1. Joseph K. Berry, MGUG 2014 Opening
Future Directions of Map Analysis and GIS Modeling
2014 Manitoba GIS User Group
Fall Conference | October 1, 2014 |  Winnipeg, Manitoba, Canada
Premise: There are three major forces driving map analysis/modeling— establishing a
map-ematical framework (SpatialSTEM), utilizing a Universal Spatial Database Key and radical changes in Raster Data Structure  
Premise: There are three major forces driving map analysis/modeling— establishing a
map-ematical framework (SpatialSTEM), utilizing a Universal Spatial Database Key
and radical changes in Raster Data Structure  
This PowerPoint with notes is posted online at—
, opening keynote presentation (35 minutes)
…related materials on the SpatialSTEM approach to include a full seminar version (50 minutes) and workshop sessions (three 2–hour sessions) are posed at…
This PowerPoint with notes and online links to further reading is posted at
Presented by Joseph K. Berry
Adjunct Faculty in Geosciences, Department of Geography, University of Denver
Adjunct Faculty in Natural Resources, Warner College of Natural Resources, Colorado State University Principal, Berry & Associates // Spatial Information Systems
— Website:
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2. Joseph K. Berry, MGUG 2014 Opening
Mapping vs. Analyzing (Processing Mapped Data)
…GIS is a Technological Tool involving —
Mapping that creates a spatial representation of an area
Display that generates visual renderings of a mapped area
Geo-query that searches for map locations having a specified classification, condition or characteristic
“Map”
(Descriptive Mapping)
“Analyze”
…and an Analytical Tool involving —
Spatial Mathematics that applies scalar mathematical formulae to account for geometric positioning, scaling, measurement and transformations of mapped data
Spatial Analysis that investigates the contextual relationships within and among mapped data layers
Spatial Statistics that investigates the numerical relationships within and among mapped data layers
(Prescriptive Modeling)
Global Positioning
System
(locate and navigate)
Remote
Sensing
(measure and classify)
Geographic Information
Systems
(map and analyze)
GPS/GIS/RS
GIS as “technical tool” versus an “analytical tool.” The white paper “Making a Case for SpatialSTEM” is posted at…
(Bottom) Geotechnology incorporates “spatial information” in a broad stroke similar to Biotechnology’s use of “biological systems” and Nanotechnology’s use of “control of matter. Geotechnology involves the three related technologies for mapping and analyzing spatial information on the surface of the earth— GPS, GIS and RS, known as the “spatial triad.”
The approach involves two dominant application arenas that emphasize Descriptive Mapping (Where is What; a Technological Tool) and Prescriptive Modeling (Why , So What and What If; an Analytical Tool) that focuses on understanding spatial patterns and relationships.
What is most important to keep in mind is that Geotechnology, like Biotechnology and Nanotechnology, is greater than the sum of its parts—GPS, GIS and RS. While these individual mapping technologies provide the enabling capabilities, it is the application environments themselves that propel geotechnology to mega status.
The SpatialSTEM approach for teaching map analysis and modeling fundamentals within a mathematical/statistical context provides a quantitative framework for spatial reasoning that resonates with science, technology, engineering and math/stat communities. The premise is that “maps are numbers first and foremost, pictures later” and we do mathematical things to mapped data for insight and better understanding of spatial patterns and relationships within decision-making contexts.
