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

Formula

Misclosure = Σ(observed values) − (required total)

Meaning

Misclosure = sum of all measured observations minus the theoretical/required sum that geometry demands

Watch Out

Sign matters! Positive misclosure = observations exceed requirement; negative = observations fall short. The correction is opposite in sign to the misclosure.

When To Use

Every time you measure angles in a triangle, horizon, or loop — calculate misclosure FIRST

Formula

Correction per observation (equal weights) = −(Misclosure) / n

Meaning

n = number of observations being corrected; negative sign reverses the misclosure direction

Watch Out

The correction is NEGATIVE of the misclosure divided by count. If misclosure is +12″, correction per angle is −12″/3 = −4″ per angle. Apply this correction to each observation.

When To Use

When all observations have equal reliability (equal weights or equal precision)

Formula

Corrected observation = Original observation + Correction

Meaning

Add the calculated correction (which carries the negative sign) to each measured value

Watch Out

Use signed arithmetic. If correction is −4″ and original angle is 60°00'10″, result is 60°00'06″.

When To Use

After calculating per-observation correction, apply to all raw measurements

Value

180°00'00″ (or 180° in decimal, 3.14159 rad)

Symbol

None; use as reference total

Quantity

Plane triangle angle sum (required)

Value

360°00'00″ (or 360° in decimal, 2π rad)

Symbol

None; use as reference total

Quantity

Horizon (station) angle sum (required)

Value

1″ to 10″ (varies with area and latitude); ε = A/(2R²) in radians, then convert to seconds

Symbol

ε

Quantity

Spherical excess (typical large triangles)

Term

Condition Equation

Example

∠A + ∠B + ∠C = 180° (plane triangle); ∠A + ∠B + ∠C = 180° + ε (spherical triangle with excess ε).

Definition

A mathematical statement enforcing a known geometric relationship that adjusted observations must satisfy exactly (e.g., triangle angles = 180°).

Term

Misclosure

Example

Triangle angles sum to 180°00'12″ instead of 180°00'00″ → misclosure = +12″.

Definition

The amount by which raw observations violate the condition equation; the error to be distributed.

Term

Figure Adjustment

Example

Adjusting three triangle angles so their sum = exactly 180°00'00″.

Definition

Process of correcting observations to satisfy geometric conditions by distributing the misclosure equally or weighted.

Term

Spherical Excess (ε)

Example

Large survey triangle with ε = 5″ → required angle sum = 180°00'05″, not 180°00'00″.

Definition

Additional angle sum in a spherical triangle due to curvature; target condition becomes 180° + ε, not 180°.

Note

This is the verification step — if adjusted angles do NOT sum to exactly 180°, there is an arithmetic error.

Equation

∠A_adj + ∠B_adj + ∠C_adj = 180°00'00″

Conditions

Plane triangle; all adjustments applied; no residual misclosure

Note

Failing to include ε is the #2 common exam error after misclosure sign.

Equation

∠A_adj + ∠B_adj + ∠C_adj = 180° + ε (spherical)

Conditions

Large triangle on Earth's surface; ε calculated from area or using spherical excess formula

Note

Heavier (more reliable) observations get smaller corrections; lighter observations get larger corrections.

Equation

w_i × c_i = constant (weighted distribution)

Conditions

Unequal observation weights w_i; corrections c_i are inversely proportional to weight

Formula

c = −(Misclosure) / n

Meaning

c = correction per observation; n = total number of observations; negative sign reverses misclosure

Watch Out

Do NOT forget the negative sign. If misclosure is +8″, each of 4 angles gets correction −8″/4 = −2″. If you write +2″, the answer is wrong.

When To Use

All observations have equal precision/reliability; is the DEFAULT in most exam problems unless stated otherwise

Formula

Adjusted value = Original value + c

Meaning

Apply the same correction to every observation

Watch Out

Arithmetic sign errors are fatal. Use a simple table: [Original] + [c] = [Adjusted].

When To Use

After computing c from misclosure/n, apply to each original measurement

Formula

Verification: Σ(Adjusted values) = Required total

Meaning

Sum of all corrected observations must equal the geometric requirement

Watch Out

If verification fails, do NOT submit the answer; find the arithmetic error first.

When To Use

ALWAYS verify at the end; if this fails, recheck c calculation or sign

Term

Equal-Weight Adjustment

Example

Three angles each get −4″ correction to eliminate a +12″ triangle misclosure.

Definition

Distribution of misclosure equally (divided by n) to all observations; assumes equal measurement precision.

Term

Equal Reliability

Example

Three theodolite angle measurements from the same instrument and operator → assume equal weights.

Definition

All observations have the same measurement uncertainty; each should carry equal weight in adjustment.

Note

All corrections are identical. This is what makes equal-weight so straightforward.

