Comprehensive GIS Study Guide: Coordinate Systems, Data Models, Cartography, and Spatial Analysis

Map Projections, Ellipsoids, Datums, and Coordinate Transformations

• Fundamental Cartographic Problem:

  • The Earth possesses a curved surface, whereas maps are flat planes.
  • Geographic Information Systems (GIS) must mathematically convert spatial information from the curved Earth onto a flat surface.

• Definition of a Map Projection:

  • A projection is the mathematical transformation utilized to transfer coordinates and features from the curved Earth onto a flat map.
  • Flattening a curved surface onto a two-dimensional plane is impossible without introducing geometric distortion.
  • Distortion alters one or more spatial properties:
  • Shape (conformality)
  • Area (equivalence)
  • Distance (equidistance)
  • Direction (azimuthality)
  • There is no single projection capable of preserving all spatial properties simultaneously; every projection represents a specific compromise.

• Ellipsoids and the Shape of the Earth:

  • The Earth is an irregular oblate spheroid rather than a perfect mathematical sphere.
  • In GIS, the physical shape of the Earth is approximated using a reference ellipsoid.
  • The standard progression from physical reality to a map follows a distinct sequence:
  • Real Earth $\rightarrow$ Approximation by reference ellipsoid $\rightarrow$ Coordinate assignment $\rightarrow$ Mathematical projection $\rightarrow$ Flat map.
  • Standard reference ellipsoids in geodetic science include:
  • Clarke 1866
  • GRS 1980
  • WGS 1984
  • Useful mnemonic: Clarke, GRS, WGS (girls, wigs, clarke).

• Datums:

  • A datum is defined as a reference ellipsoid paired with an origin offset relative to the center of the Earth:
  • $\text{Datum} = \text{Reference Ellipsoid} + \text{Offset/Positioning}$
  • The reference ellipsoid defines the geometric shape (size and oblateness).
  • The datum establishes the positioning, orientation, and geographic origin of that ellipsoid relative to Earth's center of mass or local surface.
  • Different regions and national mapping agencies frequently adopt the identical reference ellipsoid but employ different datums (different positional offsets) to optimize local surface accuracy.

• Geographic Coordinate Systems (GCS):

  • A GCS describes locations across the Earth's three-dimensional curved surface using angular coordinates.
  • Coordinates are expressed as angles measured from the center of the Earth, not as planar grid distances:
  • Latitude (angular distance north or south of the Equator)
  • Longitude (angular distance east or west of the Prime Meridian)
  • Global coordinate representation example:
  • Montreal $\approx 45^\circ\text{ N},\, 73^\circ\text{ W}$

• Projected Coordinate Systems (PCS):

  • A PCS applies mathematical formulas to project three-dimensional spherical/ellipsoidal geographic coordinates onto a two-dimensional Cartesian plane.
  • Yields planar coordinates expressed as linear measurements ($X$ and $Y$, commonly Eastings and Northings in metres or feet).
  • Planar coordinates are mandatory for key GIS analytical functions:
  • Calculating linear distances
  • Measuring surface areas
  • Performing advanced spatial analysis
  • Constructing printed or on-screen maps

• Coordinate Transformation Workflow:

  • Step 1: Real physical Earth ($\text{Geoid}$/topography).
  • Step 2: Reference Ellipsoid model ($\text{Clarke 1866}$, $\text{GRS 1980}$, or $\text{WGS 1984}$).
  • Step 3: Geographic Coordinate System ($\text{GCS}$, utilizing angular coordinates of Latitude and Longitude).
  • Step 4: Map Projection (mathematical transformation equations).
  • Step 5: Projected Coordinate System ($\text{PCS}$, utilizing planar Cartesian coordinates $X$ and $Y$).

• Significance of Layer Alignment in GIS:

  • Utilizing disparate projections generates differing spatial results across data sets.
  • When overlaying GIS layers, mismatched coordinate systems prevent layers from lining up correctly in geographic space.
  • Mismatched layer example: Overlaying a road network layer constructed in Projection A with a building footprint layer constructed in Projection B causes misregistration and invalid spatial overlay.

• Multiple-Choice Traps and Verification Rules:

  • Claim: "A projection allows curved Earth features to be represented perfectly on a flat map without distortion." $\rightarrow$ False. Distortion is mathematically inevitable.
  • Claim: "WGS 1984 is a map projection." $\rightarrow$ False. WGS 1984 is a reference ellipsoid/geodetic datum.
  • Claim: "A datum is simply an ellipsoid." $\rightarrow$ False. A datum consists of a reference ellipsoid plus an origin offset.
  • Claim: "Latitude and longitude are examples of projected coordinates." $\rightarrow$ False. Latitude and longitude are angular coordinates belonging strictly to a Geographic Coordinate System (GCS).
  • Claim: "Different projections always produce identical spatial analytical results." $\rightarrow$ False. Diverse projections distort area, length, and shape differently.
  • Claim: "A projection mathematically connects the reference ellipsoid to a flat map." $\rightarrow$ True.

