Introduction to Spatial Data

HES 505 Fall 2024: Session 3

Carolyn Koehn

How do you view the world?

…As a Series of Objects?

  • The world is a series of entities located in space.

  • Usually distinguishable, discrete, and bounded

  • Some spaces can hold multiple entities, others are empty

  • Objects are digital representations of entities

…As a Continuous Field

  • The earth is a single entity with properties that vary continuosly through space

  • Spatial continuity: Every cell has a value (including “no data” or “not here”)

  • Self-definition: the values define the field

  • Space is tessellated: cells are mutually exclusive

How did the data arise?

Spatial data as a stochastic process

\[ {Z(\mathbf{s}): \mathbf{s} \in D \subset \mathbb{R}^d} \]

There is some attribute (\(Z(\mathbf{s})\)) that we observe at a location (\(s\)). That location (\(s\)) is an element of a domain of data (\(D\)), which is a subset of real coordinate numbers (\(\mathbb{R}^d\), \(d\) = 2).

Three types of spatial data are defined by the differences in domain (\(D\)).

Areal Data

\[ {Z(\mathbf{s}): \mathbf{s} \in D \subset \mathbb{R}^d} \]

  • \(D\) is fixed domain of countable units

  • Typically involve some aggregation

Geostatistical data

\[ {Z(\mathbf{s}): \mathbf{s} \in D \subset \mathbb{R}^d} \]

Mitzi Morris

  • \(D\) is a fixed subset of \(\mathbb{R}^d\)

  • \(Z(\mathbf{s})\) could be observed at any location within \(D\).

  • Models predict unobserved locations

Point patterns

\[ {Z(\mathbf{s}): \mathbf{s} \in D \subset \mathbb{R}^d} \]

  • \(D\) is random; where \(\mathbf{s}\) depicts the location of events

Ben-Said, M. Ecol Process 10, 56 (2021).

How is the data stored?

What is a data model?

  • Data: a collection of discrete values that describe phenomena

  • Your brain stores millions of pieces of data

  • Computers are not your brain

    • Need to organize data systematically
    • Be able to display and access efficiently
    • Need to be able to store and access repeatedly

  • Data models solve this problem

2 Types of Spatial Data Models

  • Raster: grid-cell tessellation of an area. Each raster describes the value of a single phenomenon. More next week…

  • Vector: (many) attributes associated with locations defined by coordinates

The Vector Data Model

  • Vertices (i.e., discrete x-y locations) define the shape of the vector

  • The organization of those vertices define the shape of the vector

  • General types: points, lines, polygons

Image Source: Colin Williams (NEON)

Vector Challenge!

The plot below includes examples of two of the three types of vector objects. Which ones are they?

Data Carpentry: Geospatial Concepts

The Raster Data Model

  • Raster data represent spatially continuous phenomena (NA is possible)

  • Depict the alignment of data on a regular lattice (often a square)

  • Geometry is implicit; the spatial extent and number of rows and columns define the cell size

Types of Raster Data

  • Regular: constant cell size; axes aligned with Easting and Northing

  • Rotated: constant cell size; axes not aligned with Easting and Northing

  • Sheared: constant cell size; axes not perpendicular

  • Rectilinear: cell size varies along a dimension

  • Curvilinear: cell size and orientation dependent on the other dimension

Types of Raster Data

  • Continuous: numeric data representing a measurement (e.g., elevation, precipitation)

  • Categorical: integer data representing factors (e.g., land use, land cover)

What makes data (geo)spatial?

Location vs. Place

  • Place: an area having unique physical and human characteristics interconnected with other places

  • Location: the actual position on the earth’s surface

  • Sense of Place: the emotions someone attaches to an area based on experiences

  • Place is location plus meaning

  • nominal: (potentially contested) place names

  • absolute: the physical location on the earth’s surface

Describing Absolute Locations

  • Coordinates: 2 or more measurements that specify location relative to a reference system

  • Cartesian coordinate system

  • origin (O) = the point at which both measurement systems intersect

  • Adaptable to multiple dimensions (e.g. z for altitude)

Cartesian Coordinate System

Locations on a Globe

  • The earth is not flat…

Latitude and Longitude

Locations on a Globe

  • The earth is not flat…

  • Global Reference Systems (GRS)

  • Graticule: the grid formed by the intersection of longitude and latitude

  • The graticule is based on an ellipsoid model of earth’s surface and contained in the datum

Global Reference Systems

The datum describes which ellipsoid to use and the precise relations between locations on earth’s surface and Cartesian coordinates

  • Geodetic datums (e.g., WGS84): distance from earth’s center of gravity

  • Local data (e.g., NAD83): better models for local variation in earth’s surface

Global Reference Systems

Describing location: extent

  • How much of the world does the data cover?

  • For rasters, these are the corners of the lattice

  • For vectors, we call this the bounding box

Describing location: resolution

  • Resolution: the accuracy that the location and shape of a map’s features can be depicted

  • Minimum Mapping Unit: The minimum size and dimensions that can be reliably represented at a given map scale.

  • Map scale vs. scale of analysis

The earth is not flat…

Projections

  • But maps, screens, and publications are…

  • Projections describe how the data should be translated to a flat surface

  • Rely on ‘developable surfaces’

  • Described by the Coordinate Reference System (CRS)

Developable Surfaces

Projection necessarily induces some form of distortion (tearing, compression, or shearing)

Coordinate Reference Systems

  • Some projections minimize distortion of angle, area, or distance

  • Others attempt to avoid extreme distortion of any kind

  • Includes: Datum, ellipsoid, units, and other information (e.g., False Easting, Central Meridian) to further map the projection to the GCS

  • Not all projections have/require all of the parameters

The Orange Peel Analogy

A datum is the choice of fruit to use. Is the earth an orange, a lemon, a lime, a grapefruit?

A projection is how you peel your orange and then flatten the peel.

Source: Data Carpentry: Geospatial Concepts

Choosing Projections

  • Equal-area for thematic maps

  • Conformal for presentations

  • Mercator or equidistant for navigation and distance

Geometries, support, and spatial messiness

Geometries

  • Vectors store and aggregate the locations of a feature into a geometry
  • Most vector operations require simple, valid geometries

Image Source: Colin Williams (NEON)

Valid Geometries

A linestring is simple if it does not intersect

Valid polygons:

  • Are closed (i.e., the last vertex equals the first)
  • Have holes (inner rings) that inside the the exterior boundary
  • Have holes that touch the exterior at no more than one vertex (they don’t extend across a line) - For multipolygons, adjacent polygons touch only at points
  • Do not repeat their own path

Empty Geometries

  • Empty geometries arise when an operation produces NULL outcomes (like looking for the intersection between two non-intersecting polygons)

  • sf allows empty geometries to make sure that information about the data type is retained

  • Similar to a data.frame with no rows or a list with NULL values

  • Most vector operations require simple, valid geometries

Support

  • Support is the area to which an attribute applies.

  • For vectors, the attribute-geometry relationship can be:

  • constant = applies to every point in the geometry (lines and polygons are just lots of points)

  • identity = a value unique to a geometry

  • aggregate = a single value that integrates data across the geometry

  • Rasters can have point (attribute refers to the cell center) or cell (attribute refers to an area similar to the pixel) support

Types of support for vectors

Give an example of:

  • constant support
  • identity support
  • aggregate support

Spatial Messiness

  • Quantitative geography requires that our data are aligned

  • Achieving alignment is part of reproducible workflows

  • Making principled decisions about projections, resolution, extent, etc

End