Intuition The one core ideaA GPS receiver listens to radio "time stamps" from satellites in the sky and works out how far each one is by measuring how long the signal took to arrive. Because its own clock is cheap and wrong by an unknown amount, it must solve for where it is and how wrong its clock is at the same time — and that single extra unknown is why the whole subject exists.
This page is the toolbox. Before you read the parent topic , every letter and squiggle it uses should already feel like an old friend. We build each one from nothing, anchor it to a picture, and say exactly why GPS needs it.
Definition A point in 3D space( x , y , z ) one exact spot . Imagine a giant invisible box drawn around the Earth with three rulers meeting at a corner: how far East (x y z
Intuition Why GPS needs itThe whole job of GPS is to output one such triple: your ( x , y , z ) Reference Frames — ECEF and WGS84 — but for now just picture "three perpendicular rulers."
Distance-from-a-corner A point ( x , y , z )
We know the satellite's spot ( x i , y i , z i ) ( x , y , z )
Intuition WHY a square root of squares — this is just Pythagoras twiceOn flat paper, the straight-line distance between two dots is ( across ) 2 + ( up ) 2 The square root is the tool that turns "side lengths" back into a "straight-line length." Squaring first makes every gap positive (a gap of − 3 + 3
Definition The symbols here
r i true geometric distance ("range") from satellite i i label : satellite 1, 2, 3, 4...x i , y i , z i x , y , z
Why the subscript i It is an index labelling which satellite; r 3
See Time of Flight and Ranging for how a distance is physically measured rather than computed.
c c ≈ 3 × 1 0 8 fixed constant of nature.
Intuition The picture: light as a tape measure made of timeA radio pulse leaves the satellite and races toward you at c t c × t c time and multiplies by c
Worked example How big is a tiny time error?One microsecond is 1 0 − 6 c ( 3 × 1 0 8 ) ( 1 0 − 6 ) = 300 millionth of a second misplaces you by the length of three football fields. This one line is the reason the clock bias matters so much.
Definition Transmit and receive times
t t x sent the signal, stamped by its atomic clock (extremely accurate — see Clock Bias and Atomic Clocks ).t r x received it, read off our cheap receiver clock.The subscripts t x r x t ransm it and r ec eive.
Think of a runner leaving a start line at a stopwatch reading t t x t r x t r x − t t x c ( t r x − t t x ) same amount.
b clock bias b off the receiver's clock is from true GPS time, measured in seconds . If b ahead ; if negative, behind .
Intuition Why one number corrupts everything — and why that is a giftYou have one receiver clock. Whatever it is fast or slow by, it shifts every t r x same b c b same unknown appearing in every equation, one extra satellite gives you enough information to pin it down. A shared error is a solvable error.
Why is the clock error identical for all satellites There is only one receiver clock, so its offset b
ρ i = r i + c b pseudorange is the true distance r i plus the clock-caused error c b ρ "looks like a range but isn't the true one yet."
Imagine drawing a sphere around each satellite with radius equal to the pseudorange. Because of c b same amount. The true spheres would all meet at your position; the pseudo-spheres miss — until you find the b
Common mistake "Pseudorange = distance to the satellite"It feels right because it is computed as c × c b r i b
Intuition One range = one sphereIf you only know you are 20,000 km from satellite 1, you could be anywhere on a huge sphere of that radius around it. Add satellite 2: you must be on both spheres → their overlap is a circle . Add satellite 3: three spheres meet at (generally) two points , and the Earth throws one away. That is classic trilateration .
Intuition Why GPS needs afourth sphere
Three spheres pin a point if the radii are exact. But our radii are pseudoranges — all wrong by c b b Geometry says 3; the physics of clocks says 4. More at Trilateration and Multilateration .
Two spheres intersect in a... circle. Three (with exact radii) in two points.
Definition Unknowns vs equationsUnknowns: x , y , z , b Equations: one pseudorange per satellite. To solve for 4 unknowns you need at least 4 equations, hence at least 4 satellites.
Mnemonic Count the question marksEvery "?" needs its own equation. Three space coordinates plus one clock = four question marks = four satellites minimum.
Definition A vector and a unit vectorA vector is an arrow: it has a direction and a length. A unit vector is an arrow squashed to length exactly 1 direction , throwing away the length.
Intuition The line-of-sight unit vector( e i x , e i y , e i z ) i e i x , e i y , e i z "which way is the satellite, as seen from me?" The parent topic shows these are exactly r i x − x i rate of change of distance as you nudge your position is just "how much of that nudge points along the line to the satellite." Direction, not distance, is what controls the geometry.
Why the line-of-sight vector has length 1 We only care about direction ; dividing by r i
Definition A matrix as a gridA matrix is a rectangular grid of numbers, written in a big bracket. Each row here is one satellite's line-of-sight direction plus a 1 G = e 1 x e 2 x ⋮ e n x e 1 y e 2 y e n y e 1 z e 2 z e n z 1 1 ⋮ 1
Definition The symbols you'll meet withG
G ⊤ G ⊤ G ( G ⊤ G ) − 1 inverse — the matrix "undo" operation, like dividing.Δ a small change in : Δ x x Δ ρ
Intuition Why a matrix at all?With four satellites you have four intertwined equations. A matrix is just the bookkeeping that solves all four at once . When there are more than four satellites, the system is over-determined and you fit a best compromise using Least Squares Estimation ; the ( G ⊤ G ) − 1 G ⊤
What does the column of 1's in G The clock-bias direction — every satellite feels b
Definition Randomness of measurements
Noise = the small random wobble in every real measurement.σ standard deviation , the typical size of that wobble.σ UERE Covariance = a table describing how the errors in several numbers spread and lean together. Propagating it through the solution is done with Covariance Propagation .
Intuition The picture: a fuzzy dot, not a sharp pointBecause of noise, your computed position isn't a crisp dot but a small fuzzy cloud. σ DOP (Dilution of Precision) is the number that says how much the satellite geometry stretches that fuzz — good geometry keeps it a tight round blob, bad geometry smears it into a long streak.
Which factor is pure geometry, DOP or σ UERE DOP — it depends only on satellite directions, not on measurement quality.
Transmit and receive times
Four unknowns need four satellites
Spheres and intersections
Unit line of sight vectors
Noise sigma and covariance
Everything downstream — the linearised solve, the DOP formula, the Kalman Filter in GNC that later fuses these fixes over time — rests on the twelve pieces above.
Can you state, from memory, what each symbol means and why GPS needs it? Reveal to check.
( x , y , z ) The receiver's unknown position — three perpendicular ruler-distances; the final answer GPS outputs.
r i True geometric distance to satellite i
c Speed of light, 3 × 1 0 8 time into distance .
t t x , t r x Transmit time (satellite's atomic clock) and receive time (our cheap clock); their difference times c
b Receiver clock bias in seconds — one shared unknown that corrupts every measurement by the same c b
ρ i Pseudorange = true range r i c b
Why 4 satellites, not 3 Four unknowns x , y , z , b
Sphere picture Each range is a sphere around a satellite; their intersection is your location.
( e i x , e i y , e i z ) Unit line-of-sight vector — pure direction from satellite to receiver, length 1.
G G ⊤ G ( G ⊤ G ) − 1 Geometry matrix of directions (+clock column); its transpose-product and inverse form the least-squares solver.
Δ "A small change in" — used for nudges Δ x Δ ρ
σ UERE Single-measurement error size, and the geometry-only amplification factor; multiplied they give position error.