Online references for further study—
Geotechnology – Overview of Spatial Analysis and Statistics; Is it Soup Yet? ; What’s in a Name?; Melding the Minds of the “-ists” and “-ologists”
Making a Case for SpatialSTEM – Making a Case for SpatialSTEM; A Multifaceted GIS Community; GIS Education’s Need for “Hitchhikers”; Questioning GIS in Higher Education
(Biotechnology)
(Nanotechnology)
(Berry)
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3. Joseph K. Berry, MGUG 2014 Opening
A Mathematical Structure for Map Analysis/Modeling
Geotechnology  RS – GIS – GPS
Technological Tool
Mapping/Geo-Query (Discrete, Spatial Objects)
(Continuous, Map Surfaces) Map Analysis/Modeling
Analytical Tool
“Map-ematics”
Maps as Data, not Pictures
Vector & Raster — Aggregated & Disaggregated
Qualitative & Quantitative
Geo-registered
Analysis Frame
…organized set of numbers
Map Stack
 Matrix
of Numbers
Spatial Statistics Operations
Spatial Analysis Operations
Grid-based
Map Analysis
Toolbox
Book IV, Topic 9 for more discussion
A Map-ematical Framework
Traditional math/stat procedures can be extended into
geographic space to support
Quantitative Analysis of Mapped Data
“…thinking analytically
with maps”
ArcGIS Spatial Analyst operations
…over 170 individual “tools”
(Top) Of the spatial triad, the GIS component contains most of the spatial analysis and statistics operations for investigating the relationships within and among mapped data. While some of the basic analytics can be accomplished in vector-based software, most Map Analysis/Modeling involves grid-based software due to 1) the continuous nature of the data (map surfaces instead of discrete irregular spatial objects defined as separate point, line or polygon features), 2) the inherent consistency of the analysis frame, 3) the implicit topology relating all of the grid cells, and 4) the advanced mathematical/statistical operations available.
(Middle) Grid-based Spatial Analysis is analogous to traditional mathematics as it uses “cyclical processing of nested parentheticals.” Equation variables are sequentially defined (map layers retrieved), processed by an analytical algorithm (e.g., subtract one map layer from another on a cell-by-cell basis), store the intermediate solution (e.g., difference map) in the map stack. The cyclical process is continued (e.g., , retrieve divide store solution; retrieve multiply times100 store solution) to evaluate a “spatial equation,” such as the percent difference equation of %Diff= ((old – new) / old) …however in map algebra, the variables represent entire map layers defined by thousands of spatially organized numbers (geo-registered matrices) with the answer forming a map of the percent difference identifying where differences are high and where they are low throughout a project area.
Whereas Spatial Analysis operations are extensions of traditional scalar mathematics analyzing “geographic context,” Spatial Statistics extends traditional scalar statistics by analyzing the “numerical context” within and among map layers. The procedures account for the “spatial distribution” of map variables , as well as the “numerical distribution” used in central tendency analysis seeking to characterize the typical value– spatial statistics seeks to account for the spatial variation, such as generating cluster maps that identify where groupings of similar data patterns occur.
(Bottom) The SpatialSTEM approach and framework cross-references a wealth of existing grid-based map analysis operations used in GIS and RS (most available since the 1970s) with the seven classes of fundamental mathematical operations and the seven classes of fundamental statistical operations as shown in the lower portion of the slide. It is hoped that this recasting of map analysis/modeling procedures into the common framework/language of quantitative data analysis helps STEM disciplines better understand modern digital “maps as data, not just pictures” and the existence of a “map-ematics” that resonates with and extends non-spatial analysis and modeling techniques they currently employ– “thinking with maps” through quantitative spatial reasoning.
Slide 3, A Mathematical Structure for Map Analysis/Modeling – Moving Mapping to Map Analysis; Use Map-ematical Framework for GIS Modeling; Getting the Numbers Right
(Berry)
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4. Joseph K. Berry, MGUG 2014 Opening
Spatial Analysis Operations (Geographic Context)
GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if)
Spatial Analysis
Map Stack
Grid Layer
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 (matrix),
thereby providing a Mathematical Framework for map analysis and modeling of the
Contextual Spatial Relationships within 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: Classes of mathematical operations
Unique spatial
operations
(Top) There are two broad types of digital maps– Vector and Raster. Mapping and geo-query use a “vector” data structure that is akin to manual mapping in which discrete spatial objects (vector points, lines and polygons) form a collection of independent, irregular features to characterize geographic space (vis., a jigsaw puzzle). Grid-based map analysis and modeling procedures, on the other hand, operate on continuous map variables (raster map surfaces) composed of thousands of map values stored in geo- registered matrices.
Spatial analysis can be thought of as an extension of traditional mathematics involving the “contextual” relationships within and among mapped data layers. It focuses on geographic associations and connections, such as relative positioning, configurations and patterns among map locations.
