GELE Adjustment Computations (Least Squares) — Error Propagation, Variance-Covariance and Error EllipsesRevision Notes
Revision notes for GELE Adjustment Computations (Least Squares) — Error Propagation, Variance-Covariance and Error Ellipses. Short, focused, and designed for the week before exam day. Use these when you are already familiar with the chapter and need a quick refresh on the high-yield items Professional Regulation Commission (PRC) — Board of Geodetic Engineering tests.
Exam context
On the GELE 2026, the Adjustment Computations (Least Squares) subtest carries a "Core" weight in Professional Regulation Commission (PRC) — Board of Geodetic Engineering's pattern. Error Propagation, Variance-Covariance and Error Ellipses lands at position 5th out of 5 in the standard review order. Target score is 70% weighted average, no sub-test below 50%, and roughly a meaningful share of items come from Adjustment Computations (Least Squares) on a typical GELE paper.
Error Propagation, Variance-Covariance and Error Ellipses - Revision Notes
In geodetic surveying, every measured quantity carries an inherent uncertainty. When these raw observations are used to compute derived quantities — distances, coordinates, areas, azimuths — the uncertainty propagates forward. This chapter equips PRC Geodetic Engineer examinees with the mathematical tools to quantify, track, and interpret that uncertainty: the Law of Propagation of Variances, the Variance-Covariance matrix from least-squares adjustment, and the Error Ellipse as a graphical summary of 2-D positional uncertainty. These concepts are foundational to rigorous geodetic practice under RA 8560 (Philippine Geodetic Engineering Act of 1998) and the standards enforced by the National Mapping and Resource Information Authority (NAMRIA) using PRS92/WGS84 and PPCS/UTM coordinates.
Sections
Formulas
Example
Area A = L × W, L = 80 m (σ_L = 0.03 m), W = 40 m (σ_W = 0.02 m). ∂A/∂L = W = 40, ∂A/∂W = L = 80. σ_A² = (40)²(0.03)² + (80)²(0.02)² = 1.44 + 2.56 = 4.00 m⁴. σ_A = 2.00 m².
Formula
σ_y² = (∂f/∂x₁)² σ_x₁² + (∂f/∂x₂)² σ_x₂² + … + (∂f/∂xₙ)² σ_xₙ²
Variables
σ_y² = variance of the derived quantity y; ∂f/∂xᵢ = partial derivative of f with respect to xᵢ (sensitivity coefficient); σ_xᵢ² = variance of the i-th independent observation
Application
Used whenever a single derived quantity depends on multiple independent measured inputs. Core formula for traverse closures, computed areas, and coordinate transformations.
Example
Two angles α₁ and α₂ each measured with σ = 5". Sum angle = α₁ + α₂. σ_sum = √(5² + 5²) = √50 = 7.07" ≈ 7".
Formula
σ_(x₁ ± x₂) = √(σ₁² + σ₂²)
Variables
σ_(x₁ ± x₂) = standard deviation of the sum or difference; σ₁, σ₂ = standard deviations of the two independent quantities
Application
Adding or subtracting independently measured distances, angles, or elevations. Used in traverse leg summation and differential leveling runs.
Example
A distance measured as x = 250 m with σ_x = 0.05 m is multiplied by k = 3.5 to obtain a scaled quantity. σ_y = |3.5| × 0.05 = 0.175 m.
Formula
σ_y = |k| σ_x
Variables
k = constant scale factor; σ_x = standard deviation of the measured quantity x; σ_y = standard deviation of the scaled result
Application
Unit conversion, area computation from a scaled measurement, or any linear transformation of a single observation.
Example
A traverse is taped in 6 equal segments, each with σ = 0.01 m. σ_total = 0.01 × √6 = 0.01 × 2.449 = 0.0245 m.
Formula
σ_sum = σ √n
Variables
σ = standard deviation of one segment/observation; n = number of equal, independent segments or observations; σ_sum = standard deviation of the total sum
Application
Total error in a traverse or leveling run composed of n equal-precision segments. Applicable to tape-and-offset surveys and spirit leveling sections.
Example
Four independent distance readings each with σ = 0.02 m. σ_mean = 0.02 / √4 = 0.02 / 2 = 0.010 m.
