GELE Adjustment Computations (Least Squares) — Condition Equations and Figure AdjustmentSummary

A mathematical statement expressing a required geometric relationship that adjusted observations must satisfy exactly. Examples: (1) **Plane triangle condition**: sum of three angles = 180°00'00''. (2) **Horizon (station) condition**: angles around a point = 360°00'00''. (3) **Loop (elevation) condition**: sum of level differences = 0 (closed loop). (4) **Spherical triangle condition**: sum of angles = 180° + ε, where ε is the spherical excess in seconds of arc. Condition equations enforce geometry; they are not optional—the adjusted data must satisfy them exactly, with no residual.

The amount by which observed data violates a condition equation. Calculated as: **misclosure = sum(observed values) − (required geometric total)**. For example, if three triangle angles measure 60°00'10'', 60°00'02'', and 59°59'48'', their sum is 180°00'00'', yielding zero misclosure. If instead they sum to 180°00'12'', the misclosure is +12 arcseconds. A positive misclosure indicates the sum exceeds the requirement; negative means it falls short. The sign of the misclosure determines the direction of corrections: corrections are applied with **opposite sign** to eliminate the misclosure entirely.

When all observations have equal precision/reliability, the misclosure is distributed equally among all measurements. **Formula: correction per observation = −(misclosure) ÷ (number of observations)**. Each observation receives the same magnitude of correction, with sign opposite to the misclosure. Example: a 12-arcsecond triangle misclosure distributed over 3 angles yields −12″ ÷ 3 = −4″ correction per angle. If the misclosure was negative (−12″), each correction would be +4″. After application, adjusted observations sum exactly to the geometric requirement.

When observations have different precisions, distribute misclosure **in proportion to measurement variance or relative weight**. An observation with high uncertainty (large variance, low weight) receives a **larger correction** than a precise observation. The general principle: **correction ∝ weight × misclosure**. More precisely, correction_i = −w_i × misclosure ÷ Σw, where w_i is the weight of observation i. Weights are often inversely proportional to variance: w = 1/σ² or w = 1/length (for traverse legs). Example: if one angle's weight is 1 and another's is 2, the first gets twice the correction of the second.

For large triangles on Earth's surface (area exceeds ~1 km²), the sum of angles exceeds 180° by an amount called the **spherical excess**, denoted ε, measured in arcseconds. Calculated approximately as: **ε ≈ (area in m²) ÷ (Earth's radius in m)² × 206265 arcseconds** or **ε = A/(2R²)** in radians. For example, a 1000 m × 1000 m triangle on WGS84 (R ≈ 6,371 km) has ε ≈ 0.05 arcseconds. Larger triangles yield larger ε; a 10 km × 10 km triangle yields ε ≈ 5 arcseconds. The correct condition for a spherical triangle is: **sum of angles = 180° + ε**, not 180°.

Raw measured values, after corrections are applied, so that all geometric conditions are satisfied exactly. **Adjusted value = Observed value + Correction**. For example: observed angle = 60°00'10'', correction = −4'', adjusted angle = 60°00'06''. The sum of adjusted observations equals the geometric requirement by construction. Adjusted values are what are reported in final survey products and used in subsequent calculations (coordinate computation, area calculation, network analysis).

Misclosure is identified as the violation of a condition equation. The magnitude and sign of misclosure determine the correction strategy. Corrections, applied to raw observations, yield adjusted observations that satisfy the condition exactly. This is a sequential cause-and-effect chain: no misclosure = no correction = observations already satisfy condition; large misclosure = large correction = significant adjustment needed.

If all observations have equal reliability, use equal-weight distribution (constant correction). If observations have unequal reliability (specified by weights, variances, or lengths), use weighted distribution (larger correction to weaker observations). The method chosen depends on the input data characteristics and problem statement.

For small surveys (area < 1 km²), plane geometry applies; spherical excess ε ≈ 0 and can be ignored. For large surveys (area > 10 km²), spherical excess becomes measurable; the condition becomes 180° + ε. This relationship links survey size to the appropriate geometric model. Philippines' national GPS networks (under PRS92) operate on ellipsoidal/spherical geometry; local site surveys may use plane geometry. Problem context determines which model applies.