Vector Data Models, Geometric Properties, and Topology

• The Fundamental GIS Data Representation Problem:

  • Physical geographic reality is continuous, complex, and irregular, whereas computing systems require discrete, structured, and digital representations.
  • Digital spatial models convert real-world phenomena into discrete entities and mathematical structures.

• Definition of Vector Data:

  • The vector data model abstracts spatial phenomena as discrete geographic features or objects using precise coordinate geometry, accompanied by descriptive attributes and topological relationships.

• Primitive Vector Feature Types:

  • Point:
  • Zero dimensions ($0\text{-D}$.
  • Defined by a single $(X, Y)$ coordinate location.
  • Real-world entity examples: bus stop, school, tree, hospital.
  • Line (Arc):
  • One dimension ($1\text{-D}$.
  • Composed of an ordered sequence of connected points, exhibiting length and directional orientation.
  • Real-world entity examples: road, river, subway line.
  • Polygon:
  • Two dimensions ($2\text{-D}$.
  • Formed by a closed sequence of interconnected lines defining a bounded boundary area.
  • Real-world entity examples: building footprint, municipal park, census tract, lake.

• Calculable Geometric Properties by Vector Primitive:

  • Point geometries allow computation of: Location ($X, Y$ coordinates).
  • Line geometries allow computation of: Length, bearing (azimuth/direction), and location.
  • Polygon geometries allow computation of: Surface area, perimeter, centroid (geometric center), and spatial extent (bounding box coordinates).

• Simple Vector Data Model ("Spaghetti Model"):

  • Features are stored as independent geometric entities.
  • Model characteristics:
  • The system stores basic shape geometries (e.g., standard Shapefiles).
  • Geometries are recorded independently without explicit relationships between adjoining features.
  • Metaphor: Drawing a map using loose spaghetti noodles, where each noodle has its own individual geometry, but the system possesses no inherent knowledge of whether two noodles touch, cross, share a common boundary, or link into a connected network.
  • Advantages:
  • Straightforward data architecture; simple to generate and store.
  • Rapid rendering, query retrieval, and display processing.
  • Universally supported across multiple GIS software platforms.
  • Disadvantages:
  • Highly inefficient for modeling spatial relationships.
  • Example deficiency: If Road A physically intersects Road B, a simple vector model stores only their respective coordinate paths, failing to encode whether traversal from Road A to Road B is structurally valid.

• Topological Vector Data Model:

  • Enhances raw geometry by explicitly storing and maintaining the spatial relationships among neighboring geometric objects (points, lines, polygons).
  • Core topological building blocks:
  • Nodes: Point endpoints where arcs intersect, terminate, or originate.
  • Arcs: Directed line segments bounded by start nodes and end nodes.
  • Polygons: Enclosed geographic areas bounded by continuous rings of interconnected arcs.
  • Primary structural rules of topology:
  • Connectivity: Defines how line features link together. Two arcs connect precisely when they share a common junction node.
  • Contiguity: Defines spatial adjacency and boundary sharing between adjacent polygons. Polygons that share an arc are identified as contiguous neighbors.
  • Summary rule: Connectivity equals linear linkage at nodes; contiguity equals shared boundary contact between adjacent areas.

• Purposes and Utilities of Topology:

  • Geometry Validation: Verifies that data structures adhere to strict geometric constraints:
  • Ensures polygons are completely closed.
  • Ensures network segments connect without unintended gaps.
  • Verifies that adjacent polygons share an identical, single boundary rather than overlapping or leaving unmapped spaces.
  • Spatial Analysis Operations: Facilitates network routing, polygon adjacency tracking, contiguity modeling, and buffer operations.
  • Primary Function: To explicitly represent and analyze structural relationships between spatial entities.

• Topological Errors in Vector GIS:

  • Sliver: A tiny, unwanted extraneous polygon created when the boundaries of adjacent polygons fail to align or snap identically during digitizing or overlay.
  • Unsnapped Node: A node situated near another feature that was intended to connect, but fails to make exact geometric contact due to a digitizing gap.
  • Spike: An erroneous, sharp, thin coordinate projection protruding unnaturally from an arc or polygon boundary.
  • Unended Line (Dangle): A line feature that terminates without connecting to an intended intersection node or boundary line.