Formula
σ_mean = σ / √n
Variables
σ = standard deviation of a single observation; n = number of repeated observations; σ_mean = standard deviation of the arithmetic mean
Application
Precision of the mean of n repeated independent observations. Distinct from the sum: averaging REDUCES uncertainty, summing INCREASES it.
Example
If y = [y₁, y₂]ᵀ depends on x = [x₁, x₂, x₃]ᵀ, then J is (2×3), Σ_x is (3×3), and Σ_y = J Σ_x Jᵀ is (2×2).
Formula
Σ_y = J Σ_x Jᵀ
Variables
Σ_y = variance-covariance matrix of derived quantities (m × m); J = Jacobian matrix of partial derivatives (m × n); Σ_x = variance-covariance matrix of input observations (n × n); Jᵀ = transpose of J
Application
General matrix form of LPV. Used in least-squares adjustment to propagate the full covariance structure of observations into the covariance of adjusted parameters (e.g., coordinates).
Exam Tips
- Memorize the four special-case formulas: sum/difference, scaling, sum of n equals, and mean of n equals. These cover ~80% of board problems.
- For area problems, always identify the functional form first (e.g., A = LW or A = ½bh) before differentiating.
- When asked for 'error in the total distance of a traverse with n equal legs,' the answer is σ√n — not nσ.
- Board problems sometimes disguise LPV as 'error in computed coordinates' or 'error in computed bearing' — set up the functional model explicitly.
- If observations are stated as independent, you can drop the covariance terms and use the scalar LPV directly.
Key Points
- When a function y = f(x₁, x₂, …, xₙ) is computed from independently measured quantities xᵢ, the variance of y is obtained by weighting each source variance by the square of its partial derivative.
- The partial derivative ∂f/∂xᵢ acts as a sensitivity coefficient — it tells you how much y changes per unit change in xᵢ.
- For independent observations, cross-product (covariance) terms vanish; for correlated observations, they must be included via the matrix form.
- The scalar LPV is the foundation for all error budget calculations in traversing, leveling, coordinate geometry, and area computation.
- The matrix form Σ_y = J Σ_x Jᵀ is the general expression that handles both variances and covariances simultaneously.
- The Jacobian matrix J contains all first-order partial derivatives and has dimensions (m × n) where m = number of derived quantities, n = number of input observations.
Definitions
Term
Law of Propagation of Variances (LPV)
Definition
A mathematical rule that computes the variance (or covariance) of a function of random variables from the variances (and covariances) of those random variables using first-order Taylor series expansion.
Importance
Exam-critical: it is the theoretical basis for ALL error analysis in adjustment computations, traversing, and coordinate geometry.
Term
Sensitivity Coefficient
Definition
The partial derivative ∂f/∂xᵢ evaluated at the observation values; it quantifies how much the derived quantity changes per unit change in xᵢ.
Importance
Determines which observations contribute most to the total uncertainty. High sensitivity means that observation demands tighter precision.
Term
Jacobian Matrix (J)
Definition
An m × n matrix whose (i,j) entry is ∂yᵢ/∂xⱼ, containing all first-order partial derivatives of the derived quantities with respect to the input observations.
Importance
Essential for applying the matrix form of LPV when multiple derived quantities depend on multiple observations.
Term
Adding in Quadrature
Definition
The process of combining independent errors by summing their squares and then taking the square root: σ_total = √(σ₁² + σ₂² + …). This is the correct rule for independent random errors.
Importance
Frequently tested: errors do NOT add linearly; they add in quadrature. A common board-exam trap.
Section Title
The Law of Propagation of Variances (LPV)
Common Mistakes
- Adding standard deviations linearly (σ₁ + σ₂) instead of in quadrature √(σ₁² + σ₂²) for independent errors.
- Confusing σ_sum = σ√n (error of the SUM of n observations) with σ_mean = σ/√n (error of the MEAN of n observations).
- Forgetting to square the sensitivity coefficient: writing σ_y = (∂f/∂xᵢ)σᵢ instead of σ_y² = (∂f/∂xᵢ)²σᵢ².
- Using incorrect partial derivatives — always differentiate the full functional model, not a simplified approximation.
- Ignoring covariance terms when observations are NOT independent (e.g., repeated use of the same tape introduces correlation).
- Treating a constant multiplier k as a variable: k has no uncertainty, so only σ_y = |k|σ_x, NOT √(σ_k² + σ_x²).
