The Global Positioning System

Extended slides — Trilateration, error, and meters

CCE 114 Geomatics
Dr. Dan Ames

CCE 114 · Day 7 — The Global Positioning System, Part 2

Today's Goals

By the end of class you will have:

  • Trilaterated a position by hand from three signal delays, on paper
  • Explained the difference between triangulation and trilateration
  • Computed a positional error by combining the error budget with PDOP
  • Converted a latitude/longitude to meters and checked the answer in QGIS
  • Turned in the "Where Am I" solution on Learning Suite
CCE 114 · Day 7 — The Global Positioning System, Part 2

Where we left off Tuesday

  • The satellite and your receiver generate the same pseudorandom code at the same time
  • The receiver slides its copy until the two line up — that slide is the travel time
  • D = R × T, with R = 299,792,458 m/s
  • Every satellite you can hear gives you one distance
  • Today: turning those distances into a position

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CCE 114 · Day 7 — The Global Positioning System, Part 2

Triangulation or trilateration?

  • Triangulation measures angles from known points, and solves the triangle for the unknown position
    • This is what a total station does
  • Trilateration measures distances from known points, and intersects circles or spheres
    • This is what GPS does
  • The old lecture slides say "triangulation"; the honest word for GPS is trilateration

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CCE 114 · Day 7 — The Global Positioning System, Part 2

One measurement: somewhere on a sphere

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CCE 114 · Day 7 — The Global Positioning System, Part 2

Two measurements: somewhere on a circle

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CCE 114 · Day 7 — The Global Positioning System, Part 2

Three measurements: two points

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CCE 114 · Day 7 — The Global Positioning System, Part 2

Four measurements: and the clock

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CCE 114 · Day 7 — The Global Positioning System, Part 2

All four steps at once

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CCE 114 · Day 7 — The Global Positioning System, Part 2

Warm-up: find Air Force One

A photo of Air Force One is hidden somewhere in this room. Three "satellites" have been marked. Your ranges:

  • 169 inches from Satellite 1
  • 216 inches from Satellite 2
  • 151 inches from Satellite 3

Find it. You have five minutes.

CCE 114 · Day 7 — The Global Positioning System, Part 2

Activity: "Where Am I?"

I visited Europe and got lost.

I heard three radio stations announcing the time, but each was off from the actual time. The difference is the time the signal took to travel from that station to me:

  • Amsterdam: 2.37 × 10⁻³ seconds
  • Paris: 2.95 × 10⁻³ seconds
  • London: 3.45 × 10⁻³ seconds

Where am I?

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To turn in:

  1. Convert each delay to a distance, then draw the circles on a map of Europe
  2. Write your name and your solution on the paper
  3. Photograph it and upload to Learning Suite
CCE 114 · Day 7 — The Global Positioning System, Part 2

Step 1: delays become distances

D = R × T, R = 299,792 km/s

Station Delay (s) Distance
Amsterdam 0.00237 711 km
Paris 0.00295 884 km
London 0.00345 1,034 km

Step 2: draw a circle of that radius around each city. Step 3: the three circles meet at one point. That point is the answer.

Note the scale: a millisecond of error here is 300 km. GPS needs nanoseconds.

CCE 114 · Day 7 — The Global Positioning System, Part 2

The answer: Prague

  • Roughly 711 km from Amsterdam, 884 km from Paris, 1,034 km from London
  • Three distances from three known points, and the location falls out
  • You just did what a GPS receiver does, with radio stations instead of satellites and milliseconds instead of nanoseconds
CCE 114 · Day 7 — The Global Positioning System, Part 2

How wrong is your position?

Putting a number on GPS error

CCE 114 · Day 7 — The Global Positioning System, Part 2

The error budget, again

Source Typical error
Satellite clocks 1.5 – 3.6 m
Orbital errors < 1 m
Ionosphere 5.0 – 7.0 m
Troposphere 0.5 – 0.7 m
Receiver noise 0.3 – 1.5 m
Multipath 0.6 – 1.2 m

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  • These are independent error sources
  • Independent errors do not simply add — they combine as a root sum of squares
  • The result is then multiplied by PDOP

CCE 114 · Day 7 — The Global Positioning System, Part 2

Worked example: expected horizontal error

Take the middle of each range:

Source σ (m) σ²
Satellite clocks 2.5 6.25
Orbital 0.8 0.64
Ionosphere 6.0 36.00
Troposphere 0.6 0.36
Receiver noise 0.9 0.81
Multipath 0.9 0.81
Sum 44.87
  • Root sum of squares: √44.87 = 6.7 m
  • With PDOP = 2.5: 6.7 × 2.5 = 16.7 m
  • With PDOP = 6: 6.7 × 6 = 40 m

Same receiver, same sky, same instant — the geometry alone nearly triples the error.

