Geography 360 Principles of Cartography

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Geography 360Principles of Cartography

• May 1517, 2006

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Outlines Isarithmic map

• Two kinds of isarithmic map
• Isometric map from the true point data
  (continuous fields)
• Isopleth map from the conceptual point data
  (statistical surface)
• Three interpolation methods
• Regular control points to gridded surface (e.g.
  IDW, Kriging)
• Irregular control points to triangulated surface
  (e.g. triangulation)
• Map display of interpolated data (2.5D map
  display)
• Vertical view contour, hypsometric tint
• Oblique view fishnet map, block diagram
• Physical model

Reading Slocum chapter 14 15
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What is isarithmic map?

• It depicts continuous smooth phenomenon
• Temperature, elevation, rainfall, average day of
  sunshine, barometric pressure, depth to bedrock,
  earths topography, and statistical surface

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• 1. Two kinds of isarithmic map

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Two kinds of data

• True point data
• Data is actually measured at the point location
• e.g. The location of weather station for
  temperature map
• This kind of map is called isometric map
• Conceptual point data
• Data is collected over areas, and the map is
  constructed by interpolating given values at the
  centroid of areas
• e.g. The location of census tract for murder rate
  map
• This kind of map is called isopleth map

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Data types and isarithmic form
Dent 1999
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How isopleth map is created
Image source Electronic reading Nyerges
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Isometric or isopleth map?

• Think how data is collected

Current temperature in the US
Voting behavior in the US
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Isometric or isopleth map?
Toxic level
Demographic trends
Image source www.gis.com
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Depicting population distribution in different
map types

• Dot map total count in a large-scale mapping
• Proportional symbol map total count in a
  small-scale mapping, where the point location is
  conceptual (e.g. state centroid)
• Choropleth map standardized data of population
  (e.g. population density or cohort), where the
  goal is to compare between enumeration units
• Isopleth map standardized data of population,
  where the goal is to reveal overall trends,
  screening effects of arbitrary enumeration unit
  boundary

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Phenomenon, data, mapWorld, data structure, map
displayElevation, DEM, shaded relief

• Toxic level map
• Phenomenon toxic level
• Data point data of toxic level at sample points
• Map isometric map showing continuous fields of
  toxic level
• Demographic trends map
• Phenomenon elderly persons ( population over
  60)
• Data point data of population density at the
  centroids of enumeration units (the smaller the
  better)
• Map isopleth map showing statistical surface of
  demographic trends

The process of transformation from point into
surface?
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• 2. Three spatial interpolation methods

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Spatial interpolation

• We will call the data points (either true or
  conceptual) from which isarithmic maps are
  constructed control points (not a common term,
  but only for clarity purpose)
• Spatial interpolation basically estimates unknown
  values from known values at control points
  guesswork generates a continuous surface from
  sampled point values (which are discrete data)
  because we know the phenomenon mapped is
  continuous
• Is it valid to apply spatial interpolation to
  discrete phenomenon?
• Figure 14.1 see how the manual spatial
  interpolation works (it illustrates a linear
  method)
• Figure 14.4 see how different interpolation
  methods yield different-appearing maps how can
  we decide which method works the best given data?

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Location of weather stations
Surface map constructed from inverse distance
method
Surface map constructed from Kriging
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Spatial interpolation

• There are two ways to represent continuous
  surface - one is a regular or gridded form, and
  the other is an irregular form
• Regular control points to gridded surface
• Inverse Distance Weighted (IDW) z f (h) where
  h is distance to control points
• Kriging z f (h, v) r where v is the
  semivariogram model, and r is the residual (i.e.
  difference between model and observed value)
• Irregular control points to triangulated surface
• Triangulation z value is calculated from
  Delaunay triangle

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Inverse Distance Weighted (IDW)

• As the distance increases, you will inversely
  weight the values

Image source Bolstad 2005
See p. 274 for formula
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Kriging