Formulas
Example
After adjusting a GPS network in PRS92, σ̂₀ = 0.012 m and (AᵀPA)⁻¹ = [[4.5, 1.2],[1.2, 3.8]] cm². Σ_x̂ = (0.012)² × [[4500, 1200],[1200, 3800]] mm² = [[0.648, 0.173],[0.173, 0.547]] mm².
Formula
Σ_x̂ = σ̂₀² (AᵀPA)⁻¹
Variables
Σ_x̂ = aposteriori variance-covariance matrix of adjusted parameters; σ̂₀² = aposteriori variance of unit weight = [vᵀPv]/(n−u); A = design (observation equation) matrix; P = weight matrix; n = number of observations; u = number of unknowns
Application
Computed at the end of every least-squares adjustment to describe the precision and correlations of the adjusted geodetic coordinates or other parameters.
Example
If vᵀPv = 18.72 with n = 25 observations and u = 10 unknowns, then σ̂₀² = 18.72/15 = 1.248, σ̂₀ = 1.117. This is close to 1.0, indicating acceptable fit.
Formula
σ̂₀² = vᵀPv / (n − u)
Variables
v = residual vector (observed minus adjusted); P = weight matrix; n = number of observations; u = number of adjusted parameters; (n−u) = degrees of freedom (redundancy)
Application
Computed after adjustment to scale the cofactor matrix and to assess the overall quality of the adjustment. A value near 1.0 indicates the apriori weights were well chosen.
Example
If q_EE = 2.34 and σ̂₀ = 0.010 m, then σ_E = 0.010 × √2.34 = 0.010 × 1.530 = 0.0153 m.
Formula
Q_x̂ = (AᵀPA)⁻¹
Variables
Q_x̂ = cofactor matrix of adjusted parameters (dimensionless); A = design matrix; P = weight matrix
Application
Intermediate result used before σ̂₀² is available. Its diagonal elements q_ii are the cofactors, and σ_xᵢ = σ̂₀ √q_ii.
Exam Tips
- In board problems, σ̂₀² = vᵀPv/(n−u) is almost always tested in the context of 'compute the standard deviation of unit weight after adjustment.'
- Remember: more redundancy (larger n−u) generally leads to smaller σ̂₀² and thus smaller coordinate uncertainties.
- If P = I (equal weights), the formula simplifies to Σ_x̂ = σ̂₀² (AᵀA)⁻¹.
- The Philippine geodetic network (PRS92) references WGS84 and uses PPCS/UTM zone assignments — in network adjustment problems, coordinates are in meters (UTM Northing and Easting).
- For a 2-station network, the 2×2 covariance submatrix [[σ_E², σ_EN],[σ_EN, σ_N²]] is the starting point for the error ellipse.
Key Points
- The least-squares adjustment of an overdetermined system yields both the adjusted parameters x̂ and their quality indicator: the variance-covariance matrix Σ_x̂.
- Σ_x̂ = σ̂₀² (AᵀPA)⁻¹, where σ̂₀ is the aposteriori reference standard deviation (unit weight standard deviation), A is the design matrix, and P is the weight matrix.
- The diagonal elements of Σ_x̂ are the variances of the adjusted parameters (e.g., σ²_E, σ²_N for Easting and Northing).
- The off-diagonal elements are the covariances (e.g., σ_EN = σ_NE) indicating how the uncertainties of two parameters are correlated.
- The cofactor matrix Q_x̂ = (AᵀPA)⁻¹ is dimensionless; it becomes Σ_x̂ only after multiplication by σ̂₀².
- A well-conditioned network has a cofactor matrix with small diagonal values, indicating good geometry and redundancy.
- In PRS92/PPCS-UTM-based geodetic control networks in the Philippines, the adjusted coordinates from NAMRIA come with a full covariance matrix used to characterize network accuracy.
Definitions
Term
Variance-Covariance Matrix (Σ)
Definition
A symmetric square matrix where diagonal elements are the variances of the adjusted parameters and off-diagonal elements are the covariances between pairs of parameters. For adjusted 2-D coordinates: Σ = [[σ_E², σ_EN],[σ_EN, σ_N²]].
Importance
The complete, quantitative description of the precision and correlation structure of adjusted geodetic coordinates. Required for error ellipse computation.
Term
Covariance (σ_EN)
Definition
A measure of the joint variability of two parameters. Positive covariance means errors tend to be in the same direction; negative covariance means they tend to oppose each other; zero means independence.