Now redo it on your own paper with the low end of every range, and with PDOP = 1.5.

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CCE 114 · Day 7 — The Global Positioning System, Part 2

Where PDOP comes from

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Each range is a band, not a line. Where the bands cross is your area of uncertainty.

CCE 114 · Day 7 — The Global Positioning System, Part 2

Good geometry versus bad

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High PDOP: satellites bunched together, large area of uncertainty. Low PDOP: satellites widely spaced, small area.

CCE 114 · Day 7 — The Global Positioning System, Part 2

Which fix would you trust?

Two readings of the same point, one minute apart:

  1. 7 satellites, PDOP 5.8
  2. 5 satellites, PDOP 1.9
  3. They are equally good — more satellites always wins
  4. Not enough information
CCE 114 · Day 7 — The Global Positioning System, Part 2

Demo: from latitude/longitude to meters

CCE 114 · Day 7 — The Global Positioning System, Part 2

Degrees are angles, not distances

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  • Latitude and longitude are angles measured on an ellipsoid
  • A degree of latitude is nearly constant: about 111 km
  • A degree of longitude shrinks toward the poles:

  • At Provo, φ ≈ 40.25°, so 1° of longitude ≈ 84,960 m
  • At the pole it is zero
CCE 114 · Day 7 — The Global Positioning System, Part 2

Demo, step by step

  1. Start with a point from Tuesday's activity, in decimal degrees, full precision
  2. Paste it into tagis.dep.wv.gov/convert and read off the UTM easting and northing
  3. Utah is UTM Zone 12N, EPSG:32612
  4. Now do the same thing in QGIS: Layer → Add Layer → Add Delimited Text Layer, set the CRS of the incoming data to EPSG:4326
  5. Right-click the layer → Export → Save Features As…, target CRS EPSG:32612
  6. Open the attribute table, add geometry columns, and compare with step 2

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  • EPSG:4326 = WGS 84 lat/long, the degrees your phone gives you
  • EPSG:32612 = UTM Zone 12N, meters
  • Two different answers for the same place — same point, different units
CCE 114 · Day 7 — The Global Positioning System, Part 2

Now measure something

  • With the project CRS set to EPSG:32612, use the measure tool between two of your points — the answer is in meters
  • Switch the project CRS back to EPSG:4326 and measure again
  • QGIS will happily hand you a number in degrees. It is meaningless as a length
  • Rule of thumb: degrees for storing and sharing, projected meters for measuring

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CCE 114 · Day 7 — The Global Positioning System, Part 2

Check your work

  • Does the point land where you stood? Load a basemap and look
  • Is the longitude negative? Utah is west of Greenwich
  • Did you keep all the decimal places?
  • Does the distance between two of your points match what you paced off?
  • If the answer is off by a factor of about 1.3, you probably mixed up a degree of latitude with a degree of longitude

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CCE 114 · Day 7 — The Global Positioning System, Part 2

Errors are cumulative — including yours

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CCE 114 · Day 7 — The Global Positioning System, Part 2

Final Thoughts

  • Upload your "Where Am I" paper solution — name and answer, photographed — to Learning Suite
  • Read Chapter 5, GNSS and Coordinate Surveying, in GIS Fundamentals (Bolstad & Manson) if you have not already
  • Quiz 3 (GPS Part 1), open book on Learning Suite — due Saturday
  • Lab 3: GPS Data Collection and Importing Into QGIS — assignments page — due Saturday
  • Questions? Office hours: calendly.com/dan-ames/office-hours
CCE 114 · Day 7 — The Global Positioning System, Part 2

Reference deck. The Tuesday lecture uses the simplified "The Global Positioning System" deck, which already contains the Air Force One warm-up and the Prague activity. This deck keeps the longer material: the four-step position-fixing build, the error budget and PDOP arithmetic, and the latitude/longitude-to-meters demo, for anyone who wants the full story or for a semester with more time.

Say up front that they will be doing arithmetic today, not watching it. Have paper, calculators, rulers and a compass or a piece of string ready.

Two-minute recap. The one thing they need in hand for the next twenty minutes is D = R x T.

Worth thirty seconds because the vocabulary shows up on the quiz and in the reading. GPS never measures an angle to anything.

Four unknowns: latitude, longitude, height, and time. Four unknowns need four equations, which is why four satellites are the practical minimum. Start the build here and let it run through the next three slides.

The intersection of two spheres is a circle. Still an infinite number of candidate positions, but a much smaller infinity.

Three spheres intersect in exactly two points, and one of them is almost always absurd: out in space, or moving at an impossible speed. In practice three ranges are enough to fix a position.

The fourth range is not really about geometry, it is about the clock. Your receiver has a cheap quartz clock, not an atomic one, so its clock bias is a fourth unknown. The fourth satellite is what lets the receiver solve for it.