From a GIS Perspective, the spatial analysis operators can be classified as Focal, Local and Zonal (Tomlin) based on how the algorithms work or by Reclassify, Overlay, Distance and Neighbors (Berry) based on the spatial information generated.
(Bottom) The basic spatial analysis operations can be categorized to align with traditional mathematical concepts and operations. The first three groupings are associated with basic math, algebra and calculus, the middle three involve spatial expressions of distance, proximity, movement and connectivity, with the last category identifying unique spatial operators that do not have corollaries in traditional mathematics (red check).
Slide 4, Spatial Analysis Operations (Geographic Context) – Simultaneously Trivializing and Complicating GIS; SpatialSTEM Has Deep Mathematical Roots; Understanding Grid-based Data; Suitability Modeling
(Berry)
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5. Joseph K. Berry, MGUG 2014 Opening
Spatial Analysis Operations (Math Examples)
Joseph K. Berry, MGUG 2014 Opening
Map Calculus — Spatial Derivative, Spatial Integral
Advanced Grid Math — Math, Trig, Logical Functions
Slope draped over
MapSurface 0%
65%
Spatial Derivative
…is equivalent to the slope of the tangent plane
at any grid location
SLOPE MapSurface Fitted
FOR MapSurface_slope
Fitted Plane
Surface 3D
500’
2500’
Curve 2D
The derivative is the instantaneous “rate of change” of a function and is equivalent to the slope of the tangent line at any point along the curve
Dzxy Elevation
Advanced Grid Math
Surface Area
…increases with increasing inclination as a Trig function of the cosine of
the slope
angle
S_Area=
Fn(Slope)
ʃ Districts_Average Elevation
Spatial Integral
Surface 3D
COMPOSITE Districts WITH MapSurface
Average FOR MapSurface_Davg
MapSurface_Davg
…summarizes the values on a surface for specified map areas (Total= volume under the surface)
(Top) Grid Math and Map Algebra can be expanded into other mathematical arenas. The Calculus Derivative traditionally identifies a measure of how a mathematical function changes as its input changes by assessing the slope along a curve in 2-dimensional abstract space— calculated as the slope of the tangent line at any location along the curve. The spatial equivalent calculates a “slope map” depicting the rate of change in a continuous map variable in 3-dimensional geographic space— calculated as the slope of a “best fitted plane” at any location on the map surface.
(Middle) Advanced Grid Math includes most of the buttons on a scientific calculator to include trigonometric functions. For example, taking the cosine of a slope map of an elevation surface expressed in degrees and multiplying it times the planimetric surface area of a grid cell calculates the increased real-word surface area of the “inclined plane” at each grid location.
(Bottom) The Calculus Integral is identified as the area of a region under a curve expressing a mathematical function. The spatial integral counterpart summarizes map surface values within specified regions (volume under the surface). The data summaries are not limited to just a total but can be extended to most statistical metrics. For example, the average map surface value can be calculated for each district in a project area. Similarly, the coefficient of variation ((Stdev / Average) * 100) can be calculated to assess data dispersion about the average.
 Slide 5, Spatial Analysis Operations (Math Examples) – Map-ematically Messing with Mapped Data; Characterizing Micro-terrain Features; Reclassifying and Overlaying Maps ; Use Map-ematical Framework for GIS Modeling
S_area= cellsize / cos(Dzxy Elevation)
The integral calculates the area under the curve for any section of a function.
