GIS & Spatial Analysis

The big picture

In 1970 the geographer Waldo Tobler published a short paper about a computer movie of Detroit growing outward decade by decade, and dropped into it a sentence that has been quoted ever since: everything is related to everything else, but "near things are more related than distant things." He called it the first law of geography, half in jest. It is not a law in the sense that gravity is a law. It is a description of how the world usually behaves, and it is the reason an entire branch of software exists.

Think about what that sentence claims. Your neighbor's house price is a better predictor of your house price than a house price twelve states away. Soil at one end of a field resembles soil at the other end more than it resembles soil in the next county. Air pollution, dialect, rainfall, crime, mosquito density, broadband speed: measure any of them at two points and the closer the points, the more alike the numbers tend to be. That regularity is enormously useful, because it means location carries information. A spreadsheet that has thrown location away has thrown that information away with it.

A geographic information system, or GIS, is a database in which every record carries a location, plus the software to store, query, analyze, and draw those records. That is the whole idea. Everything else in this course is machinery built on top of it.

Key idea: A GIS is a database that knows where its records are, and knowing where lets you ask questions that no ordinary table can answer.

A map is an output; a GIS is a system

People use the two words as if they were the same thing, and they are not. A paper map of Chicago is a finished picture. You cannot ask it which parcels lie within 400 meters of an elevated train station, and you certainly cannot ask it to recount if the transit agency adds a station next year. A GIS holds the parcels and the stations as data, so the picture is only one of the things it can produce, and it can produce a different picture the moment the question changes.

The trade is this. A map is fast to read and slow to change. A GIS is slower to build and fast to re-ask. Working geographers keep both: the system does the reasoning, and a map is what you hand to the city council at the end of it.

Every feature in a GIS is stored twice over. There is geometry, the coordinates that place the thing in space, and there are attributes, the ordinary columns of a table that describe it: a fire hydrant's flow rate, install year, and last inspection date; a census tract's population, median income, and vacancy rate. The link between the two is the entire trick. Attributes without geometry are a spreadsheet. Geometry without attributes is a drawing. Together they are a GIS.

Key idea: Geometry says where, attributes say what, and joining them permanently is what separates a GIS from both a spreadsheet and a picture.

Layers: the mental model that makes everything else work

Picture a stack of clear plastic sheets on an overhead projector, all cut to the same size and lined up on the same corners. One sheet has the parcel boundaries. One has the street centerlines. One has sewer mains. One has zoning districts. One has the floodplain the Federal Emergency Management Agency has mapped. One has tree canopy traced from aerial photography. Each sheet is a layer: one theme, one geometry type, one attribute table. Because every sheet uses the same coordinate system, the sheets line up, and anything true on one sheet can be compared against anything true on another.

That stack is not a metaphor invented for teaching. It is how the software is organized, how the files are stored, and how the analysis works. Almost every real GIS task decomposes into: name the layers you need, get them into a common coordinate system, and then combine them.

Try the decomposition on a real question. A city wants to know where to require developers to build affordable housing near transit. The rule as written says: parcels within a quarter mile of a bus stop with fifteen minute headways, zoned for multifamily, at least half an acre, and not inside the one hundred year floodplain. Read that again as a shopping list of layers. Bus stops, with a headway attribute. Parcels, with zoning and acreage attributes. Floodplain polygons. Nothing else. Four layers, four operations: buffer the qualifying stops by a quarter mile, select parcels whose zoning is multifamily and whose area is at least 0.5 acres, keep the ones that fall inside the buffer, and erase the ones that intersect the floodplain. By Module 4 you will do exactly this, in that order, with real files. What matters today is the habit: a policy sentence, read carefully, is already a list of layers and operations.

Key idea: Break every spatial question into the layers it needs and the operations that combine them, and most of the analysis has designed itself before you open any software.

The five questions a GIS answers

Practitioners often sort spatial questions into five kinds, and the list is worth memorizing because it tells you which tool you are reaching for.

Question type	What you are asking	Example
Location	Where is a thing, or where are all of them?	Map every licensed child care center in the county.
Condition	What is at this place, or which places meet a condition?	Which tracts have more than 20 percent of households without a vehicle?
Trend	What has changed here, and how fast?	How much forest was cleared in this watershed between 1990 and 2020?
Pattern	Is the arrangement clustered, dispersed, or random, and where?	Are traffic crashes concentrated at particular intersections, or spread evenly?
Modeling	What would happen if something changed?	Which homes flood if the river crests two feet higher than the record?

Notice that the questions get harder down the list, and so does the honesty required. Location and condition questions have right answers you can verify by walking outside. Modeling questions produce a map that looks equally authoritative and rests on assumptions a viewer cannot see. Module 6 is largely about that gap.

The five components: it is not just software

Ask what a GIS consists of and most people say the software. A working system has five parts, and the software is the cheapest of them.

• Data. The layers themselves, plus the survey, digitizing, and cleanup that produced them. In organizational budgets this line usually dwarfs the rest.
• Software. The engine that stores geometry and runs the operations. QGIS is free, open source, and genuinely professional; Esri's ArcGIS Pro dominates United States government and utility work; PostGIS turns a PostgreSQL database into a spatial one.
• Hardware. Historically the constraint, now rarely so. A five year old laptop runs QGIS on county-scale data without complaint.
• People. Someone has to know the difference between a datum and a projection, and to notice when a result is absurd. That is what this course is for.
• Methods. The documented procedure: which layers, which version, which parameters, in which order. Undocumented analysis is not reproducible, and unreproducible analysis loses arguments.

The data line deserves a moment. Digitizing a county's sewer network from paper as-builts is months of labor. Keeping an address database current means processing every new plat and every street renaming, forever. When a GIS project fails, the failure is almost never that the software could not do the operation. It is that nobody had current, correct, documented data to run the operation on.

Key idea: Data acquisition and maintenance, not software, is the expensive and failure-prone part of a real GIS, which is why metadata and provenance are treated as first-class concerns in this field.

Why spatial data breaks ordinary statistics

Here is the deeper reason geographers built their own analytical toolkit rather than borrowing everyone else's. Most introductory statistics assumes your observations are independent: that knowing one value tells you nothing about the next. Tobler's law says spatial observations violate that assumption by construction. If you sample soil lead at 200 points across a neighborhood and the points are ten meters apart, you do not have 200 independent measurements. You have something closer to twenty, repeated. Treat them as 200 and your confidence intervals will be too narrow and your p-value will lie to you.

This property has a name, spatial autocorrelation, and it is both a nuisance and a signal. As a nuisance it invalidates naive statistics. As a signal it is the thing you often want to measure: disease clusters, crime hot spots, and pockets of high yield are all statements about autocorrelation. Lesson 14 gives you the tool for measuring it. For now, hold on to the fact that space is not an ordinary variable you can toss into a regression.

Space has a second awkward property, spatial heterogeneity: relationships change from place to place. Elevation predicts rainfall strongly in Washington State and hardly at all in Kansas. A model fitted to the whole country can be wrong everywhere in it, in different directions.

Key idea: Nearby observations are not independent and relationships shift across space, so spatial data needs methods that expect both rather than statistics that assume neither.

What this course can and cannot do

Time for a plain statement, because the alternative is wasting your time. GIS is a hands-on craft, and this is a text course. Reading it will teach you the concepts and the workflow: which coordinate system to pick and why, what a buffer really computes, how a weighted overlay is scored, why your choropleth changed its story when you switched from quantiles to equal intervals. It cannot teach you where the reproject button lives in the current version of any program, and it will not try. Interfaces change every release; the reasoning does not.

So the deal is this. Every lesson ends with an exercise written step by step in prose, using software you can actually get for nothing: QGIS, Google Earth, and the free data at data.gov, the Census Bureau, the United States Geological Survey, NASA Earthdata, and OpenStreetMap. Do them. Fluency in this field comes from the twentieth time you fix a broken projection, not from the first time you read about one. A person who has read this entire course and never opened QGIS knows what to do and cannot do it. Twenty hours of clicking on top of it, and you are employable.

Common misconceptions

• GIS means making maps. Map production is one output. The analysis, and the database underneath it, is the discipline; plenty of professional GIS work ends in a table, a number, or a decision, with no map at all.
• Google Maps is a GIS. It is a superb map viewer and routing service built on a GIS, but you cannot load your own parcel layer, buffer it, and intersect it with a floodplain. Consumption is not analysis.
• Professional GIS requires expensive software. QGIS is free and open source and does nearly everything in this course. The dominant commercial package matters for jobs, file compatibility, and support, not for whether the operations are possible.
• A GIS answer is objective because it came from a computer. Every layer embodies choices about what to record and how to categorize it, and every parameter you type is a judgment. The output inherits all of them and displays none.
• Spatial data is just data with two extra columns. Latitude and longitude in a spreadsheet are a start, but coordinates are meaningless without a datum, distances are wrong in the wrong projection, and nearby records are statistically dependent. Those three facts fill the next five lessons.

Recap

• Tobler's first law says near things are more alike than distant things, which is why location carries information worth keeping.
• A GIS is a database with geometry attached, plus the tools to query, analyze, and draw it; a map is one of its outputs.
• Every feature is geometry plus attributes, and layers of features registered to a common coordinate system stack like transparent sheets.
• Read a policy question closely and it becomes a list of layers and operations before you touch the software.
• Spatial questions come in five flavors: location, condition, trend, pattern, and modeling, and they get less verifiable in that order.
• The five components are data, software, hardware, people, and methods; data is the expensive one and the usual point of failure.
• Spatial autocorrelation and spatial heterogeneity break the independence assumptions of ordinary statistics, which is why this field has its own methods.
• This course teaches concepts and workflow. Fluency comes from doing the work in real software, and the exercises are there to make you.

Sources

1. United States Geological Survey. (n.d.). What is a geographic information system (GIS)? USGS.
2. Tobler, W. R. (1970). A computer movie simulating urban growth in the Detroit region. Economic Geography, 46, 234-240. Summary at Wikipedia.
3. Esri. (n.d.). What is GIS? Overview. Esri.
4. QGIS Project. (n.d.). Documentation and downloads. QGIS.

The big picture

On 8 September 1854 the parish committee of St James, Westminster, took the advice of a physician they mostly did not believe and unbolted the handle from a water pump on Broad Street in Soho. In the previous ten days, more than five hundred people within a few blocks of that pump had died of cholera; the outbreak eventually killed about 616. The physician was John Snow, and his argument was not chemical or microscopic. It was geographic.

Snow drew a street map of the neighborhood and marked each death as a small black bar stacked at its address. Drawn that way, the deaths were not scattered across Soho. They piled up around one pump and thinned out with distance from it, and Snow's later version of the map added a line around the area from which the Broad Street pump was the nearest pump on foot. Almost every death fell inside that line.

Key idea: Snow's map did not discover cholera's cause; it converted a list of addresses into a pattern with a center, which is exactly what spatial analysis still does.

The exceptions are what make it science

A cluster on a map is suggestive and nothing more. What raised Snow's map above suggestion were the places that broke the pattern, because he went and found out why.

Inside the death zone stood a workhouse holding several hundred inmates, and it lost only a handful of people. The workhouse had its own well and did not send anyone to Broad Street. A brewery on the same street lost nobody at all; its seventy workers had a well on site and an allowance of malt liquor, and the brewer told Snow his men rarely drank water. Meanwhile a widow in Hampstead, far outside the neighborhood and outside any plausible cloud of bad air, died of cholera after a cart brought her a bottle of Broad Street water because she had liked the taste since moving away. Her niece in Islington drank some and died too.

Read those three facts together and the pattern stops being a coincidence of geography and starts being a mechanism. Nearby people who did not drink the water lived. A distant person who did drink it died. The local curate Henry Whitehead, who set out to disprove Snow and ended up convinced, later traced the outbreak's likely start to a sick infant in a house at 40 Broad Street whose soiled napkins were rinsed into a cesspool leaking three feet from the well.

Be honest about the ending, because the tidy version is wrong. The removal of the handle came after the outbreak had already peaked. The Board of Health's own report rejected Snow's explanation and stayed with the theory that disease rose from foul air. Snow died in 1858, aged 45, without seeing his case accepted, and germ theory came later. The map won the argument eventually, not immediately.

Key idea: Spatial evidence becomes persuasive when you hunt down the exceptions and explain them; a cluster alone proves only that something is worth investigating.

Thematic mapping before computers

Snow was working in a century that had recently learned to map abstractions. In 1826 Charles Dupin shaded a map of France by department according to how many men could read, producing what is generally counted as the first choropleth map: no rivers, no roads, just a quantity spread over administrative areas and darkened by degree. It is the same graphic you will build in Lesson 13, and you will find in Lesson 14 that its flaws were built in from birth.

In 1889 Charles Booth published a map of London poverty with every street colored by the income and character of the people living on it, from black at the bottom, labeled with words like semi-criminal, to gold for the wealthy. Booth's investigators walked the city with police officers and recorded conditions street by street over seventeen years. The map is a monument and a warning. It is enormously detailed, it drove real reform, and its categories are moral judgments recorded as if they were measurements. Every layer you will ever load carries some version of that problem.

1963: the first real GIS

The word overlay stayed literal for a century. Planners drew themes on transparent sheets and stacked them on a light table, and the darkest areas were the least suitable. That works for a small site and collapses for a country.

Canada tried it at national scale. The Canada Land Inventory needed to map the capability of an enormous land area for agriculture, forestry, wildlife, and recreation, and the manual measurement of all those polygons was estimated to require hundreds of person-years. In 1963 a young geographer named Roger Tomlinson persuaded the federal government to build a computer system instead. The Canada Geographic Information System, built with IBM, gave us the name and several ideas that never left: a drum scanner that turned map sheets into digital lines, the separation of geometry from attribute tables so each could be updated without the other, and the ability to compute area and overlay themes by machine. Tomlinson is called the father of GIS with reason.

In 1965 Howard Fisher founded the Harvard Laboratory for Computer Graphics and Spatial Analysis. Its early program SYMAP printed maps on line printers by overstriking characters to make shades of gray, which sounds like a toy and was not: it put analytical cartography into universities. The lab's later ODYSSEY system worked out much of the topological vector data structure that Module 3 will teach you. In 1969 Jack and Laura Dangermond left Harvard and founded a small land-use consultancy in Redlands, California, called the Environmental Systems Research Institute. Its 1982 product ARC/INFO says the era's whole idea in its name: ARC held the geometry, INFO held the attribute tables, and the software joined them.

Key idea: The founding architectural decision of GIS, separating geometry from attributes and linking them by an identifier, was made in the 1960s and still governs every file you will open.

Eyes and coordinates: satellites and GPS

Two independent technologies arrived to feed the systems. On 23 July 1972 NASA launched the satellite later renamed Landsat 1, beginning what is now more than fifty years of continuous imaging of the Earth's land surface, unbroken through Landsat 9 in 2021. No other environmental record on Earth is both that long and that consistent, which is why Lesson 12's change detection exercise is possible at all.

The Global Positioning System reached full operational capability in 1995, but civilian receivers were deliberately degraded by a policy called Selective Availability, which held ordinary accuracy near 100 meters. On 1 May 2000 the United States switched it off, and civilian accuracy jumped roughly tenfold overnight. Suddenly anyone could collect coordinates. Within a decade the receiver was in every phone, and the flow of spatial data reversed: the public stopped only consuming maps and started producing them.

Onto the desktop, then into the browser

Through the 1990s, GIS moved from mainframes to desktops, and the price of entry fell from an institution to a workstation. Esri published the shapefile specification in 1998, an act with consequences out of all proportion to its ambition: a format designed for one program in the early 1990s is still, twenty-five years later, the thing your data will probably arrive in, with all the limits Lesson 7 will make you suffer.

Then the field cracked open. Steve Coast started OpenStreetMap in 2004 because British mapping data was expensive and locked up; it is now edited by millions of registered contributors and is the base map behind an enormous amount of software. Google acquired a company called Keyhole in 2004 and released Google Earth in June 2005, and a generation discovered they could fly to their own roof. Free and open source tools grew alongside: GDAL, the translation library that quietly reads and writes almost every spatial format in existence, PostGIS, which taught the PostgreSQL database to store geometry, and QGIS, begun by Gary Sherman in 2002 and now a full professional desktop package that costs nothing.

The break in the story came on 12 January 2010, when an earthquake destroyed much of Port-au-Prince. Haiti was barely mapped in any open dataset. Volunteers around the world traced roads, buildings, and camps from donated satellite imagery into OpenStreetMap, and within weeks the crowd-built map had become the base map used by search and rescue and relief agencies on the ground. The Humanitarian OpenStreetMap Team came out of that response and now organizes mapping ahead of disasters instead of after them.

Key idea: Cheap positioning, free imagery, open formats, and open source software moved GIS from a room in a government building to anyone with a laptop and a question.

Where the work actually is

GIS is not a discipline so much as a layer under other disciplines. A sample of what people were paid to do with it this week:

• Public health. Snow's descendants. The Johns Hopkins dashboard that the world watched from January 2020 was a GIS application; so is every county map of vaccination rates, lead service lines, or pharmacy deserts.
• Utilities and telecom. Every water main, transformer, and fiber strand lives in a spatial database with an install date and a material. When a backhoe hits something, the map decides how fast it gets fixed.
• Emergency management. Address points route the ambulance. Fire agencies model spread with terrain, fuel, and wind; evacuation planning is a network problem.
• Agriculture. Yield monitors write a point every second, and variable rate applicators read a prescription raster to change fertilizer within a single field.
• Conservation. Habitat suitability models, corridor analysis, and protected area gap analysis are all overlay problems, and Module 5 teaches the method.
• Archaeology. Airborne lidar sees ground through forest canopy. A 2018 survey of about 2,100 square kilometers of the Guatemalan lowlands identified more than 60,000 previously unrecorded Maya structures, causeways, and field systems under jungle that had hidden them for a thousand years.
• Business and logistics. Site selection, delivery routing, trade area analysis, and insurance risk pricing are all spatial, and the last of these decides what you pay.

One thread runs through the list. In every case the analyst's real work is not operating software. It is knowing what the data means, what the operation assumes, and how wrong the answer could be.

Common misconceptions

• Snow's map proved cholera was waterborne. The map organized the evidence and pointed at the pump. The proof came from the exceptions, the brewery, the workhouse, the Hampstead widow, and from Whitehead's tracing of the index case; the authorities rejected the conclusion anyway.
• Removing the pump handle ended the epidemic. Deaths were already declining as residents fled the district. The handle mattered as evidence and as precedent, not as the cure.
• GIS began with the personal computer. The first true GIS ran in 1963 on a national mainframe, and its core design choice, geometry separate from attributes, is still in your files today.
• Old maps were merely primitive versions of ours. Booth's poverty map was more detailed than most modern social data and encoded moral judgments as classes. Modern layers do the same thing more quietly.
• OpenStreetMap is amateur data. Quality varies by place, which is a real caution, but humanitarian agencies, logistics companies, and national statistical offices use it, and in many countries it is the most current road network available at any price.

