Presented by

Joseph K. Berry

Adjunct Faculty in Geosciences, Department of Geography, University of DenverAdjunct Faculty in Natural Resources, Warner College of Natural Resources, Colorado State University

Principal, Berry & Associates // Spatial Information Systems

Email: jberry@innovativegis.com — Website: www.innovativegis.com/basis

There is a “map-ematics” that extends traditional math/stat concepts and procedures for the quantitative analysis of spatial data

This workshop provides a fresh perspective on interdisciplinary instruction at the college level by combining the philosophy and approach of STEM

with the spatial reasoning and analytical power of grid-based Map Analysis and Modeling

This PowerPoint is posted at

www.innovativegis.com/basis/Courses/SpatialSTEM/

SpatialSTEM:A Mathematical Structure for Understanding, Communicating and Teaching

Fundamental Concepts in Spatial Reasoning, Map Analysis and Modeling

Workshop Session 1 — Spatial Analysis Operations

SpatialSTEM Seminar Review (big picture)

Traditional math/stat procedures can be extended into geographic space to…

“think analytically with maps”…using a map-ematics for analysis/modeling

Spatial DerivativeSpatial Derivative

Advanced Grid Advanced Grid MathMath

Spatial Integral Spatial Integral

Spatial Spatial AnalysisAnalysis““ContextualContextual” ”

Relationships Relationships

Effective Effective ProximityProximity

Optimal PathOptimal Path

Visual ExposureVisual Exposure

Spatial Spatial StatistiStatisticscs

““NumericalNumerical” ” Relationships Relationships

Surface ModelingSurface Modeling

Linking Data Linking Data Space Space ((WhatWhat)) and and

Geographic Space Geographic Space ((WhereWhere) )

Spatial ClusteringSpatial Clustering

Localized Localized CorrelationCorrelation

Nature of Grid-Nature of Grid-based Map Databased Map Data

Analysis FrameAnalysis FrameGrid Map LayerGrid Map Layer((Matrix of NumbersMatrix of Numbers))

Map StackMap Stack

Lat/Lon GridLat/Lon Grid((4 inch squares4 inch squares))

Grid Map Grid Map analogy analogy

to a Spreadsheetto a Spreadsheet

Lat/Lon Coordinates Lat/Lon Coordinates as a as a

Universal Spatial Universal Spatial DBMS KeyDBMS Key

1) 1) quantitative analysisquantitative analysis of of maps is a realitymaps is a reality

(GIS)(GIS)

2) 2) spatial relationships spatial relationships exist exist and are quantifiableand are quantifiable

(STEM)(STEM)

SpatialSTEMSpatialSTEM

“Technology Tool” vs. “Analysis Tool”

Geotechnology as 21st Century “mega-technology”

Spatial Triad (RS, GIS, GPS)

“Descriptive Mapping” vs. “Prescriptive Analysis”

Maps are organizedMaps are organizedsets of numbers first,sets of numbers first,

pictures later pictures later

Last slide has “live links” by slide #

to online materials for further study

www.innovativegis.com/basis/Courses/SpatialSTEM/

Handout, PowerPoint and Online References…also see www.innovativegis.com for the

online book Beyond Mapping III

www.innovativegis.com/basis/Courses/SpatialSTEM/

Handout, PowerPoint and Online References…also see www.innovativegis.com for the

online book Beyond Mapping III(Berry)

SpatialSTEM_case.pptSpatialSTEM_case.ppt

Composite Display COVERTYPE Map

Water Polygons#11, #12

Meadow Polygon#21

Forest Polygons#31, #32

Vector (Spatial & Attribute Tables) Raster (Grid Matrix)

Mapping/Geo-QueryMapping/Geo-Query (Discrete, Spatial Objects)(Discrete, Spatial Objects) (Continuous, Map Surfaces) (Continuous, Map Surfaces) Map AnalysisMap Analysis//ModelingModeling

Where – a Spatial Table contains X,Y coordinates delineating the location of each point, line and polygon boundaryWhat – a linked Attribute Table contains text/values indicating the classification of each spatial object

:ID#21X,YX,Y :

: :ID#21 Meadow : :

SpatialTable

AttributeTable

Where – Cell Position in the matrix determines location within a continuous, regular grid What – Cell Value in the matrix indicates the classification at that location

Vector vs. Raster Data Forms (analytical perspective)

…grid-based mapped data is “Preconditioned for AnalysisPreconditioned for Analysis” as spatial topology is inherent in the matrix(Berry)