Curve 2D
(Berry)
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6. Joseph K. Berry, MGUG 2014 Opening
Spatial Analysis Operations (Distance Examples)
Joseph K. Berry, MGUG 2014 Opening
Pythagoras
500 BC
Map Geometry — (Euclidian Proximity, Effective Proximity, Narrowness)
Plane Geometry Connectivity — (Optimal Path, Optimal Path Density)
Solid Geometry Connectivity — (Viewshed, Visual Exposure)
On + Off Road
96.0 minutes
…farthest away by truck, ATV and hiking
Off Road
Relative Barriers
Distance
Shortest straight line between two points (S,SL,2P)…
Proximity
…from a point to
everywhere (S,SL)…
Travel-Time
Surface
Effective Proximity
…not necessarily straight lines (S movement)
HQ (start)
On Road
26.5 minutes
…farthest away
by truck
Off Road
Absolute Barrier
Splash Algorithm
2000 AD
Plane Geometry
Connectivity
…like a raindrop, the
“steepest downhill
path” identifies the optimal route
(Quickest)
Farthest
(end) HQ
(start)
Truck = 18.8 min
ATV = 14.8 min
Hiking = 62.4 min
Seen if new tangent exceeds
all previous tangents
along the line of sight
Tan = Rise/Run
Rise
Run
Viewshed
Splash
Solid Geometry Connectivity
(Top) Traditional geometry defines Distance as “the shortest straight line between two points” and routinely measures it using a ruler or calculates it using the Pythagorean Theorem. Map Geometry extends the concept of distance to simple Euclidean Proximity by relaxing the requirement of just “two points” for distances to all locations surrounding a point or other map feature, such as a road. Concentric equidistance zones are established around a location or set of locations (“Splash” algorithm). This procedure is similar to the wave pattern generated when a rock is thrown into a still pond. Each ring indicates one “unit farther away”— increasing distance as the wave moves away. A further extension involves Effective Proximity, which relaxes “straight line” to consider intervening absolute and relative barriers to movement. For example, movement by truck might be constrained to roads (absolute barrier) but off-road travel by ATV or foot is affected by terrain conditions expressed as impedance to travel (relative barriers). The result is that the “shortest but not necessarily straight distance” is assigned to each grid location.
(Middle) Because a straight line connection cannot be assumed, optimal path routines in Plane Geometry Connectivity (2D space) are needed to identify the actual shortest route as the “steepest downhill path over the travel-time surface”– the twisting/turning route the wave-front propagation took to get there first. In Least Cost Path routing, the optimal path identifies the minimal cost (most preferred) route between two points considering discrete cost assigned to each intervening map location.
(Bottom) Solid Geometry Connectivity (3D space) involves line-of-sight connections that identify visual exposure among locations. The Viewshed algorithm uses a series of expanding rings (splash algorithm) to determine relative elevation differences from the viewer position to all other map locations. The ratio of the elevation difference (Rise) indicated as striped boxes to the distance away (Run) is used to determine visual connectivity (rise/run= tangent). Whenever the ratio exceeds the previous tangent ratio, the location is marked as seen (red); when it fails it is marked as not seen (grey). A Visual Exposure density surface is generated by noting the number of times each grid location is seen from a set of viewer locations. A Weighted Visual Exposure surface is generated by summing the weights assigned to each viewer location as an indicator of its relative importance.
Slide 6, Spatial Analysis Operations (Distance Examples) – Bending Our Understanding of Distance; Calculating Effective Distance and Connectivity; E911 for the Backcountry; Routing and Optimal Paths; Deriving and Using Travel-Time Maps; Deriving and Using Visual Exposure Maps; Creating Variable-Width Buffers; Applying Surface Analysis
Counts
# Viewers
270/621= 43% of the entire road network is connected
Visual Exposure
Highest
Weighted
Exposure
Sums
Viewer
Weights 
(Berry)
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7. Joseph K. Berry, MGUG 2014 Opening
Spatial Statistics Operations (Numeric Context)
GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if)
Spatial Statistics
Map Stack
Grid Layer
Spatial Statistics seeks to map the variation in a data set instead of focusing on
a single typical response (central tendency), thereby providing
a Statistical Framework for map analysis and modeling of the
Numerical Spatial Relationships within and among grid map layers
Map Analysis Toolbox
Basic Descriptive Statistics (Min, Max, Median, Mean, StDev, etc.)
