Error Propagation, Variance-Covariance and Error Ellipses - Study Notes
In geodetic surveying and adjustment computations, measurements are never perfect—every observation carries uncertainty. When we compute positions, distances, or areas from these measurements, the uncertainties propagate through our calculations. This chapter develops the mathematical framework to predict how measurement errors affect final results, organize uncertainty information in a covariance matrix, and visualize positional uncertainty as error ellipses. Understanding error propagation is essential for survey quality control, network design, and interpreting least-squares adjusted coordinates in compliance with Philippine surveying standards under RA 4374 and RA 8560.
1. Introduction to Error Propagation
Error propagation is the process of determining how measurement uncertainties flow through calculations to produce uncertainty in the final result. In practice, surveyors measure angles (in arc seconds or degrees), distances (in meters), and elevations (in meters). Each measurement has a standard deviation σ (sigma), representing its typical precision. When we combine measurements—such as computing a distance as the sum of multiple tape segments, or calculating an area from measured sides—the resulting quantity's uncertainty depends on:
- The uncertainties of each input measurement
- How each measurement influences the final result (through partial derivatives)
- Whether measurements are independent or correlated
The Law of Propagation of Variances (also called the propagation law or error law) is the fundamental tool. It states that for any function combining measurements, the output variance is a weighted sum of input variances, where weights are the squares of partial derivatives.
This is critical because:
- It prevents underestimating uncertainty in results
- It guides survey design by identifying which measurements most affect precision
- It validates whether achieved accuracy meets project specifications (critical for cadastral surveys under CA 141 and land registration per PD 1529)
- It establishes confidence in adjusted coordinates from least-squares processing
Examples
A surveyor measures a building width as 25.00 m with standard deviation σ = 0.05 m. The length is measured as 40.00 m with σ = 0.07 m. Both measurements are independent. What is the standard deviation of the measured area if we compute area as length × width?
Solution
For the function A = L × W, we use the propagation law:
σ_A² = (∂A/∂L)² σ_L² + (∂A/∂W)² σ_W²
Partial derivatives: ∂A/∂L = W = 40 m and ∂A/∂W = L = 25 m
σ_A² = (40)²(0.07)² + (25)²(0.05)² = 1600(0.0049) + 625(0.0025) = 7.84 + 1.5625 = 9.4025 m⁴
σ_A = √9.4025 = 3.07 m²
The area is 1000 m² with uncertainty of approximately ±3.07 m² (about 0.3% relative error).
A distance is measured as the sum of three tape segments: d = d₁ + d₂ + d₃. Each segment is measured with σ = 0.02 m and measurements are independent. Find the standard deviation of the total distance.
Solution
For a sum of independent measurements:
d = d₁ + d₂ + d₃
Partial derivatives: ∂d/∂d₁ = ∂d/∂d₂ = ∂d/∂d₃ = 1
σ_d² = (1)²σ₁² + (1)²σ₂² + (1)²σ₃² = 0.02² + 0.02² + 0.02² = 0.0012 m²
σ_d = √0.0012 = 0.0346 m ≈ 0.035 m
Notice: σ_d = σ√n = 0.02√3 = 0.0346 m. This illustrates the fundamental principle that errors in independent measurements add in quadrature (as squares), not linearly.
Key points
- Every measurement has uncertainty expressed as standard deviation σ or variance σ²
- Computed quantities inherit and combine the uncertainties of their inputs
- The propagation law quantifies this inheritance mathematically
- Partial derivatives weight how each input influences the output
- Error propagation is essential for quality control in geodetic work per RA 8560
2. The Law of Propagation of Variances
The Law of Propagation of Variances is the mathematical expression of how uncertainty flows through calculations. For a function y = f(x₁, x₂, ..., xₙ) of measured quantities x₁ through xₙ:
General Form:
σ_y² = (∂f/∂x₁)² σ_x₁² + (∂f/∂x₂)² σ_x₂² + ... + (∂f/∂xₙ)² σ_xₙ²
Or more compactly:
σ_y² = Σᵢ (∂f/∂xᵢ)² σ_xᵢ²
This assumes measurements are independent (uncorrelated). Each partial derivative ∂f/∂xᵢ represents the sensitivity of the function to changes in xᵢ—how much the output changes per unit change in that input. When this sensitivity is large, that measurement's uncertainty has a large impact on the result.
Special Cases (Important for Board Exams):
- Sum or Difference: For y = x₁ ± x₂
σ_y = √(σ_x₁² + σ_x₂²)
Errors in summed quantities add in quadrature, not linearly. - Scaled Quantity: For y = k·x (k is a constant)
σ_y = |k|·σ_x
A scaled measurement's uncertainty scales by the same factor. - Mean of n Equal Observations: If y = (x₁ + x₂ + ... + xₙ)/n
σ_y = σ_x/√n
The uncertainty of the mean decreases with the square root of the number of observations. This justifies repeated measurements. - Sum of n Equal Observations: If y = x₁ + x₂ + ... + xₙ
σ_y = σ_x·√n
The uncertainty of the sum increases with the square root of the number of segments (not linearly!).
Matrix Form (for Least-Squares Applications):
When applied to multiple correlated outputs, the propagation law is written as:
Σ_y = J · Σ_x · J^T
where:
- Σ_y is the variance-covariance matrix of outputs
- Σ_x is the variance-covariance matrix of inputs
- J is the Jacobian matrix of partial derivatives [∂f_i/∂x_j]
- The superscript T denotes matrix transpose
This matrix form is used in least-squares adjustment (critical for PRC exams) and preserves correlations between outputs.
Examples
A traverse angle is measured as the sum of three component angles: θ = θ₁ + θ₂ + θ₃. Each angle has standard deviation σ = 8 arc-seconds (8"). Find the standard deviation of the total angle.
