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

Formulas

Exam Tips

• Always compute the misclosure first and clearly write its sign before distributing corrections.
• Verify your answer: the sum of adjusted observations must equal the required total exactly.
• A quick check: Σ(all corrections) = −(misclosure). Use this to catch arithmetic errors.
• Board exam problems often give you adjusted values and ask for one corrected angle — set up the check equation first.

Key Points

• A condition equation is a mathematical expression that the adjusted observations must satisfy exactly. It represents a known geometric truth.
• The misclosure (also called the closure error or discrepancy) is: misclosure = Σ(observed) − (required total). It quantifies how much the raw observations violate the condition.
• A positive misclosure means the observed sum exceeds the required total; a negative misclosure means it falls short.
• The number of condition equations equals the number of redundant observations (degrees of freedom) in the figure.
• The condition-equation method is particularly efficient when the number of conditions is much smaller than the number of unknowns.
• In Philippine geodetic practice under RA 8560 (Geodetic Engineering Act), all control surveys must meet specified accuracy standards, which requires proper adjustment of observations.

Definitions

Section Title

Fundamentals of Condition Equations

Common Mistakes

• SIGN ERROR: Applying the correction with the same sign as the misclosure instead of the opposite sign. If misclosure = +12'', each correction must be −4'', not +4''.
• Forgetting that the correction is opposite in sign to the misclosure — memorize: 'misclosure is positive → corrections are negative'.
• Adding up the corrections and forgetting to verify that Σc_i = −misclosure (a quick arithmetic check).
• Confusing 'weight' with 'correction' — higher weight means smaller correction, not larger.

Formulas

Exam Tips

• If the problem says 'geodetic triangle' or gives a large area, suspect spherical excess — check if ε is provided.
• If ε is not given in a board problem and the triangle is described as 'plane', use 180°00'00'' as the required total.
• The three corrections must sum to −misclosure. Always verify this before writing your final answer.
• Memorize: spherical excess correction formula ε'' = Area(km²)/R²(km²) × 206265 — it appears in geodesy problems.

Key Points

• In a plane triangle, the three interior angles must sum to exactly 180°00'00''.
• In a spherical triangle (used for large geodetic triangles on the Earth's surface), the required sum is 180° + ε, where ε is the spherical excess.
• The spherical excess ε = Area / R², where Area is the triangle's area and R is the mean radius of the Earth (~6,371 km). In practice, ε is computed from approximate coordinates.
• For typical geodetic triangles in the Philippines (e.g., BPI triangulation nets), ε ranges from fractions of an arcsecond to several arcseconds for very large triangles.
• Under equal weights: each of the 3 angles receives c = −misclosure/3.
• The triangle condition is the most fundamental condition in triangulation networks such as the PRS92 horizontal control framework.

Definitions

Section Title

Triangle (Angle) Condition

Common Mistakes

• Using 180° as the required total for a large geodetic (spherical) triangle — always add the spherical excess ε.
• Computing the misclosure as (required − observed) instead of (observed − required). Stick to one convention consistently.
• Rounding intermediate corrections and then finding the adjusted angles don't quite sum to 180° — carry full precision until the final step.
• In spherical triangles, computing ε in radians and forgetting to convert to arcseconds before use.

Formulas

Exam Tips

• The phrase 'angles around a point' or 'complete horizon' is the trigger for the 360° condition.
• In multiple-choice board questions, quickly identify the condition (triangle = 180°, horizon = 360°, loop = 0) before computing.
• Equal-weight problems: divide the misclosure by n and flip the sign. That's the correction. Simple and fast.

Key Points

• When all the angles around a survey station are measured (completing the horizon), their sum must equal exactly 360°00'00''.
• This is called the 'horizon condition' or 'station equation'.
• It applies in triangulation when multiple directions are observed from a single station and the full circle of angles is measured.
• For n angles around a point: each correction = −misclosure/n (equal weights).
• This condition is frequently applied in adjustment of triangulation stations in the PPCS/UTM-based PRS92 horizontal control network.