Basic Classification (Reclassify, Contouring, Normalization)
Map Comparison (Joint Coincidence, Statistical Tests)
Unique Map Statistics (Roving Window and Regional Summaries)
Surface Modeling (Density Analysis, Spatial Interpolation)
Advanced Classification (Map Similarity, Maximum Likelihood, Clustering)
Predictive Statistics (Map Correlation/Regression, Data Mining Engines)
Statistical Perspective:
Unique spatial
operations
(Top) There are two broad types of digital maps– Vector and Raster. Mapping and geo-query use a vector data structure that is akin to manual mapping in which discrete spatial objects (vector points, lines and polygons) form a collection of independent, irregular features to characterize geographic space (viz., a jigsaw puzzle). Grid-based map analysis and modeling procedures, however, operate on continuous map variables (raster map surfaces) composed of thousands of map values stored in geo-registered matrices.
Spatial statistics can be thought of as an extension of traditional statistics involving the “numerical” relationships within and among mapped data layers. It focuses on 1) mapping the variation inherent in a data set rather than characterizing its central tendency and 2) summarizing the coincidence and correlation of the spatial distributions.
From a GIS Perspective, two dominant classes of spatial statistics are often identified— Surface Modeling that derives a continuous spatial distribution of a map variable from point sampled data and Spatial Data Mining that investigates numerical relationships among map variables.
(Bottom) The basic spatial statistic operations can be categorized to align with traditional non-spatial statistical concepts and operations. The first three groupings are associated with general descriptive statistics, the middle two involve unique spatial statistics operations that do not have corollaries in traditional statistics (red checks) and the final two identify classification and predictive statistics procedures.
 Slide 7, Spatial Statistics Operations (Numeric Context) – Infusing Spatial Character into Statistics; Paint by Numbers Outside the Traditional Statistics Box; Use Spatial Statistics to Map Abnormal Averages
(Berry)
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8. Joseph K. Berry, MGUG 2014 Opening
Spatial Statistics (Linking Data Space with Geographic Space)
Geo-registered Sample Data
Continuous Map Surface
Spatial Distribution
Surface Modeling techniques are used to derive a continuous map surface from discrete point data– fits a Surface to the data (maps the variation).
Roving Window (weighted average)
#1 = 4
Discrete Sample Map
Spatial
Statistics
Standard Normal Curve
Average = 22.6
Numeric Distribution
StDev =
26.2
(48.8)
Non-Spatial Statistics
In Data Space, a
standard normal curve can
be fitted to the data to identify
the “typical value” (average)
Histogram 70 60 50 40 30 20 10 80
In Geographic Space, the typical value forms a horizontal plane implying
the average is everywhere
X= 22.6
…lots of NE locations exceed Mean + 1Stdev
X + 1StDev =
= 48.8
Unusually high values
+StDev
Average
(Left side) Non-spatial statistics fits a standard normal curve in “data space” to assess the central tendency of the data as its average and standard deviation. In processing, the geographic coordinates are ignored and the typical value and its dispersion are assumed to be uniformly (or randomly) distributed in “geographic space.”
(Top right) The discrete point map locates each sample point on the XY coordinate plane and extends these points to their relative values (higher values in the NE; lowest in the NW). A roving window is moved throughout the area that weight-averages the point data as an inverse function of distance— closer samples are more influential than distant samples. The effect is to fit a continuous map surface that represents the geographic distribution of the data in a manner that is analogous to fitting a SNV curve to characterize the data’s numeric distribution. Underlying this process is the nature of the sampled data which must be numerically quantitative (measurable as continuous numbers) and geographically isopleth (numbers form continuous gradients in space).
(Lower-right ) The numerical distribution and the geographic distribution views of the data are directly linked. In geographic space, the “typical value” (average) forms a horizontal plane implying that the average is everywhere. In reality, the average is hardly anywhere and the geographic distribution denotes where values tend to be higher or lower than the average. In data space, a histogram represents the relative occurrence of each map value. By clicking anywhere on the map, the corresponding histogram interval is highlighted; conversely, clicking anywhere on the histogram highlights all of the corresponding map values within the interval. By selecting all locations with values greater than + 1SD, areas of unusually high values are located— a technique requiring the direct linkage of both numerical and geographic distributions.