Solution
For the sum of independent angles:
θ = θ₁ + θ₂ + θ₃
Using σ_θ = √(σ₁² + σ₂² + σ₃²):
σ_θ = √(8² + 8² + 8²) = √(64 + 64 + 64) = √192 = 13.86 arc-seconds
Alternatively: σ_θ = σ√n = 8√3 = 13.86"
The three angles combine to give a total angle uncertainty of about ±13.86", compared to the single measurement precision of ±8".
A surveyor needs to measure the height difference between two points. A barometer reading at point A is h_A = 450.25 m with σ = 0.10 m. A reading at point B is h_B = 425.75 m with σ = 0.12 m. Find the standard deviation of the height difference Δh = h_A - h_B.
Solution
For the difference of two independent measurements:
Δh = h_A - h_B
Using the difference formula:
σ_Δh = √(σ_A² + σ_B²) = √(0.10² + 0.12²) = √(0.01 + 0.0144) = √0.0244 = 0.156 m
The height difference is 24.50 m with an uncertainty of approximately ±0.156 m or ±156 mm. This illustrates why it's not σ_A - σ_B = 0.10 - 0.12 (which would be incorrect).
A distance baseline is measured by laying a 50 m tape 8 times end-to-end. Each tape length is measured as 50.00 m with σ = 0.015 m. What is the standard deviation of the 400 m baseline?
Solution
For a sum of n equal independent measurements:
Baseline = 8 × 50 m = 400 m
Using σ_total = σ√n:
σ_baseline = 0.015 × √8 = 0.015 × 2.828 = 0.0424 m ≈ 0.042 m or 42 mm
Alternatively, viewing this as d = d₁ + d₂ + ... + d₈:
σ_d² = 8(0.015)² = 8(0.000225) = 0.0018
σ_d = √0.0018 = 0.0424 m
The 400 m baseline has uncertainty of about ±42 mm, demonstrating that the relative precision improves: 42/400,000 ≈ 1 part in 9500.
Key points
- Variances add in quadrature for independent errors: σ² = Σ(σᵢ²), not σ = Σ(σᵢ)
- Partial derivatives ∂f/∂xᵢ weight how each input's uncertainty affects output
- For sums/differences: errors combine as √(σ₁² + σ₂² + ...)
- For products/quotients: use logarithmic derivatives for simpler calculation
- The mean of n observations has uncertainty σ/√n; the sum has uncertainty σ√n
- Matrix form Σ_y = J·Σ_x·J^T handles multiple outputs and preserves correlations
3. Variance-Covariance Matrix
In least-squares adjustment, multiple unknown parameters (such as x and y coordinates of several points) are estimated simultaneously from redundant measurements. The variance-covariance matrix (often written Σ or C) provides complete information about the uncertainty and correlation of all estimated parameters.
Definition:
For a set of estimated parameters x̂ = [x̂₁, x̂₂, ..., x̂ₙ]ᵀ (such as adjusted coordinates of multiple points), the variance-covariance matrix is:
Σ_x̂ = [σ_x̂₁² σ_x̂₁x̂₂ σ_x̂₁x̂₃ ... ]
[σ_x̂₁x̂₂ σ_x̂₂² σ_x̂₂x̂₃ ... ]
[σ_x̂₁x̂₃ σ_x̂₂x̂₃ σ_x̂₃² ... ]
[... ... ... ... ]
where:
- Diagonal elements σ_x̂ᵢ² are variances of individual parameters
- Off-diagonal elements σ_x̂ᵢx̂ⱼ are covariances between parameter pairs
- The matrix is symmetric: σ_x̂ᵢx̂ⱼ = σ_x̂ⱼx̂ᵢ
From Least-Squares Adjustment:
After a least-squares solution of the normal equations (AᵀPA)x̂ = AᵀPL, the variance-covariance matrix is calculated as:
Σ_x̂ = σ̂₀² (AᵀPA)⁻¹
where:
- σ̂₀² is the estimated variance of unit weight (reference variance), calculated from residuals
- (AᵀPA)⁻¹ is the inverse of the normal equation matrix
- A is the design (Jacobian) matrix from linearization
- P is the weight matrix
For equal weights (P = I):
Σ_x̂ = σ̂₀² (AᵀA)⁻¹
Interpretation of Covariance:
The covariance σ_x̂ᵢx̂ⱼ measures the linear relationship between two estimated parameters:
- If σ_x̂ᵢx̂ⱼ > 0: parameters tend to move together (positive correlation)
- If σ_x̂ᵢx̂ⱼ < 0: parameters move oppositely (negative correlation)
- If σ_x̂ᵢx̂ⱼ = 0: parameters are uncorrelated
Correlation Coefficient:
To compare covariances on a standard scale, the correlation coefficient is:
ρ_ij = σ_x̂ᵢx̂ⱼ / (σ_x̂ᵢ · σ_x̂ⱼ)
where -1 ≤ ρ_ij ≤ +1. A correlation of |ρ| ≈ 1 indicates strong interdependence; |ρ| ≈ 0 indicates independence.
Physical Meaning in Surveying:
In a network adjustment, high correlations between coordinates of adjacent points (e.g., ρ_x̂₁,x̂₂ near 1) indicate that errors in one point's position are highly linked to errors in the other—improving one tends to improve both. This is normal in traverses and networks. Very high correlations (>0.95) may indicate weak network geometry or near-singular equations, requiring better field measurements or tie-in points.
Examples
A small GPS network adjustment yields the following variance-covariance matrix for three points (each with x and y coordinates, so 6 parameters total). For Point 1, the matrix subblock is: Σ₁ = [[0.0004, 0.00015], [0.00015, 0.0005]] m². Calculate the standard deviations of x₁ and y₁, and their correlation coefficient.
