GELE Adjustment Computations (Least Squares) — Least Squares — Observation EquationsMemory Anchors

Tags

• definition
• principle
• formula

Topic

The Least-Squares Principle

Concept

Least squares minimizes the weighted sum of squared residuals (∑wv²)

Anchor Id

A1

Difficulty

easy

Memory Aid

Imagine you are a basketball coach and each player's missed free throw is a 'residual.' A clumsy big man (low weight) missing by 2 meters matters less than your star shooter (high weight) missing by 2 centimeters. You square the misses so big blunders get punished extra hard, then you minimize the total pain. That is least squares — punish blunders, respect precision, and minimize the total weighted suffering.

Anchor Type

analogy

Why It Works

The sports analogy connects abstract minimization to a concrete, emotionally resonant scenario. Squaring = extra punishment for big misses. Weights = importance of each player/observation. Filipino students who follow the PBA or UAAP will immediately relate.

Example Usage

Board exam asks: 'What does least squares minimize?' Think of the basketball coach → minimize weighted squared misses → ∑wv².

Recall Trigger

Think: 'Star shooter vs. clumsy big man = weighted residuals'

Tags

• definition
• formula
• sign convention

Topic

Observation Equations — Residual Definition

Concept

Residual v = computed minus observed (v = Ax̂ − l)

Anchor Id

A2

Difficulty

easy

Memory Aid

Remember 'CO': Computed minus Observed = residual. Think of CO as 'Company Owes' — the company (computed value) owes something relative to what was actually observed. The 'debt' is the residual v.

Anchor Type

mnemonic

Why It Works

The acronym CO is short, the phrase 'Company Owes' gives it a story, and the sign convention (computed − observed) is baked into the imagery.

Example Usage

When setting up residual equations, always write v = Ax̂ − l (computed − observed), never flip the sign. Trigger: 'CO, computed first.'

Recall Trigger

CO → Computed minus Observed → v

Tags

• definition
• matrix
• linearization

Topic

Design Matrix A

Concept

The design matrix A contains partial derivatives of each observation function with respect to each unknown

Anchor Id

A3

Difficulty

medium

Memory Aid

The design matrix A is like a Philippine government org chart (organigram). Each row is one barangay (observation). Each column is one department (unknown parameter). The cell entry tells you how sensitive that barangay is to that department's actions — that is a partial derivative. If a barangay has zero in a column, that department does not affect it at all.

Anchor Type

analogy

Why It Works

Org charts are familiar to Filipino students from civics and public administration contexts. The row = observation, column = unknown mapping is made concrete.

Example Usage

Setting up a leveling network: each level route is one row; each benchmark correction is one column; the entry is +1 or −1 depending on direction.

Recall Trigger

Org chart → rows are observations, columns are unknowns, entries are sensitivities (partials)

Tags

• formula
• matrix
• normal equations

Topic

Normal Equations

Concept

Normal equations: (AᵀPA)x̂ = AᵀPl

Anchor Id

A4

Difficulty

medium

Memory Aid

Chant the formula as: 'ATPA times x̂ equals ATPl' — say it like a rap beat: 'A-T-P-A, solve for x, A-T-P-L, that's the text!' The pattern AᵀP_A on the left and AᵀP_l on the right is symmetric: both start with AᵀP, but one ends in A and the other in l.

Anchor Type

acronym

Why It Works

Rhythmic chunking (rap-style repetition) leverages phonological memory loops. The parallel structure (both sides start with AᵀP) makes it self-checking.

Example Usage

On the board exam, when asked to write the normal equations, start with AᵀP on both sides — left gets A appended, right gets l appended.

Recall Trigger

Rap beat: 'A-T-P-A … A-T-P-L'

Tags

• formula
• matrix inversion
• solution

Topic

Least-Squares Solution

Concept

Solution: x̂ = (AᵀPA)⁻¹ AᵀPl

Anchor Id

A5

Difficulty

medium

Memory Aid

Remember the solution as 'INVERT, then APPLY': Invert the normal equation matrix (AᵀPA)⁻¹, then apply it to the right-hand side AᵀPl. The phrase: 'INVERT the boss, APPLY to the workers' — (AᵀPA) is the boss (normal matrix), AᵀPl is the workers (data-driven vector).