Slide 8, Spatial Statistics Operations (Linking Data Space with Geographic Space) – Spatial Interpolation Procedures and Assessment; Linking Data Space and Geographic Space; Babies and Bath Water; Making Space for Mapped Data
(Berry)
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9. Joseph K. Berry, MGUG 2014 Opening
Spatial Statistics Operations (Data Mining Examples)
Joseph K. Berry, MGUG 2014 Opening
Slope
(Percent)
Map Clustering:
Elevation
(Feet)
X axis = Elevation (0-100 Normalized)
Y axis = Slope (0-100 Normalized)
Elevation vs. Slope Scatterplot
Data Space
Slope draped
on Elevation
Slope
Elev
Geographic Space
Data Pairs +
Plots here in…
Data
Space
Cluster 1
Low,Low
Cluster 2
High,High +
Geographic
Space
Advanced Classification (Clustering)
Entire Map
Extent
Spatially Aggregated Correlation
Scalar Value – one value represents the overall non-spatial relationship between the two map surfaces
…where x = Elevation value and y = Slope value
and n = number of value pairs
r =
…1 large data table
with 25rows x 25 columns =
625 map values for map wide summary
Predictive Statistics (Correlation)
Map Correlation:
Slope
(Percent)
Elevation
(Feet)
Roving Window
Localized Correlation
Map Variable – continuous quantitative surface represents the localized spatial relationship between the two map surfaces
…625 small data tables
within 5 cell reach =
81map values for localized summary
r = .432 Aggregated
As overview, Spatial Variable Dependence can be defined as what occurs at a location in geographic space is related to 1) the conditions of that variable at nearby locations, termed Spatial Autocorrelation (intra-variable dependence) used in surface modeling to generate a continuous map surface from discrete point data, or 2) the conditions of other map layer variables at that location, termed Spatial Correlation (inter-variable dependence) used in spatial data mining to characterize spatial relationships among map layers.
(Top) Map clustering uses spatial correlation to identify the location of inherent groupings of two or more map layers. For example, pairs of elevation and slope values are partitioned in data space into groups (called clusters) so that the value pairs in the same cluster are more similar to each other (minimal differences) than to those in other clusters while at the same time the difference between the clusters are maximized. The results is the optimal partitioning of data into the specified number of clusters considering two or more map surface layers.
(Bottom) Map correlation assesses the degree of dependency among the same maps of elevation and slope. Spatially “aggregated” correlation involves solving the standard correlation equation for the entire set of paired values to represent the overall relationship as a single metric. Like the statistical average, this value is assumed to be uniformly (or randomly) distributed in “geographic space” forming a horizontal plane.
“Localized” spatial correlation, on the other hand, maps the degree of dependency between the two map variables by successively solving the standard correlation equation within a roving window to generate a continuous map surface. The result is a map representing the geographic distribution of the spatial dependency throughout a project area indicating where the two map variables are highly correlated (both positively, red tones; and negatively, green tones) and where they have minimal correlation (yellow tones).
“Localized T-statistic” (Map Comparison) is generated by using a “roving window” to collect surrounding difference values that are evaluated using the standard “paired” T-test equation…
TStatistic = dMean / ( dStdev / Sqrt(n) )
…for a T-statistic map whose values are compared to the critical T-test metric to create a T-test map that identify those locations that have a significant statistical difference.
The “bottom-line” is that map variables can be substituted for scalar values in traditional equations to generate spatial solutions for most (all?) mathematical and statistical procedures– map analysis and modeling are direct extensions of traditional techniques..
 Slide 9, Spatial Statistics Operations (Data Mining Examples) – Characterizing Patterns and Relationships; Analyzing Map Similarity and Zoning; Discover the “Miracle” in Mapping Data Clusters
(Berry)
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10. Joseph K. Berry, MGUG 2014 Opening
Grid-based Map Data (geo-registered matrix of map values)
2.50 Latitude/Longitude Grid (140mi grid cell size)
Coordinate of first grid cell is 900 N 00 E
Analysis
Frame
(Matrix)
300 90
The Latitude/Longitude grid forms a continuous surface for geographic referencing where each grid cell represents a given
portion of the earth’ surface.