Solution
From the 2×2 submatrix for Point 1:
σ_x̂₁² = 0.0004 m² → σ_x̂₁ = √0.0004 = 0.020 m = 20 mm
σ_ŷ₁² = 0.0005 m² → σ_ŷ₁ = √0.0005 = 0.0224 m = 22.4 mm
Covariance: σ_x̂₁ŷ₁ = 0.00015 m²
Correlation coefficient:
ρ_x̂₁ŷ₁ = 0.00015 / (0.020 × 0.0224) = 0.00015 / 0.000448 = 0.335
Interpretation: The x and y coordinates of Point 1 have a weak positive correlation (ρ ≈ 0.34), meaning their errors are somewhat linked but not strongly dependent. This is typical in a well-designed network where points are connected through various directions.
In a 1-D leveling traverse, four bench marks (BM1 to BM4) are adjusted. The variance-covariance matrix for elevations is:
Σ = [[0.0010, 0.0008, 0.0006, 0.0004],
[0.0008, 0.0020, 0.0018, 0.0016],
[0.0006, 0.0018, 0.0030, 0.0028],
[0.0004, 0.0016, 0.0028, 0.0040]] m²
Find the standard deviations of all elevations and explain the pattern.
Solution
Standard deviations (diagonal square roots):
σ_BM1 = √0.0010 = 0.0316 m = 31.6 mm
σ_BM2 = √0.0020 = 0.0447 m = 44.7 mm
σ_BM3 = √0.0030 = 0.0548 m = 54.8 mm
σ_BM4 = √0.0040 = 0.0632 m = 63.2 mm
Pattern: Uncertainties increase from BM1 to BM4. This is typical in a leveling traverse where the first point (BM1) is fixed as a reference, and uncertainty propagates and accumulates downstream. Each successive point inherits the errors from all previous segments.
Notice also that covariances decrease as points move further apart: σ_BM1,BM4 = 0.0004 < σ_BM1,BM2 = 0.0008. Distant points are less correlated.
Key points
- Variance-covariance matrix is square, symmetric, and contains complete uncertainty information
- Diagonal elements are variances σ² of individual parameters; off-diagonals are covariances
- From least-squares: Σ_x̂ = σ̂₀² (AᵀPA)⁻¹; the inverse normal matrix scaled by reference variance
- Covariance σ_ij > 0 means parameters move together; < 0 means they move oppositely
- Correlation coefficient ρ_ij ∈ [-1, 1] standardizes covariance for comparison
- High correlation (>0.95) flags weak geometry; circular/uncorrelated errors indicate good design
4. Error Ellipses and Confidence Regions
A 2-D position (such as the latitude-longitude or x-y coordinates of a survey point) has uncertainty in two directions. The error ellipse (or standard error ellipse) is a graphical representation of this 2-D uncertainty, showing the region in which the true position is likely to lie with a specified confidence level.
Geometric Derivation:
For a 2-D position with coordinates (x, y) adjusted via least-squares, the 2×2 variance-covariance submatrix is:
Σ_pos = [[σ_x², σ_xy ],
[σ_xy, σ_y² ]]
This matrix can be diagonalized by rotating coordinates to align with its eigenvectors (principal directions). The eigenvalues λ₁ and λ₂ of Σ_pos are the variances in these principal directions; their square roots are the semi-major and semi-minor axes of the error ellipse.
Calculation of Error Ellipse Parameters:
- Semi-axes a and b (standard error ellipse, ~39% confidence):
a = √λ₁ (semi-major axis)
b = √λ₂ (semi-minor axis)
where λ₁ ≥ λ₂ Orientation angle α (azimuth of semi-major axis):
tan(2α) = 2σ_xy / (σ_x² - σ_y²)or from eigenvector of λ₁.
- Eigenvalues (if you don't use a calculator):
λ₁, λ₂ = [(σ_x² + σ_y²) ± √((σ_x² - σ_y²)² + 4σ_xy²)] / 2
Confidence Scaling:
The standard error ellipse (semi-axes a and b) corresponds to approximately 39% confidence (1σ in 2-D). For higher confidence levels, scale the semi-axes by a factor k:
- 68% confidence (1σ): k ≈ 1.51
- 95% confidence (2σ): k ≈ 2.45
- 99% confidence (3σ): k ≈ 3.04
So the 95% error ellipse has semi-axes a₉₅ = 2.45a and b₉₅ = 2.45b.
Interpretation and Survey Quality:
- Circular ellipse (a ≈ b): Uncertainty is equal in all directions. Indicates well-conditioned geometry with balanced precision in both x and y (e.g., point well-triangulated from multiple directions).
- Elongated ellipse (a >> b): Large uncertainty along one direction, small along another. Indicates weak geometry in one direction—the survey network is "loose" perpendicular to the major axis. Common in long narrow traverses or intersections from limited directions.
- Orientation of major axis: Points toward the direction of poorest geometry. In a traverse extending east-west, the major axis typically points north-south (perpendicular to the traverse direction).
- Small ellipse: High precision; achieved through redundancy, good equipment, and careful field procedures.
- Large ellipse: Poor precision; indicates need for more measurements, higher-quality instruments, or network redesign.
Application in Philippine Surveying:
Under RA 8560 and for projects involving:
- Cadastral surveys (CA 141): Error ellipses must fit within property boundaries and coordinate limits
- Control networks (RA 4374): Error ellipses define positional tolerance and network accuracy classes
- Land titles (PD 1529): Monument positions must have error ellipses within acceptable limits for legal certainty
Surveyors must compute and report error ellipses for all adjusted points, especially control monuments and property corners.
Examples
A GPS-adjusted point has a variance-covariance matrix:
Σ = [[0.0004, 0.00006],
[0.00006, 0.0009]] m²
Calculate the standard error ellipse (semi-axes a and b) and the orientation angle. Then find the 95% error ellipse.