Anchor Type

mnemonic

Why It Works

Two-step action verbs (INVERT, APPLY) are easier to sequence than pure algebra. The boss/worker metaphor encodes the matrix relationship.

Example Usage

After forming normal equations, the answer is always: invert the left-side matrix, multiply by the right-side vector. That is x̂.

Recall Trigger

INVERT the boss → APPLY to workers → x̂

Tags

• formula
• definition
• redundancy

Topic

Redundancy / Degrees of Freedom

Concept

Redundancy (degrees of freedom) = n − u (observations minus unknowns)

Anchor Id

A6

Difficulty

easy

Memory Aid

Story: Engineer Maria has n = 10 GNSS baseline observations to solve u = 6 coordinate unknowns in a PRS92 network. She has 4 'leftover' observations she did not strictly need. Those 4 extras are her redundancy — her safety net. Without them, she cannot check her work. 'n minus u gives you your safety net' — always n − u.

Anchor Type

micro_story

Why It Works

A relatable Filipino engineering scenario with a named character makes the formula personal and memorable. The 'safety net' metaphor encodes why redundancy matters.

Example Usage

12 observations, 8 unknowns → redundancy = 12 − 8 = 4. You need at least 1 redundancy to adjust.

Recall Trigger

Maria's safety net: n minus u = redundancy

Tags

• formula
• statistical check
• variance

Topic

Reference Variance

Concept

Reference variance: σ̂₀² = vᵀPv / (n − u)

Anchor Id

A7

Difficulty

medium

Memory Aid

Think of σ̂₀² as the teacher's grade curve. After the exam (adjustment), the teacher checks: 'Did I make the test too hard or too easy?' vᵀPv is the total weighted pain (sum of weighted squared residuals). Dividing by n − u (degrees of freedom) gives the average pain per extra observation. If the result ≈ 1, the test was fair (weights were correct). If large, the test was too hard (or there's a blunder).

Anchor Type

analogy

Why It Works

Filipino students are very familiar with grade curves and teacher assessments. The metaphor maps perfectly: total error / degrees of freedom = fairness check.

Example Usage

After adjustment, compute σ̂₀² = vᵀPv/(n−u). If ≈ 1.0, weighting was realistic. If >> 1, suspect blunders.

Recall Trigger

Teacher's grade curve: vᵀPv divided by (n − u)

Tags

• formula
• weighted mean
• special case

Topic

Single-Unknown Case — Weighted Mean

Concept

Single unknown = weighted mean: x̂ = Σwᵢlᵢ / Σwᵢ

Anchor Id

A8

Difficulty

easy

Memory Aid

Picture a traditional Filipino weighing scale (timbangan) with multiple fruit vendors placing their price quotes (observations) on pans. The heavier the vendor's reputation (weight wᵢ), the more their quote tilts the balance. The equilibrium point of the scale is the weighted mean — the least-squares answer for one unknown.

Anchor Type

visual_association

Why It Works

The timbangan is a culturally familiar object in Philippine markets. The physical equilibrium metaphor directly encodes the weighted mean formula.

Example Usage

Three level routes give elevations 152.34, 152.30, 152.36 m with weights 0.5, 1.0, 0.25. Compute: (0.5×152.34 + 1.0×152.30 + 0.25×152.36)/(0.5+1.0+0.25) = 152.32 m.

Recall Trigger

Timbangan (weighing scale) in the palengke → weighted mean

Tags

• definition
• weight
• inverse variance

Topic

Weight Matrix P

Concept

Weight is inversely proportional to variance: w = 1/σ² (or w ∝ 1/K for leveling routes)

Anchor Id

A9

Difficulty

medium

Memory Aid

A precise GPS receiver (low variance σ²) is a trustworthy witness in court — you give it HIGH weight. A rusty barometer (high variance) is an unreliable witness — LOW weight. Weight is the inverse of unreliability. The less variance, the more you trust it, the higher the weight: w = 1/σ².