Recap

• Snow's 1854 dot map located a cholera outbreak around one pump; his 1855 version added a walking-distance boundary that contained nearly every death.
• The workhouse, the brewery, and the Hampstead widow were the exceptions that turned a cluster into a causal argument.
• Dupin's 1826 literacy map and Booth's 1889 London poverty map show thematic mapping, and its baked-in judgments, long before computers.
• The Canada Geographic Information System, begun in 1963 under Roger Tomlinson, was the first true GIS and separated geometry from attributes.
• The Harvard Lab, Esri's ARC/INFO in 1982, Landsat from 1972, and GPS with Selective Availability switched off in 2000 built the modern stack.
• OpenStreetMap, Google Earth, GDAL, PostGIS, and QGIS moved the field into browsers and onto free desktops; the 2010 Haiti response showed what that made possible.
• GIS is working equipment in public health, utilities, emergency management, agriculture, conservation, archaeology, and logistics.

Sources

1. Encyclopaedia Britannica. (2025). John Snow, British physician. Britannica.
2. UCLA Fielding School of Public Health. (n.d.). John Snow site: the Broad Street pump outbreak. UCLA.
3. United States Geological Survey. (n.d.). Landsat missions: fifty years of imaging the Earth. USGS.
4. National Coordination Office for Space-Based PNT. (n.d.). GPS: the Global Positioning System. GPS.gov.
5. OpenStreetMap Foundation. (n.d.). About OpenStreetMap. OpenStreetMap.

The big picture

In a cattle pasture in Osborne County, Kansas, a brass disk sits in concrete under a small fence. It is called Meades Ranch, and from 1927 until the middle of the 1980s every published coordinate in the United States traced back to it. The North American Datum of 1927 fixed a mathematical model of the Earth's shape to the real planet at that one point, and the entire national survey hung off it like a net pinned to a wall with a single nail.

That is a strange thing to do, and it explains a class of error that still costs people money. If you pull a coordinate off a 1970s map and type it into a phone, the phone will happily drop a pin, and the pin can be seventy meters from where the surveyor meant. Nothing is broken. The two numbers were measured against two different models of the Earth.

Key idea: A latitude and longitude is meaningless on its own. It is meaningful only with respect to a stated datum, which is the model of the Earth's shape and how that model is anchored to the ground.

Latitude and longitude are angles, not distances

Start with what the numbers are. Latitude is the angle north or south of the equator, from 0 at the equator to 90 at each pole. Longitude is the angle east or west of the prime meridian through Greenwich, from 0 to 180 in each direction. They are angles measured at the center of the Earth, which is why they behave strangely as distances.

Old records write angles in degrees, minutes, and seconds. There are 60 minutes in a degree and 60 seconds in a minute, exactly as in an hour. Converting to decimal degrees is division and addition:

30 degrees 16 minutes 02 seconds north becomes 30 + (16 / 60) + (2 / 3600) = 30 + 0.26667 + 0.00056 = 30.2672 north.

Going west or south, do the same arithmetic and then make the result negative. 97 degrees 44 minutes 35 seconds west becomes 97 + 0.73333 + 0.00972 = 97.7431, then negated: -97.7431. Those two numbers land on the Texas State Capitol. Drop the minus sign, as thousands of people do every year, and you land at 97.7431 east, which is in eastern Tibet. The sign error is not subtle in its consequences and is completely silent in its symptoms.

Two more traps live here. First, order. Humans say latitude first; the GeoJSON format specifies longitude first, as do many programming libraries, because they think in x then y. Half of all beginner mapping bugs are a swapped pair. If your points land in the Indian Ocean off the coast of Africa, at the intersection of the equator and the prime meridian, you have either swapped the pair or lost the values entirely. That spot is called Null Island for exactly this reason. Second, degrees are not a unit of length.

What a degree is worth on the ground

A degree of latitude is close to constant, about 111 kilometers, because meridians all run pole to pole. A degree of longitude shrinks as the meridians converge, following the cosine of the latitude: 111.32 kilometers times the cosine of your latitude.

Latitude	Cosine	One degree of longitude	One degree of latitude
0, the equator	1.000	111.3 km	110.6 km
30, New Orleans or Cairo	0.866	96.4 km	110.9 km
45, Minneapolis or Milan	0.707	78.7 km	111.1 km
60, Oslo or Anchorage	0.500	55.7 km	111.4 km
80, northern Greenland	0.174	19.3 km	111.7 km

This table is the reason you must never compute a distance or draw a buffer in unprojected degrees. A one-degree square in Norway is half as wide as a one-degree square in Ecuador and contains half the ground. Software will cheerfully return an answer in degrees, and the answer will be wrong by a factor that depends on how far north you are working.

The same table gives you a precision rule worth memorizing. Each decimal place you add to a coordinate divides the ground distance by ten.

Decimal places	Approximate ground precision	What it resolves
2	1.1 km	A town
3	110 m	A large city block
4	11 m	A building
5	1.1 m	A doorway
6	0.11 m	A survey mark
7 and beyond	Under 1 cm	Nothing your data supports

When a dataset gives you fifteen decimal places, it is reporting the precision of a floating point number, not the accuracy of a measurement. A phone GPS fix is good to a few meters on a good day. Storing that as 30.267194336891 is a claim you cannot back.

Key idea: Degrees are angles whose ground value changes with latitude, and reported decimal places describe storage precision, never measurement accuracy.

Three surfaces: terrain, geoid, ellipsoid

To attach numbers to a planet you first need a model of its shape, and the Earth offers three surfaces that all disagree.

The terrain is the real, rocky, mountainous surface. It is impossible to describe with an equation.

The geoid is the surface where gravity has the same potential everywhere, roughly the level the oceans would settle to if you cut canals through the continents. It is smooth compared with terrain but still lumpy, because gravity varies with the density of what lies beneath. The geoid rises to about 85 meters above the reference ellipsoid near Iceland and drops to about 106 meters below it in the Indian Ocean south of Sri Lanka. Water follows the geoid, which is why it is the reference for heights and for anything about flooding.

The ellipsoid is a mathematical solid: a sphere squashed at the poles. The modern standard, GRS80, gives the Earth an equatorial radius of 6,378,137 meters and a polar radius about 21 kilometers shorter, a flattening of about 1 part in 298. It has no lumps, which makes it the surface you can actually compute with. Latitude and longitude are defined on the ellipsoid.

Now the practical consequence. A GPS receiver naturally measures ellipsoidal height, the distance above that mathematical solid. A benchmark or a flood map uses orthometric height, the height above the geoid, which is what people mean by elevation above sea level. They are related by the geoid height N, and the relationship is one line: orthometric height equals ellipsoidal height minus geoid height. In the conterminous United States the geoid sits roughly 20 to 35 meters below the ellipsoid, so a receiver reporting a raw ellipsoidal height can be about 30 meters off from the elevation on the map. That is the difference between dry and flooded.

Key idea: The ellipsoid is computable, the geoid is where water goes, and mistaking one height for the other introduces tens of meters of vertical error.

Datums, and why yours does not match mine

A datum is an ellipsoid plus the recipe for tying it to the actual planet: where its center sits, how it is oriented, and which measurements defined that. Two families exist.

A local datum picks an ellipsoid that fits one region well and anchors it at a surface point. NAD27 used the Clarke 1866 ellipsoid pinned at Meades Ranch. It fit North America nicely and fit nowhere else, and its center is not the center of the Earth.

A geocentric datum puts the ellipsoid's center at the Earth's center of mass, which satellites can measure directly. WGS84, the datum GPS broadcasts in, and NAD83, the North American standard from 1986, are both geocentric.

Here is what that costs you in practice.

• NAD27 to NAD83 moves coordinates by tens of meters, and by more than 100 meters in parts of the western United States. Any coordinate from a pre-1990 map or deed needs transforming, and the shift is not a constant you can subtract; it varies across the country and must be computed with a grid-based transformation.
• NAD83 to WGS84 agreed to about a meter when NAD83 was defined. They have since drifted apart, because NAD83 is fixed to the North American tectonic plate and moves with it, while WGS84 tracks a global frame that stays put relative to the Earth as a whole. The plate carries North America roughly 2 centimeters a year, so the gap is now on the order of 1 to 2 meters. Invisible on a county map. Fatal when you are locating a gas line, a property corner, or a utility pole.

Australia made the same problem impossible to ignore. The continent moves about 7 centimeters a year, so by 2017 official Australian coordinates fixed in 1994 were nearly two meters from where satellites said the ground was. The country shifted its national datum on 1 January 2017 to close the gap. When self-driving cars and centimeter-grade agriculture arrive, a moving continent stops being a geodesist's curiosity.

The United States has been preparing a modernized reference system to replace NAD83 and NAVD88 for years, and the schedule has slipped more than once; check the National Geodetic Survey for the current state before you plan work around it.

Heights have datums too

Elevation carries the same problem one dimension down. The National Geodetic Vertical Datum of 1929 and the North American Vertical Datum of 1988 differ across the conterminous United States by a few tenths of a meter up to more than a meter, varying by location. Coastal work adds tidal datums, where mean lower low water, mean sea level, and mean higher high water can differ by several meters in a place with a big tide range. A sea level rise map is a statement about a vertical datum before it is a statement about the ocean, and papers have been rewritten because two agencies used different ones.

The single most common beginner mistake

Learn this distinction now and you will save yourself a bad afternoon.

To transform or reproject data is to recompute every coordinate so that the feature stays in the same physical place while the numbers change to suit a new system. To define or assign a coordinate system is to tell the software what the existing numbers already mean, changing no coordinate at all.

Use transform when the data is correctly labeled and you need it in a different system. Use define only when the data arrived with no label, or a wrong one, and you know from the metadata what it truly is. Misuse produces two signature failures. Define the wrong system and your county drifts hundreds of kilometers into the ocean while every coordinate value stays intact. Transform data whose label was already wrong and you apply a correct calculation to a false premise, moving everything by a small, plausible, hard-to-spot amount, which is worse.

In practice, coordinate systems are referenced by EPSG codes, short integers from a public registry: 4326 is WGS84 geographic, 4269 is NAD83 geographic, 3857 is the Web Mercator used by online basemaps, and 32614 is UTM zone 14 north on WGS84. When someone asks what projection your data is in, an EPSG number is a complete answer and a vague phrase is not.

Key idea: Transforming recomputes coordinates to preserve position; defining relabels coordinates and moves nothing. Choose the wrong one and your data lands in the ocean or, worse, one meter from the truth.

Common misconceptions

• Latitude and longitude are a universal address. They locate a point on a specific ellipsoid tied down a specific way. The same numbers on NAD27 and NAD83 are different places on the ground.
• WGS84 and NAD83 are the same thing. They were built to agree in the 1980s and have drifted 1 to 2 meters apart in North America because one is pinned to a moving plate. For survey and utility work that gap matters.
• More decimal places means a more accurate location. Precision is how finely you wrote the number down; accuracy is how close it is to the truth. Fifteen decimals on a three-meter GPS fix is false confidence stored in a database.
• The GPS elevation is the elevation. Receivers measure height above the ellipsoid. Height above sea level requires a geoid model, and in North America the correction is roughly 20 to 35 meters.
• A degree is a distance. A degree of longitude is 111 kilometers at the equator and 19 kilometers in northern Greenland. Any buffer or distance computed in raw degrees inherits that error.
• If the layers line up on screen, the coordinate systems must be right. Software reprojects on the fly for display, which can hide a mislabeled dataset until the moment you run an analysis or export the result.

Recap

• Coordinates are angles on a model of the Earth, and mean nothing without the datum that defines that model.
• Convert degrees-minutes-seconds by dividing minutes by 60 and seconds by 3600, then negate for west and south.
• A degree of latitude is about 111 km; a degree of longitude is 111.32 km times the cosine of latitude, which is why degrees are useless as a distance unit.
• Decimal places map to ground precision: four places is about 11 meters, five is about 1 meter, six is about 11 centimeters.
• Terrain, geoid, and ellipsoid are three different surfaces; heights above the last two differ by tens of meters.
• NAD27 is a local datum anchored at Meades Ranch; NAD83 and WGS84 are geocentric and now differ by 1 to 2 meters because plates move.
• Vertical datums matter as much as horizontal ones, especially for flooding and coastal work.
• Transforming recomputes coordinates; defining relabels them. EPSG codes name systems unambiguously.

Sources

1. National Oceanic and Atmospheric Administration. (n.d.). What is a geodetic datum? National Ocean Service. NOAA.
2. National Geodetic Survey. (n.d.). Geodesy, datums, and the National Spatial Reference System. NOAA. NOAA NGS.
3. IOGP. (n.d.). EPSG geodetic parameter dataset registry. EPSG.
4. Wikipedia contributors. (2025). Geocentric Datum of Australia. Wikipedia.
5. United States Geological Survey. (n.d.). Maps, imagery, and publications: coordinate systems. USGS.

The big picture

In 1569 Gerardus Mercator printed a world map on eighteen copperplate sheets that assembled into a wall map roughly two meters across, and put the reason for it in the title: it was corrected and adapted for the use of navigators. His trick was arithmetic. He stretched the spacing between parallels of latitude by exactly as much as the meridians were being stretched apart at that latitude, so that at any point the map stretched equally in both directions.

That single choice bought sailors something no other map offered. A line of constant compass bearing, a rhumb line, plots on Mercator's map as a straight line. Lay a ruler between Lisbon and Havana, read the angle, hold that heading, and you arrive. For three centuries that was worth any price in accuracy elsewhere, and the price was enormous.

Key idea: Every projection is a deliberate trade. Mercator bought straight compass courses and paid for them with area distortion that grows without limit toward the poles.

Why you cannot win

Peel an orange and try to flatten the peel without tearing or stretching it. You cannot. In 1827 Carl Friedrich Gauss proved why, in a result he liked well enough to call the remarkable theorem: the curvature of a surface is intrinsic to it, so a sphere and a plane cannot be mapped onto each other without deformation. Not with cleverness, not with more computing power, not ever.

So flattening the Earth always destroys something. Cartographers name four properties a map might preserve:

• Area. Equal-area, or equivalent, projections keep the relative sizes of regions correct. Every choropleth, every density calculation, every statement about how much land burned belongs here.
• Shape and angle. Conformal projections preserve angles locally, so small shapes look right and any two lines cross the map at their true angle. Navigation, surveying, and large-scale local mapping live here.
• Distance. Equidistant projections keep distances true, but only along certain lines or from certain points, never everywhere.
• Direction. Azimuthal projections keep bearings true from the map's center point outward.

The rule that matters: no projection can be both conformal and equal-area. That is not an engineering limitation waiting to be solved. It is a mathematical impossibility. Any map that shows continents at correct relative size is stretching their shapes, and any map with correct shapes is misrepresenting sizes somewhere.

Nicolas Tissot gave the field a way to see this. Imagine drawing tiny circles all over the globe, then projecting them. On a conformal map the circles stay circles but swell or shrink. On an equal-area map they keep their area but squash into ellipses. Those projected circles, called Tissot's indicatrix, are the standard visual audit of a projection's damage.

Key idea: Preserving area and preserving shape are mathematically exclusive, so the question is never which projection is accurate but which distortion your analysis can tolerate.

Mercator's distortion, in numbers

You can compute Mercator's error yourself with one trigonometric function, and doing so once permanently changes how you look at a world map.

On a normal Mercator projection, the east-west stretch at latitude L is 1 divided by the cosine of L. Because the projection is conformal, the north-south stretch must be the same. Area is length times length, so the area is exaggerated by the square of that factor: 1 divided by the cosine squared of the latitude.

Latitude	Cosine	Linear stretch	Area exaggeration
0 (equator)	1.000	1.0	1.0 times
30	0.866	1.15	1.3 times
45	0.707	1.41	2.0 times
60	0.500	2.00	4.0 times
70	0.342	2.92	8.5 times
80	0.174	5.76	33 times
85	0.087	11.5	132 times
90 (pole)	0.000	infinite	infinite

Now apply it. Greenland has an area of about 2.17 million square kilometers and sits roughly between 60 and 83 degrees north, centered near 72 degrees. Africa covers about 30.3 million square kilometers and straddles the equator, so its average exaggeration is close to 1. The cosine of 72 degrees is 0.309; squared, that is 0.0955; one divided by that is about 10.5. So Mercator inflates Greenland roughly tenfold while leaving Africa nearly alone.

Africa is genuinely about 14 times the size of Greenland. On Mercator, Greenland's ten-times inflation cuts the visible ratio to about 1.3 to 1, which is why generations of students have believed the two are comparable.

Run the same arithmetic on Alaska and Brazil for a second shock. Alaska is about 1.72 million square kilometers, centered near 64 degrees north; the cosine of 64 is 0.438, squared is 0.192, so the exaggeration is about 5.2. Multiply: Alaska appears as roughly 8.9 million square kilometers of map. Brazil covers 8.5 million square kilometers near the equator and appears at nearly its true size. On Mercator they look like twins. In reality Brazil is about five times larger.

Key idea: Mercator's area error is one divided by the cosine squared of the latitude, which is 4 times at 60 degrees and 33 times at 80, so any polar or high-latitude area read off a Mercator map is fiction.

The argument about Peters, told fairly

In 1973 the German filmmaker Arno Peters called a press conference to present a world map he said corrected the Eurocentrism of Mercator. His map showed every country at its correct relative area, and by his account it restored visual justice to the tropics, where most of the world's people and most of its formerly colonized nations live. Aid agencies, church organizations, and UNESCO adopted it, and it is still sold as the Peters projection.

Cartographers responded with two concrete objections rather than a defense of Mercator. The first was priority: the projection was published in 1855 by the Scottish clergyman James Gall, and it is now usually called Gall-Peters. The second was that equal-area world maps were not scarce. Mollweide published one in 1805, Lambert in 1772; the profession had dozens, most of which distorted shapes less brutally than Gall-Peters does, where equatorial landmasses are stretched vertically into taffy and high latitudes squashed flat. In 1989 seven North American geographic organizations passed a joint resolution urging that rectangular world maps in general, Mercator and Gall-Peters alike, be abandoned for general reference.

What survives the argument is the part worth keeping. Mercator on a classroom wall does misinform, not from conspiracy but from a navigation tool being used as a picture of the world. And equal-area is the correct requirement for any map making a claim about how much. The specific projection Peters promoted is one of the weaker ways to satisfy that requirement. Winkel Tripel, adopted by National Geographic in 1998 after a decade with Robinson, splits the difference as a compromise projection that is neither conformal nor equal-area but keeps everything reasonably close. Equal Earth, published in 2018 by Bojan Savric, Tom Patterson, and Bernhard Jenny, gives true areas with pleasant shapes and has been widely adopted since, including by NASA for global datasets.

Choosing on purpose

Three questions decide a projection, in order.

What does the analysis need? If you will compute area, use an equal-area projection, full stop. If you need correct local shapes and angles, as in surveying, engineering plans, or navigation, use a conformal one. If everything depends on distance and direction from a single site, a radio tower or an earthquake epicenter, use azimuthal equidistant centered on that point.

What shape is your study area? Regions wide east to west in the middle latitudes suit a conic projection, which is why Lambert conformal conic and Albers equal-area conic are the workhorses for the United States, Europe, and China. Regions long north to south suit a transverse cylindrical projection, which is exactly what UTM is. Round or polar regions suit an azimuthal projection touching the middle of the area.