• Shapefile Data Architecture:

  • An industry-standard simple vector format.
  • Non-topological: Shapefiles do not inherently store or encode explicit topology, connectivity, or boundary-sharing rules.

• Vector Model Selection Framework:

  • Use a vector model when geographic reality is composed of discrete, bounded objects.
  • Evaluate operational requirements:
  • When linear connectivity, network flow, or polygon contiguity analysis is required: Implement a Topological Vector Model.
  • When storage simplicity, platform interoperability, and display performance are required without complex spatial relationship modeling: Implement a Simple Vector Model.

Attribute Data Organization and Measurement Levels

• Dual Nature of Geographic Information:

  • GIS data comprises two coupled components:
  • Spatial Data: Answers "WHERE is it?" by defining geographic locations and geometries ($X, Y$ coordinates, lines, boundaries).
  • Attribute Data: Answers "WHAT is it like?" by recording non-spatial characteristics, properties, and values attached to geographic features.

• Structure of the GIS Attribute Table:

  • Attribute information is stored in relational database tables linked directly to geometry primitives:
  • Row (Record): Corresponds to exactly one geographic feature entity (e.g., Feature ID $1$).
  • Column (Field): Corresponds to one specific measured attribute or variable (e.g., Population).
  • Cell: Contains one individual attribute value for a specific feature and field (e.g., $15{,}000$).
  • Database distinction trap: Fields represent vertical columns; records represent horizontal rows.

• Attribute Table Case Examples:

  • Montreal Neighbourhood Attribute Table:
  • Record A: Population = $15{,}000$, Average income = $\$45{,}000$, Land use = Residential
  • Record B: Population = $8{,}000$, Average income = $\$62{,}000$, Land use = Commercial
  • Record C: Population = $20{,}000$, Average income = $\$38{,}000$, Land use = Residential
  • Numbered Feature ID Attribute Table:
  • ID $1$: Population = $15{,}000$, Income = $45{,}000$, Land use = Residential
  • ID $2$: Population = $8{,}000$, Income = $62{,}000$, Land use = Commercial

• The Four Levels of Measurement (Stevens' Scales):

  • Mathematical hierarchy progression: Nominal $\rightarrow$ Ordinal $\rightarrow$ Interval $\rightarrow$ Ratio.
  • Moving up the hierarchy increases the mathematical operations permissible on the underlying data.

• 1. Nominal Scale (Name/Category):

  • Data represent qualitative categories, labels, or names.
  • Possesses NO inherent numerical ranking, hierarchy, or mathematical magnitude.
  • Example: Land use classes (Residential, Commercial, Industrial).

• 2. Ordinal Scale (Ranked Order):

  • Data represent ordered categories establishing relative ranking.
  • Mathematical differences between successive ranks are unknown, uneven, and not equal.
  • Examples:
  • Suitability/density levels: Low, Medium, High.
  • Condition assessments: Poor, Fair, Good, Excellent.

• 3. Interval Scale (Equal Intervals without Absolute Zero):

  • Ordered numeric data exhibiting precisely equal intervals between units, but possessing NO true (non-arbitrary) zero point.
  • Zero does not indicate the total absence of the measured phenomenon.
  • Multiplicative ratios are mathematically invalid.
  • Example: Temperature in degrees Celsius ($^\circ\text{C}$):
  • The change from $20^\circ\text{C}$ to $30^\circ\text{C}$ is an interval difference of $10^\circ\text{C}$.
  • The change from $30^\circ\text{C}$ to $40^\circ\text{C}$ is an identical interval difference of $10^\circ\text{C}$.
  • The intervals are uniform and equal.
  • However, $0^\circ\text{C}$ does not represent an absence of temperature.
  • Consequently, $40^\circ\text{C}$ cannot be described as "twice as hot" as $20^\circ\text{C}$.

• 4. Ratio Scale (Equal Intervals with True Absolute Zero):

  • Numerical data possessing both uniform intervals and a true, physical absolute zero point.
  • Zero denotes the complete absence of the measured property.
  • Multiplicative statements, division, and ratios are fully valid mathematically.
  • Examples:
  • Distance: $20\text{ km}$ is mathematically twice as long as $10\text{ km}$; $0\text{ km}$ indicates zero distance.
  • Population: A tract with $20{,}000$ people contains twice the population of one with $10{,}000$ people; $0$ people indicates zero population.
  • Surface Area: Measurable in square units where $0$ signifies absence of area.