Anchor Type

analogy

Why It Works

The court-witness analogy captures the concept of trustworthiness inversely scaling with uncertainty. Filipino legal dramas make this relatable.

Example Usage

Level route of 2 km vs. 8 km: w₁ = 1/2 = 0.5, w₂ = 1/8 = 0.125. Shorter route = more weight.

Recall Trigger

Trustworthy witness = high weight; unreliable witness = low weight → w = 1/σ²

Tags

• condition
• redundancy
• system requirement

Topic

Condition for Adjustment

Concept

Need n > u to perform least-squares adjustment (redundancy must be positive)

Anchor Id

A10

Difficulty

easy

Memory Aid

Story: A judge (least squares) can only render a verdict if there is more evidence (observations n) than suspects (unknowns u). With exactly as many suspects as clues (n = u), you get a unique but unverifiable answer — no error checking possible! You need at least ONE extra clue (n > u) to catch inconsistencies. Board exam trap: n = u means no adjustment, just a unique solution.

Anchor Type

micro_story

Why It Works

The court-case narrative is engaging and culturally familiar. The 'extra clue = redundancy' link is logically tight and memorable.

Example Usage

If a problem has 6 observations and 6 unknowns, state: 'No adjustment possible; unique solution, zero redundancy.'

Recall Trigger

Judge needs more clues than suspects: n > u

Tags

• matrix
• weight
• independence

Topic

Weight Matrix P — Structure

Concept

Weight matrix P is diagonal for independent (uncorrelated) observations

Anchor Id

A11

Difficulty

medium

Memory Aid

Picture a Filipino chess board. Only the diagonal squares are lit up — everything off-diagonal is dark. That is the weight matrix P for independent observations: a diagonal blaze of weights, zero everywhere else. If observations are correlated, some off-diagonal squares light up too.

Anchor Type

visual_association

Why It Works

The chess-board image is iconic and spatially encodes the mathematical structure of a diagonal matrix in a memorable way.

Example Usage

When observations are independent, build P = diag(w₁, w₂, …, wₙ). No cross terms.

Recall Trigger

Chess board diagonal only lit → P is diagonal for independent obs

Tags

• process
• linearization
• Taylor series

Topic

Linearization

Concept

Linearization: observation equations are linearized using Taylor series (first-order partials)

Anchor Id

A12

Difficulty

hard

Memory Aid

Linearization is like approximating a hilly mountain trail (curved function) with a series of short flat boardwalks (tangent lines). Each boardwalk is only valid near its starting point — your approximate values x₀. The partial derivatives in matrix A are the slopes of those boardwalks. Far from x₀, the approximation breaks down — you need iteration.

Anchor Type

analogy

Why It Works

The mountain-trail-to-boardwalk image makes the mathematical concept of Taylor linearization tangible and visually vivid.

Example Usage

In a trilateration network, observed distances are nonlinear functions of coordinates. Linearize around approximate coords x₀, iterate until corrections are negligible.

Recall Trigger

Mountain trail → flat boardwalks → linearization via partials

Tags

• formula
• definition
• sign convention

Topic

l Vector (Observed Minus Computed)

Concept

The vector l = observed minus computed at approximations (l = ℓ − f(x₀))

Anchor Id

A13

Difficulty

medium

Memory Aid

l is the 'Leftovers' vector — what is left over after you subtract what your approximate model predicts from what you actually measured. 'l = Leftovers = Observed minus Computed-at-x₀.' Note: l and v have opposite sign: l is o−c, v is c−o.

Anchor Type

mnemonic

Why It Works

The 'leftovers' metaphor is universally relatable (Filipino culture values using all food — leftovers are the remainder). The sign contrast with v is highlighted.