Solution
Step 1: Extract parameters
σ_x² = 0.0004 m²
σ_y² = 0.0009 m²
σ_xy = 0.00006 m²
Step 2: Calculate eigenvalues
Trace = σ_x² + σ_y² = 0.0004 + 0.0009 = 0.0013
Determinant = σ_x²σ_y² - σ_xy² = (0.0004)(0.0009) - (0.00006)² = 0.00000036 - 0.0000000036 = 0.0000003564
λ₁, λ₂ = [0.0013 ± √(0.0013² - 4×0.0000003564)] / 2
= [0.0013 ± √(0.00000169 - 0.00000143)] / 2
= [0.0013 ± √(0.00000026)] / 2
= [0.0013 ± 0.000510] / 2
λ₁ = (0.0013 + 0.000510) / 2 = 0.0009050 m²
λ₂ = (0.0013 - 0.000510) / 2 = 0.0003950 m²
Step 3: Calculate standard error ellipse semi-axes
a = √λ₁ = √0.0009050 = 0.0301 m = 30.1 mm
b = √λ₂ = √0.0003950 = 0.0199 m = 19.9 mm
Step 4: Calculate orientation angle
tan(2α) = 2σ_xy / (σ_x² - σ_y²) = 2(0.00006) / (0.0004 - 0.0009) = 0.00012 / (-0.0005) = -0.24
2α = -13.5° → α = -6.75° or equivalently α ≈ 83.25° from east
Step 5: 95% confidence error ellipse (scale by k = 2.45)
a₉₅ = 2.45 × 30.1 = 73.7 mm
b₉₅ = 2.45 × 19.9 = 48.8 mm
Conclusion: The standard error ellipse has semi-major axis 30.1 mm and semi-minor axis 19.9 mm, oriented at about 83° from east (nearly north). The point has slightly better precision in the north-south direction (smaller semi-minor axis = 19.9 mm) and slightly poorer precision east-west. The 95% confidence ellipse is approximately 74 mm by 49 mm.
Two points in a leveling network have the following 2×2 covariance submatrices:
Point A: Σ_A = [[0.0001, -0.00005],
[-0.00005, 0.0001]] m²
Point B: Σ_B = [[0.0004, 0.00008],
[0.00008, 0.0001]] m²
Compare the error ellipses of Points A and B. Which has better geometry and why?
Solution
Point A:
σ_x² = 0.0001, σ_y² = 0.0001, σ_xy = -0.00005 (negative covariance)
Tr_A = 0.0002
Det_A = (0.0001)² - (-0.00005)² = 0.00000001 - 0.0000000025 = 0.0000000075
λ₁_A = [0.0002 + √(0.0002² - 4×0.0000000075)] / 2 = [0.0002 + √(0.00000004 - 0.00000003)] / 2
= [0.0002 + 0.0000100] / 2 ≈ 0.000105 m²
λ₂_A = 0.000095 m²
a_A = √0.000105 = 0.0102 m = 10.2 mm
b_A = √0.000095 = 0.0097 m = 9.7 mm
Ratio: a_A/b_A = 10.2/9.7 ≈ 1.05 (nearly circular)
Point B:
σ_x² = 0.0004, σ_y² = 0.0001, σ_xy = 0.00008
Tr_B = 0.0005
Det_B = (0.0004)(0.0001) - (0.00008)² = 0.00000004 - 0.0000000064 = 0.0000000336
λ₁_B = [0.0005 + √(0.0005² - 4×0.0000000336)] / 2 = [0.0005 + √(0.00000025 - 0.000000134)] / 2
= [0.0005 + √(0.000000116)] / 2 = [0.0005 + 0.000341] / 2 ≈ 0.0004205 m²
λ₂_B = [0.0005 - 0.000341] / 2 ≈ 0.0000795 m²
a_B = √0.0004205 = 0.0205 m = 20.5 mm
b_B = √0.0000795 = 0.0089 m = 8.9 mm
Ratio: a_B/b_B = 20.5/8.9 ≈ 2.30 (elongated)
Conclusion: Point A has much better geometry. Its error ellipse is nearly circular (ratio 1.05), indicating equal precision in x and y—the point is well-constrained from multiple directions. Point B's ellipse is significantly elongated (ratio 2.30), indicating weaker constraint in one direction (likely the x-direction, since σ_x is larger). Point B may lie on a traverse or be intersected from a limited range of angles.
Key points
- Error ellipse visualizes 2-D positional uncertainty; semi-axes from eigenvalues of covariance matrix
- Standard ellipse is ~39% confidence; scale by k ≈ 2.45 for 95% confidence
- Circular ellipse (a ≈ b) indicates well-conditioned geometry; elongated indicates weak direction
- Orientation angle α (azimuth of major axis) points to the direction of poorest geometry
- Error ellipses are required in Philippine surveys (RA 8560, CA 141, PD 1529) for regulatory compliance
- Eigenvalue calculation: λ = [(σ_x² + σ_y²) ± √((σ_x² - σ_y²)² + 4σ_xy²)] / 2
5. Practical Applications and Least-Squares Integration
In professional geodetic practice, error propagation, variance-covariance matrices, and error ellipses are integrated throughout the least-squares adjustment workflow. Understanding this workflow is essential for the PRC Geodetic Engineer Licensure Examination.
Survey Design Phase:
Before fieldwork begins, surveyors perform pre-analysis using error propagation to predict whether the proposed network will achieve required accuracy. By computing how measurement uncertainties (σ for angles, distances) will propagate through the network geometry, engineers can:
- Determine how many measurements are needed
- Choose measurement methods (total station, GPS, level rod)
- Design adequate redundancy
- Estimate expected error ellipses
For example, to ensure a traverse's endpoint has error ellipse ≤ 0.05 m in both directions, one might increase taping frequency or add cross-ties.