What do others expect? If your state publishes in a State Plane zone and every agency you exchange data with uses it, that convention has real value.

Projection	Preserves	Standard use
Mercator	Shape and angle, badly wrong on area	Marine navigation, and nothing else
Transverse Mercator	Shape, within a narrow north-south band	UTM zones, national grids
Lambert conformal conic	Shape and angle across mid-latitude bands	Aeronautical charts, State Plane east-west zones
Albers equal-area conic	Area	United States thematic and statistical maps, EPSG 5070
Azimuthal equidistant	Distance and direction from the center	Radio range, seismic distance, the United Nations emblem
Winkel Tripel	Nothing exactly; balances all errors	General world reference maps
Equal Earth	Area, with acceptable shapes	Global thematic mapping since 2018

Web Mercator, the one you use every day

Every major online basemap draws in Web Mercator, registered as EPSG 3857. It exists because Mercator is conformal and square-friendly: north stays up everywhere, the world fits a square tile scheme that quarters neatly at each zoom level, and shapes look right at street scale where users spend their time. It truncates at about 85.05 degrees north and south, because going further would require infinite height, which is why online maps have no Antarctica worth speaking of and why the Arctic looks so vast.

It also has a quirk that offends geodesists: it applies spherical formulas to ellipsoidal coordinates, which is fine for display and can be off by tens of kilometers if you treat it as a survey-grade system. So: perfectly good for showing things, unacceptable for measuring them. Measure Greenland's area in Web Mercator and the software will hand you a number roughly ten times too large, without complaint. The habit to build is simple. Reproject to an equal-area system before any area or density calculation, then reproject the result back for display if you like.

Key idea: Web Mercator is a display system. Compute areas and distances in a projection chosen for your region, then map the answer in whatever looks good.

Common misconceptions

• Some projection is finally accurate. Gauss proved otherwise in 1827. Every flat map distorts; you choose which property to protect.
• Mercator was designed to make Europe look important. It was designed in 1569 so that a compass bearing would plot as a straight line, which is a genuine and hard problem. The harm came from hanging a navigation chart on a classroom wall for four hundred years.
• The Peters projection is a modern correction. The same projection was published by James Gall in 1855, and equal-area world maps with better shapes existed decades earlier. The critique of Mercator was fair; the specific replacement was not new or best.
• Projection only affects how the map looks. It changes every computed area, length, and angle. An area calculation in Web Mercator can be wrong by a factor of ten, and the software will not warn you.
• Conformal means shapes are right everywhere. Conformality is local. Angles at a point are preserved; the outline of a continent spanning many degrees is still deformed.
• Choosing a projection is a cartographer's aesthetic decision. It is an analytical decision that determines whether your numbers are true, made before the map exists.

Recap

• A curved surface cannot be flattened without distortion; Gauss proved it in 1827.
• Projections may preserve area, shape and angle, distance, or direction, but never all of them, and never both area and shape.
• Mercator exaggerates area by one over the cosine squared of latitude: 2 times at 45 degrees, 4 at 60, 33 at 80.
• Greenland is inflated about tenfold, making it look comparable to Africa, which is actually 14 times larger; Alaska and Brazil look equal though Brazil is five times bigger.
• Gall-Peters is equal-area and badly distorts shape; Winkel Tripel and Equal Earth are the modern general-purpose choices.
• Choose by analytical need, then by the shape of the study area, then by convention: equal-area conic for United States statistics, conformal for local engineering, azimuthal equidistant for distance from a point.
• Web Mercator is for display only; reproject before measuring anything.

Sources

1. Encyclopaedia Britannica. (2025). Map projection. Britannica.
2. Wikipedia contributors. (2025). Mercator projection. Wikipedia.
3. Wikipedia contributors. (2025). Equal Earth projection. Wikipedia.
4. NOAA Office of Coast Survey. (n.d.). Nautical charts and chart projections. NOAA.

The big picture

Unfold a United States Geological Survey topographic quadrangle and read the small print in the bottom margin. It says 1:24,000, and it says the sheet covers 7.5 minutes of latitude by 7.5 minutes of longitude, which is somewhere between 49 and 70 square miles depending on how far north you are. About 55,000 of these sheets, drawn one at a time over decades, were needed to cover the lower 48 states.

Now the number that matters and that almost nobody reads. Under the old National Map Accuracy Standards, a map at that scale promises that 90 percent of well-defined points fall within one fiftieth of an inch of their true position on the sheet. One fiftieth of an inch at 1:24,000 is 480 inches on the ground, which is 40 feet, about 12 meters. That is the guarantee. The stream on the paper may be twelve meters from the stream in the field, and the map is still meeting its specification.

Key idea: Every dataset has a scale, and its scale sets a hard ceiling on the questions it can honestly answer.

Projected coordinates: from angles to meters

Lesson 3 left you with angles. Angles are awkward: they need trigonometry to become distances, and their ground value changes with latitude. So most analysis happens in a projected coordinate system, where positions are plain x and y measured in meters or feet on a flat grid. Distance becomes the Pythagorean theorem again, area becomes multiplication, and buffers become honest circles.

The vocabulary is worth pinning down. The x coordinate is called the easting, the y coordinate the northing. Because negative coordinates invite sign errors, projected systems add a false easting and sometimes a false northing, constants large enough to keep every value in the region positive. A false easting is not a fudge factor. It is a deliberate offset stated in the system's definition.

UTM: sixty narrow windows on the world

The Universal Transverse Mercator system solves the distortion problem by refusing to cover much at a time. It slices the world into 60 zones, each 6 degrees of longitude wide, running from 80 degrees south to 84 degrees north, and gives each zone its own transverse Mercator projection with a central meridian down its middle.

Finding your zone is arithmetic you can do in your head. Add 180 to your longitude, divide by 6, drop the remainder, add 1. For Austin, Texas, at -97.7431: add 180 to get 82.2569, divide by 6 to get 13.71, take the whole number 13, add 1, and Austin is in zone 14, northern hemisphere. As an EPSG code that is 32614 on WGS84, or 26914 on NAD83.

Two constants define the zone's grid. The central meridian is given a false easting of 500,000 meters, so eastings run roughly from 160,000 to 840,000 within the zone and never go negative. Northings count meters from the equator; in the southern hemisphere a false northing of 10,000,000 meters is added so those stay positive too. That is why a southern hemisphere northing near 6,000,000 and a northern one near 4,000,000 can both be valid and must never be mixed.

Then the clever part. The projection is not tangent to the central meridian but slightly secant, with a scale factor of 0.9996 applied at the center. That makes the center slightly too small, exactly 1.0 correct along two lines about 180 kilometers either side of it, and slightly too large at the zone edges. Spreading the error like that holds distortion to about 1 part in 2,500 anywhere in the zone, which is 40 centimeters per kilometer. For most mapping and engineering that is nothing.

The price is seams. A project spanning two zones has two grids that do not join, and coordinates from adjacent zones cannot be compared without reprojection. States that straddle a zone boundary, and many do, live with this constantly. When someone hands you eastings and northings without a zone number, you have been handed nothing: the same pair of numbers is valid in all sixty zones and names sixty different places.

Key idea: UTM trades global coverage for low distortion by working in 6-degree slices, so a UTM coordinate without its zone and hemisphere is unusable.

State Plane: distortion under one part in ten thousand

The United States went further in the 1930s. The Coast and Geodetic Survey built the State Plane Coordinate System with a design target that surveyors could work with: distortion under 1 part in 10,000, roughly half a foot per mile. Meeting that meant many small zones, more than 120 across the states, with big states split into several.

The zone shapes follow the same logic as any projection choice. States wider east to west than north to south, such as Tennessee, use Lambert conformal conic zones. States taller than wide, such as Illinois, use transverse Mercator zones. Alaska adds an oblique Mercator zone for the panhandle, whose axis runs diagonally.

State Plane carries a trap worth knowing before it costs you a day. Many zones are published in feet, and there are two feet. The US survey foot is 1200/3937 of a meter; the international foot is exactly 0.3048 meters. They differ by 2 parts per million, which sounds harmless until you remember that State Plane northings run into the millions. At a northing of 3,000,000 feet the two definitions disagree by about 6 feet. The National Institute of Standards and Technology retired the US survey foot at the end of 2022, but decades of data were captured in it, so the ambiguity will outlive everyone reading this. If a dataset is in State Plane feet, find out which foot.

Scale, and why the words feel backwards

Map scale is a ratio: one unit on the map equals so many of the same units on the ground. A representative fraction of 1:24,000 means one inch on paper is 24,000 inches on the Earth, and it does not matter whether you measure in inches, centimeters, or thumbs.

Convert it and it becomes useful. At 1:24,000, one inch equals 24,000 inches, which is 2,000 feet. In metric, one centimeter equals 24,000 centimeters, which is 240 meters. At 1:100,000, one centimeter is exactly one kilometer, which is why that scale is popular.

The vocabulary trips everyone once. Large scale means a large fraction, so 1:1,200 is large scale: a small area shown in great detail. Small scale means a small fraction, so 1:1,000,000 is small scale: a huge area with little detail. A site plan is large scale. A world map is small scale. It feels backwards because we think about the area covered, but the term describes the fraction.

Scale	One centimeter equals	Typical use	Detail level
1:1,200	12 m	Site plans, utility as-builts	Individual manholes
1:24,000	240 m	USGS topographic quadrangle	Buildings, streams, trails
1:100,000	1 km	County and regional planning	Road networks, main streams
1:1,000,000	10 km	State and national overviews	Major highways and cities
1:10,000,000	100 km	Continental reference	Countries and capitals

Generalization: what small scale does to the truth

Shrink the scale and features cannot simply shrink with it, or they would vanish. Cartographers apply generalization, a set of deliberate distortions with names: selection, dropping features entirely; simplification, removing vertices from a wiggly line; smoothing; aggregation, merging many buildings into a built-up area; and displacement, physically moving features apart so both remain visible.

Do the arithmetic on displacement and it stops being abstract. The thinnest line a printer or screen renders legibly is around half a millimeter. At 1:1,000,000, half a millimeter on the map is 500 meters on the ground. A two-lane road is about 10 meters wide, so its symbol is fifty times too wide, and where a road and a railway run side by side through a canyon, the cartographer must shove them apart to keep both visible. At that scale, the position of a road can be off by hundreds of meters by design, and the map is not wrong. It is generalized.

This is why scale is a property of the data, not just the display. Zooming into 1:100,000 hydrography until it fills your screen at 1:2,000 does not add creeks that were never digitized. The pixels get bigger; the information does not.

Key idea: Generalization removes, merges, smooths, and displaces features on purpose, so data captured at one scale carries built-in errors that no amount of zooming will remove.

Resolution, precision, and accuracy are four different things

These words get used interchangeably and mean entirely different things.

• Scale is the ratio between map and ground, and for vector data it records the detail at which features were captured.
• Resolution is the size of the smallest thing represented. For a raster it is the cell size: a 30 meter Landsat pixel cannot show a driveway. For vector data the analogous idea is the minimum mapping unit, the smallest polygon the producer bothered to record.
• Precision is how finely a measurement is recorded: the number of decimal places, or how repeatable your instrument is.
• Accuracy is how close the value is to the truth. A GPS unit that reports the same wrong position to nine decimals every time is precise and inaccurate.

Resampling a 30 meter raster to 5 meter cells produces a file thirty-six times larger containing not one additional fact. The cells are smaller; the resolution, in the sense of what can be seen, is unchanged. The same illusion is why a blurry photograph enlarged is still blurry.

Modern positional accuracy is reported by the National Standard for Spatial Data Accuracy, which states a root mean square error and a radius at 95 percent confidence rather than the old pass-or-fail rule. Either way, the point stands: ask what accuracy the data claims before you use it to decide where a fence goes.

Common misconceptions

• Large scale means a big area. It means a big fraction and therefore a small area in great detail. 1:1,200 is large scale; 1:1,000,000 is small.
• Zooming in increases detail. Zooming changes display scale only. Data captured at 1:100,000 stays 1:100,000 no matter how large it is drawn.
• UTM coordinates are globally unique. The same easting and northing occur in every one of the 60 zones and in both hemispheres. Without the zone, the numbers are meaningless.
• The 0.9996 scale factor is an error. It is deliberate. Shrinking the central meridian slightly balances the distortion across the zone instead of letting it all pile up at the edges.
• A foot is a foot. The US survey foot and the international foot differ by 2 parts per million, which is about 6 feet at a State Plane northing of 3 million.
• More decimal places means better data. Precision is not accuracy. A 1:24,000 map is accurate to roughly 12 meters however many decimals its digitized coordinates carry.

Recap

• Projected systems replace angles with meters or feet on a flat grid, using false eastings to keep values positive.
• UTM divides the world into 60 zones 6 degrees wide; find yours by adding 180 to the longitude, dividing by 6, and adding 1.
• UTM applies a 0.9996 scale factor at the central meridian to spread distortion, holding error near 1 part in 2,500 across the zone.
• State Plane uses over 120 small zones to hold distortion under 1 part in 10,000, and its foot units come in two incompatible flavors.
• Scale is a fraction: 1:24,000 means one inch equals 2,000 feet, and large scale means small area with high detail.
• Generalization deliberately drops, merges, smooths, and displaces features as scale decreases, so small-scale data carries built-in positional error.
• Scale, resolution, precision, and accuracy are four separate properties, and a 1:24,000 quadrangle is accurate to about 12 meters.

Sources

1. United States Geological Survey. (n.d.). Topographic maps and the US Topo product. USGS.
2. National Geodetic Survey. (n.d.). State Plane Coordinate System. NOAA. NOAA NGS.
3. National Institute of Standards and Technology. (2020). US survey foot: revised unit conversion factors. NIST.
4. Wikipedia contributors. (2025). Universal Transverse Mercator coordinate system. Wikipedia.

The big picture

Around 1951 Lewis Fry Richardson was testing whether countries with longer shared borders went to war more often, and he needed border lengths. Looking them up, he found that Spain gave its border with Portugal as 987 kilometers while Portugal gave the same border as 1,214 kilometers. The Netherlands and Belgium disagreed about theirs by a similar margin. Nobody was lying. They had measured with different ruler lengths.

Benoit Mandelbrot picked this up in a 1967 paper in Science with a title that is also the whole problem: how long is the coast of Britain? Walk it with 200 kilometer dividers and you get about 2,400 kilometers. Walk it with 50 kilometer dividers, which follow bays your longer dividers stepped across, and you get about 3,400. Shrink the ruler further and the number keeps climbing, because a coastline reveals new wiggles at every scale.

Here is what that means for you as soon as you open a file. The length of a coastline is not a fact about the coast. It is a fact about the dataset. When your software reports a perimeter to two decimal places, it is reporting how many vertices someone digitized, dressed up as a measurement of the world.

Key idea: Vector features are sampled approximations of continuous shapes, so lengths and perimeters are properties of the data's capture scale rather than properties of the world.

Three geometry types, and why the choice is yours

The vector data model represents the world as discrete features with sharp edges, each stored as coordinates.

• A point is a single coordinate pair: a well, a crash, a bird sighting, an address.
• A line, more precisely a polyline, is an ordered list of coordinates connected by straight segments: a road centerline, a stream, a power line. Curves are approximated by many short segments.
• A polygon is a closed ring of coordinates enclosing an area: a parcel, a lake, a census tract, a burn scar.

Two refinements matter in practice. A polygon may have interior rings, holes, which is how you store a lake with an island in it or a county containing an independent city. And a feature may be multipart: Hawaii is one state and one row in the attribute table, whose geometry contains several separate polygons. Multipart features surprise beginners who count rows and think they have counted islands.

Now the part that trips people up. Nothing in the world is inherently a point. A city is a point on a world map and a polygon on a county map. A river is a line at 1:100,000 and a polygon with two banks at 1:2,000. A road is a line for routing and a polygon for computing impervious surface. Geometry type is a modeling decision driven by scale and by the question you intend to ask, and the same phenomenon is legitimately stored both ways in different layers of the same project.

Key idea: Geometry type encodes a decision about scale and purpose, not a truth about the feature, so ask what the layer is for before judging how it is stored.

Vertices, and the arithmetic of detail

Every vector feature is just a list of vertices. More vertices means a closer approximation, a bigger file, and slower analysis. Less means the opposite. Two operations manage this trade.

Simplification removes vertices while keeping the general shape. The standard method, published by David Douglas and Thomas Peucker in 1973, draws a straight line between a feature's endpoints, finds the vertex farthest from that line, and keeps it only if it is farther than a tolerance you choose, then recurses on the two halves. Set the tolerance to 50 meters on a county boundary layer and file size can fall by 90 percent with no visible change on a state map. Set it too high and neighboring polygons that shared a boundary get simplified independently and pull apart, leaving gaps and overlaps along a line that used to be shared. Simplify a boundary layer with a topology-aware tool, not a plain one.

Densification does the reverse, inserting vertices along long segments. It matters when reprojecting: a straight line between two vertices 500 kilometers apart in one projection is not a straight line in another, so a state boundary can visibly bow after reprojection unless it was densified first.

Topology: how features know about each other

Consider two adjacent counties. In the simplest storage, called spaghetti, each county is a closed ring of coordinates, and the shared boundary is stored twice: once clockwise in county A, once counterclockwise in county B. Nothing records that they are the same line. If an editor moves one vertex, the counties overlap or a gap opens between them, and the file will not object.

Topology is the explicit storage of spatial relationships: which line segments bound which polygons, which lines connect at which nodes, what contains what. The shared boundary is stored once, and both counties reference it, so moving a vertex moves it for both.

The three relationships that matter are:

• Adjacency: which polygons share an edge. Needed for spatial statistics such as Moran's I in Lesson 14.
• Connectivity: which lines join at which nodes. Without it there is no routing, because the software cannot tell a genuine intersection from two lines that merely cross on screen.
• Containment: which features fall inside which polygons.

Even in formats that store spaghetti, you will meet topology as a set of rules the data must satisfy. The classics: polygons in a coverage must not overlap and must not leave gaps; lines must not have dangles, endpoints not connected to anything, which appear as an undershoot when a digitized street stops short of the intersection or an overshoot when it crosses past. A stream network with three undershoots will route water into a dead end, and the map will look perfect.

The commonest topological wound is the sliver. Two agencies digitize the same county boundary independently. Intersect the layers and along that shared line you get several hundred long, thin polygons two centimeters wide, each with a real area and a full attribute record. They will pollute every count and area sum downstream until you find and remove them, usually with a rule such as delete polygons under a threshold area or with a very high perimeter-to-area ratio.

Key idea: Topology stores relationships rather than repeating coordinates, which is what makes routing possible and slivers, gaps, and dangles preventable.

When geometry is simply invalid

Some geometries break the rules of what a polygon is. A ring that crosses itself, called a bowtie, has no well-defined inside. A ring that does not close, or an interior ring outside its exterior ring, is nonsense. Software often draws these anyway and then fails partway through an intersect with a message about invalid geometry.

The workflow is always the same: run the validity check, read where the errors are, then run the fix tool, then check again. Do this the day you receive a layer, not the afternoon a deadline is due. In QGIS the tools are Check Validity and Fix Geometries; in a PostGIS database the equivalent test is a validity function applied to the geometry column. The habit matters more than the button.