Example Usage

If a measured distance is 250.342 m and the distance computed from approximate coordinates is 250.315 m, then l = 250.342 − 250.315 = 0.027 m.

Recall Trigger

l = Leftovers: observed minus computed at approximations

Tags

• special case
• arithmetic mean
• equal weights

Topic

Special Case — Unweighted Mean

Concept

Unweighted case (all weights equal) → simple arithmetic mean

Anchor Id

A14

Difficulty

easy

Memory Aid

When all the weights are set to one, The arithmetic mean gets done! No heavy, no light — all equal play, The mean of the data shows the way. But give some more trust, give others less — Then weighted mean handles the rest!

Anchor Type

rhyme

Why It Works

Rhyming couplets exploit phonological memory. The rhyme encodes both the equal-weight case (arithmetic mean) and the unequal-weight case (weighted mean) as a contrast pair.

Example Usage

If all observations have equal precision (same σ), set all wᵢ = 1. Then x̂ = Σlᵢ/n = arithmetic mean.

Recall Trigger

Rhyme: 'All weights equal one → arithmetic mean is done'

Tags

• statistical check
• quality control
• blunder detection

Topic

Reference Variance as Quality Check

Concept

Checking σ̂₀² ≈ 1 validates the weight system

Anchor Id

A15

Difficulty

medium

Memory Aid

Story: Quality Control Inspector Noel is checking a bridge bolt factory. After production (adjustment), he measures a sample and computes the average deviation. If it matches the machine's claimed precision (σ̂₀² ≈ 1), the machine is calibrated correctly. If the deviation is way off, the machine lied about its precision — just as bad weights in a least-squares network produce σ̂₀² >> 1, signaling the weight model is wrong or there is a blunder.

Anchor Type

micro_story

Why It Works

Quality control is an engineering concept Filipino students encounter in multiple subjects. The factory inspector character makes the abstract statistical test concrete.

Example Usage

After adjustment, if σ̂₀² = 4.7, conclude that weights were too optimistic (precisions overestimated) or a blunder exists — re-examine the network.

Recall Trigger

Inspector Noel's QC check: σ̂₀² ≈ 1 means calibrated correctly

Tags

• formula
• model
• parametric form

Topic

Fundamental Observation Equation

Concept

Observation equation form: v = Ax̂ − l (the fundamental parametric model)

Anchor Id

A16

Difficulty

medium

Memory Aid

Walk through the Geodetic Engineering building at your university. At the ENTRANCE: see a 'v' (residual) written on the door — you always start with residuals. Inside Room 1 (A): the DESIGN MATRIX room — full of partial derivatives. In Room 2 (x̂): the UNKNOWNS room — the adjusted parameters. At the EXIT (l): the LEFTOVERS bin — observed minus computed. The equation is the path: v = A·x̂ − l.

Anchor Type

method_of_loci

Why It Works

Method of loci (memory palace) encodes algebraic structure into spatial memory, which is processed by a different brain system and is highly persistent.

Example Usage

During an exam, mentally 'walk the building' to reconstruct v = Ax̂ − l before setting up any least-squares problem.

Recall Trigger

Walk the GE building: entrance v → Room A → Room x̂ → exit l

Tags

• formula
• weight
• leveling

Topic

Leveling Route Weights

Concept

Leveling route weight ∝ 1/K where K = route length in km

Anchor Id

A17

Difficulty

easy

Memory Aid

A longer commute from Quezon City to Taguig (long route) is more tiring and error-prone than a short trip within Makati (short route). The shorter the commute (K), the more reliable the level route, the higher the weight. Weight = 1/commute distance. Short trip → light pack → high trust → high weight.

Anchor Type

analogy

Why It Works

Metro Manila commuting is a daily reality for many Filipino students. The length-of-commute metaphor maps perfectly onto route-length weighting.

Example Usage

Level route A: K = 2 km → wA = 0.5. Level route B: K = 4 km → wB = 0.25. Route A gets more weight.

