy = 12. The design matrix relates observations to the unknown PPCS coordinates. After adjustment, densified points are available for subsequent surveys (e.g., DEM generation, land use mapping) conforming to RA 4374 standards.

A structure (bridge, dam) is surveyed at two epochs (Year 1 and Year 2). Each epoch has n = 30 measurements determining u = 15 coordinates. Separate adjustments yield two sets of coordinates. The difference, tested using the covariance matrices from both adjustments, reveals whether deformation has occurred (e.g., subsidence, settlement). This requires understanding both the adjustment itself and the cofactor matrix for error propagation.

To convert coordinates from WGS84 to PRS92, a Helmert 7-parameter transformation (3 translations, 3 rotations, 1 scale) is fitted to n = 20 control points observed in both systems. u = 7 parameters. Redundancy = 13. The observation equations relate transformed coordinates to measured coordinates. After least-squares adjustment, the 7 parameters are refined; σ̂₀² validates the transformation's quality. This is essential for Philippine cadastral and engineering surveys.

In autonomous surveying systems (e.g., UAV-based topography), n = 500 sensor measurements (IMU, lidar, camera) update u = 50 unknown trajectory points. Least-squares adjustment fuses all sensor data optimally, producing a precise trajectory and a reference variance that indicates sensor health. This demonstrates the versatility of observation equations in contemporary geodetic engineering.