• Measurement Level Summary Rules:

  • Nominal: Categorical names only.
  • Ordinal: Ordered ranking only.
  • Interval: Meaningful order plus equal intervals, but lacks an absolute zero.
  • Ratio: Meaningful order, equal intervals, plus an absolute zero.

Cartographic Principles, Map Scale, Generalization, and Symbolization

• The Conceptual Nature of Maps:

  • A map is a graphic representation of spatial and geographic information.
  • Maps fundamentally abstract and simplify reality to communicate specific spatial relationships, distributions, and phenomena.
  • Maps cannot depict every real-world entity; cartographers selectively filter which features to preserve and decide how they are symbolized.

• Definition and Mechanics of Map Scale:

  • Scale is defined as the mathematical ratio between a distance measured on the map and the corresponding distance on the ground in the real world:
  • $\text{Scale} = \frac{\text{Distance on Map}}{\text{Distance in Real World}}$
  • Scale categories:
  • Large-Scale Map: Displays a relatively small geographic area with a large amount of detail (e.g., $1:5{,}000$ neighborhood plan).
  • Small-Scale Map: Displays a relatively large geographic area with a small amount of detail (e.g., $1:5{,}000{,}000$ continental map).
  • Mathematical fraction comparison:
  • Comparing $1:10{,}000$ and $1:1{,}000{,}000$:
    • $\frac{1}{10{,}000} > \frac{1}{1{,}000{,}000}$
  • Because $\frac{1}{10{,}000}$ is a larger numerical fraction, it represents the larger scale.
  • Core mnemonic: Large scale = large detail; small scale = small detail.

• Methods of Expressing Map Scale:

  • Representative Fraction (RF): Dimensionless ratio where one unit on the map equals the stated number of identical units on the ground (e.g., $1:50{,}000$, meaning $1\text{ cm}$ represents $50{,}000\text{ cm}$, or $1\text{ inch}$ represents $50{,}000\text{ inches}$).
  • Verbal Scale: Explicit text description using mixed measurement units (e.g., $1\text{ cm} = 1\text{ km}$).
  • Graphic (Bar) Scale: A calibrated ruler or bar printed directly onto the map layout.
  • Primary mechanical advantage: If a map is photographically or digitally enlarged or reduced, the graphic scale bar changes size proportionally alongside the map, remaining accurate.

• Cartographic Generalization:

  • When moving from large scale to small scale, physical space on the page decreases relative to geographic coverage.
  • Fine-grained geographic elements must be abstracted, simplified, aggregated, or omitted through generalization.
  • Example: A high-resolution street map of Montreal displays minor local streets, alleys, and individual lanes. On a national map of Canada, displaying minor urban streets produces illegible clutter; only major national highways are retained.

• Symbology and Visual Encodings:

  • GIS features link geometric primitives to visual symbols:
  • Points denote discreet locations.
  • Lines denote linear pathways or boundaries.
  • Polygons denote enclosed areal regions.
  • Attribute values control the visual variables (color, size, shape, texture) assigned to symbols to communicate data patterns.

• Data Classification Impact:

  • Altering the mathematical thresholds or classification schemes used to group numeric data fundamentally alters the visual spatial pattern perceived by the reader.
  • Example: Binning median neighbourhood household income into classes ($\$0\text{--}30\text{k}$, $\$30\text{--}50\text{k}$, $\$50\text{--}70\text{k}$, and $\$70\text{k}+$) influences the spatial cluster boundaries perceived by an observer.

• Subjectivity and Non-Neutrality in Cartography:

  • Maps are representations of reality, not objective reality itself.
  • The cartographic process involves deliberate, human choices:
  • Data selection and exclusion.
  • Classification method and class boundaries.
  • Graphic symbology, font hierarchy, and color palettes.
  • Selected scale and map projection.
  • These choices influence the message communicated to the map reader, embedding political, social, and ideological implications.

Raster Data Models, Structure, and Resolution

• Definition of Raster Data:

  • A raster data model represents geographic reality via an array of contiguous, tessellated square cells (pixels) structured in regular rows and columns.
  • Particularly suited for representing continuous surfaces and spatial fields:
  • Elevation
  • Ambient air temperature
  • Satellite and aerial imagery
  • Continuous surface modeling

• Categorical Capability of Raster Models:

  • Raster models are not limited exclusively to continuous numerical data.
  • Rasters effectively encode categorical (discrete) classes using integer codes linked to a lookup table:
  • Cell Value $1$ = Forest
  • Cell Value $2$ = Urban
  • Cell Value $3$ = Water
  • In categorical rasters, stored integers act as nominal classification labels rather than direct mathematical quantities.