Each geometric condition has a specific required total: plane triangle → 180°; spherical triangle → 180° + ε; horizon closure → 360°; level loop → 0. The misclosure formula is always the same (observed sum minus required total), but the required total varies by condition type. Identifying the correct condition type is the first step in any problem.

Condition-adjusted observations feed into subsequent geodetic computations: adjusted triangle angles → coordinate differences via sine rule; adjusted traverse angles and distances → final coordinates; adjusted loop elevations → benchmark heights. All downstream work depends on condition consistency. Publishable survey products (maps, digital datasets, title documents per CA 141) must use condition-adjusted data.

In Philippine triangulation networks (e.g., GPD primary network under NAMRIA), triangles are the basic geometric units. Raw field angles from theodolite observations rarely sum exactly to 180° (or 180° + ε for large triangles). Before using triangle angles to compute coordinates, the three angles are adjusted so their sum equals exactly 180° (or 180° + ε). This ensures all subsequent coordinate calculations are geometrically consistent and satisfy the triangle constraint. Example: a 50 km × 50 km triangle observed in Luzon has angles summing to 180°00'05.2''; after distributing the 5.2″ misclosure equally among three angles (−1.73″ per angle), the adjusted angles sum to exactly 180° + ε, where ε ≈ 0.78″ for this area.

When multiple lines radiate from a survey station (e.g., a turntable observation point), angles between adjacent lines must sum to exactly 360°. Raw measurements typically yield 360°00'08'' or 359°59'52''. The misclosure is distributed equally (or by weight) among the angles so the sum becomes exactly 360°. This is routine in transit-traverse networks and theodolite observations of building facades (e.g., RA 4374 boundary surveys). Neglecting horizon closure introduces inconsistencies that accumulate through the network.

In differential leveling, a loop departing from a known benchmark, passing through intermediate points, and returning to the starting point must have zero net height change (misclosure = 0). In practice, misclosure appears due to instrumental errors, settlement, and reading errors. The misclosure is distributed proportional to leveling distance (weighted by leg length); longer legs receive proportionally larger corrections. This is standard practice for Philippine height networks and is required by PD 1529 (land survey standards). Example: a 10 km leveling loop closes with +0.08 m misclosure; distributed over 10 km, the rate is −0.008 m/km, correcting each 1 km segment by −0.008 m.

A traverse loop (sequence of measured distances and angles forming a closed polygon) rarely closes perfectly. The misclosure in latitude (ΔN) and departure (ΔE) is calculated and distributed. For simple cases (equal weights), distribute equally among all legs. For more rigorous work (unequal leg reliabilities), distribute proportional to leg length or measurement variance. This is mandated by CA 141 (Cadastral Act) and PD 1529 for any boundary survey. Adjusted coordinates of boundary points must yield a closed polygon; misclosure must be eliminated before titling.

Modern Philippine surveys using GNSS (GPS under WGS84 datum, transitioning to PRS92) record multiple baseline vectors forming a network. Baselines must satisfy geometric closure conditions (if three points form a triangle, the three baseline vectors must close). The condition-equation approach (simplified version presented here) forms the conceptual foundation for full network least-squares adjustment. Understanding condition equations and misclosure distribution prepares engineers to work with commercial software (e.g., Trimble, Leica) that enforces closure constraints automatically.

Condition-equation misclosure serves as a quality-control diagnostic. A large misclosure flags measurement errors or equipment malfunction. Example: if a triangle's angle misclosure is 2 arcminutes (instead of expected 10 arcseconds), the surveyor suspects a gross error in one measurement (perhaps a wrong angle reading or instrument pointing error). Small misclosures (few arcseconds to arcminutes) are normal and handled by adjustment; large misclosures trigger remeasurement. This quality-assurance step is essential for professional practice and PRC licensure standards.

PRC Geodetic Engineer Licensure Examination board-style problems frequently present raw field data with misclosure and ask for adjusted values. The standard approach: (1) identify the condition type (triangle, horizon, loop), (2) calculate misclosure, (3) determine weights or assume equal, (4) compute correction per observation, (5) apply corrections, (6) verify sum equals geometric requirement. Mastering this workflow is essential for scoring high marks on the adjustment/least-squares section of the exam.