The attribute table is the other half of the layer

Every vector layer carries a table: one row per feature, one column per field, plus the geometry. Field types are the usual database ones, and choosing them badly costs you later.

Type	Holds	Trap
Integer	Whole numbers, counts, codes	Storing a ZIP or FIPS code as integer destroys leading zeros
Real or double	Measurements, rates	Never test two reals for exact equality
Text or string	Names, codes, categories	Fixed widths truncate silently in older formats
Date	Calendar dates	Some formats keep the date and drop the time
Boolean	True or false	Not supported in every format; often stored as 0 and 1

One distinction deserves its own paragraph because it silently ruins analyses. Null is not zero. Zero rainfall means a gauge recorded no rain. Null means no gauge reported. Average a column where the nulls were filled with zeros and you have invented a drought. When you receive data, look for the code the producer used for missing values, which is often a null, sometimes an empty string, and depressingly often -9999 or -999. Feed -9999 into a mean and the result is not merely wrong, it is absurd, which at least makes it easy to catch.

Joins: attaching a table to a map

Most of the interesting attributes you want are not in the spatial layer. Boundaries come from one agency and statistics from another, and you connect them with a join: match rows in a table to features in a layer using a shared key.

In the United States that key is usually a FIPS code or GEOID. States get two digits, 48 for Texas and 06 for California. Counties get five: the state's two plus three of their own. Census tracts get eleven, adding six more. The codes nest, which means you can aggregate counties into states by taking the first two characters, a trick worth remembering.

Here is the workflow, and then the four ways it fails.

1. Download the boundary layer, for example census tracts from the Census Bureau's TIGER/Line files, and find its key field, usually named GEOID.
2. Download the data table, for example median household income by tract, and find its key field.
3. Confirm the two keys are the same type and same format.
4. Join the table to the layer on that field, then check how many rows matched.
5. Export to a new file, because in most software a join is a live view that disappears when the project closes.

The four failures, in order of how often they waste an afternoon:

• Leading zeros gone. A spreadsheet opened the CSV, saw 06075 as a number, and stored 6075. Alabama, Alaska, Arizona, Arkansas, California, Colorado, and Connecticut all begin with 0, so exactly the alphabetical front of the country breaks. Fix by importing the key column as text before anything else touches it.
• Type mismatch. One key is text, the other integer. The join runs and matches zero rows, usually without an error worth reading.
• Wrong vintage. Census geography changes. Tract boundaries from 2010 and data tabulated for 2020 tracts share a field name and do not describe the same polygons. Match the vintage.
• Silent partial match. The join succeeds for 2,000 of 2,300 rows and the map looks fine, with 300 quietly empty features. Always count matched records and look at the unmatched ones. They are frequently a real signal, such as water-only tracts with no population.

A join can be one to one, one table row per feature, or one to many, several rows per feature, as when a parcel has three owners of record. A one-to-many join duplicates the geometry once per matching row, so a layer of 1,000 parcels can become 2,400 features. If your feature count jumped after a join, this is why.

Key idea: Joins are where most GIS projects actually break, and the fixes are mundane: keys as text, matching vintages, and always counting how many rows matched.

Common misconceptions

• A river is a line. A river is whatever the layer's purpose requires. Line for network tracing, polygon for area and width, point for a gauging station.
• The perimeter reported by the software is the true perimeter. It is the perimeter of the digitized approximation. Finer data gives a longer coastline, forever.
• Simplification is harmless cleanup. Simplifying adjacent polygons separately tears shared boundaries apart. Use topology-preserving tools on any layer whose features touch.
• If two lines cross on screen they are connected. Without a shared node, they are two lines passing at different notional heights. Routing will refuse to turn there, correctly by its own logic.
• Null and zero are interchangeable. One is a measured absence, the other is an absence of measurement, and confusing them corrupts every average you compute.
• A join that produced a map worked. Partial joins draw beautifully. Count the matches every time.

Recap

• Vector data stores discrete features as coordinates: points, polylines, and polygons, with interior rings and multipart features as refinements.
• The coastline paradox means measured length depends on the ruler, so lengths and perimeters describe the data's capture scale.
• Douglas-Peucker simplification cuts vertices to a tolerance; use topology-aware versions on layers whose polygons share boundaries.
• Topology stores adjacency, connectivity, and containment explicitly, preventing slivers, gaps, dangles, undershoots, and overshoots.
• Invalid geometries such as self-intersecting rings must be detected and fixed before analysis, not during it.
• The attribute table carries typed fields; null is not zero, and codes like -9999 mean missing.
• Joins connect tables to features by key, usually FIPS or GEOID, and fail through lost leading zeros, type mismatches, wrong vintages, and unnoticed partial matches.

Sources

1. United States Census Bureau. (n.d.). TIGER/Line shapefiles. Census Bureau.
2. United States Census Bureau. (n.d.). Understanding geographic identifiers (GEOIDs). Census Bureau.
3. Wikipedia contributors. (2025). Coastline paradox. Wikipedia.
4. QGIS Project. (n.d.). QGIS documentation: working with vector data. QGIS.

The big picture

A single Landsat scene covers about 185 kilometers by 180 kilometers of ground. Divide by the 30 meter cell size and you get 6,167 columns by 6,000 rows, which is roughly 37 million cells. Each cell holds a 16 bit number, so that is about 74 megabytes. For one band. The satellite records eleven, and a delivered scene runs near a gigabyte before you have analyzed anything at all.

Now halve the cell size to 15 meters. You do not double the file; you quadruple it, because you double both rows and columns. That single relationship, storage scaling with the square of resolution, governs every decision in raster work: what you can download, what you can process on a laptop, and how long you will wait.

Key idea: A raster is a grid of numbers, and halving the cell size multiplies the data by four, so resolution is a budget you spend rather than a setting you maximize.

What a raster actually is

Where vector data stores features, the raster model stores a rectangular grid of cells, each holding one value. That value can be an elevation in meters, a reflectance, a temperature, a rainfall total, or a code standing for a land cover class.

The georeferencing is startlingly simple, which is why rasters are fast. A raster does not store a coordinate per cell. It stores six numbers: the coordinate of the upper-left corner, the cell width, the cell height, and two rotation terms that are almost always zero. From those, the position of any cell is arithmetic. The cell in row 400, column 250 of a grid whose upper-left corner is at easting 500,000, northing 4,300,000 with 30 meter cells sits at easting 500,000 plus 250 times 30, and northing 4,300,000 minus 400 times 30. Note the minus: rows count downward from the top while northings increase upward, which is why the vertical cell size is conventionally stored as a negative number and why an upside-down raster is a classic symptom of a hand-built world file.

Rasters come in two flavors, and confusing them causes real damage.

• Continuous rasters hold measurements that vary smoothly: elevation, slope, temperature, reflectance, distance. Interpolating between neighboring cells is meaningful.
• Discrete or categorical rasters hold codes: 41 for deciduous forest, 21 for developed open space. The numbers are labels, not quantities. The average of forest and water is not a wetland.

Every raster also needs a NoData value, a flag meaning this cell has no measurement, distinct from a cell whose measurement happens to be zero. Sea level and no data are both easily written as 0, and the results of confusing them look plausible.

Resampling: the operation that quietly ruins categorical data

Any time a raster is reprojected, snapped to a different grid, or changed in resolution, its values must be recomputed onto new cell centers. Three methods dominate.

Method	How it works	Use for
Nearest neighbor	Copies the value of the closest input cell	Categorical data, always; preserves the exact codes
Bilinear	Weighted average of the 4 surrounding cells	Continuous data such as elevation or reflectance
Cubic convolution	Weighted fit through 16 surrounding cells	Continuous data where visual smoothness matters

The rule has no exceptions: never use bilinear or cubic on a categorical raster. Average land cover class 41 with class 11 and the software hands back 26, which is a real class code for a completely unrelated cover type. The map will look fine. Every subsequent count will be wrong, and the errors will cluster along class boundaries, which is exactly where analyses tend to care.

Key idea: Resample continuous rasters with bilinear or cubic and categorical rasters with nearest neighbor, because averaging class codes invents categories that were never there.

Raster or vector? Ask what the phenomenon is like

Consideration	Vector	Raster
Suits	Discrete objects with edges: parcels, roads, wells	Continuous fields: elevation, temperature, imagery
Attributes	Many fields per feature	One value per cell per band
Precision	Coordinates as precise as captured	Limited by cell size; boundaries become staircases
Overlay	Geometric intersection, computationally heavy	Cell-by-cell arithmetic, very fast
Storage	Scales with feature count and vertices	Scales with area divided by cell size squared
Networks	Natural, with topology	Awkward; requires cost surfaces

You will convert between them constantly. Rasterizing a polygon layer turns parcels into a grid, and a cell that is 60 percent inside a parcel is assigned wholly in or wholly out, which is why rasterized boundaries have a staircase edge and why rasterized areas differ slightly from vector areas. Vectorizing a raster traces cell boundaries into polygons and produces exactly those staircases as geometry, which is honest but ugly; smoothing afterward is cosmetic and moves lines.

The formats you will actually meet

Format	What it is	Watch out for
Shapefile (.shp)	The universal vector exchange format since the 1990s	At least three files that must travel together, a 2 GB limit, field names capped at 10 characters, weak null support, one geometry type per file
GeoPackage (.gpkg)	An OGC standard: one SQLite file holding many vector layers, tiles, and tables	Almost none; this is the sensible modern default
GeoJSON (.geojson)	Plain text vector, readable by humans and browsers	Requires WGS84 coordinates by the standard; files grow enormous and have no spatial index
KML / KMZ	Google Earth display format, always in WGS84	Built for presentation, not analysis; styling is mixed into the data
GeoTIFF (.tif)	The standard self-describing raster format	Check the compression and whether it is tiled; a cloud optimized GeoTIFF allows reading a window without downloading the whole file
NetCDF / HDF	Multidimensional arrays with time and depth, standard for climate and satellite products	Needs its own vocabulary; a variable may hold decades of daily grids
LAS / LAZ	Lidar point clouds, one record per return	Enormous; usually processed into a DEM before use
CSV with coordinates	A table with latitude and longitude columns	Carries no coordinate system; you must state it on import, and column order tricks are common

The shapefile deserves special contempt and continued use. It was designed for one product in the early 1990s and became the lingua franca anyway. Its ten-character field name limit is the source of an entire genre of mystery: export a layer with a field called POPULATION_2020 and it silently becomes POPULATIO, and if you also had POPULATION_2010 you now have POPULATIO and POPULATI_1, and next week nobody knows which is which. Use GeoPackage for your own work and export to shapefile only when someone insists.

Metadata: the difference between data and a liability

A file named parcels_final_v3_USE_THIS.shp on a shared drive, with no accompanying documentation, is not an asset. Nobody can tell what year it represents, who made it, what the field ZONE_C means, or whether it may be published.

Formal metadata standards exist for this: the Federal Geographic Data Committee's content standard, long used across United States agencies, and the international ISO 19115 family. They are thorough and tedious, and if a full record is not going to happen, write a plain README beside the file answering these seven questions:

1. What is this, in one sentence, and what does one feature or one cell represent?
2. Who produced it and when, and what period does it describe? Data date and publication date are different.
3. What coordinate reference system, by EPSG code?
4. How was it captured, at what scale or resolution, and what accuracy is claimed?
5. What does every attribute field mean, including the codes and the missing-value convention?
6. What processing has been applied since it left the source?
7. What is the license, and may it be redistributed?

That last question is not decoration. United States federal works are generally in the public domain, which is why USGS and Census data can be used freely. OpenStreetMap is licensed under the Open Database License, which requires attribution and imposes share-alike terms on derived databases. Some state portals publish data you may view but not redistribute. Reading the license takes two minutes and prevents the sort of problem that arrives by letter.

Key idea: Data without documented source, date, coordinate system, field definitions, and license is unusable by anyone else and, within a year, by you.

Where the free data actually is

Source	What you get
data.gov	A catalog pointing at tens of thousands of United States federal, state, and local datasets
Census Bureau TIGER/Line and data.census.gov	Every boundary in the country, plus the demographic tables that join to them
USGS EarthExplorer and The National Map	Landsat archives back to 1972, aerial photography, elevation from the 3D Elevation Program, hydrography, land cover
NASA Earthdata and Worldview	MODIS, VIIRS, and mission data products, with browse imagery for nearly every day since 2000
Copernicus Data Space	Sentinel-2 optical imagery at 10 meters with a revisit of about five days, free
NOAA	Weather, climate normals, bathymetry, coastal lidar, tidal datums
OpenStreetMap	Global roads, buildings, and points of interest, often the most current source available at any price
Natural Earth	Public domain small-scale world basemap layers at three generalization levels
State, county, and city open data portals	Parcels, zoning, permits, crashes, trees, streetlights, usually the most detailed data that exists

The practical order for a new project: start local, because a county portal beats a national dataset for anything local; go federal for elevation, imagery, hydrography, and demographics; use OpenStreetMap for roads and buildings where official data is missing; use Natural Earth for anything at continental scale or smaller.

Common misconceptions

• Higher resolution is always better. Storage grows with the square of resolution, and a 1 meter grid of a 30 meter measurement is precision without information.
• Resampling method is a detail. Bilinear resampling of categorical data invents class codes that never existed, and does so invisibly.
• NoData is the same as zero. Zero elevation is sea level. NoData is the absence of a measurement. Mixing them puts oceans in your mountains.
• Shapefile is a single file. It is at least three that must travel together, plus the .prj that tells you what the coordinates mean. Email one and the recipient gets nothing usable.
• Free data is unrestricted data. Public domain, CC-BY, and share-alike licenses impose different duties, and OpenStreetMap's share-alike terms reach derived databases.
• Metadata is bureaucracy. It is the answer to what year is this and what is field ZONE_C, questions your future self will ask within a month.

Recap

• A raster is a grid georeferenced by an origin, a cell size, and rotation terms, so cell positions are computed rather than stored.
• Storage scales with the square of resolution: halving cell size quadruples the file.
• Continuous rasters hold measurements and resample with bilinear or cubic; categorical rasters hold codes and require nearest neighbor.
• Vector suits discrete objects and networks; raster suits continuous fields and fast cell arithmetic; conversion between them costs precision at the edges.
• GeoPackage is the sensible modern default; shapefile persists despite the 2 GB limit and 10-character field names; GeoTIFF, and its cloud optimized form, is the raster standard.
• Metadata must state what, who, when, which CRS, how captured, what the fields mean, what processing was applied, and the license.
• Free data flows from data.gov, the Census Bureau, USGS, NASA, Copernicus, NOAA, OpenStreetMap, Natural Earth, and local portals; check the license before redistributing.

Sources

1. United States Geological Survey. (n.d.). EarthExplorer data portal. USGS.
2. United States Geological Survey. (n.d.). The National Map: elevation, hydrography, and land cover. USGS.
3. NASA. (n.d.). Earthdata: open access to NASA Earth observation data. NASA.
4. Open Geospatial Consortium. (n.d.). GeoPackage encoding standard. OGC.
5. OpenStreetMap Foundation. (n.d.). Copyright and license. OpenStreetMap.

The big picture

Federal law adds prison time to a drug offense committed within 1,000 feet of a school. That is a buffer, written into a statute before anyone drew it. When New Jersey actually mapped its zones in the 2000s, adding the schools, the day care centers, the parks, and the public housing that the law also covered, the circles merged into a single sheet across the dense cities: most of the land area of places like Newark and Camden fell inside a zone. An enhancement that applies nearly everywhere distinguishes nearly nothing, and several states redrew their laws once they saw the map.

That is the lesson in miniature. A buffer is trivial to compute and the hardest thing about it is knowing what the number means. This lesson covers the two operations you will run more than any others: selecting the features you care about, and drawing a distance around them.

Key idea: The analysis is easy; specifying it correctly is the work. Distance thresholds and selection rules encode policy decisions that only look like technical parameters.

Selection by attribute: SQL with a map attached

An attribute query filters rows by their values, using the same expression language a database uses. In QGIS it lives under Select by Expression; the syntax is essentially a SQL WHERE clause.

A simple one: ZONING = 'R1' AND ACRES > 0.5

The operators are the familiar set: equals, not equals, greater than, less than, and their combinations. Text values need straight quotes; numbers do not. Four refinements do most of the remaining work:

• IN replaces long chains of OR. ZONING IN ('R1','R2','R3') is much easier to read and harder to get wrong than three ORs.
• LIKE does pattern matching with the percent sign as a wildcard. STREETNAME LIKE '%AVE%' catches every avenue.
• IS NULL tests for missing values. This is the one people get wrong: FIELD = NULL is never true, because null is not a value that equals anything, including itself. You must write IS NULL and IS NOT NULL.
• NOT negates, and is where the second common error lives.

The error worth burning into memory is operator precedence. AND binds more tightly than OR, exactly as multiplication binds more tightly than addition. So this query:

ZONING = 'R1' OR ZONING = 'R2' AND ACRES > 1

does not mean what almost everyone intends. It is read as: R1 parcels of any size, plus R2 parcels over an acre. Every tiny R1 lot in the city comes along. What you meant needs parentheses:

(ZONING = 'R1' OR ZONING = 'R2') AND ACRES > 1

Use parentheses even when you are sure you do not need them. This mistake produces a result that looks reasonable, and nothing in the map says the count is three times too high.

Key idea: AND binds tighter than OR, and null equals nothing, so parenthesize every mixed condition and test missing values with IS NULL.

Selection by location: which relationship do you mean?

A spatial query selects features by their relationship to features in another layer. The relationships have precise definitions, and choosing among them is a decision about policy, not about software.

Relationship	Selects features that	Typical use
Intersects	Share any point at all, including a boundary touch	The broadest catch; parcels touched by a flood zone in any degree
Within (completely inside)	Fall entirely inside the other feature	Parcels wholly inside a district
Contains	Wholly enclose the other feature	Tracts that fully contain a park
Touches	Share a boundary but no interior	Neighboring parcels, adjacency lists
Crosses	Pass through, as a line through a polygon	Pipelines crossing wetlands
Within a distance of	Lie closer than a stated distance	A buffer without making the buffer

Now watch how much rides on the choice. A county wants parcels affected by the 100-year floodplain. Using intersects, a 40 acre parcel with one square meter of its corner in the floodplain is selected, and the owner receives a letter. Using within, that parcel is excluded, and so is a parcel that is 95 percent flooded but whose driveway pokes out. Neither is right in general. The correct rule usually needs a threshold, such as parcels where more than 10 percent of the area or any part of the building footprint falls inside the zone, which requires an overlay and an area calculation, and that is Lesson 9.

What you must not do is pick a relationship because it produced a number you liked. Write the rule down before you run it.

Buffers: the workhorse

A buffer creates a new polygon covering everything within a stated distance of a feature. Around points you get circles; around lines, a corridor with rounded or squared ends; around polygons, a fattened version of the original.

Four options change the result more than beginners expect.