Recall Trigger

Short Makati trip = high weight; long QC-to-Taguig = low weight → w = 1/K

Tags

• matrix
• formula
• normal equations

Topic

Normal Equation Matrix

Concept

AᵀPA is the normal equation matrix (also called the coefficient matrix of normal equations)

Anchor Id

A18

Difficulty

medium

Memory Aid

Chunk it as three layers: A-transpose wraps around P in the middle, then A again on the right. Like a sandwich: A (top bread) | P (filling) | A (bottom bread) = AᵀPA. The sandwich is square and symmetric — always! It must be invertible for a unique solution.

Anchor Type

chunking

Why It Works

Food chunking (sandwich) is a universal mnemonic. The symmetric sandwich structure also encodes the mathematical property that AᵀPA is symmetric positive definite.

Example Usage

When computing normal equations, assemble the AᵀPA sandwich first, then compute its inverse. A 2×2 sandwich for 2 unknowns.

Recall Trigger

Sandwich: A-top | P-filling | A-bottom = AᵀPA

Tags

• process
• iteration
• nonlinear

Topic

Iteration in Nonlinear Cases

Concept

Iteration is needed when the observation functions are nonlinear

Anchor Id

A19

Difficulty

hard

Memory Aid

Story: Surveyor Ben is locating a buried BM using GPS in a dense forest near the NAMRIA office. His first coordinate estimate is rough. He runs the least-squares adjustment, gets corrections Δx, updates x₀ ← x₀ + Δx, then runs it again. Each cycle, the corrections shrink. When corrections are smaller than 0.001 m (sub-millimeter), he stops — convergence achieved. Iteration = refining the approximation until the boardwalks match the mountain.

Anchor Type

micro_story

Why It Works

A named character in a recognizable Philippine setting (NAMRIA context) makes the iterative process vivid. The convergence criterion (small corrections) is explicitly included.

Example Usage

For nonlinear geodetic networks (trilateration, GPS baseline adjustment), always iterate: update approximations, recompute A and l, solve, repeat until corrections converge.

Recall Trigger

Ben updates x₀ each cycle until Δx < tolerance → iteration

Tags

• classification
• method comparison
• concept

Topic

Parametric vs. Condition Method

Concept

The parametric (observation equation) method vs. the condition equation method

Anchor Id

A20

Difficulty

hard

Memory Aid

Parametric method (observation equations): You write equations FROM THE OBSERVATIONS, solving for the UNKNOWNS directly. It is like filling out a BIR tax form — you enter your known data and compute your tax liability (unknowns). Condition method: You write equations that the ADJUSTED OBSERVATIONS must satisfy, using correlates. It is like writing the rules of a budgeting game before you play — constraints first, then find compatible values. Board exam: the parametric method is more common and easier to set up.

Anchor Type

analogy

Why It Works

The BIR tax form is a universally recognized Filipino experience. The contrast between the two methods is encoded as two distinct familiar activities.

Example Usage

Most board exam problems use the parametric/observation equation method. Set up v = Ax̂ − l for each observation.

Recall Trigger

BIR form = parametric (observations → unknowns); budgeting rules = condition method

Formula

v = Ax̂ − l

Mnemonic

VAL: V equals A-hat minus L. 'VAL always starts the party' — Residual (V), design matrix times unknowns (A-x̂), minus leftovers (L). Or: 'Verkada Ang Lahat' — V = A·x̂ − l.

When To Use

This is the fundamental observation equation. Write one row per observation when setting up any least-squares problem in parametric form.

What Each Part Means

v = vector of residuals (computed minus observed); A = design matrix (partials of observations w.r.t. unknowns); x̂ = vector of adjusted parameter corrections; l = vector of observed minus computed at approximate values (the 'leftovers').

Formula

(AᵀPA)x̂ = AᵀPl

Mnemonic

The ATPA ATPL chant: 'A-T-P-A solves for x, A-T-P-L is the text.' Both sides start with AᵀP — left gets A, right gets l. Remember: left is the coefficient, right is the constant.

