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

Concept

Law of Propagation of Variances

Sub Concepts

• Partial Derivative Weighting
• Sum and Difference Rule
• Scaling Rule
• Sum of n Equal Observations Rule
• Matrix Form (Jacobian Method)

Relationship To Central

Foundation for understanding how errors in measured quantities flow into computed results

Concept

Variance-Covariance Matrix

Sub Concepts

• Diagonal Elements (Parameter Variances)
• Off-Diagonal Elements (Covariances)
• Least-Squares Variance-Covariance Formula
• Reference Variance (σ₀²)
• Normal Matrix Inverse

Relationship To Central

Mathematical structure that captures both individual uncertainties and correlations in adjusted survey parameters

Concept

Error Ellipse

Sub Concepts

• Standard Error Ellipse (≈39% confidence)
• Semi-Major and Semi-Minor Axes
• Ellipse Orientation Angle
• Eigenvalue Decomposition
• Confidence Scaling Factors (95%, 99%)

Relationship To Central

Geometric visualization of 2-D positional uncertainty in survey coordinates (E, N in PPCS/UTM)

Concept

Error Sources and Independence

Sub Concepts

• Systematic Errors (eliminated by design/correction)
• Random Errors (propagate via variance formulas)
• Correlated vs. Independent Measurements
• Assumption Validation in Propagation

Relationship To Central

Prerequisite understanding for propagation—errors combine only if independent

Concept

Practical Applications in Philippine Surveying

Sub Concepts

• Traverse Closure and Leg Errors
• Trilateration Position Uncertainty
• Traverse Network Geometry Assessment
• WGS84/PRS92 Transformation Uncertainties
• Land Titling Surveys (RA 4374, RA 8560)
• Compliance with Accuracy Standards (PD 1529, CA 141)

Relationship To Central

Real-world contexts where error propagation and ellipses guide survey design and quality assessment

Concept

Board-Exam Problem-Solving Strategies

Sub Concepts

• Identify Independent vs. Correlated Terms
• Apply Correct Propagation Formula
• Compute Partial Derivatives Accurately
• Construct and Interpret Covariance Matrices
• Calculate Ellipse Parameters from Eigenvalues
• Avoid Common Pitfalls (Linear Addition, Wrong Scaling)

Relationship To Central

Critical techniques for scoring on PRC Licensure Examination questions

To

Variance-Covariance Matrix

From

Law of Propagation of Variances

Strength

strong

Relationship

The law provides the mathematical mechanism (partial derivatives, Jacobian) that populates the elements of the variance-covariance matrix in a least-squares adjustment.

To

Error Ellipse

From

Variance-Covariance Matrix

Strength

strong

Relationship

The 2×2 covariance submatrix (σx², σy², σxy) directly determines the error ellipse parameters (semi-axes a and b, orientation angle θ) via eigenvalue decomposition.

To

Error Ellipse

From

Law of Propagation of Variances

Strength

strong

Relationship

Error propagation rules compute individual measurement variances, which feed into the Jacobian-based propagation to produce the positional covariance, which defines the ellipse.

To

Law of Propagation of Variances

From

Error Sources and Independence

Strength

strong

Relationship

The assumption of independence among measured quantities is a prerequisite for applying the standard propagation formula; correlated errors require the matrix form.

To

Practical Applications in Philippine Surveying

From

Variance-Covariance Matrix

Strength

strong

Relationship

The covariance matrix output from least-squares adjustment directly informs compliance with Philippine accuracy standards (PD 1529, CA 141) and fitness assessment for land titling (RA 4374, RA 8560).

To

Practical Applications in Philippine Surveying

From

Error Ellipse

Strength

moderate

Relationship

The error ellipse provides a visual, intuitive assessment of survey quality and directional uncertainty, essential for communicating fitness-for-purpose to land offices and stakeholders.

To

Law of Propagation of Variances

From

Board-Exam Problem-Solving Strategies

Strength

moderate

Relationship

Effective exam strategies include identifying independent vs. correlated terms, selecting the correct propagation formula (sum-difference, Jacobian, scaling), and avoiding the common pitfall of linear error addition.

To

Variance-Covariance Matrix

From

Board-Exam Problem-Solving Strategies

Strength

moderate

Relationship

Exam questions often require constructing or interpreting covariance matrices, checking positive-definiteness, and extracting submatrices for specific parameters.

To

Error Ellipse

From

Board-Exam Problem-Solving Strategies

Strength

moderate

Relationship

Licensure exam problems commonly ask students to calculate ellipse parameters (a, b, θ) from covariance matrices and interpret what the shape reveals about survey geometry.

To

Error Sources and Independence

From

Practical Applications in Philippine Surveying

Strength

moderate

Relationship

Philippine survey design standards (PD 1529, CA 141, RA 4374, RA 8560) specify which error sources must be controlled and how to validate independence assumptions in field protocols.

To

Survey Applications — Traverse Networks

From

Error Ellipse

Strength

strong

Relationship

Error ellipses at each traverse station reveal the propagation of leg errors along the network; elongation indicates weak network geometry in that direction.

To

Survey Applications — Trilateration

From

Error Ellipse

Strength

moderate

Relationship

In distance-only positioning (trilateration), the error ellipse from distance variances shows how measurement accuracy translates to position uncertainty in horizontal E-N coordinates.

To

Survey Applications — WGS84/PRS92 Transformations

From

Variance-Covariance Matrix

Strength

moderate

Relationship

Transformation between global (WGS84) and local (PRS92, PPCS/UTM) frames propagates coordinate uncertainties; the covariance matrix structure is preserved through affine transformations.