Analysis Frame

Col 3, Row 22Col 3, Row 22

The Analysis Frame provides consistent “parceling” needed for map analysis and extends discrete Points, Lines and Polygons…

Col 1, Row 1Origin

Col 1, Row 1Origin

3D Surface(stair steps)

Grid

A Grid Map consists of a matrix of numbers with a value indicating the characteristic/condition at each grid cell location—

Raster(continuous)

Grid Data Structure (fundamental organizational concepts)

Contour Lines

Intervals

Vector (discrete)

Elevation Contour Interval

2300-2500’

Elevation Contour Interval

2300-2500’

3D Surface(gradient)

…to continuous Map Surfaces

Map Stack

Map Stack

…forming a set of geo-registered Map Layers or “Map Stack”

Data listing for aMap Stack

Drill-down

(Berry)

Exercise 1-1) Map Analysis Framework (grid-based data structure and example analysis)

Companion Materials on the Map Analysis CD-ROM

(or Flash Drive)

MapCalc Learner Installation

Note: there lots of additional materials on the CD such as additional readings, exercises, color figures, and software to include a fully

functional demo version of Surfer software a very popular and versatile package for Surface Modeling of field data and 3D graphics.

I have found that an Elevation map and the Slope operation are ideal for demonstrating grid map data structure, display and analysis capabilities—most (all?) students have a good understanding of terrain as a surface and

terrain steepness as a concept (particularly if they ski, hike or climb).

Display Tools …the following “hands-on” experience is designed to gain

familiarity with grid-based data structure and display of map surfaces

ContourLines Contour Intervals

3D Lattice

3D Grid

…pushes up each cell to its stored value (blocky)…pushes the nodes of the wireframe to the stored value (smooth)

(Berry)

Example Map Analysis …the following “hands-on” experience is designed to gain familiarity accessing map analysis operations by

calculating a Slope map from the Elevation surface that identifies a continuous surface of terrain steepness (the Spatial Derivative)

Slopemap(Spatial Derivative)

Slopemap draped over Elevation Steep

Gentle

Map Calculus (Spatial Derivative)

GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if)

Spatial Analysis extends the basic set of discrete map features (points, lines and polygons) to map surfaces that represent continuous geographic space as a set of contiguous grid cells,and thereby provides a mathematical/statistical framework for analyzing and modeling the

Contextual Spatial Relationshipswithin and among grid map layers

Map Analysis Toolbox

Basic GridMath & Map Algebra ( + - * / )Advanced GridMath (Math, Trig, Logical Functions)Map Calculus (Spatial Derivative, Spatial Integral)

Map Geometry (Euclidian Proximity, Effective Proximity, Narrowness)Plane Geometry Connectivity (Optimal Path, Optimal Path Density)

Solid Geometry Connectivity (Viewshed, Visual Exposure)Unique Map Analytics (Contiguity, Size/Shape/Integrity, Masking, Profile)

Mathematical Perspective:

Map StackGrid Layer

Spatial Analysis Operations (Geographic Context)

(Berry)

Basic GridMath & Map Algebra ( + - * / )

Advanced GridMath (Math, Trig, Logical Functions)

Basic Grid Math and Map Algebra Example (Precision Agriculture)

(Berry)

Exercise 1-2) Using Grid Math and Map Algebra Operations (calculating difference and percent change maps)

Note: other grid map surfaces for two time periods can be substituted, such as pollution levels or barometric pressure or wildlife activity or sales revenue or home values. The

objectives of the exercise is to experience visualization of surface data (map variables) and executing a simple grid math operation then extending the processing to spatially

evaluating a familiar algebraic equation—map variables are just organized sets of numbers.

There are numerous exercise “tweaking” possibilities depending on the discipline and class environment involved– we can discuss in Workshop Session 3. My only caveat is to use hypothetical data and not attempt to download

and geo-register real data (or worse, local data) unless you have considerable GIS experience.