When To Use

After assembling A, P, and l, multiply out to form normal equations. Solve for x̂ using matrix inversion.

What Each Part Means

Aᵀ = transpose of design matrix; P = diagonal weight matrix (inverse variances); A = design matrix; x̂ = unknown corrections; l = observed-minus-computed vector. AᵀPA is the normal matrix (square, symmetric). AᵀPl is the right-hand side vector.

Formula

x̂ = (AᵀPA)⁻¹ AᵀPl

Mnemonic

'INVERT THE BOSS, APPLY TO WORKERS': (AᵀPA)⁻¹ = invert the boss; AᵀPl = the workers (data). Combine: x̂ = boss⁻¹ × workers.

When To Use

Use after forming normal equations. For a 2×2 or 3×3 system, invert manually. For larger systems, use Gaussian elimination or a calculator.

What Each Part Means

(AᵀPA)⁻¹ = inverse of the normal matrix (the 'coefficient of unknowns'); AᵀPl = transformed observation vector; x̂ = least-squares adjusted parameter corrections.

Formula

σ̂₀² = vᵀPv / (n − u)

Mnemonic

'VPV over freedom': V-P-V (residuals weighted and summed) divided by degrees of freedom (n minus u). Think: 'VPV / DF = fairness score.' The numerator vᵀPv is the weighted sum of squared residuals (total punishment); the denominator n − u is how many redundant checks you had.

When To Use

Always compute after adjustment to check the quality of the solution. Compare to expected value (should be ≈ 1 for a properly weighted, blunder-free network).

What Each Part Means

v = vector of final residuals; P = weight matrix; vᵀPv = weighted sum of squared residuals (a scalar); n = number of observations; u = number of unknowns; n−u = degrees of freedom (redundancy).

Formula

x̂ = Σwᵢlᵢ / Σwᵢ (weighted mean, single unknown)

Mnemonic

'Timbangan Formula': Sum of (weight × reading) on top, sum of weights on bottom. It is literally how you balance a scale — heavier weight pulls the average toward its reading. 'W-L over W' — weights times levels, divided by weights.

When To Use

Only when there is ONE unknown observed multiple times (e.g., multiple level routes to one benchmark, repeated angle measurements at one station). This is the most common board-exam application.

What Each Part Means

wᵢ = weight of i-th observation; lᵢ = i-th observed value; Σwᵢlᵢ = sum of weighted observations; Σwᵢ = sum of all weights; x̂ = most probable (least-squares) value.

Formula

w ∝ 1/K (leveling), w = 1/σ² (general)

Mnemonic

'Short and Light vs. Long and Heavy': Short route (small K) → small denominator → large weight (light, easy to trust). Long route (large K) → large denominator → small weight (heavy, hard to trust). General: variance (σ²) is the denominator — less variance = more weight.

When To Use

Assign weights before computing weighted mean or setting up P matrix. For leveling networks, use w = c/K where c is a constant (often set to make one route have weight 1).

What Each Part Means

K = route length in km (for differential leveling); σ² = variance of the observation; w = weight assigned to the observation. For leveling, accumulated error grows with distance, so longer routes get less trust.

Formula

r = n − u (redundancy / degrees of freedom)

Mnemonic

'n minus u = your safety net r': n observations MINUS u unknowns = r redundant checks. No net if n = u. Big net if n >> u. 'Safety Net Size = n minus u'.

When To Use

Before any adjustment: count n and u. If n ≤ u, stop — you cannot adjust. Always state the redundancy in your solution for full marks.

What Each Part Means

n = total number of independent observations; u = number of unknown parameters; r = redundancy (degrees of freedom). If r = 0: unique solution, no checking. If r < 0: under-determined, no solution.

Chain Title

Steps to Solve a Least-Squares Problem (Parametric Method)

Recall Test

Without looking, list the 10 steps of the parametric least-squares procedure. Can you name what you compute at step 8 and 9?