Importance
Non-zero covariance means the error ellipse is tilted relative to the coordinate axes. Ignoring it leads to an incorrect (axis-aligned) ellipse.
Term
Cofactor Matrix (Q_x̂)
Definition
The scaled version of the covariance matrix, Q_x̂ = Σ_x̂ / σ̂₀² = (AᵀPA)⁻¹. It is dimensionless and represents the relative precision structure of the adjusted parameters.
Importance
Used in network design and pre-analysis before actual observations are collected to predict expected precision.
Term
Aposteriori Variance of Unit Weight (σ̂₀²)
Definition
The estimated variance computed from the residuals after adjustment: σ̂₀² = vᵀPv/(n−u). Scales the cofactor matrix into actual physical units (e.g., m²).
Importance
A critical quality indicator. Values significantly different from 1.0 signal mismodeling, blunders, or incorrect apriori standard deviations.
Term
Design Matrix (A)
Definition
The matrix of partial derivatives (coefficients) relating observations to unknown parameters in the linearized observation equation system: l = Ax + v.
Importance
Directly determines both the adjusted parameters and their covariance matrix. Poor network geometry manifests as a near-singular AᵀPA.
Section Title
Variance-Covariance Matrix in Least-Squares Adjustment
Common Mistakes
- Confusing the cofactor matrix Q with the covariance matrix Σ — they differ by the factor σ̂₀².
- Using σ₀² (apriori) instead of σ̂₀² (aposteriori) when scaling the cofactor matrix after adjustment.
- Forgetting that (n−u) in the denominator of σ̂₀² is the redundancy (degrees of freedom), not n alone.
- Assuming the covariance matrix is diagonal (σ_EN = 0) without verifying — correlated parameters require the full off-diagonal terms for correct ellipse calculation.
- Inverting AᵀPA numerically without checking its condition number — an ill-conditioned matrix signals weak geometry and unreliable covariances.
Formulas
Example
σ_E² = 0.0036 m², σ_N² = 0.0064 m², σ_EN = 0.0020 m². Mean = (0.0036+0.0064)/2 = 0.0050. Δ = (0.0036−0.0064)/2 = −0.0014. λ = 0.0050 ± √(0.0014² + 0.0020²) = 0.0050 ± √(0.00000196 + 0.0000040) = 0.0050 ± 0.00244. λ₁ = 0.00744 m², λ₂ = 0.00256 m².
Formula
λ = [(σ_E² + σ_N²)/2] ± √{[(σ_E² − σ_N²)/2]² + σ_EN²}
Variables
λ₁, λ₂ = eigenvalues of the 2×2 covariance matrix (λ₁ ≥ λ₂); σ_E², σ_N² = variances of Easting and Northing; σ_EN = covariance between Easting and Northing
Application
Direct formula to compute eigenvalues of the 2×2 position covariance matrix without full matrix diagonalization. The ± gives λ₁ (larger, major axis) and λ₂ (smaller, minor axis).
Example
From the previous example: a = √0.00744 = 0.0863 m, b = √0.00256 = 0.0506 m. Standard error ellipse: semi-axes 86.3 mm × 50.6 mm.
Formula
a = √λ₁, b = √λ₂
Variables
a = semi-major axis of the standard error ellipse (m); b = semi-minor axis of the standard error ellipse (m); λ₁, λ₂ = eigenvalues of Σ_pos
Application
Gives the physical dimensions of the standard error ellipse (39% confidence level). Multiply by appropriate scale factor for higher confidence levels.
Example
Using σ_E² = 0.0036, σ_N² = 0.0064, σ_EN = 0.0020: θ = 0.5 × arctan[2(0.0020)/(0.0036−0.0064)] = 0.5 × arctan[0.0040/(−0.0028)] = 0.5 × arctan(−1.4286) = 0.5 × (−55.0°) = −27.5° from the E-axis.
Formula
θ = (1/2) arctan[2σ_EN / (σ_E² − σ_N²)]
Variables
θ = angle of the major axis measured from the E-axis (or from N-axis, depending on convention); σ_EN = covariance; σ_E², σ_N² = variances
Application
Determines the orientation of the error ellipse. When σ_EN = 0 (no correlation), θ = 0° and the ellipse axes align with the coordinate axes.