Equation

c_1 = c_2 = c_3 = ... = c_n = −M/n (where M = misclosure)

Conditions

Equal weights; all observations measured with same precision

Formula

c_i = −(Misclosure) × (σ_i² / Σσ_j²)

Meaning

c_i = correction for observation i; σ_i² = variance (inverse of weight) for observation i; sum in denominator is total variance

Watch Out

This is an INVERSE relationship: LARGER variance → LARGER correction. Weaker measurements corrected more. Do NOT mix up weight and variance signs.

When To Use

When observations have unequal precision or explicit weights are given; more uncertain observations get larger corrections

Formula

Weight w_i = 1 / σ_i² or w_i = constant / σ_i²

Meaning

Weight is inversely proportional to variance; heavier (lower variance) observations have higher weight

Watch Out

If σ values are given, calculate w = 1/σ². If weights are given directly, use them. Do NOT confuse weight with variance.

When To Use

Converting standard deviation or uncertainty to weight for weighted adjustment

Formula

c_i = −(Misclosure) × (w_i / Σw_j)

Meaning

Alternative form using weights directly: correction proportional to weight fraction

Watch Out

Verify that Σw_i = n (approximately) or is normalized; adjust the formula if weights are not normalized.

When To Use

When weights are given instead of standard deviations

Term

Weighted Adjustment

Example

One angle measured with ±1″ precision, another with ±2″; the ±2″ observation gets 4× larger correction (variance ratio 4:1).

Definition

Distribution of misclosure proportional to observation variances (or inverse of weights); accounts for unequal measurement reliability.

Term

Variance (σ²)

Example

Observation A: σ² = 1″², Observation B: σ² = 4″² → B is 4× less reliable; gets 4× larger correction.

Definition

Measure of measurement uncertainty; larger variance = less reliable observation.

Term

Weight (w)

Example

If w_A = 4, w_B = 1, then A is 4× more reliable and gets 1/4 the correction of B.

Definition

Inverse of variance; w = 1/σ². Higher weight = more reliable observation.

Note

Quick check: if one observation is twice as uncertain, its correction should be 4× larger (variance is squared).

Equation

c_i / c_j = σ_i² / σ_j² (ratio of corrections equals ratio of variances)

Conditions

Same misclosure, different observation precisions

Note

This verifies the correction distribution; must sum to exactly the negative misclosure.

Equation

Σ(c_i × w_i) = −Misclosure (verification for weighted adjustment)

Conditions

All weights normalized or proportional

Formula

Required sum = 180°00'00″ (plane) or 180° + ε (spherical)

Meaning

For plane triangles, exactly 180°; for spherical, add the spherical excess ε

Watch Out

Forgetting spherical excess (ε) for large triangles is a common exam error. Always check the problem statement for 'spherical' or large area.

When To Use

SET this as the reference before calculating misclosure

Formula

Misclosure = (∠A_obs + ∠B_obs + ∠C_obs) − (180° + ε)

Meaning

Sum of three measured angles minus the required total (including excess if spherical)

Watch Out

Pay attention to sign. If observed sum is 180°00'15″ and required is 180°00'00″, misclosure = +15″ (positive). Correction will be −15″/3 = −5″ per angle.

When To Use

Calculate misclosure FIRST; this determines the magnitude and direction of correction

Formula

c = −(Misclosure) / 3

Meaning

For three angles with equal weight, divide misclosure by 3 and reverse sign

Watch Out

This is the most direct formula in triangle problems. Three angles always → divide by 3.

When To Use

Equal-weight adjustment of a plane or spherical triangle

Formula

Spherical excess ε (seconds) = A / (2R²) × (180 × 3600 / π)

Meaning

A = triangle area in m²; R = Earth radius (≈ 6,371 km); ε in arc-seconds

Watch Out

This formula requires consistent units (area in m², radius in m). The result is very small for typical survey triangles (few arc-seconds).

When To Use

When area is given and excess must be calculated; rarely required for adjustment itself, but good to know

Value

180°00'00″ or 180°

Symbol

Target for condition equation

Quantity

Plane triangle angle sum

Value

6,371.0 km (mean); 6,378.137 km (equatorial)

Symbol

R

Quantity

Earth radius (WGS84)

Value

0.5″ to 2″

Symbol

ε

Quantity

Spherical excess (typical survey triangle, ~1 km sides, area ~0.5 km²)

Term

Plane Triangle

Example

Local building survey triangle with sides ~500 m.

Definition

Triangle on a flat surface; angle sum = 180°00'00″; used for small surveys (< 100 km typical sides).

Term

Spherical Triangle

Example

Regional geodetic network triangle with sides 10 km or larger; ε ≈ 1″ to 10″.

Definition

Triangle on Earth's curved surface; angle sum = 180° + ε (spherical excess); used for large surveys (sides > 1 km).