Evaluating and Algebraic Equation …the following “hands-on” experience spatially evaluates the

standard Percent Change equation to generate a map of percent change (spatially disaggregated

solution) instead of a single change value (spatially aggregated solution)

Calculating a Difference Map …the following “hands-on” experience is designed to

demonstrate an application of a basic Grid Math operation (minus) on two map variables

Operators - basic mathematical operations, i.e., Sum, DifferenceCommon - mathematical functions, i.e., Exponent, Square RootLogical - relational operators, i.e., AND, ORGrid - statistical functions, i.e., Maximum, MinimumTrig - trigonometric functions, i.e., Sine, Tangent, Cosine

…most of the math functions on a pocket calculator

Crop Yield Maps …the following “hands-on” experience is designed to gain familiarity with the yield map

surfaces for a farmer’s field

(Berry)

LOCATIONS

1 Ranch2 Cabin3 Thisplace4 Thatplasce5 Anotherplace

Ranch

Cabin

Three Measurement Methods

• Analog– align ruler and count the implied grid spaces

• Mathematical– count the grid spaces along the sides of an implied right triangle and solve

c2 = a2 + b2

• Algorithmic– “SPLASH”

a

b

c Pythagorean Theorem

Propagating Distance “Waves” (simple and effective proximity)

Distance is the “shortest straight line between two points”

…Simple Proximity relaxes “two points” requirement

…Effective Proximity relaxes the “straight line” requirement by considering intervening relative and absolute barriers to movement

…left with the modern concept of distance as the set of shortest not necessarily straight paths among locations—”Proximity Surfaces”

(Berry)

How far?

a

b

A series of wavefronts (ripples) form concentric halos of one grid space increments. This procedure extends the concept of Distance to that of Proximity

Distance—S, SL, 2PProximity—S, SL

…like nailing the ruler and spinning it

Variable-Width Buffers (simple vs. uphill)

Simple Buffer– “as-the-crow-flies” proximity to the road; no absolute or relative barriers are considered; dark blue line indicates the full simple buffer reach (polygon)

Clipped Buffer– simple proximity for just the land areas

Uphill Buffer– simple proximity to the road for just the areas that are uphill from the road; absolute barrier (uphill only– absolutely no downhill steps)

DownhillDownhill

(Berry)

Map Geometry (Euclidian Proximity, Effective Proximity)

Exercise 1-3) Map Geometry (Euclidian Proximity, Effective Proximity)

Simple Proximity …the following “hands-on” exercise provides experience in calculating “simple” proximity surfaces (as the crow flies) first from a single point and then from a set lines.

…from Turnout1 to everywhere

…from Roads to everywhere

Effective Proximity …the following exercise is designed to provide experience in calculating “effective” proximity surfaces (as the crow walks).

…from Turnout1 just along roads

…hiking from Roads to

everywhere

(Berry)

Calculating Visual Connectivity (sequentially assessing the tangent)

Seen if new tangent exceeds all previous tangents along the line of sight—

Absolute Barrier <SurfaceMap>

At <Viewer_heightValue> Thru <Screens_heightMap>) Onto <Target_heightMap>…like proximity, Visual Connectivity starts somewhere

(starter cell) and moves through geographic space by steps (wave front)– like a lighthouse beam…

Radiate – visual connectivity is calculated by a series of “waves” that carry the tangent to beat

Viewshed

Simply– at least one viewer location “sees” a map location (binary seen or not seen)

Visual Exposure

Completely – number of “viewers” that see each location (relative density)

Weighted – viewer cell value is added (relative importance)

(Berry)

Visual Exposure (simple sum and weighted sum)

A Visual Exposure map identifies how many times each location is seen from an “extended eyeball” composed of numerous viewer

locations, such as a road network (simple sum)

Completely – number of “viewers” that see each location (relative density)

In calculating a Weighted Visual Exposure map, different road

types are weighted by the relative number of cars per unit of time …the total “number of cars” replaces

the “number of times seen” for each grid location (weighted sum)

Weighted – viewer cell value is added (relative importance)

(Berry)

Variable-Width Buffers (line-of-sight connectivity)

Line-of-Sight Buffer– identifies all land locations (clipped) within 250m that can be seen from the road…

250m “viewshed” of the road

Line-of-Sight Exposure– notes the number of times each location in the buffer is seen

Line-of-Sight Noise– locations hidden behind a ridge or farther away from a source (road) greatly decrease noise levels.

(Berry)

Solid Geometry Connectivity (Viewshed, Visual Exposure)

Exercise 1-4) Solid Geometry Connectivity (Viewshed, Visual Exposure)

…Viewshed from Turnout1 to everywhere

…Viewshed from Roads to

everywhere

Viewshed Analysis …the following exercise is designed to provide experience in calculating the Viewshed of one or more viewing locations (binary seen/not seen by at least one viewer location).

…Visual Exposure from Roads to

everywhere

Viewshed Analysis …the following exercise is designed to provide experience in calculating the Viewshed of one or more viewing locations (binary seen/not seen by at least one viewer location).

(Berry)