Data Collection:
Field measurements are recorded with their nominal precisions (specifications of instruments, methods). These become the input σ values for error propagation.
Adjustment Computation:
Least-squares adjustment solves the normal equations (AᵀPA)x̂ = AᵀPL:
- Design matrix A includes partial derivatives (Jacobian) from linearized measurement equations
- Weight matrix P embodies measurement precisions: P = Σ_L⁻¹ where Σ_L is the input variance-covariance matrix
- Solution yields adjusted parameters x̂ and residuals v̂
- Reference variance σ̂₀² = (v̂ᵀPv̂) / (degrees of freedom) = Σ(pᵢv̂ᵢ²) / (redundancy)
- Parameter covariance matrix: Σ_x̂ = σ̂₀²(AᵀPA)⁻¹
Quality Control:
The σ̂₀ value indicates overall fit:
- σ̂₀ ≈ 1.0: measured precisions were realistic; adjustment is well-behaved
- σ̂₀ > 1.5: either input precisions were overestimated or unmodeled errors/blunders exist
- σ̂₀ < 0.7: input precisions were underestimated (conservative estimates) or network is over-constrained
Residuals v̂ᵢ are checked against their standard errors σ̂(v̂ᵢ); outliers flag blunders.
Report Output:
A professional adjustment report includes:
- Adjusted coordinates with standard deviations σ_x̂, σ_ŷ for each point
- Variance-covariance matrix (often truncated to key points)
- Correlation matrix showing dependencies between parameters
- Error ellipses for each adjusted point (plotted on maps)
- Overall network accuracy assessment per RA 4374 standards
Compliance with Philippine Standards:
- RA 4374 (Philippine Geometric Datum): Networks must be tied to national reference frame (WGS84 via PRS92)
- RA 8560 (Geodetic Engineering Law): All cadastral and control surveys must undergo least-squares adjustment with published uncertainty estimates
- CA 141 (Property Boundaries): Property corner positions must have error ellipses within specified limits (typically <0.05 m for residential property, <0.10 m for larger parcels)
- PD 1529 (Land Registration): Title certificates must include or reference monumentation and accuracy of survey positions
Examples of Practical Interpretation:
- Traverse Survey: A 2 km traverse is adjusted. Point at 1 km from start has σ_x = 0.030 m, σ_y = 0.015 m, and σ_xy = 0.002 m². Error ellipse is elongated (wider perpendicular to traverse direction). This is expected because lateral control is weaker than along-traverse control. To improve, install cross-ties every 0.5 km.
- GPS Network: Five control points adjusted with multi-day occupations. Reference variance σ̂₀ = 0.85 (< 1.0) indicates conservative input precisions from manufacturer specs. Actual field results were slightly better than expected. Report is acceptable.
- Leveling Network: Bench marks over 50 km circuit show errors growing from 10 mm at start to 50 mm at end. Error ellipse for end point is 45 mm × 8 mm (elongated vertically). This reflects cumulative error in leveling and suggests the network should be re-observed or supplemented with GPS heights.
Common Board-Exam Scenarios:
- Compute σ of a computed quantity (distance, area, angle) from field measurements
- Interpret a variance-covariance matrix subblock (extract standard deviations, correlation)
- Calculate error ellipse semi-axes and orientation from a 2×2 covariance block
- Explain what high correlation (ρ > 0.9) indicates about network geometry
- Compare error ellipses to project requirements (CA 141 tolerances, RA 8560 accuracy classes)
- Determine if a blunder-tainted measurement is detectable (residual > 3σ_v̂)
Examples
A land surveyor measures a rectangular parcel: Length L = 150.00 m (σ = 0.04 m), Width W = 80.00 m (σ = 0.03 m). A corner is established by intersection of two lines from known points. The corner's adjusted x-coordinate is 2500.15 m (σ_x = 0.025 m), and y-coordinate is 3000.20 m (σ_y = 0.020 m), with covariance σ_xy = 0.0003 m².
Part (a): Calculate the area and its standard deviation.
Part (b): Calculate the standard error ellipse of the corner.
Part (c): Explain whether this error ellipse complies with CA 141 for a residential lot (tolerance 0.05 m).
Solution
Part (a): Area and standard deviation
A = L × W = 150 × 80 = 12,000 m²
Using propagation law:
σ_A² = (∂A/∂L)²σ_L² + (∂A/∂W)²σ_W²
where ∂A/∂L = W = 80, ∂A/∂W = L = 150
σ_A² = (80)²(0.04)² + (150)²(0.03)² = 6400(0.0016) + 22500(0.0009) = 10.24 + 20.25 = 30.49 m⁴
σ_A = √30.49 = 5.52 m²
The area is 12,000 ± 5.52 m² (relative error ≈ 0.046%, which is excellent for cadastral work).
Part (b): Standard error ellipse
Σ = [[0.000625, 0.0003 ],
[0.0003, 0.0004 ]] m² (note: σ_x² = 0.025² = 0.000625)
Eigenvalues:
Tr = 0.001025, Det = (0.000625)(0.0004) - (0.0003)² = 0.00000025 - 0.00000009 = 0.00000016
λ₁, λ₂ = [0.001025 ± √(0.001025² - 4×0.00000016)] / 2
= [0.001025 ± √(0.00000105 - 0.00000064)] / 2
= [0.001025 ± √(0.00000041)] / 2
= [0.001025 ± 0.000640] / 2
λ₁ = (0.001025 + 0.000640) / 2 = 0.0008325 m²
λ₂ = (0.001025 - 0.000640) / 2 = 0.0001925 m²
a = √0.0008325 = 0.0289 m = 28.9 mm
b = √0.0001925 = 0.0139 m = 13.9 mm
Orientation: tan(2α) = 2(0.0003) / (0.000625 - 0.0004) = 0.0006 / 0.000225 = 2.667
2α = 69.4° → α ≈ 34.7° from east
Part (c): Compliance assessment
CA 141 tolerance for residential property: 0.05 m (50 mm) in both x and y (independent error limits) or within a 50 mm radius circle.