Term

Spherical Excess (ε)

Example

Triangle with area 100 km² → ε ≈ 3″ (rough approximation).

Definition

Additional angle sum in a spherical triangle; proportional to area and inversely to Earth radius squared.

Note

This is the MANDATORY verification step. Must hold exactly (within rounding to nearest 0.1″).

Equation

∠A_adj + ∠B_adj + ∠C_adj = 180°00'00″

Conditions

Plane triangle; all equal-weight corrections applied

Note

Same verification, but target is shifted by ε. Both sides must balance.

Equation

∠A_adj + ∠B_adj + ∠C_adj = 180° + ε

Conditions

Spherical triangle; excess ε pre-calculated

Formula

Required sum = 360°00'00″

Meaning

Angles around a single point always sum to 360° by definition

Watch Out

This is analogous to a closed polygon. Do NOT confuse with triangle (180°).

When To Use

Whenever correcting multiple angles (3 or more) observed from one station

Formula

Misclosure = Σ(observed angles) − 360°

Meaning

How much the sum of all station angles exceeds or falls short of 360°

Watch Out

Common exam trick: 4 angles might sum to 360°00'08″ → misclosure = +8″. Students sometimes forget to subtract from 360°.

When To Use

Calculate misclosure after summing all angles from the station

Formula

c = −(Misclosure) / n, where n = number of angles at station

Meaning

Equal correction to each angle; n is the count of angles (could be 3, 4, 5, etc.)

Watch Out

Use the COUNT of angles, not the number of sides (they differ by 1 in closed polygons). If you measure 4 angles, n = 4.

When To Use

Equal-weight distribution among station angles

Value

360°00'00″ or 360°

Symbol

Reference total

Quantity

Horizon angle sum (required)

Term

Horizon Condition

Example

Observer at station P measures 4 angles to surrounding points A, B, C, D; ∠APB + ∠BPC + ∠CPD + ∠DPA = 360°.

Definition

All angles measured around a single point sum to exactly 360°.

Term

Station Misclosure

Example

Four angles sum to 360°00'06″ → station misclosure = +6″.

Definition

The angular closure error at a single observation point; usually a few arc-seconds.

Note

Verification must hold exactly. If it does not, recheck arithmetic.

Equation

∠_1_adj + ∠_2_adj + ... + ∠_n_adj = 360°00'00″

Conditions

n angles at a station; all equal-weight corrections applied

Formula

Linear misclosure (traverse) = √[(ΣΔx_obs − 0)² + (ΣΔy_obs − 0)²]

Meaning

Magnitude of closure error in a closed traverse; ΔX and ΔY are easting and northing increments

Watch Out

This is a VECTOR misclosure (2D). Do NOT just add ΣΔx + ΣΔy; calculate the magnitude (hypotenuse).

When To Use

After summing all course increments in a closed traverse; should equal zero but rarely does

Formula

Correction c_i = −(Misclosure) × (Length_i / Σ(Lengths))

Meaning

Distribute linear misclosure in proportion to course lengths (longer courses get more correction)

Watch Out

Misclosure is negative in sign (reverse direction of closure error). Multiply by the length fraction to get per-course correction.

When To Use

Equal-length-weight method: each course corrected proportional to its measured length

Formula

Elevation misclosure (level loop) = ΣΔh_obs − 0

Meaning

In a closed level loop, sum of elevation differences should return to starting elevation; misclosure is the failure

Watch Out

Sign: if final elevation is above start, misclosure is positive (upward). Correction reverses this.

When To Use

Leveling loop closure check

Formula

c_i = −(Elevation misclosure) × (L_i / Σ(L))

Meaning

Distribute elevation misclosure proportional to sight length (longer sights = more correction)

Watch Out

In leveling, longer sight distances have larger potential errors; proportional distribution is standard.

When To Use

Leveling loop adjustment with equal-weight assumption

Value

1:1000 to 1:10000 (better surveys are 1:5000 or tighter)

Symbol

R_closure

Quantity

Relative closure spec (traverse)

Value

0.05–0.20 m (if closure spec 1:1000 → expect ~0.10 m or less)

Symbol

M_closure

Quantity

Typical linear misclosure (100 m traverse)

Term

Closed Traverse

Example

Polygon survey with 5 courses: A→B→C→D→E→A; if all courses are measured, the loop must close.

Definition

A sequence of measured courses that return to the starting point; ΣΔx = 0, ΣΔy = 0, ΣΔz = 0 exactly.

Term

Closure Error (Misclosure)

Example

Traverse closes 0.15 m short in easting, 0.08 m over in northing → linear closure error = √(0.15² + 0.08²) ≈ 0.17 m.

Definition

The linear or angular deviation from exact closure in a traverse or level loop.