• Distance units. The buffer is computed in the units of the layer's coordinate system. If the layer is in unprojected degrees and you type 1000, you have asked for a thousand-degree buffer, which is nonsense that some software will attempt. If you type 0.01 degrees at latitude 35, you get about 1.11 kilometers north to south and only 0.91 kilometers east to west: an ellipse pretending to be a circle. Reproject to a projected system in meters or feet first. Always.
• Variable distance. Buffer distance can come from a field, so a pipeline layer can buffer 50 feet for a 6 inch line and 300 feet for a 36 inch line in a single operation.
• Multi-ring. A set of nested buffers at 500, 1,000, and 2,000 meters is how impact zones and noise contours are usually presented.
• Dissolve. This is the one that produces wrong numbers, discussed next.

Why you must dissolve before you sum

Here is arithmetic worth doing once by hand.

One 1,000 foot buffer around a point is a circle of radius 1,000 feet. Its area is pi times 1,000 squared, which is 3,141,593 square feet. An acre is 43,560 square feet, so one circle covers 72.1 acres.

A city has 50 schools. Buffer them all without dissolving and the software returns 50 separate polygons. Sum the area field and you get 50 times 72.1, which is 3,607 acres. Report that as the area of the city under the school zone rule and you have overstated it badly, because schools cluster and their circles overlap; every overlap is counted twice, or three times where three circles meet. Run the same buffer with dissolve enabled and you get one merged polygon whose true area might be 2,100 acres. The difference, in a dense city, is routinely 30 to 50 percent.

The rule generalizes: whenever you will sum an area or count a population, dissolve overlapping buffers first. When you want to know which school a location falls near, keep the buffers separate so the identity survives. The choice depends on the question, and getting it wrong produces a number that is plausible, defensible-sounding, and inflated.

Key idea: Overlapping buffers double-count area and population, so dissolve before summing and keep them separate only when you need to know which source feature is near.

What a circle quietly assumes

A buffer measures straight-line distance, as the crow flies. People are not crows.

Draw a quarter-mile buffer around a light rail station in a city cut by an interstate. The circle sweeps happily across eight lanes of freeway and includes several blocks whose residents, to reach the station, must walk half a mile to an overpass and half a mile back. The buffer says they live 400 meters away. Their walk is 1.8 kilometers. Every transit-oriented development study that used a plain circle has this error baked in, and the correction is a network service area, which Lesson 10 builds.

Three more assumptions ride along. Distance from what: a setback measured from a property line and one measured from a building envelope are different rules with the same name. Distance through what: noise and pollution buffers ignore the hill, the wall, and the wind. And distance as a proxy for exposure: a circle treats someone at 999 feet as fully affected and someone at 1,001 feet as unaffected, when the real relationship decays smoothly. When the underlying effect is gradual, a distance surface or a decay function, both of which you will build in Module 5, describes it better than any circle.

Putting it together

Most real tasks are a query, then a buffer, then another query. The transit example from Lesson 1 in operations:

1. Select by attribute on the bus stops layer: HEADWAY_MIN <= 15. Say 240 of 3,100 stops qualify.
2. Reproject to the appropriate State Plane or UTM zone if not already there.
3. Buffer those stops by 1,320 feet, a quarter mile, with dissolve on, because you will be measuring area.
4. Select by attribute on parcels: ZONING IN ('MF1','MF2','MF3') AND ACRES >= 0.5
5. Select by location from that selection: parcels that intersect the dissolved buffer.
6. Remove parcels intersecting the floodplain layer.
7. Count the survivors, sum their acreage, and write down every threshold you chose.

That last step is not paperwork. When someone asks why a particular parcel is on the list, the honest answer is a chain of stated rules, and the chain is the analysis.

Common misconceptions

• A buffer is a simple objective distance. The distance encodes a policy judgment, measured from a chosen reference, in a chosen unit, ignoring barriers.
• Intersects and within are interchangeable. One catches a parcel touched at a corner; the other requires total containment. On a floodplain analysis the two can differ by thousands of properties.
• FIELD = NULL finds empty records. It never matches anything. Null is the absence of a value and requires IS NULL.
• AND and OR can be mixed freely. AND evaluates first. Without parentheses, your query silently selects a different set than the one you described in the meeting.
• Summing buffer areas gives the area covered. Only if the buffers do not overlap. Dissolve first, or you will inflate the answer by tens of percent.
• A quarter-mile circle is a quarter-mile walk. Circles ignore freeways, rivers, fences, and one-way stairs. Network distance is usually 20 to 60 percent longer than straight-line distance in a real street grid.

Recap

• Attribute selection is a SQL WHERE clause: use IN for lists, LIKE for patterns, IS NULL for missing values, and parentheses for any mixed AND and OR.
• Spatial selection requires choosing a relationship, and intersects, within, contains, and touches give materially different answers.
• Buffers are computed in the layer's coordinate system units, so reproject out of degrees before buffering.
• One 1,000 foot circle covers 72.1 acres; fifty undissolved circles sum to 3,607 acres and overstate the truth wherever they overlap.
• Dissolve buffers before summing area or population; keep them separate when you need to know which feature is nearby.
• Straight-line buffers ignore barriers, and a quarter-mile circle around a station is not a quarter-mile walk.
• Real analyses chain query, buffer, and spatial selection, and every threshold in the chain must be written down.

Sources

1. Wikipedia contributors. (2025). Drug-free school zone. Wikipedia.
2. Wikipedia contributors. (2025). Buffer analysis. Wikipedia.
3. QGIS Project. (n.d.). Documentation: vector geoprocessing and selection tools. QGIS.
4. NOAA Office for Coastal Management. (n.d.). Digital Coast: data, tools, and coastal flood analysis. NOAA.

The big picture

In 1969 the landscape architect Ian McHarg published Design with Nature, and the method inside it was made of paper. To choose a route for the proposed Richmond Parkway on Staten Island, his team drew each factor on its own transparent sheet: slopes, forests, water, historic sites, residential land, wildlife habitat. Areas of high value or high cost were shaded dark. Then they stacked the sheets on a light table.

Where many sheets were dark, little light came through, and that corridor was expensive to build through in every sense. Where the stack was pale, the route did least damage. McHarg's team read the answer off the light table and recommended a corridor. The technique has an unglamorous name, sieve mapping, and it is the direct ancestor of every suitability model in this course. What computers changed is that the sheets became layers, the shading became numbers, and the answer became reproducible.

Key idea: Overlay combines layers so that each output piece knows everything true about it from every input, which is what turns a stack of maps into an answer.

The overlay family, by what each one returns

Every overlay operation cuts geometry where layers meet. They differ in what geometry survives and which attributes come along, and that second half is what people forget.

Operation	Geometry returned	Attributes returned	Use when
Clip	The part of the input inside the clip layer	Input layer only	Cutting a national roads layer down to one county
Intersect	Only where both layers overlap	Both layers, joined on each piece	How much of each parcel is in each flood zone
Union	Everything from both layers, cut where they overlap	Both, with nulls where a layer is absent	Building a complete mosaic of all combinations
Erase (difference)	The input minus the overlapping area	Input layer only	Removing wetlands and floodplains from developable land
Symmetrical difference	Everything except the overlap	Both, with nulls	Finding disagreement between two versions of a boundary
Dissolve	Adjacent features merged by a shared attribute value	Chosen field, plus optional summary statistics	Counties to states, parcels to zoning districts

Clip and intersect look identical on screen and are not the same operation. Clip is a cookie cutter: the output is the input layer's features, trimmed, carrying the input's attributes and nothing else. Intersect is a combination: the output carries fields from both layers, so each resulting piece can tell you both the parcel number and the flood zone. If your analysis needs to know what the other layer said, clip has already thrown that away.

Here is the intersect payoff in numbers. A 100 acre parcel overlaps the 100-year floodplain. Intersect them and you get a new polygon of 32 acres carrying both PARCEL_ID and FLOOD_ZONE. Divide by the original area and you have the fact the county actually needs: this parcel is 32 percent in the floodplain. No selection by location can produce that number, because selection returns whole features, not the pieces.

Key idea: Selection returns whole features; overlay cuts them into pieces that carry the attributes of every layer involved, which is how you compute how much rather than merely whether.

Two cautions before the worked problem

Project first. Every area number an overlay produces comes from the coordinate system the data is in. Run this in Web Mercator and your acreages are wrong by a latitude-dependent factor, as Lesson 4 showed. Reproject everything to one appropriate projected system before the first operation.

Expect slivers. Intersect two layers that were digitized independently and the shared boundaries will produce hundreds of hairline polygons. Clean them out with a rule, such as delete polygons under 0.01 acres, and state the rule in your notes.

A worked site selection, with the count at every step

A county has asked where a 20 acre community solar array could go. The rules, agreed before any software opens, are: at least 20 usable contiguous acres; zoned agricultural or industrial; within one mile of a three-phase distribution line; outside the 100-year floodplain; outside mapped wetlands; not within 500 feet of an existing residence; and on ground under 5 percent slope. Seven criteria, six layers, one funnel.

Step	Operation	Parcels remaining
0	All parcels in the county	41,300
1	Select by attribute: ACRES >= 20	2,845
2	Select by attribute: ZONING IN ('AG','I1','I2')	1,190
3	Select by location: intersects the dissolved 1 mile buffer on three-phase lines	412
4	Erase the floodplain, then keep parcels still holding 20 contiguous acres	178
5	Erase wetlands, same size test	151
6	Erase the dissolved 500 foot buffer around residences, same size test	62
7	Intersect with the under-5-percent slope polygons, same size test	34

Thirty-four candidate sites out of 41,300 parcels. Now the three things a careful analyst says about that number.

First, before or after matters enormously. Step 1 tested parcel acreage; step 4 retested after cutting the floodplain out. Of the 412 parcels that entered step 4, 233 had any land left outside the floodplain, but only 178 still held 20 contiguous acres. The other 55 were reduced to fragments. Had you applied the size test only at the start, as beginners do, you would have carried 55 unusable parcels through to the final list and put them in front of a developer.

Second, contiguous is doing real work. Erasing a stream and its floodplain from the middle of a parcel often leaves two pieces, say 12 acres and 11. The total is 23 acres and the parcel qualifies on a naive area sum, but you cannot build one 20 acre array on it. The fix is mechanical: after every erase, convert multipart features to single part, so each fragment becomes its own row, then apply the area test to the fragments. If you skip this step, your model will systematically favor exactly the parcels that are chopped up.

Third, report the sensitivity. Rerun the funnel with the distribution line buffer at half a mile instead of one mile and the final count falls from 34 to 9. Rerun with the residential setback at 1,000 feet instead of 500 and it falls to 21. Neither number is more true than the other. A result that swings by a factor of four on a threshold nobody voted on is not a finding, it is a scenario, and presenting the scenario alone is where GIS earns its bad reputation. Present the table of thresholds and outcomes.

Key idea: A site selection is a chain of thresholds, and a competent analysis reports how the answer moves when the thresholds do.

Dissolve, the quiet workhorse

Dissolve merges adjacent features that share an attribute value. Dissolve a parcels layer by ZONING and you get one polygon per zoning district instead of 41,300 lots. Dissolve counties by STATE_FIPS and you get states.

The part that gets overlooked is that dissolve can carry summary statistics. Dissolving census blocks by tract can sum population, so the tract polygons come out with the correct totals attached. Dissolving parcels by owner name can sum acreage and count parcels, which is how you find that one entity holds 4,100 acres across 47 separate lots. That question is asked constantly in reporting, land conservation, and antitrust work, and it is a dissolve with a sum.

What overlay cannot tell you

Those 34 parcels are candidates, not answers. The map is silent about whether any owner will sell, whether the price is plausible, whether the soil holds a foundation, whether the interconnection queue at the substation is five years long, and whether the neighbors will fill a hearing room. It is also silent about its own age: if the parcels layer is three years old, some of those sites now have houses on them.

Say this out loud when you hand over the result. A shortlist that arrives with its assumptions attached invites the right argument, about the thresholds, which is a conversation the analysis can survive. A shortlist that arrives looking like an answer invites the wrong one, about whether GIS can be trusted at all.

Common misconceptions

• Clip and intersect are the same. Clip keeps only the input's attributes; intersect keeps both layers' attributes on each piece, which is what lets you compute proportions.
• Overlay order does not matter. For a pure chain of intersects and erases the final geometry is the same, but any size or count test applied between steps changes the answer, and running the cheapest filters first can turn hours into minutes.
• Total remaining acreage is what counts. Two fragments of 12 and 11 acres do not host a 20 acre facility. Explode multipart features and test the pieces.
• Overlay output is clean. Independently digitized boundaries generate slivers on every intersect. Set a minimum area rule and document it.
• A site suitability map is a recommendation. It is the set of places not yet excluded by the criteria you chose. Ownership, cost, and politics are not in the layers.
• More criteria means a better model. Each criterion multiplies the assumptions and shrinks the result. Ten weak criteria can eliminate every site while feeling rigorous.

Recap

• Overlay cuts geometry where layers meet and, crucially, carries attributes from the inputs onto the resulting pieces.
• Clip trims and keeps one layer's attributes; intersect keeps both; union keeps everything; erase removes; dissolve merges by value.
• Intersecting a parcel with a floodplain yields the acreage in each zone, and therefore the percentage, which selection alone cannot give.
• Reproject to an appropriate projected system before any overlay, and plan to remove slivers afterward.
• A worked funnel took 41,300 parcels to 34 candidate solar sites through seven documented steps.
• Testing size before exclusions rather than after inflated the result by 55 parcels; contiguity requires exploding multipart features.
• Thresholds drive the answer: one mile to half a mile cut the result from 34 sites to 9, so sensitivity belongs in the report.

Sources

1. Wikipedia contributors. (2025). Design with Nature. Wikipedia.
2. Federal Emergency Management Agency. (n.d.). Flood Map Service Center. FEMA.
3. National Renewable Energy Laboratory. (n.d.). Solar resource data, tools, and siting analysis. NREL.
4. QGIS Project. (n.d.). Documentation: vector overlay algorithms. QGIS.

The big picture

For more than a decade, a rented farmhouse outside Potwin, Kansas received a steady stream of visitors: sheriff's deputies, federal agents, angry people looking for stolen phones, and process servers. The family living there had done nothing. A database company that maps internet addresses to locations needed a fallback coordinate for any address it could place only within the United States, and it picked the country's rough geographic center, rounding it to 38.0000 north, 97.0000 west. That rounded point landed on the farm. Roughly 600 million addresses resolved there. When a journalist traced the story in 2016, the company moved its default to the middle of a lake.

Nobody made an error in the ordinary sense. A default value, chosen for convenience, became a location, and once it was a location it was treated as evidence. This lesson covers the four operations that attach features to each other through position: measuring proximity, routing through networks, joining by location, and turning addresses into coordinates. All four are ordinary tools. All four manufacture false certainty when nobody checks.

Key idea: Every operation in this lesson produces a coordinate or a distance that looks equally solid whether it was measured, interpolated, or invented as a fallback.

Three ways to measure how far

Euclidean distance is the straight line, computed with the Pythagorean theorem in a projected coordinate system. It is what buffers use.

Manhattan distance is the sum of the horizontal and vertical legs, named for a street grid. Take two addresses separated by 600 meters east and 800 meters north. Euclidean distance is the square root of 600 squared plus 800 squared, which is exactly 1,000 meters. Manhattan distance is 600 plus 800, which is 1,400 meters, forty percent farther. Nobody walks the hypotenuse through the buildings.

Network distance is the real answer: the shortest path along actual streets, sidewalks, or rails, respecting one-way restrictions and barriers. In a well-connected grid, network distance typically runs 1.2 to 1.4 times the straight-line distance. Where a river, a rail yard, or a freeway intervenes, the ratio can exceed 3. Measure both for the same city and the ratio itself is a map of how badly connected a neighborhood is, which is a finding in its own right.

Thiessen polygons, also called Voronoi polygons, divide space so that every location belongs to whichever point is nearest. Snow's 1855 walking-distance boundary was an early hand-drawn version, and Alfred Thiessen formalized the idea in 1911 to weight rain gauges by the area each represented. They are quick and genuinely useful for asking which facility is nearest, provided you accept two assumptions they make silently: that travel is straight-line and that facilities have unlimited capacity. The nearest emergency room to your house may be the wrong one to drive to if it is full or across a river.

Key idea: Straight-line distance is a convenience, network distance is the truth, and the ratio between them tells you which neighborhoods are cut off.

Networks: edges, nodes, and impedance

A routable network is a topological structure. Edges are street segments; nodes are the junctions where they connect; and each edge carries an impedance, the cost of traversing it. Choosing that impedance chooses the answer: length gives the shortest route, travel time gives the fastest, and elevation gain gives the route a cyclist would actually pick. The same network answers all three differently.

The network must also encode restrictions: one-way flags, turn restrictions such as no left turn, barriers, and height or weight limits for trucks. Miss the one-way flags and your router will confidently send an ambulance the wrong way up a boulevard.

The classic algorithm, published by Edsger Dijkstra in 1959, works outward from the origin like water filling a maze: repeatedly take the cheapest node reached so far, look at its neighbors, and record a better cost if you found one. When the destination is reached at the cheapest node, the path is optimal. Everything modern is a refinement for speed.

From that core the standard tools follow:

• Closest facility: which of these 14 fire stations reaches this address fastest.
• Service area or isochrone: every place reachable within four minutes of this station. Draw a four-minute drive polygon next to a 1.5 mile circle around the same station and they will not resemble each other: the polygon stretches along arterials and collapses in cul-de-sacs.
• Origin-destination matrix: travel time from every one of 800 blocks to every one of 20 clinics, the input to any access study.
• Location-allocation: given demand and candidate sites, which three sites minimize average travel for the population served.
• Routing with many stops: the vehicle routing problem that governs deliveries, snowplows, and meter readers.

Spatial joins: joining by where instead of by key

Lesson 6 joined tables using a shared code. A spatial join transfers attributes between layers using position, which is what you need when no common key exists.

The commonest form is point in polygon. You have 1,240 traffic crashes as points and 88 census tracts as polygons. A spatial join with a count summary attaches to each tract the number of crashes falling inside it. Other summaries work the same way: mean speed limit, maximum severity, sum of damages.

Three details decide whether the result is trustworthy.

• Join type. One-to-one with a summary gives one row per polygon and is what you want for mapping. One-to-many gives a row per matched pair, which multiplies your features. Join-to-nearest attaches the closest feature within a search radius, which is how each address gets its nearest station.
• Boundary cases. A crash recorded at an intersection often sits exactly on a tract boundary. Software applies a tie-break rule, and you should know what it is, because in some cities a large share of incidents are coded to intersections and therefore to boundaries.
• What the count means. A tract with 140 crashes and a tract with 6 may differ in danger or may differ only in traffic volume and size. Counts require normalization before they mean anything, which is Lesson 14 and the single most important idea in the last module.

Geocoding: the least accurate step in most projects

Geocoding converts an address into coordinates. Most spatial data about people enters a GIS this way, and it is the weakest link in the chain.

The process has three stages. First the address is parsed and normalized: 123 N. Main Street Apartment 4B becomes a house number, a directional, a street name, a type, and a unit, with abbreviations standardized. Then it is matched against reference data. Then coordinates are assigned, and how they are assigned determines the accuracy.