Error ellipse semi-axes: a = 28.9 mm, b = 13.9 mm
The 95% confidence ellipse (scale by 2.45):
a₉₅ = 2.45 × 28.9 ≈ 70.8 mm
b₉₅ = 2.45 × 13.9 ≈ 34.0 mm
Conclusion: At 95% confidence, the major axis (70.8 mm) exceeds the CA 141 tolerance of 50 mm. This corner position does NOT comply with residential property standards. The surveyor should either:
- Perform additional measurements to improve precision (more traverse legs, closer control points)
- Upgrade to RTK-GPS with better precision
- Classify the property as agricultural/non-critical (tolerance 0.10 m), if applicable
- Establish a higher-order control monument near the property and re-survey from it
A least-squares adjustment of a 1 km control traverse yields variance-covariance matrix (extracted 2×2 blocks for two points):
Point T1 (start, fixed): σ_x = 0, σ_y = 0 (given coordinates)
Point T2 (1st station): Σ_T2 = [[0.0001, 0.00005 ],
[0.00005, 0.0004 ]] m²
Point T3 (end, 1 km from start): Σ_T3 = [[0.0004, 0.0001 ],
[0.0001, 0.0010 ]] m²
Also, σ̂₀ = 1.08 (reference variance) and degrees of freedom = 12.
Part (a): Compare the standard deviations of T2 and T3. Explain the pattern.
Part (b): Calculate error ellipses for both points and interpret their shapes.
Part (c): Explain whether σ̂₀ = 1.08 is acceptable and what it suggests about field measurements.
Solution
Part (a): Standard deviations
Point T2:
σ_x(T2) = √0.0001 = 0.0100 m = 10 mm
σ_y(T2) = √0.0004 = 0.0200 m = 20 mm
Point T3:
σ_x(T3) = √0.0004 = 0.0200 m = 20 mm
σ_y(T3) = √0.0010 = 0.0316 m = 31.6 mm
Pattern: Uncertainties grow from T2 to T3 as distance from fixed point T1 increases. In a traverse, errors accumulate: each successive station inherits uncertainties from all previous measurements. T2 is 50% of the way (500 m), T3 is at the end (1000 m), so T3's errors are larger. Standard deviations approximately double, consistent with √n growth in cumulative error.
Part (b): Error ellipses
Point T2:
Σ_T2 = [[0.0001, 0.00005 ],
[0.00005, 0.0004 ]]
Tr = 0.0005, Det = (0.0001)(0.0004) - (0.00005)² = 0.00000004 - 0.0000000025 = 0.0000000375
λ₁_T2 = [0.0005 + √(0.0005² - 4×0.0000000375)] / 2 = [0.0005 + √(0.00000025 - 0.00000015)] / 2
= [0.0005 + √(0.0000001)] / 2 = [0.0005 + 0.000316] / 2 ≈ 0.0004158 m²
λ₂_T2 ≈ 0.0000842 m²
a_T2 = √0.0004158 = 0.0204 m = 20.4 mm
b_T2 = √0.0000842 = 0.0092 m = 9.2 mm
Ratio: a_T2/b_T2 ≈ 2.22 (moderately elongated)
Point T3:
Σ_T3 = [[0.0004, 0.0001 ],
[0.0001, 0.0010 ]]
Tr = 0.0014, Det = (0.0004)(0.0010) - (0.0001)² = 0.0000004 - 0.00000001 = 0.0000003900
λ₁_T3 = [0.0014 + √(0.0014² - 4×0.0000003900)] / 2 = [0.0014 + √(0.00000196 - 0.00000156)] / 2
= [0.0014 + √(0.00000040)] / 2 = [0.0014 + 0.000633] / 2 ≈ 0.001017 m²
λ₂_T3 ≈ 0.000383 m²
a_T3 = √0.001017 = 0.0319 m = 31.9 mm
b_T3 = √0.000383 = 0.0196 m = 19.6 mm
Ratio: a_T3/b_T3 ≈ 1.63 (moderately elongated, less so than T2)
Interpretation: Both ellipses are elongated (a > b), indicating the traverse direction provides better constraint than the perpendicular direction—typical for a linear traverse. T2's ellipse is more elongated (2.22) because it's influenced by the fixed point T1 in one direction but has grown in the perpendicular direction. T3's ratio (1.63) is less extreme because lateral error is now substantial in both directions after 1 km. T3's ellipse is notably larger overall (31.9 × 19.6 mm vs. 20.4 × 9.2 mm for T2), reflecting cumulative error growth.
Part (c): Reference variance assessment
σ̂₀ = 1.08 is slightly above 1.0, suggesting:
✓ Field measurements were slightly worse than nominal specifications, OR
✓ Input precisions were slightly overestimated, OR
✓ Minor unmodeled errors exist (e.g., temperature effects on tape, slight systematic angle bias)
With σ̂₀ = 1.08 and degrees of freedom = 12:
χ² statistic = (n - m) σ̂₀² = 12 × 1.08² ≈ 13.99
For 12 d.f., the 90% confidence limit is χ²(0.90, 12) ≈ 18.5, so σ̂₀ = 1.08 falls well within acceptable range.
Conclusion: The adjustment is acceptable and well-behaved. A σ̂₀ of 1.08 is quite good and indicates the survey was executed competently. No action is required, but the surveyor should note that field precision was very slightly below specifications—likely due to normal field variability, not instrumental defects. Report and publish the adjusted coordinates and error ellipses; they are reliable for cadastral use under RA 8560.