Example
Standard ellipse a = 0.0863 m, b = 0.0506 m. At 95% confidence: a₀.₉₅ = 2.45 × 0.0863 = 0.211 m, b₀.₉₅ = 2.45 × 0.0506 = 0.124 m.
Formula
Confidence ellipse semi-axes: a_p = c_p × a, b_p = c_p × b
Variables
a_p, b_p = semi-axes at probability level p; c_p = confidence scale factor (c₀.₃₉ = 1.00 for standard ellipse, c₀.₆₃ = 1.41, c₀.₉₅ = 2.45, c₀.₉₉ = 3.03)
Application
Scaling the standard error ellipse to the desired confidence level for reporting geodetic coordinate accuracy in NAMRIA / PPCS-UTM survey reports.
Example
σ_E = 0.010 m, σ_N = 0.015 m. CEP ≈ 1.177 × √[(0.000100 + 0.000225)/2] = 1.177 × √(0.0001625) = 1.177 × 0.01275 = 0.0150 m = 15.0 mm.
Formula
CEP ≈ 1.177 × √[(σ_E² + σ_N²)/2]
Variables
CEP = Circular Error Probable (radius of 50% probability circle, in meters); σ_E, σ_N = standard deviations of Easting and Northing (for near-equal precisions)
Application
Quick estimate of positional accuracy in GNSS/GPS survey reports. Used in Philippine geodetic network accuracy specifications.
Exam Tips
- Eigenvalue formula for 2×2: memorize λ = mean ± √(half-difference² + covariance²). This is frequently tested.
- The 39% vs. 68% distinction for the standard error ellipse is a high-frequency board-exam question. Know it cold.
- Scale factor table to memorize: 39% → 1.00; 95% → 2.45; 99% → 3.03. Do not mix with 1-D z-scores.
- An elongated error ellipse in a board-exam scenario with a question 'what does this mean?' — answer: weak geometry in the direction of the major axis; more observations needed.
- In PPCS/UTM problems, Easting = x-axis (E) and Northing = y-axis (N). The angle θ in the ellipse formula is measured from E (or from N by convention — verify the convention stated in the problem).
- If σ_EN = 0 is given or implied, the eigenvalues simplify to λ₁ = max(σ_E², σ_N²) and λ₂ = min(σ_E², σ_N²) — a common simplification in board problems.
Key Points
- A 2-D position error cannot be represented by a single number; it requires an ellipse that shows the direction and magnitude of maximum and minimum uncertainty.
- The error ellipse is derived from the 2×2 variance-covariance submatrix of the adjusted position: Σ_pos = [[σ_E², σ_EN],[σ_EN, σ_N²]].
- The semi-axes (a and b) of the ellipse equal the square roots of the eigenvalues λ₁ ≥ λ₂ of Σ_pos: a = √λ₁, b = √λ₂.
- The orientation angle θ of the major axis with respect to the E-axis (or N-axis) is computed from the eigenvectors of Σ_pos.
- The standard error ellipse corresponds to approximately 39% probability (not 68% — that is the 1σ value for a 1-D normal distribution).
- To construct the 95% confidence ellipse, scale the standard axes by c₀.₉₅ ≈ 2.45 (derived from the chi-squared distribution with 2 degrees of freedom).
- A circular error ellipse (a ≈ b) indicates well-conditioned geometry (position equally well-determined in all directions).
- An elongated error ellipse (a >> b) flags weak geometry in the direction of the major axis — the network needs additional observations in that direction.
- The Circular Error Probable (CEP) is the radius of a circle containing 50% probability and equals approximately 1.177 × (a + b)/2 for near-circular ellipses.
Definitions
Term
Error Ellipse
Definition
A geometric representation of 2-D positional uncertainty. Its center is the adjusted position, its semi-axes are proportional to the square roots of the eigenvalues of the covariance matrix, and its orientation is given by the eigenvectors.
Importance
The standard tool for visualizing and reporting coordinate accuracy in geodetic surveys, GPS network adjustments, and cadastral plans under PD 1529.
Term
Eigenvalue (λ)
Definition
A scalar associated with a square matrix such that Av = λv for some non-zero vector v (the eigenvector). For a 2×2 symmetric covariance matrix, the two eigenvalues give the variances along the principal axes of the error ellipse.
Importance
The mathematical foundation for computing error ellipse dimensions. Board exams may ask to compute eigenvalues from a given 2×2 covariance matrix.