Term

Loop Misclosure Distribution

Example

A 100 m course in a 500 m loop gets 20% of the total misclosure correction.

Definition

Method of correcting course increments to satisfy closure; typically proportional to course length.

Note

Must hold exactly within rounding. If not, recheck correction calculations.

Equation

ΣΔx_adj = 0, ΣΔy_adj = 0 (verification for closed traverse)

Conditions

All course increments adjusted; loop closed

Note

Same verification principle applied to leveling.

Equation

ΣΔh_adj = 0 (verification for leveling loop)

Conditions

All elevation differences adjusted; loop closed vertically

Values

• 3 angles
• 180°00'00″
• ±10″ (typical)
• Equal-weight (most common) or weighted by angle precision

Property

Plane Triangle

Values

• 3 angles
• 180° + ε
• ±10″ (after subtracting ε)
• Equal-weight or weighted; ε must be pre-calculated

Property

Spherical Triangle

Values

• n ≥ 3 angles
• 360°00'00″
• ±15″ (typical)
• Equal-weight: divide by n angles

Property

Horizon (Station)

Values

• Multiple courses
• ΣΔx = 0, ΣΔy = 0
• 0.05–0.20 m (depends on perimeter)
• Proportional to course length (linear distribution)

Property

Closed Traverse

Values

• Multiple elevation diffs
• ΣΔh = 0
• 0.01–0.05 m (depends on distance)
• Proportional to sight distance or number of setups

Property

Level Loop

Values

• All observations have same precision; no weights given
• Observations have different standard deviations or explicit weights; precision varies

Property

When to use

Values

• c = −(Misclosure) / n
• c_i = −(Misclosure) × (σ_i² / Σσ_j²) or c_i = −(Misclosure) × (w_i / Σw_j)

Property

Formula for correction

Values

• Identical for all observations
• Varies: weaker observations (higher variance) get larger corrections

Property

Correction magnitude

Values

• Straightforward; most common problem type
• Requires variance or weight values; more calculation

Property

Exam difficulty

Values

• All measurements equally reliable; distribute error equally
• Some measurements more trustworthy; less correction to reliable observations

Property

Physical interpretation

Values

• Forgetting to negate the misclosure
• Always: c = −(Misclosure) / n. If misclosure +12″, correction is −12″/3 = −4″ per angle.
• If original angle is 60°00'10″ + (−4″) = 60°00'06″. With wrong sign: 60°00'14″ (makes angle larger, wrong direction).

Property

Wrong sign for correction

Values

• Confusing plane and spherical triangles; ε is small and easy to overlook
• For large triangles (sides > 1 km), ALWAYS check if excess must be included. Required sum is 180° + ε, not 180°.
• ε = 5″; observed sum = 180°00'18″. Misclosure = 180°00'18″ − (180°00'05″) = 13″, not 18″.

Property

Forgetting spherical excess ε

Values

• Confusing number of observations with something else (perimeter, angle count)
• Count the observations being corrected. For 3 triangle angles: divide by 3. For 4 station angles: divide by 4.
• Three angles, misclosure +9″ → correction per angle is −9″/3 = −3″, NOT −9″/4 or other value.

Property

Dividing by wrong number in equal-weight

Values

• Rushing; assuming calculation is correct without checking
• ALWAYS verify: Sum of adjusted values must equal the required total (180°, 360°, ε, 0, etc.).
• If adjusted angles sum to 180°00'02″ instead of 180°00'00″, there is an error. Go back and recheck.

Property

Not verifying the result

Values

• Both are common conditions; easy to mix up under exam stress
• Horizon = angles around ONE POINT = 360°. Triangle = three angles of a polygon = 180°. Read problem carefully.
• Four angles at a station sum to 360°00'08″ → misclosure +8″. (NOT 180° condition.)

Property

Confusing horizon (360°) with triangle (180°)

Values

• Forgetting that w = 1/σ², so larger σ → smaller w, but correction c ∝ σ²
• Higher variance = less reliable = LARGER correction. If σ_B = 2σ_A, then σ_B² = 4σ_A² → c_B = 4 c_A.
• Observation A (σ = 1″) and B (σ = 2″) with misclosure +12″ over 2 obs. c_A = −12″ × (1/5) = −2.4″, c_B = −12″ × (4/5) = −9.6″.

Property

Mixing variance and weight

Values

• Adding ΣΔx + ΣΔy instead of using Pythagorean theorem; treating as 1D instead of 2D
• Traverse misclosure is a VECTOR. M = √((ΣΔx)² + (ΣΔy)²). Always use distance formula, not sum.
• ΣΔx = +0.12 m, ΣΔy = +0.09 m → M = √(0.12² + 0.09²) = √(0.0225) = 0.15 m. (NOT 0.21 m from adding.)

Property

Linear misclosure confusion in traverse