Conceptual Spreadsheet (73 x 144)
#Rows= 73 #Columns= 144
…each 2.50 grid cell is
about 140mi x 140mi
18,735mi2
…from Lat/Lon
“crosshairs to grid cells”
that contain map values indicating characteristics or conditions at each location
Lat/Lon
(Top) The most fundamental and ubiquitous grid form is the Latitude/Longitude coordinate system that enables every location on the Earth to be specified by a pair of numbers. The Prime Meridian and Equator serve as base references for establishing the angular movements expressed in degrees of Longitude (X; east/west movement) and Latitude (Y; north/south movement) of an imaginary vector from the center of the earth passing through any location on the surface of the earth. Longitude coordinates vary from 0 (Greenwich, England) to 180 decimal degrees E and W; Latitude coordinates vary from 0 (Equator) to 90 decimal degrees N and S. The intersecting Lat/Lon lines form a grid covering the globe that progressively becomes more refined with increasing decimal places-- double-precision binary floating-point has from significant decimal digits precision, resulting in less than .5 foot maximum locational precision for the Lat/Lon grid.
(Bottom) The Lat/Lon grid can be conceptualized as a large spreadsheet with a map value entered into the table at each grid space location. In turn, additional map layers could be stored as separate spreadsheet pages to form a map stack for analysis. While a spreadsheet form is useful to conceptualize the Lat/Lon grid it is an impractical storage mechanism except for small grid maps of significantly less than 100r x 100c= 10,000 cells.
Grid-based Mapped Data (Matrix of Map Values) – Organizing Geographic Space for Effective Analysis; To Boldly Go Where No Map Has Gone Before; Beware the Slippery Surfaces of GIS Modeling ; Explore Data Space
The easiest way to conceptualize a grid map is as an Excel spreadsheet with each cell in the table corresponding to a Lat/Lon grid space (location) and each value in a cell representing the characteristic or condition (information) of a mapped variable occurring at that location.
…maximum Lat/Lon decimal degree resolution is a four-inch square
anywhere in the world
(Berry)
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11. Joseph K. Berry, MGUG 2014 Opening
Universal Spatial Db Key (developing spatially-aware databases)
Joseph K. Berry, MGUG 2014 Opening
Spatially Keyed
data in the cloud
Lat/Lon serves as a Universal dB Key
for joining data tables based on location
…Spatially Keyed data in the cloud are downloaded and configured to the
Analysis Frame defining the Map Stack
…like a faucet spewing data
“What” = Data Value
“Where” = Lat/Lon cell
Conceptual Organization
Elevation Surface
Spreadsheet
30m Elevation
(99 columns x 99 rows)
Database Table
Geographic
Space
Grid
“Where”
RDBMS Organization
Data Space
Each column (field) represents a single map layer with the values in the rows indicating the characteristic
or condition at each grid cell location (record)
“What”
Keystone
Concept
2D Matrix 1D Field
(Bottom) Grid map values can be stored as “fields” in a data forming a long string of values; often in lexicographical order (left-to-right by rows beginning at the top as you would read a book) to match a computer programmer’s view of the ordering of matrices. In the example, the string of numbers is stored in inverted lexicographical order with the origin in the SW corner (instead of the NW corner) to match a mathematician's view of a graphical plot. Each field in the table represents a different map variable (theme), each record represents a grid cell location in the matrix and each map value represents the characteristic/condition recorded (map category).
(Top) The records in data tables in any database (in a project folder, or anywhere in a hard drive, or on a LAN connection or even in the cloud) can be joined based on their Lat/Lon coordinates. The implied grid can be flexibly defined to encompass cell sizes of varying resolution and the values sharing a grid space are assumed to be spatially coincident– the level of “slop” used to join records being left to the judgment of the user. Frequently used grid cell sizes include 1m, 10m, 30m, 90m and 1km with the user specifying the origin and #row/#columns to define the analysis window to use for a particular area of interest AOI extent). Also, the user specifies how to process the data if two or more data values “fall” into the same grid cell location– sum, average, max, min, etc.