Definitions

Section Title

Horizon (Station) Condition

Common Mistakes

• Confusing the horizon condition (360°) with the triangle condition (180°) — the target depends on what geometry you are closing.
• When only some angles around a point are measured (not a full horizon), the target is not 360° — do not apply the horizon condition unless all angles are measured.
• Forgetting that the corrections must sum to −misclosure (e.g., four angles, misclosure = +8'', four corrections each = −2'', sum = −8''. Check: −8'' = −(+8''). ✓)

Formulas

Exam Tips

• Leveling problems: weight is proportional to 1/length. Correction is proportional to length (longer section = bigger correction).
• A quick way to remember: 'Long section, long walk, more chance of error, more correction.'
• For PRC board problems, if section lengths are given, immediately set up the ratio c_i = −f_h × (L_i/ΣL).
• Always state the tolerance check (f_h ≤ k√L) before adjustment — board problems may ask if the fieldwork is acceptable.

Key Points

• In a closed differential leveling loop, the algebraic sum of the measured elevation differences (ΔH) must equal zero: Σ(ΔH) = 0.
• The misclosure of a level loop is: f_h = Σ(ΔH_observed) − 0 = Σ(ΔH_observed).
• For leveling, corrections are distributed in proportion to the length of each section (longer sections get larger corrections) — this is an inverse-weight distribution since variance is proportional to length.
• For a traverse loop, the linear misclosure is distributed proportionally to the length of each course.
• Weighted distribution general rule: longer/weaker observation → larger correction.
• This approach is standard practice in Philippine vertical control densification under NAMRIA guidelines and RA 8560.

Definitions

Section Title

Loop (Level/Traverse) Condition and Weighted Distribution

Common Mistakes

• In leveling, distributing the misclosure equally (as if equal weights) instead of proportionally to section length — this is the most common error in this topic.
• Forgetting that level loop corrections use length as the proportionality factor (not number of setups, unless specifically stated).
• Getting the direction of ΔH corrections wrong — if f_h = +8 mm, all ΔH values must be reduced, so corrections are negative.
• Not checking that the sum of all corrections equals exactly −f_h after computation.

Formulas

Exam Tips

• Board exam problems almost always involve equal-weight distribution — memorize c = −misclosure/n.
• For the triangle condition: write out the sum, subtract 180° (or 180° + ε), divide by 3, flip sign. That's it.
• For leveling loops: list the sections, their lengths, compute ΣL, then apply c_i = −f_h(L_i/ΣL) to each.
• Practice computing in degrees-minutes-seconds arithmetic — careless DMS arithmetic is a major source of lost marks.
• After adjustment, always add up the corrected values to confirm they sum to the required total.

Key Points

• Figure adjustment applies condition equations simultaneously to an entire geometric figure (triangle, quadrilateral, etc.) containing multiple conditions.
• The number of conditions = redundant observations = total observations − minimum required.
• For a simple triangle: 3 angles measured, 2 needed to compute the 3rd → 1 condition (the angle sum condition).
• For a quadrilateral with diagonals: typically 4 triangle conditions + 1 side condition = 5 conditions for 8 measured angles.
• In the PRC board examination, figure adjustment problems most commonly involve: (1) triangle angle condition, (2) horizon condition, (3) leveling loop distribution.
• Always follow the systematic procedure: (a) identify the condition, (b) compute misclosure, (c) compute corrections, (d) apply corrections, (e) verify closure.

Definitions

Section Title

Figure Adjustment — Worked Board-Style Problems

Common Mistakes

• Applying only one condition when multiple conditions exist in a figure (e.g., adjusting only the triangle condition but forgetting the side condition in a quadrilateral).
• Not following the systematic 5-step procedure — skipping the verification step leads to undetected arithmetic errors.
• Mixing up the sign convention mid-problem — establish 'misclosure = observed − required' at the start and stick to it throughout.