Key points
- Pre-analysis uses error propagation to predict network accuracy before fieldwork
- Least-squares adjustment yields Σ_x̂ = σ̂₀²(AᵀPA)⁻¹ and error ellipses from 2×2 submatrices
- Reference variance σ̂₀ ≈ 1.0 indicates realistic input precisions; >1.5 flags problems
- Error ellipses must comply with CA 141 (property boundaries) and RA 4374 (control network) limits
- Correlation matrix and ellipse shapes reveal geometry quality and measurement balance
- Residuals checked against standard errors to detect blunders per RA 8560 protocols
6. Common Board-Exam Pitfalls and Quick Reference
As you prepare for the PRC Geodetic Engineer Licensure Examination, here are critical mistakes and clarifications to keep in mind:
Pitfall 1: Adding Errors Linearly Instead of Quadrature
Wrong: σ_total = σ₁ + σ₂
Right: σ_total = √(σ₁² + σ₂²) for independent errors
Why: Individual measurement errors can partially cancel. The correct combination accounts for this cancellation probabilistically.
Pitfall 2: Confusing σ√n (Sum) with σ/√n (Mean)
Wrong: "The sum of n measurements has error σ/√n"
Right:
- Sum of n equal measurements: σ_sum = σ√n (error grows)
- Mean of n equal measurements: σ_mean = σ/√n (error shrinks)
Why: The sum accumulates all errors; the mean averages them out.
Example: Three angle measurements, each σ = 6"
- Sum: σ_sum = 6√3 ≈ 10.4" (larger)
- Mean: σ_mean = 6/√3 ≈ 3.46" (smaller)
Pitfall 3: Incorrect Partial Derivatives
Wrong: For A = L × W, using ∂A/∂L = 1
Right: ∂A/∂L = W, ∂A/∂W = L
Quick check: Dimensions must be correct. ∂A/∂L has units of length (width), not dimensionless.
Pitfall 4: Forgetting the Jacobian Square
Wrong: σ_y = (∂f/∂x)σ_x
Right: σ_y² = (∂f/∂x)² σ_x²
The variance formula has squared derivatives; always square before summing.
Pitfall 5: Misinterpreting Covariance Sign
Wrong: "Negative covariance means bad correlation"
Right:
- Negative covariance: parameters move oppositely (one increases, other decreases)
- Positive covariance: parameters move together
- Either can be acceptable; sign just indicates the type of linkage
Example: In a simple 1-D leveling network, adjacent bench marks have positive covariance (if BM1 is set too high, BM2 error is in same sense). In a closed traverse, opposite corner errors may be negatively correlated.
Pitfall 6: Confusing Standard Error Ellipse with Confidence Ellipse
Standard ellipse (1σ, ~39% confidence):
- Semi-axes: a, b (not scaled)
- Use when comparing geometry quality
95% confidence ellipse (~95% probability true point is inside):
- Semi-axes: 2.45a, 2.45b (scaled)
- Use for tolerances, compliance checks (CA 141, RA 4374)
Pitfall 7: Wrong Orientation Angle Interpretation
Wrong: "α is the bearing of the major axis" (without clarification)
Right: α is the counterclockwise angle from the east (x-axis) to the major axis, in the local coordinate system. In UTM/PPCS, convert to true bearing or grid bearing as needed.
Pitfall 8: Ignoring Correlation in Confidence Intervals
Wrong: Treating σ_x and σ_y as completely independent when computing 2-D tolerance
Right: Use full 2×2 covariance matrix; correlation affects the actual coverage probability.
For example, if σ_x = 0.05 m, σ_y = 0.05 m but ρ = 0.95, the 2-D error is NOT a 0.05√2 ≈ 0.071 m circle. The highly correlated errors create an elongated ellipse.
Pitfall 9: Misremembering Eigenvalue Formula
Wrong: λ = (Tr ± √Det)
Right: λ = [(Tr ± √(Tr² - 4Det)) / 2]
The discriminant is Tr² - 4Det, not just Det.
Pitfall 10: Not Checking Units
Wrong: Mixing σ in mm with covariance in m²
Right: Ensure all σ and matrix entries use consistent units (all meters, all mm, etc.)
Quick Reference Table:
| Situation | Formula | Example |
|---|---|---|
| Single measurement y=f(x) | σ_y² = (df/dx)² σ_x² | Area A = s²: σ_A = 2sσ_s |
| Sum of n equal measurements | σ_sum = σ√n | Three 50 m segments, σ=0.02 m: σ_total = 0.02√3 = 0.0346 m |
| Mean of n measurements | σ_mean = σ/√n | Same: σ_mean = 0.02/√3 = 0.0115 m |
| Sum/difference of 2 measurements | σ = √(σ₁² + σ₂²) | Two angles 8", 6": σ = √(64+36) = 10" |
| Error ellipse semi-axes | a,b = √λ₁, √λ₂ (eigenvalues) | Covariance [[0.0004, 0.00015], [0.00015, 0.0005]]: solve 2×2 determinant |
| 95% confidence scaling | k ≈ 2.45 | Standard a=30mm → 95% ellipse: 2.45×30 ≈ 74 mm |
| Correlation coefficient | ρ = σ_xy / (σ_x σ_y) | σ_xy = 0.0003, σ_x = 0.02, σ_y = 0.02: ρ = 0.75 |
Board Exam Strategy:
- Always show work: Examiners give partial credit for correct method even if arithmetic is slightly off
- State assumptions: "Assuming independent measurements..." or "Using the standard error ellipse (39% confidence)..."
- Check reasonableness: Does the answer make physical sense? (e.g., error should increase along a traverse)
- Quote standards: Reference RA 8560, CA 141, or relevant Philippine survey specs to show professional awareness
- Sketch ellipses: A rough sketch of an error ellipse, even if not perfectly to scale, demonstrates understanding and can earn partial credit
Examples
A point is adjusted via least-squares with variance-covariance matrix:
Σ = [[0.0009, -0.0002 ],
[-0.0002, 0.0004 ]] m²
Part (a): Extract standard deviations and correlation coefficient.