• Similar to IDW in that
• A grid is overlaid on top of control points, and
  the goal is to derive values at a grid point from
  control points
• Values at a grid are determined by values at
  nearby control points weighted by inverse
  distance
• Different from IDW in that
• It builds the model of spatial autocorrelation
  from known values (called semivariogram), and
  the weights are determined such that observed
  values are best fitted into the specified model
• By model-fitting mechanism, the estimated values
  are supposed to reflect the spatial structure of
  given data it also provides the way to validate
  the weights (e.g. standard error of the estimate)

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Triangulation

• Unlike IDW and Kriging, triangulation honors
  control points
• Triangulation helps determine the edge from which
  values are interpolated
• So how do we determine the best edges to work on?
• It works this way (see Figure 14.3B at p. 274)
• Draw Thiessen polygon from control points
• Thiessen polygon equally divides the area of
  influence to each control point
• Connect control points at neighboring Thiessen
  polygons to result Delaunay triangle
• Delaunay triangle minimizes the length of edges
  formed by control points
• Compare imaginary triangle IDE to ICE which is
  smaller?

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Thiessen polygon

• Creates an area of equal influence given point
  locations

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Discussion review questions

• Which spatial interpolation method do you think
  can handle discontinuity? (e.g. lake as flat
  plane instead of U-shaped gutter)
• Which spatial interpolation method do you think
  will produce inconsistent results depending on
  parameters chosen (e.g. control points
  considered) compared to others?
• Which spatial interpolation method do you think
  is considered a optimal method?

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Which method to choose?

• Triangulation honors the control point data, and
  can handle discontinuity (e.g. ridge, lake)
• Pros it works well when control points are
  critical points
• Cons angular contour
• Inverse distance fast, simplicity of method
• Pros easy to understand
• Cons deterministic method (no uncertainty
  handling mechanism)
• Kriging most rigorous method provided that the
  model is properly specified
• Pros stochastic method (uncertainty handling),
  reflects overall spatial structure of data
• Cons complexity of method, sensitive to model
  specification

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Spatial interpolation in ArcGIS- Creating the
right surface map -

• Spatial Analyst
• Create contour from DEM
• Be aware of a wide array of parameters to choose
  from
• Geostatistical Analyst
• Provides exploratory tools for choosing the right
  parameters or models, including cross validation
  methods

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Image capture from Spatial Analyst
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• 3. Map display of interpolated data

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2.5D vs. 3D phenomenon

• So far we have worked on 2D map display of
  spatial entities (no height dimension)
• Now we move on to 3D map display of spatial
  entities
• What we commonly refer to as 3D map display can
  depict two categories as follows
• 2.5D phenomenon (e.g. elevation) z value is
  single-valued Color plate 14.1 depicts height
  above a zero point
• Z value is replaced by a single value of the
  theme mapped
• e.g. Prism map showing population density
• True 3D phenomenon (e.g. geological profile) z
  value is multi-valued Color plate 4.1 depicts
  geological materials underneath the earths
  surface
• Different values can be assigned to each (x,y,z)
• e.g. geological materials vary by (x,y,z)

Read Slocum p. 57
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Displaying the interpolated data

• Vertical view
• From Gods eye view 90
• Contour lines (Figure 14.16A)
• Hypsometric tints (Figure 14.16B)
• Hill shading (shaded relief) (Color plate 15.3,
  Color plate 15.2C)
• Oblique view
• From Birds eye view 090
• Fishnet map (Figure 14.16C)
• Block diagrams (Figure 15.17)
• Physical model

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Contour lines

• Each contour line depicts the same elevation

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Hypsometric tint

• Space between contour lines is color-coded
• Can be either classed or unclassed

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Hill shading (shaded relief)

• Illuminate earths topography with imaginary
  light source

Cardinal direction of light source? What do you
think determines the reflectance values of pixel
in digital image?
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Which map display is this?
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Oblique view

• Fishnet map Block diagram

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Oblique view

• Fishnet map Fishnet map draped image

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Physical model

• 3D map representation of 3D