The classic reference data is a street centerline file with address ranges: this segment runs from 100 to 198 on the even side. To place 150, the geocoder interpolates. If the segment is 400 meters long, the fraction is 150 minus 100 over 198 minus 100, which is 0.51, so the point goes 204 meters along the segment, then is offset perpendicular by a fixed amount, often 10 meters, to put it on the correct side of the street.

Notice everything that assumes. It assumes house numbers are evenly spaced along the block, which fails wherever lot sizes vary. It assumes the range is complete and correct. It puts every unit of a 300 unit apartment building at one point, and the whole of a corner lot at whichever street it is addressed on. It cannot know about the subdivision built last year that is not in the reference file. Interpolated geocodes in cities are typically off by tens of meters and in rural areas by hundreds; when the address is matched only to a ZIP code centroid, the error can be kilometers.

Key idea: Interpolated geocoding assumes evenly spaced addresses along a complete street range, and every violation of that assumption becomes silent positional error.

Reading a geocoding result honestly

Every geocoder returns more than coordinates. Keep those extra fields and use them.

Match type	Typical accuracy	Safe uses
Rooftop or parcel point	A few meters	Anything, including parcel-level analysis
Street range interpolation	Tens of meters urban, hundreds rural	Neighborhood and tract analysis
Street or block centroid	A block or more	Coarse mapping only
ZIP or postal centroid	Hundreds of meters to kilometers	State and regional summaries only
City or default fallback	Meaningless as a location	Nothing; exclude and count separately

A vendor reporting a 98 percent match rate may have matched 78 percent to rooftops and 20 percent to postal centroids. That second group will stack hundreds of points on a handful of coordinates, and on a hot spot map those stacks look exactly like real clusters. The Potwin farm is the same failure at national scale.

So the workflow: geocode, then map the results colored by match type. Look for improbable stacks. Check whether unmatched records share a characteristic, since new construction, rural routes, and non-English street names fail disproportionately, and dropping them silently biases the study toward established, urban, English-named places. Report the match rate and the match type distribution in your methods, every time.

Common misconceptions

• Distance is distance. Straight-line, Manhattan, and network distances differ by 40 percent or more for the same pair of points, and only one of them describes a trip.
• Thiessen polygons show service areas. They assume straight-line travel and unlimited capacity. A river or a full hospital breaks both assumptions.
• The shortest route is the fastest route. Only if impedance is distance. Change impedance to travel time and the path changes, which is why routing engines ask what you are optimizing.
• A high geocoding match rate means good coordinates. Match rate counts matches of any quality. The distribution of match types is the number that matters.
• Point-in-polygon counts are comparable across polygons. Not until they are normalized by population, area, or exposure. Raw counts mostly map where people are.
• Unmatched addresses are random. They cluster in new subdivisions, rural areas, and neighborhoods whose addressing the reference file handles poorly, so dropping them shifts your findings.

Recap

• Euclidean, Manhattan, and network distances answer different questions; a 600 by 800 meter offset is 1,000 meters straight-line and 1,400 by the grid.
• Network distance usually runs 1.2 to 1.4 times straight-line distance and far more where barriers intervene.
• Thiessen polygons assign every location to its nearest point, assuming straight-line travel and no capacity limits.
• Networks are edges, nodes, impedance, and restrictions; Dijkstra's 1959 algorithm underlies closest facility, service areas, matrices, and location-allocation.
• Spatial joins transfer attributes by position; watch join type, boundary tie-breaks, and the need to normalize counts.
• Geocoding interpolates along address ranges and assumes evenly spaced, correctly recorded house numbers.
• Keep and report match type; postal centroids and default coordinates manufacture clusters that look real.

Sources

1. United States Census Bureau. (n.d.). Census Geocoder: free address matching service. Census Bureau.
2. Wikipedia contributors. (2025). MaxMind, including the Potwin, Kansas default-coordinate case. Wikipedia.
3. Wikipedia contributors. (2025). Dijkstra's algorithm. Wikipedia.
4. Wikipedia contributors. (2025). Voronoi diagram. Wikipedia.
5. OpenStreetMap Foundation. (n.d.). Routing engines and network data. OpenStreetMap.

The big picture

In a 1983 doctoral dissertation, C. Dana Tomlin did something unglamorous that has outlasted every piece of software written since. He proposed that all the things you might do to a grid fall into a small number of kinds, gave them names, and showed that complicated analyses are just those few operations chained together. He called the system map algebra, and once you know the four categories you can look at any raster workflow and see its skeleton.

The idea is that a raster behaves like a variable. If elevation and slope are grids covering the same ground with the same cell size, then elevation plus slope is a grid. So is slope times 2, or the condition slope less than 5. Analysis becomes arithmetic performed on millions of cells at once, and the software does it in a fraction of the time an equivalent vector overlay would take.

Key idea: In map algebra a raster is a variable, so analysis becomes arithmetic on aligned grids, and every complex model decomposes into local, focal, zonal, and global operations.

The four kinds of raster operation

Type	Output cell depends on	Examples
Local	The value at that same cell, in one or more grids	Add, multiply, reclassify, conditional tests, weighted overlay, NDVI
Focal	The neighborhood around that cell	3 by 3 mean smoothing, focal maximum, slope, aspect, hillshade, edge detection
Zonal	All cells sharing a zone defined by another layer	Mean elevation per watershed, total population per district, maximum slope per parcel
Global	The entire grid	Euclidean distance surface, cost distance, flow accumulation, viewshed

Two of these deserve a sentence each because they are underused. A zonal statistic is the raster answer to a spatial join: hand it a polygon layer of census tracts and a temperature grid, and it returns the mean temperature per tract, ready to map. A global distance surface gives every cell its distance to the nearest feature of some kind, which is a continuous version of a buffer and far more useful when the effect you are modeling fades with distance rather than stopping at a line.

Reclassification: turning measurements into judgments

Suitability models cannot add degrees Celsius to percent canopy to people per square kilometer. The units are incommensurable. So each input is reclassified onto a common ordinal scale, usually 1 through 5, where 5 means most suitable for the purpose at hand.

This step is where the values enter the model. Deciding that a summer surface temperature above 37 degrees scores 5 and 30 to 32 degrees scores 1 is a judgment about what matters, made by you, in a dialog box, in about four seconds. It will not appear anywhere on the final map. Write your breaks down in the methods, always.

A worked weighted suitability model

A city wants to know where to plant street trees first. Four layers are available, all resampled to a common 30 meter grid: summer land surface temperature, existing canopy cover, population density, and impervious surface. The team assigns weights that sum to 1.

Criterion	Why it matters	Weight
Summer land surface temperature	Cooling benefit is greatest where it is hottest	0.35
Existing canopy (inverted)	Add trees where there are fewest	0.25
Population density	More residents means more people cooled	0.20
Impervious surface	Heat storage and stormwater runoff	0.20

Three cells, with raw values, reclassified scores, and the arithmetic spelled out.

Cell	Temperature	Canopy	Population density	Impervious
A, a warehouse district	38.5 C, score 5	12 percent, score 4	1,100 per sq km, score 2	82 percent, score 4
B, a dense apartment corridor	33.0 C, score 3	30 percent, score 2	12,000 per sq km, score 5	60 percent, score 4
C, a leafy outer suburb	30.2 C, score 1	46 percent, score 1	900 per sq km, score 1	22 percent, score 2

Multiply each score by its weight and add:

Cell A: 0.35 times 5, plus 0.25 times 4, plus 0.20 times 2, plus 0.20 times 4, equals 1.75 plus 1.00 plus 0.40 plus 0.80, which is 3.95.

Cell B: 0.35 times 3, plus 0.25 times 2, plus 0.20 times 5, plus 0.20 times 4, equals 1.05 plus 0.50 plus 1.00 plus 0.80, which is 3.35.

Cell C: 0.35 plus 0.25 plus 0.20 plus 0.40, which is 1.20.

The warehouse district wins. Now change one thing, and change nothing about the data. Suppose the parks department argues that the point of the program is people, not pavement, so population density should carry 0.35 and temperature 0.20. Recompute:

Cell A: 0.20 times 5, plus 0.25 times 4, plus 0.35 times 2, plus 0.20 times 4, equals 1.00 plus 1.00 plus 0.70 plus 0.80, which is 3.50.

Cell B: 0.20 times 3, plus 0.25 times 2, plus 0.35 times 5, plus 0.20 times 4, equals 0.60 plus 0.50 plus 1.75 plus 0.80, which is 3.65.

The ranking flipped. Same four layers, same reclassification, two defensible weightings, and a different neighborhood gets the trees. This is not a flaw in the method; it is the method telling the truth about where the decision actually lives. The weights are a policy statement written as decimals, and the map is downstream of them.

Key idea: A weighted suitability map is a picture of its weights, so publish the weight table beside the map and show at least one alternative weighting.

Constraints are not criteria

One distinction saves models from absurdity. A criterion is something you prefer more or less of, and it gets a score and a weight. A constraint is something absolutely forbidden, and it gets a mask of 0 and 1, multiplied against the final result.

You cannot plant a tree in the middle of a runway, on open water, or on someone's roof. Giving those cells a low score is wrong, because a weighted sum lets a high score elsewhere compensate; a runway with terrible canopy and extreme heat can score well. Multiplying by a 0 mask ends the argument: any cell that is water, runway, or building becomes 0 and stays 0 no matter how attractive it looks on every other criterion.

A second caution about arithmetic on ordinal scores. Scores of 1 through 5 are ranks, and averaging ranks pretends the gap from 1 to 2 equals the gap from 4 to 5. Everyone in the field does it anyway, because the alternative is heavier machinery, but be aware that the numbers are ordered labels wearing the costume of measurements.

Interpolation: from scattered points to a continuous surface

Now the opposite problem. You have measurements at points, 31 rain gauges, 84 wells, 12 air quality monitors, and you need a value everywhere. The justification for even trying is Tobler's law from Lesson 1: nearby places resemble each other, so a measurement carries information about its surroundings.

Thiessen polygons are the crudest answer: every location takes the value of the nearest sample. The surface is blocky, with cliffs at the polygon boundaries, but it makes no assumptions beyond nearest is best.

Inverse distance weighting is the everyday workhorse. Each sample contributes to the estimate in proportion to 1 divided by its distance raised to a power p, usually 2. Work one example completely.

Three samples surround the location you want to estimate: sample A at 100 meters with value 20, sample B at 200 meters with value 30, and sample C at 400 meters with value 60.

Sample	Distance	Value	Weight with p = 2	Weight times value
A	100 m	20	0.0001	0.002
B	200 m	30	0.000025	0.00075
C	400 m	60	0.00000625	0.000375
Total			0.00013125	0.003125

Divide the weighted total by the weight total: 0.003125 divided by 0.00013125 gives 23.8. The nearest sample dominates: even though one neighbor reads 60, the estimate sits close to the 20 next door.

Now set p to 1, so weights are 1 over distance rather than 1 over distance squared. The weights become 0.01, 0.005, and 0.0025, summing to 0.0175, and the weighted values sum to 0.5. The estimate becomes 28.6. Same data, same location, a difference of five units, produced entirely by a parameter most people leave at the default.

IDW has three properties worth memorizing. It is an exact interpolator: at a sample location it returns that sample's value. It never predicts outside the range of the data, so in this example nothing below 20 or above 60 can ever appear, which means IDW cannot find a peak or a valley between your samples. And it produces bull's-eyes: isolated concentric rings around each point, especially with a high power and sparse data.

Kriging, and what it buys

Kriging begins where IDW stops asking questions. Instead of assuming distance decay follows 1 over distance squared, it measures the actual spatial structure of your data and derives the weights from it.

The tool is the semivariogram. Take every pair of samples, plot how different their values are against how far apart they are, and average within distance bins. Typically the plot rises: close pairs are similar, distant pairs are not. Three features get names.

• The nugget is the value the curve appears to hit at zero distance. In theory two samples at the same place should be identical, so a nugget above zero measures measurement error plus variation finer than your sampling.
• The sill is the plateau, the overall variance of the data, reached when points are far enough apart to be unrelated.
• The range is the distance at which the curve reaches the sill. Beyond the range, a sample tells you nothing about a location. If your range is 800 meters and your gauges are 5 kilometers apart, no method can honestly interpolate between them.

Kriging fits a model to that curve and uses it to compute weights that are optimal in a least-squares sense. It also returns something no other common method does: a variance surface, an estimate of its own uncertainty at every cell, high far from samples and low near them. Mapping that surface next to the prediction is the single most honest thing you can do with interpolated data.

The cost is assumptions. Kriging assumes the spatial structure is roughly the same across the study area, a condition called stationarity, and that you have enough samples to fit a variogram, in practice several dozen at minimum and preferably a hundred or more. Feed it 11 wells and it will still produce a beautiful smooth surface with confidence intervals, and those intervals will be fiction.

Key idea: Every interpolation invents the values between your samples; IDW invents them with a fixed rule, kriging invents them with a rule fitted to the data and reports how unsure it is.

Common misconceptions

• Weighted overlay is objective because it is arithmetic. The weights and the reclassification breaks are opinions. Change them and the top-ranked site changes, as it did above.
• Constraints can be handled with a low score. A weighted sum lets other criteria compensate. Absolute exclusions must be multiplied in as a 0 mask.
• More criteria makes a better model. Each one adds assumptions and correlated information; temperature and impervious surface largely measure the same thing, so including both quietly double-weights pavement.
• Interpolation reveals the true surface. It produces a plausible surface consistent with your samples and your chosen method. Two methods on identical data give different maps.
• IDW with a higher power is more accurate. Higher powers make the surface more local and more bull's-eyed. The power is a modeling choice to be justified, not a quality dial.
• Kriging is the advanced option, so it is better. Kriging needs enough samples and a meaningful variogram. With sparse or wildly non-stationary data it delivers false precision dressed in statistics.

Recap

• Map algebra treats grids as variables; operations are local, focal, zonal, or global.
• Reclassification converts incommensurable units onto a common ordinal scale, and the breaks are judgments that must be documented.
• A weighted suitability score is the sum of weight times score; in the worked example the top cell changed when the weights moved from heat-first to people-first.
• Criteria get weights; constraints get 0 and 1 masks that no other criterion can override.
• IDW weights samples by 1 over distance to a power; with p = 2 the worked estimate was 23.8 and with p = 1 it was 28.6.
• IDW is exact, never predicts beyond the data range, and produces bull's-eyes around isolated points.
• Kriging fits a semivariogram with a nugget, sill, and range, derives optimal weights, and returns a variance surface, at the price of stationarity assumptions and a real sample size.

Sources

1. Wikipedia contributors. (2025). Map algebra. Wikipedia.
2. Wikipedia contributors. (2025). Inverse distance weighting. Wikipedia.
3. Wikipedia contributors. (2025). Kriging. Wikipedia.
4. Multi-Resolution Land Characteristics Consortium. (n.d.). National Land Cover Database, including impervious surface and canopy products. MRLC.
5. QGIS Project. (n.d.). Documentation: raster analysis and interpolation. QGIS.

The big picture

In February 2000 the Space Shuttle Endeavour flew for eleven days with a 60 meter mast extended from its cargo bay. Radar antennas at both ends bounced signals off the ground and compared the returns, and from the tiny differences in phase the Shuttle Radar Topography Mission computed the height of the land beneath it. In under two weeks it mapped the elevation of nearly every land surface between 60 degrees north and 56 degrees south, about 80 percent of the Earth's land. That dataset, eventually released worldwide at roughly 30 meter resolution, is why anyone anywhere can now compute a slope map for free.

A digital elevation model is just a raster whose cell values are heights. Everything in the first half of this lesson, slope, aspect, hillshade, viewsheds, streams, and watersheds, is derived from that one grid by arithmetic. Nothing else is measured.

Key idea: Terrain analysis derives many products from a single elevation grid, so every error in the DEM propagates into all of them at once.

Which elevation model do you have?

Three terms get used loosely and mean different things.

• A digital surface model (DSM) records the top of whatever is there: tree canopy, roofs, bridges. It is what the first radar or lidar return sees.
• A digital terrain model (DTM), often called bare earth, records the ground with vegetation and structures removed.
• DEM is used for either, which is exactly why you must check.

Run a flood model on a DSM and the water will pond on rooftops and flow along the top of the forest. Run a viewshed on a bare-earth DTM in a city and every building becomes invisible, so your cell tower appears to see the whole downtown. Read the metadata before you compute.

Where the data comes from: in the United States, the 3D Elevation Program publishes a seamless 1/3 arc-second model, about 10 meters, and lidar-derived 1 meter data over a growing share of the country. Globally, SRTM at 1 arc-second, the ASTER global model, and the Copernicus DEM cover nearly everything at about 30 meters.

Slope, computed by hand

Slope is the rate of elevation change. The standard method examines the eight neighbors of each cell and fits a plane. Take a three by three window from a 10 meter DEM, with north at the top:

	West	Center	East
North row	60	61	62
Middle row	50	51	52
South row	40	41	42

The east-west gradient weights the middle row double and divides by eight times the cell size: the east column total is 62 plus twice 52 plus 42, which is 208; the west column total is 60 plus twice 50 plus 40, which is 200. The difference, 8, divided by 8 times 10 meters, gives a gradient of 0.1, meaning the land rises one meter for every ten meters east.

The north-south gradient does the same with rows: the south row total is 40 plus twice 41 plus 42, which is 164; the north row total is 60 plus twice 61 plus 62, which is 244. The difference is minus 80, divided by 80, giving minus 1.0: the land drops a full meter for every meter you travel south.

Combine them with the Pythagorean theorem: the square root of 0.1 squared plus 1.0 squared is the square root of 1.01, which is 1.005. The slope is the arctangent of that, 45.1 degrees, or 100.5 percent. Sanity check: over the 20 meters from the north row to the south row the land falls 20 meters, which is a one-to-one drop, and a one-to-one drop is 45 degrees. The arithmetic agrees with the intuition, which is the point of doing it once.

Try a gentler window where all three rows read 50, 51, 52. Now the north-south gradient is zero and the east-west gradient is 0.1, so the slope is the arctangent of 0.1, which is 5.7 degrees, or 10 percent.

Percent slope	Degrees	Feels like
2	1.1	Barely perceptible; drainage minimum for paving
5	2.9	Accessible ramp limit under many building codes
10	5.7	A noticeable hill on a bicycle
25	14.0	A steep street; near the limit for most construction
50	26.6	Hard hiking; erosion risk on bare soil
100	45.0	Scrambling terrain

Percent and degrees are not interchangeable, and a 100 percent slope is 45 degrees, not vertical. Every year someone reports a hillside as dangerously steep because a number in percent was read as degrees.

One more caution that matters for regulation. Slope depends on the cell size you computed it from. A 30 meter DEM averages away the short steep pitches, so its maximum slopes are systematically gentler than the same terrain measured from 1 meter lidar. An ordinance saying no construction above 25 percent slope means different things on different data, and the enforcement question is not what the hill is but which raster the county used.

Key idea: Slope is computed from a fitted plane over neighboring cells, so its values depend on DEM resolution, and percent and degrees must never be confused.