Part (b): Calculate the standard error ellipse (1σ).
Part (c): Calculate the 95% confidence ellipse.
Part (d): Interpret the negative off-diagonal element.
Solution
Part (a): Standard deviations and correlation
σ_x = √0.0009 = 0.0300 m = 30 mm
σ_y = √0.0004 = 0.0200 m = 20 mm
σ_xy = -0.0002 m²
Correlation:
ρ = σ_xy / (σ_x σ_y) = -0.0002 / (0.0300 × 0.0200) = -0.0002 / 0.0006 = -0.333
Part (b): Standard error ellipse eigenvalues
Tr = 0.0009 + 0.0004 = 0.0013
Det = 0.0009 × 0.0004 - (-0.0002)² = 0.00000036 - 0.00000004 = 0.00000032
λ = [0.0013 ± √(0.0013² - 4×0.00000032)] / 2
= [0.0013 ± √(0.00000169 - 0.00000128)] / 2
= [0.0013 ± √(0.00000041)] / 2
= [0.0013 ± 0.000640] / 2
λ₁ = (0.0013 + 0.000640) / 2 = 0.0009700 m²
λ₂ = (0.0013 - 0.000640) / 2 = 0.0003300 m²
a = √0.0009700 = 0.0311 m = 31.1 mm
b = √0.0003300 = 0.0182 m = 18.2 mm
Ellipse ratio: a/b = 31.1/18.2 ≈ 1.71 (moderately elongated)
Orientation:
tan(2α) = 2σ_xy / (σ_x² - σ_y²) = 2(-0.0002) / (0.0009 - 0.0004) = -0.0004 / 0.0005 = -0.8
2α = -38.66° → α ≈ -19.33° or (180° - 19.33°) = 160.67° from east
Part (c): 95% confidence ellipse
a₉₅ = 2.45 × 31.1 = 76.2 mm
b₉₅ = 2.45 × 18.2 = 44.6 mm
Part (d): Interpretation of negative covariance
The negative value σ_xy = -0.0002 indicates that x and y errors are negatively correlated with strength ρ = -0.333. Specifically:
- When the true x-coordinate is underestimated (adjusted x is too small), the true y-coordinate tends to be overestimated (adjusted y is too large)
- When x is overestimated, y tends to be underestimated
In surveying terms: This often occurs in surveys with uneven geometry. For example, in an intersection from two known stations:
- If the angle from Station A is slightly off, it biases x in one direction
- If the angle from Station B compensates, it biases y in the opposite direction
- The result: negative correlation
This is not a defect—it's a natural result of the survey geometry. The weak correlation (|ρ| = 0.33) indicates the x and y errors are somewhat independent, which is reasonable for a well-designed network.
Key points
- Errors combine in quadrature (√Σσ²), not linearly (Σσ), for independent measurements
- Sum of n measurements: σ = σ√n (error grows). Mean: σ = σ/√n (error shrinks).
- Variance formula has squared partial derivatives: σ_y² = (∂f/∂x)²σ_x²
- Standard error ellipse (~39%) vs. 95% confidence ellipse (scale by k≈2.45)
- Negative covariance indicates opposite correlation; still acceptable depending on context
- Eigenvalue formula: λ = [(Tr ± √(Tr² - 4Det))/2], not (Tr ± √Det)
- Always verify units consistency throughout calculations
- For board exams: show work, state assumptions, reference Philippine standards
Diagrams
Error Propagation, Covariance, and Error Ellipses—Concept Overview
This mind map shows the hierarchical structure of concepts in error propagation and uncertainty analysis. It connects the three main pillars—error propagation law, variance-covariance matrix, and error ellipse—and links them to practical surveying applications.
Least-Squares Adjustment Workflow: From Field Measurements to Error Ellipses
This flowchart traces the complete least-squares adjustment workflow, from field measurements through to error ellipse calculation and compliance checking. It shows decision points (reference variance check, specification compliance) and links to Philippine surveying standards.
Survey Geometry Quality Assessment via Error Ellipse Shape
This state diagram maps error ellipse shapes (aspect ratios) to geometric quality assessments and decision outcomes. It guides surveyors in interpreting error ellipses and deciding whether a network meets Philippine standards.
In summary
Error propagation, variance-covariance matrices, and error ellipses are the mathematical foundations for understanding measurement uncertainty in geodetic surveys and least-squares adjustments. The Law of Propagation of Variances (σ_y² = Σ(∂f/∂xᵢ)²σ_xᵢ²) quantifies how measurement errors combine in calculations. For practical applications, errors add in quadrature, not linearly—a critical distinction for board exams. The variance-covariance matrix (Σ_x̂ = σ̂₀²(AᵀPA)⁻¹) captures both the uncertainty of individual parameters and their correlations after least-squares adjustment. The error ellipse, derived from the eigenvalues and eigenvectors of a 2×2 covariance submatrix, visualizes 2-D positional uncertainty. A circular ellipse indicates well-conditioned geometry; an elongated ellipse flags weak measurement geometry in one direction. Understanding these concepts is essential for survey design, quality control, and compliance with Philippine standards (RA 4374 for control networks, CA 141 for property boundaries, RA 8560 for cadastral surveys, and PD 1529 for land registration). In the PRC Geodetic Engineer Licensure Examination, expect questions requiring calculation of propagated errors, interpretation of variance-covariance matrices, and evaluation of error ellipses against project specifications. Always work in SI units, show partial derivatives clearly, remember that σ of a sum is σ√n (not σ/√n), and quote relevant Philippine laws to demonstrate professional competency.