• Three Fundamental Properties of Raster Cells:

  • Location: Explicit spatial coordinates derived from row and column positions within the grid matrix.
  • Size (Spatial Resolution): The physical surface area on the ground represented by the width and height of an individual cell.
  • Value: The numerical attribute, measurement, or categorical class code assigned to that specific cell.

• Spatial Resolution Dynamics:

  • Spatial resolution is defined directly by cell dimension:
  • $1\text{ m}$ resolution: Each individual cell covers an area of $1\text{ m} \times 1\text{ m} = 1\text{ m}^2$.
  • $30\text{ m}$ resolution (e.g., Landsat satellite imagery): Each cell covers $30\text{ m} \times 30\text{ m} = 900\text{ m}^2$.
  • Smaller cell dimensions equate to higher spatial resolution, finer detail, and greater data volume.
  • Larger cell dimensions equate to lower spatial resolution, coarser representation, and reduced detail.
  • Practical constraint: A $30\text{ m}$ Landsat pixel is well suited for identifying regional deforestation, regional crop distributions, or lake boundaries, but cannot resolve narrow urban bike paths or individual structures.

• Grid Architecture and Coordinate Addressing:

  • Raster cells are ordered systematically along a grid of horizontal rows ($R$) and vertical columns ($C$).
  • The geometric origin of a GIS raster matrix is situated at the upper-left corner.
  • Cells are indexed as $(C, R)$, denoting the intersection of column $C$ and row $R$.

• Cell address application: In a matrix with rows $R_1, R_2, R_3$ and columns $C_1, C_2, C_3, C_4, C_5$, Cell $G$ located at row $2$, column $2$ is designated as coordinate $(2, 2)$.

• Cell value storage: An elevation raster matrix contains discrete height values in metres (e.g., cells recording $105\text{ m}, 110\text{ m}, 115\text{ m}, 120\text{ m}, 125\text{ m}, 130\text{ m}, 135\text{ m}$), where each cell holds an explicit elevation reading.

• Resolution comparison:

  • High resolution employs smaller cells, capturing high spatial variability, crisp edge boundaries, and detailed geometry.
  • Low resolution employs larger cells, averaging localized variation, producing blocky boundaries, and sacrificing fine details.

• Raster Advantages and Limitations:

  • Advantages:
  • Simple, uniform data architecture based on an array matrix.
  • Optimal structure for recording continuously varying surfaces and gradients.
  • Directly compatible with remote sensing imagery and map algebra computations.
  • Limitations:
  • High-resolution grids generate massive file sizes and require substantial computational storage.
  • Linear features and vector boundaries appear pixelated or blocky.
  • Sub-pixel features smaller than cell dimensions disappear completely or become averaged out.

GIS Data Querying (Attribute vs. Spatial Queries)

• Attribute Query Principles:

  • Queries features based strictly on non-spatial values recorded in the attribute table.
  • Does not evaluate geometric position, physical proximity, or spatial orientation.
  • Examples:
  • Find all parcels where $\text{Household Income} < \$20{,}000$.
  • Select census units where $\text{Total Population} > 50{,}000$.
  • Filter zoning polygons where $\text{Land Use} = \text{"Residential"}$.

• Spatial Query Principles:

  • Queries features based on physical locations, topological intersections, or geometric distances relative to other geographic features.
  • Evaluates geometric coordinates and spatial relationships rather than descriptive tabular fields.
  • Examples:
  • Select residential parcels located within a distance of $200\text{ m}$ from a transit station.
  • Identify agricultural zones that physically intersect a designated river layer.
  • Select all water wells completely contained inside a specific municipal boundary.

• Compound Queries Combining Spatial and Attribute Criteria:

  • Complex GIS analysis integrates attribute constraints with geometric spatial relationships simultaneously.
  • Worked scenario: Identify all census tracts where median household income is below $\$20{,}000$, the English-speaking population exceeds $50\%$, AND the tract boundary lies within $200\text{ m}$ of a railway station.
  • Condition 1 (Attribute criteria): $\text{Income} < 20{,}000$
  • Condition 2 (Attribute criteria): $\text{English\_percentage} > 50$
  • Condition 3 (Spatial criteria): $\text{Distance to Railway Station} \le 200\text{ m}$

• Boolean Logic Operators in GIS Queries:

  • AND: Returns true only when all specified conditions are met simultaneously (intersection of criteria).
  • OR: Returns true when at least one of the specified conditions is met (union of criteria).
  • XOR (Exclusive OR): Returns true when exactly one of the conditions is met, evaluating to false if neither or both conditions are satisfied.