Geography 360Principles of Cartography

May 15~17, 2006

Outlines: Isarithmic map

1. Two kinds of isarithmic map• Isometric map from the true point data (continuous fields)• Isopleth map from the conceptual point data (statistical

surface)

2. Three interpolation methods• Regular: control points to gridded surface (e.g. IDW, Kriging)• Irregular: control points to triangulated surface (e.g.

triangulation)

3. 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

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 collectedCurrent 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, mapWorld, 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?

Surface map constructed from inverse distance method

Surface map constructed from Kriging

Location of weather stations

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

Image source: Bolstad 2005

See p. 274 for formula

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 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 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– 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