Term
Standard Error Ellipse (SEE)
Definition
The error ellipse corresponding to the 1-sigma (standard deviation) level in 2-D, which encloses approximately 39% of the probability mass of the bivariate normal distribution — NOT 68% as in the 1-D case.
Importance
Frequently misunderstood: the 1σ ellipse is only 39% — not 68%. This is a classic board-exam trick question.
Term
Confidence Ellipse
Definition
An error ellipse scaled to contain a specified probability (e.g., 95% or 99%). Obtained by multiplying the standard ellipse semi-axes by the appropriate chi-squared-based scale factor.
Importance
Required when reporting geodetic coordinates to clients or government agencies under NAMRIA / RA 8560 accuracy standards.
Term
Circular Error Probable (CEP)
Definition
The radius of a circle centered on the adjusted position that contains 50% of the probability. Commonly used in GPS accuracy specifications and military mapping.
Importance
Appears in GNSS accuracy specifications and in Philippine military and navigation mapping products referencing WGS84.
Term
Well-Conditioned Geometry
Definition
A network geometry where the design matrix A results in a well-conditioned normal matrix AᵀPA, producing a nearly circular error ellipse (a ≈ b) with small axes, indicating the position is well-determined in all directions.
Importance
Practical implication: elongated ellipses should trigger additional observations in the weak direction before submitting the survey to NAMRIA or the LRA (Land Registration Authority under PD 1529).
Section Title
The Error Ellipse
Common Mistakes
- Claiming the standard error ellipse corresponds to 68% confidence — it corresponds to ~39% in 2-D.
- Using c = 2.0 for 95% confidence — the correct factor for 2-D is c₀.₉₅ ≈ 2.45 (from χ² with 2 DOF), not 1.96 (the 1-D z-value).
- Computing the ellipse orientation angle incorrectly — using arctan(σ_EN/σ_E²) instead of arctan[2σ_EN/(σ_E² − σ_N²)] and forgetting the factor of 1/2.
- Assuming the error ellipse axes always align with the coordinate (E,N) axes — they only do so when σ_EN = 0.
- Forgetting that a = √λ₁ (square root of eigenvalue), not λ₁ itself.
- Confusing 'small error ellipse' with 'accurate position' without verifying σ̂₀² ≈ 1.0 — a small ellipse from a biased adjustment is misleading.
Formulas
Example
L = 120.00 m, σ_L = 0.05 m; W = 60.00 m, σ_W = 0.03 m. σ_A = √[(60×0.05)² + (120×0.03)²] = √[(3.0)² + (3.6)²] = √[9.0 + 12.96] = √21.96 = 4.69 m².
Formula
σ_A = √[(W σ_L)² + (L σ_W)²]
Variables
A = LW = rectangular area; L, W = length and width; σ_L, σ_W = their standard deviations
Application
Standard board-exam area problem. Partial derivatives: ∂A/∂L = W, ∂A/∂W = L.
Example
A first-order leveling line of 25 km with σ = 3 mm/km. σ_total = 3 × √25 = 3 × 5 = 15 mm = 0.015 m.
Formula
σ_Δh(total) = σ_per_km × √L_km
Variables
σ_per_km = standard deviation per km of leveling (mm/km); L_km = total length of leveling line in km; σ_Δh(total) = total standard deviation of the height difference
Application
Spirit leveling error accumulation formula used in geodetic leveling networks under NAMRIA standards.
Exam Tips
- Read each problem carefully to identify: (1) the functional model, (2) the independent variables and their uncertainties, (3) what is being asked (σ or σ²).
- For traverse problems: error in closure = σ_single_leg × √(number of legs) if all legs have equal precision.
- Always double-check: is the question asking for standard deviation (σ) or variance (σ²)? Board exams test both.
- Time management: LPV scalar problems take ~2 min; matrix/eigenvalue problems take ~5 min. Budget accordingly.
- Philippine cadastral survey problems (PD 1529, CA 141) often involve area computation — know the error propagation for both rectangular and triangular areas.
Key Points
- Worked examples bridge theory and exam performance. Practice recognizing the functional model from the problem statement before differentiating.
- Always state the formula, identify all variables, substitute values, and verify units.
- For PRC board problems, partial credit is awarded for correct setup even if arithmetic is wrong — show all work.
- Typical problem types: error of sum/difference, error of a product (area), error of n-segment sum, covariance matrix extraction, error ellipse axes and orientation.