Aspect and hillshade

Aspect is the compass direction the slope faces, from 0 to 360 degrees, with flat cells flagged separately. In the northern hemisphere south-facing slopes receive the most sun, so they are warmer and drier, melt snow first, and support different vegetation than the north-facing slope of the same ridge. Aspect drives vineyard siting, solar potential, avalanche forecasting, and habitat modeling.

Hillshade simulates illumination from a light source to make the terrain readable. The conventional defaults are an azimuth of 315 degrees, the northwest, and an altitude of 45 degrees above the horizon. The northwest is not aesthetic preference. Human vision assumes light comes from above and slightly left, and illuminating a hillshade from the southeast triggers relief inversion: valleys pop out as ridges and ridges sink into valleys, and the effect is so strong that many people cannot see it correctly even when told. Try it once, flip the azimuth to 135, and watch the landscape turn inside out.

Viewsheds

A viewshed marks every cell visible from an observer point, by tracing a line from the observer to each cell and checking whether intervening terrain rises above the sightline. It answers where a cell tower has coverage, which houses will see the wind turbines, and which parts of a battlefield are exposed.

Its inputs are the whole analysis. Observer height above ground, target height, and maximum radius all change the result substantially; a 30 meter tower sees far more than a person standing at the same coordinate. And the assumptions are strong: a bare-earth DEM ignores forests and buildings entirely, so an urban viewshed computed on a DTM is nonsense. Visibility is treated as binary, though something barely visible on the horizon is not the same as something filling the view. Over long distances, Earth curvature and atmospheric refraction matter and must be enabled in the tool.

Water: filling, flowing, accumulating

Hydrologic analysis is a fixed sequence, and each step is a raster operation you already know.

1. Fill sinks. A sink is a cell lower than all its neighbors, so water entering it cannot leave. Most are artifacts of interpolation and noise, and filling them raises each to the level of its lowest outlet. Be careful: some sinks are real, such as karst depressions, kettle lakes, and playas, and filling them erases genuine hydrology.
2. Flow direction. The standard D8 method sends each cell's water to whichever of its eight neighbors gives the steepest descent, coded as one of eight directions. It is simple and slightly wrong: real water spreads, and on a broad flat hillside D8 forces it into a single artificial line.
3. Flow accumulation. Count how many cells drain into each cell, following the direction grid. Ridge cells have accumulation near zero; a major river's cells have accumulation in the millions.
4. Stream definition. Threshold the accumulation grid: every cell above the threshold becomes stream. Here is the judgment call. Set the threshold at 100 cells and you get a dense fur of channels including gullies that flow twice a year; set it at 10,000 and you get only major creeks. The threshold sets your drainage density, so state it.
5. Watershed delineation. Pick a pour point, typically a gauging station or a culvert, and the tool walks the flow direction grid upstream to find every cell that drains to it. That set is the watershed.

Key idea: A delineated stream network is a function of a threshold you chose, so two analysts can produce different drainage densities from identical elevation data.

What satellites actually measure

Remote sensing means measuring something without touching it, and for our purposes it means recording reflected or emitted electromagnetic energy. Four resolutions describe any sensor:

• Spatial: ground size of a pixel. Landsat 30 meters, Sentinel-2 10 meters, MODIS 250 to 1,000 meters, commercial satellites under a meter.
• Spectral: how many bands and how narrow. Landsat has about a dozen; hyperspectral sensors have hundreds.
• Radiometric: how finely brightness is recorded. 8 bit data has 256 levels; 12 bit has 4,096, which matters in shadows and on snow.
• Temporal: revisit interval. Landsat every 16 days, Sentinel-2 about every 5 days with two satellites, MODIS daily.

You cannot have all four at once, because a sensor collecting fine detail over a narrow swath cannot also revisit daily. Choosing imagery is choosing which resolution matters for the question.

The band that changes everything is the near infrared, just beyond what your eye can see. Chlorophyll absorbs red light strongly for photosynthesis, while the internal structure of a healthy leaf scatters near infrared strongly. So a thriving plant is dark in red and bright in near infrared, and the gap between those two numbers is a measure of plant health.

NDVI, computed by hand

The Normalized Difference Vegetation Index, developed with data from the first Landsat in the early 1970s, is near infrared minus red, divided by near infrared plus red. Work four surfaces with plausible reflectance values.

Surface	Red	Near infrared	Numerator	Denominator	NDVI
Dense healthy crop	0.08	0.50	0.42	0.58	0.72
Stressed or sparse grass	0.18	0.30	0.12	0.48	0.25
Bare soil	0.22	0.28	0.06	0.50	0.12
Open water	0.05	0.02	-0.03	0.07	-0.43

The index runs from minus 1 to plus 1. Water is negative because it absorbs near infrared. Bare soil sits near 0.1 to 0.2. Vigorous vegetation runs from about 0.6 upward. Clouds and snow are bright in both bands and land near zero, which is why cloud masking comes before any index.

Why divide rather than subtract? Because the ratio partially cancels illumination. A hillside in partial shadow reflects less in both bands, so a plain difference would report it as less vegetated, while the ratio largely removes the effect. That is the reason nearly every remote sensing index is a normalized difference, including those for water, burn severity, and built-up area.

Change detection, and the four ways it lies

Subtracting one date's index from another is arithmetic. Making the difference mean something requires four conditions.

• Same or cross-calibrated sensors. Two instruments measure the same ground differently. Compare Landsat 5 to Landsat 8 without correction and you have measured the satellites.
• Radiometric and atmospheric correction. Haze, sun angle, and season change apparent brightness. Use surface reflectance products, which have this correction already applied.
• Matched phenology. A June image against an October image will show enormous vegetation loss across a deciduous forest that is simply autumn. Compare anniversary dates.
• Co-registration. If the two images are misaligned by half a pixel, every field edge, road, and shoreline produces a bright ring of false change. Check alignment on a fixed feature such as a road intersection before differencing.

Done properly, the technique is extraordinary. NASA's imagery of the Aral Sea, which covered about 68,000 square kilometers in 1960 and whose eastern basin dried completely in 2014, is a change detection anyone can browse; so are burn severity maps, urban growth studies, and the annual deforestation records that now drive policy. Fifty years of consistent Landsat coverage means the baseline exists for almost anywhere on Earth.

Common misconceptions

• A DEM is a DEM. Surface models include trees and buildings; terrain models do not. Choosing wrong floods rooftops or makes skyscrapers transparent.
• Percent slope and degrees are similar numbers. A 100 percent slope is 45 degrees. Confusing them turns a steep hill into a cliff on paper.
• Slope is a property of the land. It is a property of the land as sampled at a cell size, and finer DEMs report steeper maxima.
• Hillshade azimuth is decorative. Lighting from the southeast inverts perceived relief, turning valleys into ridges for most viewers.
• A viewshed shows what you can see. On bare-earth data it ignores every tree and building, and it treats a speck on the horizon as equal to a full view.
• Stream networks come from the data. They come from a flow accumulation threshold that you chose, and the choice sets the drainage density.
• NDVI measures how green something looks. It measures the contrast between red absorption and near infrared reflection, which is why plastic turf reads as bare ground and a dark conifer forest reads as vigorous.

Recap

• Terrain products all derive from one elevation grid, so DEM error propagates everywhere at once.
• Distinguish surface models from bare-earth terrain models before running floods or viewsheds.
• Slope from a three by three window: gradients east-west and north-south, combined by Pythagoras, then arctangent; the worked case gave 45.1 degrees.
• Slope values depend on cell size, which makes percent-slope ordinances resolution dependent.
• Aspect drives sun, snow, and habitat; hillshade is conventionally lit from 315 degrees to avoid relief inversion.
• Hydrology runs fill, flow direction by D8, flow accumulation, a stream threshold you must justify, and watershed delineation from a pour point.
• Sensors trade spatial, spectral, radiometric, and temporal resolution against each other.
• NDVI is near infrared minus red over their sum, running from about minus 0.4 for water to above 0.6 for vigorous vegetation.
• Change detection requires calibrated sensors, atmospheric correction, matched seasons, and precise co-registration.

Sources

1. United States Geological Survey. (n.d.). 3D Elevation Program (3DEP). USGS.
2. NASA. (n.d.). Earthdata: Shuttle Radar Topography Mission and elevation products. NASA.
3. United States Geological Survey. (n.d.). Landsat missions: bands, surface reflectance, and spectral indices. USGS.
4. NASA Earth Observatory. (n.d.). World of Change: the shrinking Aral Sea. NASA Earth Observatory.
5. NASA. (n.d.). Worldview: browse global satellite imagery by date. NASA Worldview.

The big picture

In 1869 a retired French engineer named Charles Joseph Minard published a chart of Napoleon's 1812 march on Moscow. A tan band leaves the Polish border, its width proportional to the size of the army: 422,000 men. The band narrows as it crosses Russia. At Moscow it turns black for the retreat, and along the bottom of the sheet runs a temperature scale, so you can read the plunging cold beneath the shrinking army. Where the black band crosses the Berezina river it drops by a third in the width of a fingernail. At the left edge it ends at 10,000.

Six variables on one sheet: army size, position in two dimensions, direction of travel, date, and temperature. No color beyond two tones, no decoration, and it can be read in ten seconds by someone who has never seen a chart before. Minard's map is the standard against which cartographic design is measured, and what makes it work is not artistry. It is that every mark on it encodes something, and the most important thing is the most visible.

Key idea: Design is not decoration applied to finished analysis; it decides what a reader sees first, and therefore what the map says.

Purpose, then hierarchy

Two kinds of map cover most work. A reference map shows where things are: a street map, a topographic sheet, a nautical chart. A thematic map makes an argument about a distribution: unemployment by county, crash density by intersection, forest loss by decade. Reference maps balance many features evenly. Thematic maps must not.

Visual hierarchy is the deliberate ordering of what the eye reaches first, second, and third. On a thematic map the theme comes first, geographic context second, and everything else third. The tools are contrast in size, in lightness, and in saturation, plus the figure-ground separation that makes the subject area feel like an object sitting on a background.

The most common failure in student and professional work alike is a base map that outranks the data. If the county boundaries are drawn in heavy black while the data classes are pale pastels, the reader's eye reads a boundary map with some coloring behind it. Fix it by pushing everything that is not the subject toward gray: thin the lines, lighten the labels, desaturate the basemap. When in doubt, print the map, tape it to a wall, and walk six paces away. Whatever you still see is your hierarchy, whether you designed it or not.

Visual variables: match the mark to the measurement

In 1967 Jacques Bertin catalogued the ways a mark on a page can vary: position, size, shape, value meaning lightness, color hue, orientation, and texture. His insight was that these are not interchangeable, because the eye reads them differently. Some are perceived as ordered, some only as different.

Visual variable	Perceived as	Correct use
Position	Ordered and quantitative	Everything; the strongest encoding there is
Size	Ordered, roughly quantitative	Counts and totals, as proportional symbols
Value (lightness)	Ordered	Rates, densities, and any sequential quantity
Color hue	Different, not ordered	Categories: land cover, party, zoning
Shape	Different, not ordered	Categories of point features
Orientation and texture	Weakly different	Secondary categories, hatching for overlays

From that table falls the rule broken most often in modern GIS: hue is for categories, lightness is for quantities. Mapping a rate with a rainbow ramp fails because red, green, and blue have no inherent order, so the reader has to consult the legend for every polygon. Rainbow ramps also invent boundaries where the data is smooth, because the eye perceives sharp transitions at yellow and cyan, and they collapse into mush for a colorblind reader and in grayscale.

Match the map type to the data as well. A choropleth shades predefined areas and requires a rate or ratio, never a raw count. Proportional symbols scale a circle by a value and are the correct choice for counts. Dot density scatters dots representing a quantity within each area. Isarithmic maps draw contours through a continuous surface. Kernel density makes a smooth intensity surface from points. A cartogram distorts area itself by the variable, which is honest about population but hard to read.

Key idea: Lightness and size are read as ordered, hue and shape as merely different, so a quantity mapped by hue forces the reader to decode a legend instead of seeing a pattern.

Classification: four schemes, one dataset, four stories

Here is the demonstration that changes how people read maps. Ten counties, one variable, cases per 100,000 residents, sorted:

2, 3, 3, 4, 5, 6, 7, 9, 18, 44

Mean 10.1, standard deviation about 12.1. Now classify into five classes four different ways.

Scheme	Class breaks	Counties per class	What the map looks like
Equal interval	Range 42 divided by 5, so breaks at 10.4, 18.8, 27.2, 35.6	8, 1, 0, 0, 1	Almost the entire state in the lowest class with one dark outlier
Quantile	Two counties per class	2, 2, 2, 2, 2	An evenly varied map that looks full of pattern
Natural breaks	Groups at the gaps: 2 to 4, 5 to 7, 9, 18, 44	4, 3, 1, 1, 1	The clusters actually present in the data
Standard deviation	Mean 10.1, breaks at 10.1 plus or minus 12.1 and 24.2	8 below the mean, 1, 0, 1 above two deviations	Deviation from average rather than raw level

Read across that table slowly, because nothing in the data changed. Under equal interval, eight counties are indistinguishable and the map says one place is a crisis. Under quantiles, the county with 3 cases and the county with 4 cases land in different classes and appear meaningfully different, while the county with 18 shares a class with the county with 44, which has two and a half times the rate. Natural breaks reproduces the actual structure but its breaks are unfamiliar numbers that are hard to justify to a public audience and change if you add one county.

None of the four is wrong. They answer different questions, and the choice belongs in your methods.

• Equal interval for evenly spread data, and for any set of maps that must be compared, because the breaks are simple to state.
• Quantile when relative rank is the point, accepting that it will always fill every class regardless of whether real differences exist.
• Natural breaks for a single skewed distribution when you want the classes to follow the data's own clumps.
• Standard deviation when the story is about being above or below average.
• Manual breaks when a threshold has external meaning: a regulatory limit, the poverty line, a doubling.

One rule is not negotiable. When you make a series of maps to compare, the class breaks must be identical across all of them. Reclassifying each year separately guarantees that every map looks about the same and the trend disappears, which is how a decade of steady improvement can be published as a decade of no change.

Four to seven classes is the practical range. Beyond seven, readers cannot reliably distinguish shades, and the extra precision is decorative.

Key idea: Class breaks are an argument, not a setting, and identical data classified four ways produces four different maps of the same state.

Color, chosen for readers who are not you

Three families cover nearly every thematic need, and Cynthia Brewer's ColorBrewer schemes remain the reliable source for all three.

• Sequential: one hue running light to dark, for data with a low-to-high order. Light means less. This is so consistent across cultures that violating it confuses everyone.
• Diverging: two hues meeting at a neutral middle, for data with a meaningful midpoint such as zero change, the national average, or a break-even point. Setting the neutral color anywhere other than the true midpoint is a way of lying with a legend.
• Qualitative: distinct hues of similar lightness for categories, with no implied order.

Then design for the audience you actually have. Around 8 percent of men of northern European ancestry and about half a percent of women have some form of color vision deficiency, most commonly in distinguishing red from green. A red-to-green map is unreadable for roughly one man in twelve. The robust fix is not to avoid color but to ensure that lightness alone carries the ordering. If your scheme still reads correctly printed in grayscale, it will read correctly for nearly every colorblind viewer too. Convert your map to grayscale once before publishing; it takes ten seconds and catches most of these failures.

Two more habits. Keep saturated, dark colors for the small, important marks rather than the large background areas, because a fully saturated field exhausts the eye and swamps everything drawn on top of it. And remember that color carries cultural meaning: red for danger and blue for water are so entrenched that mapping water in red will simply be misread, no matter how clear the legend.

The elements, and which ones you actually need

The traditional list is title, legend, scale bar, north arrow, source, and inset. In practice, keep the ones that do work.

• Title: what, where, and when. Unemployment Rate is not a title; Unemployment Rate by County, Ohio, September 2024 is.
• Legend: with units and the exact quantity, including the denominator. Cases per 100,000 residents, not Cases.
• Scale bar: yes for large-scale maps; it becomes meaningless on a small-scale world map where scale varies across the sheet.
• North arrow: only when north is not up or the projection makes orientation ambiguous. On an ordinary north-up county map it is decoration.
• Source and date: always, including the data's vintage and the projection for anything at regional scale or smaller.

Common misconceptions

• Design is what you do after the analysis. Symbolization and classification are analytical choices that determine what conclusion a reader draws.
• A rainbow ramp shows more detail. Hue has no perceived order, so a rainbow forces legend lookups, invents false edges, and fails in grayscale and for colorblind readers.
• Natural breaks is the objective option. It optimizes within-class variance for the data at hand, and its breaks shift when the data changes, which makes comparison across maps impossible.
• Quantile classification is fair because classes are equal. It guarantees a full-looking map whether or not real differences exist, and it can split near-identical values across classes.
• Every map needs a north arrow and a scale bar. Both are meaningless or misleading on some maps. Include elements that do work.
• Color choice is taste. Sequential, diverging, and qualitative are determined by the data's structure, and accessibility is a requirement rather than a preference.

Recap

• Minard's 1869 map encodes six variables and remains the standard because its hierarchy is exact.
• Thematic maps need a hierarchy where the theme dominates and base data recedes.
• Bertin's visual variables split into ordered (position, size, lightness) and unordered (hue, shape); quantities need the ordered ones.
• Choropleths need rates; counts belong in proportional symbols or dot density.
• Ten counties classified four ways produced four different maps: 8-1-0-0-1 under equal interval, 2-2-2-2-2 under quantiles, and the data's real clumps under natural breaks.
• Fix class breaks across any series of maps meant to be compared.
• Use sequential, diverging, and qualitative schemes for their matching data structures, and let lightness carry the ordering so the map survives grayscale and color vision deficiency.
• Titles state what, where, and when; legends state units and denominators; include elements only when they do work.

Sources

1. Wikipedia contributors. (2025). Charles Joseph Minard. Wikipedia.
2. Brewer, C. A., and Harrower, M. (n.d.). ColorBrewer: color advice for cartography. ColorBrewer.
3. Wikipedia contributors. (2025). Jacques Bertin and the visual variables. Wikipedia.
4. Wikipedia contributors. (2025). Jenks natural breaks optimization. Wikipedia.
5. Encyclopaedia Britannica. (2025). Cartography. Britannica.

The big picture

In 1950 the sociologist William Robinson took the 1930 United States census and computed the same relationship twice. Across the 48 states, the correlation between the share of residents who were foreign born and the share who were illiterate was about minus 0.53: states with more immigrants had less illiteracy. Then he computed it across the 97 million individuals themselves. There the correlation was plus 0.12: an immigrant was, individually, more likely to be illiterate than a native-born person.

Same census, opposite conclusions. Neither number is an error. Immigrants had settled disproportionately in states with strong school systems and literate native populations, so the state-level pattern reflected where immigrants went, not what immigrants were. Robinson's paper named the mistake of reading the first number as a fact about people: the ecological fallacy.

Every map in this lesson is a version of that trap. Aggregate data has properties that individual data does not, and a choropleth is aggregate data with a pretty face.

Key idea: A relationship measured between areas is a fact about areas, and inferring from it what is true of individuals inside those areas is a fallacy with a body count in policy.