Leave this up while students work. A single measurement places the observer somewhere on the sphere; two place them on a circle; three place them at one of two points; knowing you are also on the surface of the Earth, a fourth sphere, kills one of them. Additional measurements remove clock error and atmospheric error.

Instructor setup: tape the Air Force One picture somewhere in the room and mark three fixed reference points as satellites 1, 2 and 3, then measure the distances with a tape before class and put the real numbers on this slide. The numbers shown are from the 2021 running of this activity: Satellite 1 = the light on the zoom camera, Satellite 2 = the light above row 2 left, Satellite 3 = the BYU sticker on the front control panel. For Clyde 254 the recorded set was 221 inches from the projector lens on the ceiling, 260 inches from the screw on the bottom of the front left speaker, and 149 inches from the angled smoke detector. Hand out tape measures and let them swing arcs. The point is that they will physically do what three spheres do.

This is the graded activity, worked on paper. Give them a printed map of Europe with a scale bar, or let them use a web map. Twelve to fifteen minutes. Circulate and check that they converted seconds to kilometers before drawing anything.

Check the arithmetic with them: 0.00237 s x 299,792 km/s = 710.5 km. The original speaker notes list 711, 885 and 1035 km, and label the first one "miles" by mistake; they are all kilometers.

Reveal only after they have committed to an answer. Ask what a fourth station would have added: a check on their own clock. Ask what would happen if all three stations were in a line.

Ask why they do not just add: because it would be extraordinarily unlucky for every source to be at its maximum, in the same direction, at the same instant. The root sum of squares is the standard way to combine independent uncertainties, and they have seen it in statics and in measurements courses.

Give them three minutes to redo it with the low values: sqrt(1.5^2 + 0.5^2 + 5^2 + 0.5^2 + 0.3^2 + 0.6^2) is about 5.3 m, times 1.5 is about 8 m. That spread, 8 m to 40 m, is the honest answer to "how accurate is GPS?"

Left: a single range is really a fuzzy band, because the measured time is uncertain. Right: two or more fuzzy bands intersect in a patch, not a point. The shape of that patch is set entirely by the angles between the satellites.

Same range uncertainty in both pictures. The only difference is where the satellites are. This is why your fix improves if you wait ten minutes, and why it is bad in a canyon: you only see the strip of sky overhead.

Answer B. Satellite count matters only up to a point; geometry is what multiplies your error. Five well-spread satellites beat seven clustered ones. This is why field receivers report PDOP and why survey planning software predicts it hours ahead.

Do the cosine live. Then ask the trap question: "what is the distance between 40.2500, -111.6500 and 40.2600, -111.6600?" If anyone answers using degrees, that is the lesson.

Run this live and let them follow along. If the network is down, the tagis site is the only online piece; the QGIS half works offline. This is the same workflow they need for Lab 3.

QGIS does apply an ellipsoidal correction when the project ellipsoid is set, so the measure tool may still give a sensible meter value in a geographic CRS. Show both, and make the point that the field calculator and most geoprocessing tools will not do that for you.

111,320 / 84,960 = 1.31, which is the signature of that particular mistake at this latitude. Worth naming so they recognize it in the lab.

Close the loop on the instrument-versus-user point from Tuesday. System and other flaws are under about 9 meters; user error can be plus or minus a kilometer. Everything in today's checklist is about the second number.

Confirm the Saturday due dates against Learning Suite before class. Remind them that Lab 3 uses the points they collected on Tuesday, so anyone who missed the activity needs to go collect three points.

Conversion notes (2026-09-02): sources were "GPS and Triangulation.pptx" (2025) slides 1-5, 13-15, 22, 24-26 and, for the four-step position-fixing build, "GPS basics.pptx" (2024) slides 33-36, a legacy Trimble real-time-surveying deck whose pale blue slide background was whitened when the figures were extracted. Hidden source slide 5 (the classroom distances for the Air Force One activity) was kept deliberately as an instructor backup: the three distances are on the activity slide and the 2021 satellite locations plus the Clyde 254 alternative are in its speaker notes — the instructor should re-measure for the actual room. Hidden slide 13 ("d1 d2 d3") was dropped as a bare shape overlay with no context. No ArcGIS screenshots appear in either source deck, so no QGIS re-shoots are needed; the QGIS workflow on the "Demo, step by step" and "Now measure something" slides was written for this deck. New material not in the sources, written to cover the assigned topics: the triangulation-versus-trilateration slide, the root-sum-of-squares error computation and its worked example, the PDOP quiz, and the whole latitude/longitude-to-meters demo section. The Prague distances are computed from the source delays and match the numbers in the original speaker notes; the original note labels the Amsterdam distance "miles", which is a typo for kilometers.