Definitions
Term
Functional Model
Definition
The mathematical equation relating the derived quantity (e.g., area, coordinate, azimuth) to the measured observations. Must be established before applying LPV.
Importance
The starting point for ALL error propagation problems. Incorrect model = incorrect error estimate.
Term
Error Budget
Definition
A tabular or analytical breakdown of all uncertainty contributions to a derived quantity, showing how each source contributes and their combined effect via LPV.
Importance
Required in geodetic survey design to ensure the final product meets accuracy specifications (e.g., NAMRIA Order 1/2/3 control point requirements).
Section Title
Board-Exam Worked Problems
Common Mistakes
- In leveling problems, forgetting that the formula is σ√L (proportional to √distance), not σ√n (proportional to √number of setups) unless the number of setups is specified.
- Mixing units: σ_per_km in mm/km must be converted consistently before computing σ_total in mm or m.
- In the error ellipse eigenvalue formula, forgetting to compute the full discriminant √{[(σ_E²−σ_N²)/2]² + σ_EN²} — dropping the σ_EN² term when σ_EN is small but not zero.
Connections
- Error Propagation → Least-Squares Adjustment: LPV is used both before adjustment (to predict precision) and after adjustment (to propagate residual uncertainty into derived quantities from adjusted parameters).
- Variance-Covariance Matrix → Error Ellipse: The 2×2 position submatrix of Σ_x̂ is the direct input to all error ellipse computations — the two concepts are inseparable.
- Error Ellipse → Geodetic Network Design: Predicted error ellipses (pre-analysis using simulated observations) guide the design of optimal geodetic control networks compliant with NAMRIA Order 1/2/3 standards.
- LPV → Traverse Surveying: Error of closure in a traverse is computed by propagating individual leg uncertainties forward — a direct application of the sum rule σ_sum = σ√n.
- LPV → Differential Leveling: Height difference errors propagate as σ√L (per-km standard deviation × √length in km), governing NAMRIA's allowable misclosure tolerances.
- Covariance Matrix → Relative Accuracy: The covariance between two control points determines the relative error ellipse for the baseline between them — critical for cadastral surveys under PD 1529 and CA 141.
- Error Ellipse → GNSS Processing: Modern GPS/GNSS software outputs covariance matrices and error ellipses for all adjusted positions — the theory here is directly applied in field geodetic work under PRS92/WGS84.
- Eigenvalues → Principal Component Analysis: The eigenvalue decomposition of the covariance matrix is the mathematical bridge between geodetic statistics and principal component analysis used in deformation monitoring.
- Weight Matrix P → Covariance Matrix: P = σ₀² Σ_obs⁻¹ — the weight matrix is the inverse of the scaled observation covariance matrix, directly linking measurement precision to the final adjusted coordinate precision.
- σ̂₀² → Hypothesis Testing: The aposteriori variance of unit weight is used in chi-squared and F-tests to verify whether the adjustment is statistically acceptable — a prerequisite for reporting geodetic results under RA 8560.
Exam Strategy
For PRC Geodetic Engineer board examinations in Adjustment Computations (Least Squares), follow this prioritized approach: (1) MASTER the scalar LPV formula and its four special cases (sum, scaling, n-sum, mean) — these appear in nearly every exam set and are worth 3–5 points each. (2) UNDERSTAND the structure of Σ_x̂ = σ̂₀²(AᵀPA)⁻¹ and be able to extract variances and covariances from a given matrix — 2–3 questions typically. (3) PRACTICE the eigenvalue formula for the 2×2 covariance matrix and the error ellipse axis/orientation formulas — these are medium-difficulty but high-value questions. (4) MEMORIZE the critical probability facts: standard ellipse = 39%, 95% scale factor = 2.45, 99% scale factor = 3.03. (5) AVOID the classic traps: adding errors linearly, using 1-D confidence factors in 2-D, confusing σ√n with σ/√n. (6) CONNECT to Philippine context: when problems mention cadastral surveys (PD 1529, CA 141), think area error propagation; when problems mention GPS networks (RA 8560, PRS92/WGS84, PPCS/UTM), think covariance matrices and error ellipses; when problems mention leveling (NAMRIA standards), think σ√L formula. (7) For time management: allocate 2 min for simple LPV problems, 5 min for eigenvalue/ellipse problems, and 7 min for full adjustment + covariance problems. Show all formula setups even if computation time is short — partial credit is awarded in the Geodetic Engineer board exam.