Normalization: the fix that most maps skip

Start with the simplest and most consequential error in thematic mapping.

County A has 900,000 residents and 450 cases of a disease. County B has 20,000 residents and 30 cases. Map the raw counts and County A is fifteen times darker; it looks like the epicenter and County B disappears. Now divide.

County	Population	Cases	Rate per 100,000	Ranking
A	900,000	450	50	Fifteen times more cases
B	20,000	30	150	Three times the rate

Both statements are true and they point opposite directions. County A needs more clinicians; County B has the problem. A choropleth of raw counts is, in almost every case, a map of where the people are, drawn expensively.

Then a harder question: which denominator? The choice is not obvious and it is not neutral.

• Per resident answers what is a person's risk here. It breaks in places whose daytime population dwarfs their residents. A beach town with 3,000 residents and 40,000 summer visitors will post a spectacular drowning rate per capita that describes tourism, not danger to residents.
• Per unit of exposure is usually better where you can get it: crashes per million vehicle miles traveled rather than per capita, infections per test performed, burglaries per household.
• Per unit area gives density, which answers a different question, how concentrated, and is right for things like impervious surface or housing units.
• Per unit of opportunity matters for anything about capacity: patients per physician, students per classroom.

State the denominator in the legend, always. Cases per 100,000 residents is a legend entry. Cases is not.

The small numbers problem

Here is the trap that follows immediately after you correctly normalize.

A county of 2,000 people with 2 cases has a rate of 100 per 100,000. One more case, which could be a single family, takes it to 150. A county of 900,000 needs 450 more cases to move that far. Rates computed on small populations are wildly unstable, so the extremes of any county rate map, both the highest and the lowest, tend to be the least populated counties.

This is not a subtlety. Map United States county rates for a rare cancer and the darkest counties are sparse and rural, which invites theories about pesticides and well water. Map the lowest rates and they are also sparse and rural. The pattern is variance, not cause. Small denominators produce extreme values in both directions, and any map that classifies by raw rate will select for tiny places at both ends.

The defenses are ordinary: suppress or flag rates computed on populations below a stated threshold, aggregate small units into larger ones, pool several years of data, or use a smoothing method that pulls unstable local rates toward the regional average. Whichever you choose, say so.

Key idea: Rates from small populations swing wildly, so the extremes of a rate map are often a map of small denominators rather than of anything happening on the ground.

The modifiable areal unit problem

Your data almost never arrives in the units the phenomenon occurs in. Disease happens to bodies; you get counties. Traffic happens on road segments; you get census tracts. Those units were drawn for other purposes, and they are modifiable: they could have been drawn otherwise. That produces two distinct effects.

The scale effect: the same underlying data aggregated to different sized units gives different results. Correlations computed between variables generally strengthen as units get bigger, because aggregation averages away individual variation. A relationship that is weak at the block level can look overwhelming at the state level, and nothing changed but the containers.

The zonation effect: units of the same size, drawn with different boundaries, also give different results. Stan Openshaw and Peter Taylor demonstrated this in 1979 by taking 99 Iowa counties and grouping them into 12 districts in an enormous number of different ways, then computing the correlation between the percentage of elderly residents and the percentage voting Republican for each grouping. The correlations they could produce ranged from strongly negative to strongly positive. Same counties, same votes, same ages, any answer you liked.

The zonation effect has a familiar political name. Gerrymandering is the deliberate exploitation of the modifiable areal unit problem: a party with 55 percent of a state's votes can be drawn into eight of ten seats or three of ten, using the same voters, by packing opponents into a few districts and cracking the rest across many. When you hear that a map is neutral because it was drawn by a computer, remember that the computer optimizes whatever objective it was given, and the objective is the map.

There is no solution, only honesty. Report the units you used, be suspicious of a finding that appears at one scale, and where you can, test at two or three levels of aggregation and report whether the result survives.

Spatial autocorrelation, measured

Lesson 1 introduced the fact that nearby things resemble each other. Now measure it, because measuring it is how you tell a real cluster from an eye-catching arrangement of colors.

Moran's I compares each area's departure from the overall mean with the departures of its neighbors. When high values sit next to high values and low next to low, the products are positive and I is positive. When high sits next to low, I goes negative. The statistic runs roughly from minus 1 to plus 1, and under complete randomness its expected value is not zero but minus 1 divided by (n minus 1), which for a few hundred areas is very close to zero.

Moran's I	Pattern	Example
Near plus 1	Strong clustering of similar values	Income by tract in most cities
Near 0	No spatial structure; values scattered at random	Lottery winners per county
Near minus 1	Dispersion; high and low alternate	Competing facilities spaced to avoid each other

Before the statistic can be computed, you must define neighbors, and that definition is a modeling choice as consequential as any weight in Lesson 11. Rook contiguity counts areas sharing an edge; queen contiguity also counts those sharing only a corner. K nearest neighbors fixes the count at, say, six. A distance band counts everything within a stated radius. In a county grid these give different neighbor sets and therefore different values of I. Report the weights you used.

Significance is usually assessed by permutation: shuffle the values randomly among the areas several hundred times, compute I each time, and see where your observed value falls in that distribution. If your I sits far out in the tail, the arrangement is unlikely to be random.

Global statistics hide local truth

Moran's I is a single number for an entire map, which means a city with an intense cluster of poverty on one side and pure randomness everywhere else can return a modest global I that describes neither region.

Local indicators fix this. Local Moran's I, introduced by Luc Anselin in 1995, computes a statistic for every area and sorts significant ones into four kinds: high values surrounded by high (a hot spot), low surrounded by low (a cold spot), and the two spatial outliers, high surrounded by low and low surrounded by high. Those outliers are often the most interesting features on the map, and a global statistic cannot see them at all. The Getis-Ord Gi statistic does a similar job and is what most hot spot mapping tools run underneath.

One caution that gets ignored constantly. Running a local test on 3,000 counties at the 5 percent significance level will flag about 150 of them as significant purely by chance. Hot spot maps of large areas are therefore always partly noise, and a correction for multiple comparisons, or at minimum a sentence acknowledging the issue, belongs in the write-up.

Key idea: A global autocorrelation statistic tells you whether structure exists; local statistics tell you where, and both depend on a neighbor definition you chose.

A checklist for reading any thematic map

1. Is the variable a count or a rate, and if a rate, what is the denominator?
2. What are the class breaks, and what happens to the pattern under a different scheme?
3. What areal units are these, who drew them, and for what purpose?
4. Are the extreme areas also the least populated ones?
5. What is the data's date, and what is its positional and attribute accuracy?
6. What is not shown: excluded records, suppressed cells, unmatched geocodes?
7. If it claims a cluster, was that tested, with which neighbor definition?

Common misconceptions

• A choropleth of counts is fine if the counts are accurate. Accurate counts shaded by area still map population. Normalize, and name the denominator in the legend.
• Per capita is the neutral denominator. It is one choice among several, and it misdescribes places with large daytime or visitor populations.
• Extreme counties reveal real risk. With small populations, both extremes are usually variance. Suppress, aggregate, pool years, or smooth, and say which.
• Choosing units is a technical step. Scale and zonation change results, sometimes reversing the sign of a correlation. It is the analysis, not the setup.
• State-level relationships describe people. That is precisely Robinson's ecological fallacy, and it produced opposite conclusions from the same census.
• A hot spot map shows where the problem is. It shows where a statistic exceeded a threshold, under a neighbor definition you chose, before correcting for thousands of simultaneous tests.

Recap

• Robinson's 1950 comparison gave minus 0.53 between states and plus 0.12 between individuals, defining the ecological fallacy.
• Counts map population; rates map risk. County A with 450 cases had a third the rate of County B with 30.
• The denominator should match the exposure: vehicle miles for crashes, tests for infections, households for burglaries.
• Small populations produce unstable rates, so both extremes of a county rate map tend to be the smallest counties.
• The modifiable areal unit problem has a scale effect and a zonation effect; Openshaw and Taylor produced correlations from strongly negative to strongly positive from one Iowa dataset.
• Gerrymandering is the zonation effect applied deliberately.
• Moran's I measures clustering from about minus 1 to plus 1 and requires an explicit definition of neighbors.
• Local Moran's I and Getis-Ord Gi locate hot spots, cold spots, and spatial outliers, subject to the multiple comparisons problem.

Sources

1. Wikipedia contributors. (2025). Ecological fallacy, including Robinson (1950). Wikipedia.
2. Wikipedia contributors. (2025). Modifiable areal unit problem. Wikipedia.
3. Wikipedia contributors. (2025). Moran's I. Wikipedia.
4. United States Census Bureau. (n.d.). American Community Survey: guidance on estimates and margins of error. Census Bureau.
5. Centers for Disease Control and Prevention. (n.d.). Data and statistics: rate calculation and small area estimates. CDC NCHS.

The big picture

In November 2017 the fitness company Strava published an updated global heat map: roughly a billion recorded runs and rides, aggregated into glowing lines showing where people exercise. It was beautiful, and it was published as aggregate data with no names attached.

In January 2018 a 20-year-old Australian university student named Nathan Ruser looked at the parts of the map covering Syria, Iraq, and Afghanistan. In countries where almost nobody jogs with a GPS watch, the map was nearly black, except for small bright loops in the middle of the desert. Each loop was a perimeter run inside a military base, some of them not publicly acknowledged. Nobody's individual track was published. The pattern gave away the location, the layout, and in some cases the patrol routes.

Nothing was hacked. Aggregate location data, published in good faith, turned out to carry information nobody intended to release, because location is not an attribute like any other.

Key idea: Location data resists anonymization, because a pattern of places is itself an identifier even when every name has been removed.

How a web map actually works

Every slippy map you have ever dragged is built the same way. The world is rendered in advance into square tiles, conventionally 256 pixels on a side, arranged in a pyramid of zoom levels. Zoom level 0 is one tile containing the entire world. Each level down divides every tile into four, so level 1 has 4 tiles, level 2 has 16, and level z has 4 raised to the power z. By level 19 there are more than 200 billion, which is why they are generated on demand and cached rather than all made in advance.

A tile is addressed by three numbers, its zoom level and its column and row, which is why tile URLs look like a path ending in z, x, and y. The projection is Web Mercator, chosen because it makes the world a square that quarters evenly and keeps north up. Ground resolution at the equator is about 156,543 meters per pixel at zoom 0, halving at each level: roughly 150 meters per pixel at zoom 10 and about 0.3 meters per pixel at zoom 19.

Two kinds of tile exist. Raster tiles are images rendered on the server, simple and fixed: to change the styling you must regenerate them. Vector tiles ship the geometry itself in a compact binary form and let the client draw it, which means one tile set can be restyled instantly, rotated, labeled in different languages, and rendered crisply on any screen. Nearly all modern basemaps use vector tiles.

To put your own data on a map, the practical ladder is short. Small datasets, up to a few thousand features, can be loaded as GeoJSON directly. Larger ones need vector tiles or a server. Standard services from the Open Geospatial Consortium let one organization publish layers that another consumes live: WMS delivers rendered images, WMTS delivers tiles, and WFS delivers the features themselves. The common client libraries, Leaflet, MapLibre, and OpenLayers, are free and open source, and a working interactive map is genuinely about thirty lines of code.

Open data, and reproducible work

Module 3 listed the sources. The habit that matters is what you do with them.

A GIS analysis done by clicking through dialog boxes cannot be repeated, not by a colleague and not by you in six months. Everything in this course can be scripted: QGIS records a chain of operations as a Processing model you can rerun with different inputs; Python with GeoPandas, rasterio, and the GDAL library covers vector, raster, and format conversion; R with sf and terra does the same; and PostGIS runs spatial queries in SQL against a real database. Pick one and start recording your workflows in it. The moment someone asks what happens if we use half a mile instead of one mile, the scripted analyst answers in ninety seconds and the clicking analyst loses an afternoon.

Locational privacy

In 2013 researchers led by Yves-Alexandre de Montjoye analyzed fifteen months of mobile phone records for 1.5 million people, containing only the time and the cell tower for each call. No names, no addresses. They found that four approximate points in time and space were enough to uniquely identify 95 percent of individuals in the dataset. Home and work alone narrow you to a handful of people; add two more places and it is just you.

Apply that to ordinary GIS work. A geocoded home address in a health dataset is an identifier, however carefully names were stripped. Publishing patient locations as points on a map, even without labels, discloses the addresses to anyone who can georeference an image, and people have done exactly that from published figures. The rule is unambiguous: exact coordinates of residences are personal data.

The protections, in rough order of strength:

• Aggregate to areas large enough to contain a minimum number of people, and suppress cells below that threshold. Simple, effective, and it costs you spatial detail.
• Geomask: displace each point randomly within a radius, ideally a radius that adapts to population density so rural points move further than urban ones. Preserves broad patterns; defeats address recovery.
• Perturb the values rather than the locations, adding calibrated statistical noise. The United States Census Bureau adopted a formal version of this, differential privacy, for the 2020 census, which set off a real and unresolved argument with the people who need accurate small-area counts.

All three trade accuracy for protection, and there is no configuration that gives you both. Decide deliberately, document the method, and be honest that a protected dataset is less precise. Note also that agencies increasingly publish their own aggregated products precisely so that outsiders do not need the raw points.

Key idea: Four location points identify almost anyone, so protecting location means aggregating, masking, or perturbing, and every method costs precision you must disclose.

The power of maps

The cartographer Denis Wood argued in 1992 that maps work by presenting an interested selection of the world as if it were simply the world. That is not a complaint about dishonest mapmakers. It is a description of what a map is: a set of choices about what to include, what to name, what to aggregate, and what to shade, made by someone, for a purpose.

The clearest American example is a set of maps from 1935 to 1940. The Home Owners' Loan Corporation graded neighborhoods in more than 200 cities from A to D for mortgage risk. Grade D areas were outlined in red, and the accompanying area descriptions named the residents' race and national origin explicitly as the reason. Those maps guided decades of lending. The University of Richmond's Mapping Inequality project has digitized them, so you can open your own city and compare.

What makes this more than history is what shows up in current data. Formerly D-graded neighborhoods still have measurably less tree canopy and more pavement, and multi-city studies have found them several degrees Celsius hotter on summer afternoons than formerly A-graded neighborhoods in the same cities. A map made in 1937 is legible in a thermal image made in 2020. When you build the tree-planting suitability model from Lesson 11, you are working in that inheritance whether or not you name it.

The point generalizes without drama. Base maps show different national borders depending on which country you view them from, because the companies serving the tiles comply with local law. Place names encode who did the naming. Census categories are political artifacts. Counter-mapping and participatory GIS exist because communities found that appearing on an official map, with their own boundaries and their own names, changes what can be argued. Knowing this does not make maps untrustworthy. It makes them documents, and the right questions about a document are who made it, for whom, from what, and what is missing.

Where this takes you

Geospatial work sits inside other fields more often than it stands alone, and the job titles reflect that.

Role	Daily work	Skills that matter
GIS technician	Digitizing, editing, maintaining authoritative layers	Accuracy, topology, patience, documentation
GIS analyst	Answering questions with the operations in this course	Query, overlay, statistics, cartography, writing
Geospatial developer	Building web maps, data pipelines, and services	Python or JavaScript, PostGIS, APIs, version control
Remote sensing analyst	Imagery classification, change detection, monitoring	Physics of sensors, indices, machine learning, cloud processing
Cartographer	Designing maps for publication and products	Typography, color, generalization, editorial judgment
Drone or survey technician	Collecting imagery and control points in the field	Licensing, such as the FAA Part 107 certificate in the United States, and photogrammetry

Local government is the largest employer of GIS staff in most countries, followed by utilities, environmental consulting, transportation, insurance, logistics, defense and intelligence, technology firms, and humanitarian organizations. For current pay and outlook, look up cartographers and photogrammetrists, and surveying and mapping technicians, in the Bureau of Labor Statistics Occupational Outlook Handbook; the figures are revised annually and any number printed in a course text is stale within the year.

Three pieces of advice worth more than a certificate. First, GIS plus a domain beats GIS alone: the people who do best know hydrology, or epidemiology, or transportation planning, or land conservation, and use spatial tools inside it. Second, a portfolio outperforms credentials: three or four documented projects, each with the question, the data sources, the method, the map, and an honest statement of limitations, will do more in an interview than any list of software. Third, learn to script, because it is the difference between running an analysis and being able to run it again.

And the closing that this course owes you, stated as plainly as it was in Lesson 1. You now know what to do. You know why a datum mismatch shifts your data a meter, why area computed in Web Mercator is fiction, why an undissolved buffer inflates a total, why class breaks change a story, and why four location points identify a person. None of that becomes skill until you open the software and do it badly a few times. Pick a question you actually care about in the place you actually live, get the data, make the map, write down what you are unsure about, and then make it again. That is the entire method, and it does not end.

Common misconceptions

• Aggregated data is anonymous. Strava published only aggregates and revealed base perimeters. Patterns of place are identifying.
• Removing names is deidentification. Four spatio-temporal points identified 95 percent of 1.5 million people. A geocoded home address is personal data.
• Privacy protection is free. Aggregation, masking, and noise all reduce accuracy, and the 2020 census argument is exactly about that cost.
• Web maps show the same world everywhere. Borders and names vary by the jurisdiction you view from, because tile providers comply with local law.
• Saying maps carry interests means maps cannot be trusted. It means asking who made this, for whom, from what data, and what is missing, which is how any document is read.
• You need a specialist degree to work in GIS. Domain knowledge plus demonstrated projects carries more weight; the technical entry point is a free download and public data.

Recap

• Web maps are pyramids of tiles addressed by zoom, column, and row, drawn in Web Mercator, with vector tiles now dominant because they can be restyled on the client.
• Publish small data as GeoJSON, larger data as vector tiles or OGC services such as WMS, WMTS, and WFS.
• Script your analyses in QGIS models, Python, R, or PostGIS so they can be rerun and audited.
• Four spatio-temporal points identified 95 percent of 1.5 million mobile phone users; exact residence coordinates are personal data.
• Protect location by aggregating with minimum counts, geomasking, or adding calibrated noise, and disclose the accuracy cost.
• Maps encode interests: HOLC redlining maps from the 1930s still correlate with tree canopy and summer heat today.
• Careers span technician, analyst, developer, remote sensing, cartography, and field collection, mostly inside local government, utilities, and consulting.
• GIS plus a domain, a documented portfolio, and the ability to script are what move a career; and the skill itself only comes from doing the work in real software.

Sources

1. de Montjoye, Y.-A., Hidalgo, C. A., Verleysen, M., and Blondel, V. D. (2013). Unique in the crowd: the privacy bounds of human mobility. Scientific Reports, 3, 1376. Nature.
2. Nelson, R. K., Winling, L., et al. (n.d.). Mapping Inequality: redlining in New Deal America. University of Richmond Digital Scholarship Lab. University of Richmond.
3. United States Bureau of Labor Statistics. (n.d.). Occupational Outlook Handbook: cartographers and photogrammetrists. BLS.
4. Leaflet contributors. (n.d.). Leaflet: an open source JavaScript library for interactive maps. Leaflet.
5. United States Census Bureau. (n.d.). Decennial census: methodology and disclosure avoidance. Census Bureau.