Memory Chain

Story chain: 'Identify Maria (1-2), Linearize her Path (3-4), Pack her Weights (5), Form the Normal Sandwich (6), Invert the Boss (7), Find the Residual Leftovers (8), Check the QC Score (9), Update and Repeat (10).' Each bold word is an action step.

Items To Remember

• 1. Identify unknowns (u) and observations (n); confirm n > u
• 2. Choose approximate values x₀ for the unknowns
• 3. Linearize each observation equation; compute partials → build matrix A
• 4. Compute l = observed − computed at x₀
• 5. Assign weights; build diagonal P matrix
• 6. Form normal equations: N = AᵀPA, t = AᵀPl
• 7. Solve: x̂ = N⁻¹ t
• 8. Compute residuals: v = Ax̂ − l
• 9. Compute reference variance: σ̂₀² = vᵀPv/(n−u)
• 10. Update x₀ ← x₀ + x̂; iterate if nonlinear

Chain Title

Board-Exam Traps in Least Squares (Common Pitfall Sequence)

Recall Test

Name 4 common board-exam pitfalls in least-squares adjustment from memory. What does n = u imply?

Memory Chain

Acronym: 'SPONWS' — Sign (v), Sign (l opposite), P diagonal, nequals u no-go, One unknown = mean, Watch σ̂₀². Remember: 'SPONWS — Sponge the Errors Away from the Board Exam!'

Items To Remember

• Sign of v: computed minus observed (not reversed)
• Sign of l: observed minus computed (opposite of v)
• P is diagonal only if observations are independent
• n = u means NO adjustment (zero redundancy)
• Weighted mean only works for ONE unknown
• σ̂₀² ≈ 1 is the check, not σ̂₀² = 0

Chain Title

Conditions for a Valid Least-Squares Adjustment

Recall Test

State 5 conditions required for a valid least-squares adjustment. What happens if AᵀPA is singular?

Memory Chain

Remember 'MORE-NICE-RANDOM-FULL-INVERT': More obs than unknowns → Independent obs → Random normal errors → Full rank A → Invertible AᵀPA. 'MORE NICE RANDOM FULL INVERT = a perfect least-squares setup!'

Items To Remember

• n > u (more observations than unknowns)
• Observations must be independent (or correlation accounted for in P)
• Errors must be random and normally distributed
• A must have full column rank (rank = u)
• AᵀPA must be invertible (non-singular)

Chain Title

Types of Weights Used in Philippine Geodetic Practice

Recall Test

How do you assign weights to (a) leveling routes, (b) repeated angle measurements, and (c) GNSS baselines?

Memory Chain

Story: 'LRAGE weights: Level routes by Length, Repeated measurements by Number, Angles by Variance, GNSS by Covariance, Equal weights give Average.' L-R-A-G-E = 'LRAGE — Large Surveys Need Good Effort to Weight Properly!'

Items To Remember

• Leveling routes: w = 1/K (K = km)
• Repeated measurements: w = n (number of repetitions)
• Angle observations: w = 1/σ² (σ from set statistics)
• GNSS baselines: w = 1/σ² from covariance matrix
• Equal precision: w = 1 for all (arithmetic mean)

Chain Title

The Four Key Matrices/Vectors in Observation Equations

Recall Test

Name all 5 key matrices/vectors. State the dimension of each in terms of n (observations) and u (unknowns).

Memory Chain

Acronym: 'APLXv' — A-P-L-X-v. Chant: 'All Pupils Learn X Variables' where X = x̂ and V (residuals) = the verdict. Dimensions: A is n-by-u, P is n-by-n, l and v are n-by-1, x̂ is u-by-1.

Items To Remember

• A — Design matrix (partials, n×u)
• P — Weight matrix (diagonal, n×n)
• l — Observed minus computed vector (n×1)
• x̂ — Unknown corrections vector (u×1)
• v — Residual vector (n×1)