Quick Review Questions
Two traverse legs are measured independently: leg 1 has σ₁ = 0.04 m and leg 2 has σ₂ = 0.03 m. What is the standard deviation of the total distance (leg 1 + leg 2)?
Sum of independent quantities: errors add in quadrature, NOT linearly. Linear sum would give 0.07 m (incorrect). The correct answer 0.05 m is the Pythagorean combination — note that 3-4-5 right triangle relationship appears here.
A rectangular land parcel (under PD 1529) has L = 100 m (σ_L = 0.04 m) and W = 50 m (σ_W = 0.02 m). Compute the standard deviation of the computed area A = LW.
Functional model A = LW. Partial derivatives: ∂A/∂L = W = 50, ∂A/∂W = L = 100. Apply LPV: σ_A² = (50)²(0.04)² + (100)²(0.02)² = 4.00 + 4.00 = 8.00 m⁴. σ_A = √8.00 = 2.83 m².
A traverse consists of 9 equal segments, each taped with σ = 0.015 m. What is the standard deviation of the total traverse length?
Sum of n equal, independent observations: σ_sum = σ√n. Do NOT multiply by n (that would be 0.135 m — wrong). The √n growth reflects the statistical accumulation of random errors in independent segments.
The mean of 9 independent distance readings (each σ = 0.015 m) is computed. What is the standard deviation of the mean?
This is the INVERSE of the previous question. Averaging n observations REDUCES uncertainty by 1/√n. Contrast: sum → σ√n (increases); mean → σ/√n (decreases). Both formulas are derived from LPV applied to the sum and mean functional models respectively.
The 2×2 covariance matrix of an adjusted PPCS/UTM position is Σ = [[0.0025, 0.0015],[0.0015, 0.0049]] m². Compute the semi-axes of the standard error ellipse.
Apply the eigenvalue formula for a 2×2 symmetric matrix. Mean = (σ_E² + σ_N²)/2 = 0.0037. Half-difference = (σ_E² − σ_N²)/2 = −0.0012. Discriminant = √(0.0012² + 0.0015²) = 0.00192. Semi-major axis a = √λ₁ = 75.0 mm, semi-minor axis b = √λ₂ = 42.2 mm.
What probability level does the standard error ellipse (semi-axes a and b without scaling) correspond to in a 2-D bivariate normal distribution?
This is a critical distinction. In 1-D, the ±1σ interval contains 68.3% of the distribution. In 2-D, the analogous '1σ ellipse' contains only about 39.3% because you are dealing with a bivariate distribution. To achieve 95% confidence in 2-D, scale the axes by c₀.₉₅ ≈ 2.45.
What does an elongated error ellipse at a geodetic control point indicate about the survey network?
The major axis points in the direction of maximum positional uncertainty. This occurs when the network has poor azimuthal coverage — e.g., all baselines running E-W would produce an elongated ellipse in the N-S direction. The remedy is to add observations (GPS baselines, distances, or angles) from directions that are roughly perpendicular to the major axis.
In a least-squares adjustment with n = 20 observations and u = 8 unknowns, vᵀPv = 24.0. Compute σ̂₀ and interpret the result.
The degrees of freedom (redundancy) = n − u = 12. σ̂₀ = 1.414, which is significantly greater than 1.0. This suggests the apriori weights were overestimated (observations were assumed more precise than they actually were), or there are undetected blunders in the data. Investigation is warranted before accepting the adjustment.
A geodetic leveling line of 36 km has a standard deviation of 4 mm per km. What is the total standard deviation of the height difference?
This is the σ√L form of error propagation for leveling, analogous to σ√n for n equal segments. The standard deviation grows as the square root of the line length, reflecting the random accumulation of instrument and refraction errors over distance.
The covariance σ_EN between Easting and Northing coordinates is found to be negative. What does this imply for the error ellipse orientation?
Non-zero covariance means the two coordinates are statistically correlated — errors tend to be anti-correlated (when E error is positive, N error tends to be negative). This produces an ellipse tilted relative to the E-N axes. The orientation formula θ = ½ arctan[2σ_EN/(σ_E²−σ_N²)] gives the angle; negative σ_EN tilts the major axis into the 2nd/4th quadrants of the E-N plane.
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