Transcript Geography 360 Principles of Cartography

Geography 360
Principles of Cartography
May 15~17, 2006
Outlines: Isarithmic map
1.
Two kinds of isarithmic map
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2.
Isometric map from the true point data (continuous fields)
Isopleth map from the conceptual point data (statistical surface)
Three interpolation methods
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3.
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)
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Vertical view: contour, hypsometric tint
Oblique view: fishnet map, block diagram
Physical model
Reading: Slocum chapter 14 & 15
What is isarithmic map?
• It depicts continuous smooth phenomenon
• Temperature, elevation, rainfall, average day of sunshine,
barometric pressure, depth to bedrock, earth’s
topography, and statistical surface
1. Two kinds of isarithmic map
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
Data types and isarithmic form
Dent 1999
How isopleth map is created
Image source: Electronic reading Nyerges
Isometric or isopleth map?
• Think how data is collected
Current temperature in the US
Voting behavior in the US
Isometric or isopleth map?
Toxic level
Demographic trends
Image source: www.gis.com
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
Phenomenon, data, map
World, data structure, map display
Elevation, 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?
2. Three spatial interpolation methods
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?
Location of weather stations
Surface map constructed from
inverse distance method
Surface map constructed from
Kriging
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
Inverse Distance Weighted (IDW)
• As the distance increases, you will inversely
weight the values
See p. 274 for
formula
Image source:
Bolstad 2005
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)
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?
Thiessen polygon
• Creates an area of equal influence given
point locations
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?
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
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
Image capture from Spatial Analyst
3. Map display of interpolated data
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 singlevalued; 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 earth’s surface
– Different values can be assigned to each (x,y,z)
– e.g. geological materials vary by (x,y,z)
Read Slocum p. 57
Displaying the interpolated data
• Vertical view
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From God’s 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 Bird’s eye view: 0~90°
– Fishnet map (Figure 14.16C)
– Block diagrams (Figure 15.17)
• Physical model
Contour lines
• Each contour line depicts the same
elevation
Hypsometric tint
• Space between contour lines is color-coded
• Can be either classed or unclassed
Hill shading (shaded relief)
• Illuminate earth’s topography with imaginary
light source
Cardinal direction of light source?
What do you think determines the reflectance values of pixel in digital image?
Which map display is this?
Oblique view
• Fishnet map
Block diagram
Oblique view
• Fishnet map
Fishnet map +
draped image
Physical model
• 3D map representation of 3D