Within this construct, Lat/Lon coordinates move from cross-hairs for precise navigation (intersecting lines) to a continuous matrix of spaces covering the globe for consistent data storage (grid cells). The recognition of a universal spatial key coupled with spatial analysis/statistics procedures and GPS/RS technologies provides a firm foothold “to boldly go where no map has gone before.”
Grid-based Mapped Data (Universal Spatial Key) – The Universal Key for Unlocking GIS’s Full Potential; Thinking Outside the Box
Lat/Lon as a
Universal Spatial Key
Once a set of mapped data is stamped with its Lat/Lon
“Spatial Db Key”
…it can be
linked to any other database table with spatially tagged records
without the explicit storage of a fully expanded grid layer—
All of the
spatial relationships are implicit
in the relative Lat/Lon positioning
in the raster grid.
(Berry)
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12. Data Structures & Analytics Revisit Geo-reference
GIS Development Cycle (…where we’re heading)
Radically new
Data Structures & Analytics
Square
(4 sides)
2D Planar
(X,Y Data)
Cartesian Coordinates
GIS Evolution
Revisit Analytics
(2020s)
Cube
(6 squares)
3D Solid
(X,Y,Z Data)
Revisit Geo-reference
(2010s)
Future Directions
Contemporary GIS
GeoWeb
(2000s)
Hexagon
(6 sides)
Pentagonal
Dodecahedral
(12 pentagons)
Map Analysis (1990s)
Spatial dB Mgt (1980s)
The Early Years
In the 1970s the research and early applications centered on Computer Mapping (display focus) that yielded to Spatial Data Management (data structure/management focus) in the next decade as we linked digital maps to attribute databases for geo-query. The 1990s centered on GIS Modeling (analysis focus) that laid the groundwork for whole new ways of assessing spatial patterns and relations, as well as entirely new applications such as precision agriculture and geo- business.
Today, GIS is centered on the GeoWeb and Mobile Devices (mapping focus) which brings us full circle to our beginnings. While advances in virtual reality and 3D visualization can “knock-your-socks-off” they represent incremental progress in visualizing maps that exploit dramatic computer hardware/software advances. The truly geospatial innovation waits the next re-focusing on data/structure and analysis.
The bulk of the current state of geospatial analysis relies on “static coincidence modeling” using a stack of geo-registered map layers. But the frontier of GIS research is shifting focus to “dynamic flows modeling” that tracks movement over space and time in three-dimensional geographic space. But a wholesale revamping of data structure is needed to make this leap.
In addition to the changes in the processing environment, contemporary maps have radical new forms of display beyond the historical 2D planimetric paper map. Today, one expects to be able to drape spatial information on a 3D view of the terrain. Virtual reality can transform the information from pastel polygons to rendered objects of trees, lakes and buildings for near photographic realism. Embedded hyperlinks access actual photos, video, audio, text and data associated with map locations. Immersive imaging enables the user to interactively pan and zoom in all directions within a display.
The impact of the next decade’s evolution (2010s) will be huge and shake the very core of GIS—the Cartesian coordinate system itself …a spatial referencing concept introduced by mathematician Rene Descartes 400 years ago.
The current 2D square for geographic referencing is fine for “static coincidence” analysis over relatively small land areas, but woefully lacking for “dynamic 3D flows.” It is likely that Descartes’ 2D squares will be replaced by hexagons (like the patches forming a soccer ball) that better represent our curved earth’s surface …and the 3D cubes replaced by nesting polyhedrals for a consistent and seamless representation of three-dimensional geographic space. This change in referencing extends the current six-sides of a cube for flow modeling to the twelve-sides (facets) of a polyhedral—radically changing our algorithms as well as our historical perspective of mapping (2020s).
The new geo-referencing framework provides a needed foothold for solving complex spatial problems, such as intercepting a nuclear missile using supersonic evasive maneuvers or tracking the air, surface and groundwater flows and concentrations of a toxic release. While the advanced map analysis applications coming our way aren’t the bread and butter of mass applications based on historical map usage (visualization and geo-query of data layers) they represent natural extensions of geospatial conceptualization and analysis …built upon an entirely new set analytic tools, geo- referencing framework and a more realistic paradigm of geographic space.
4D GIS (XYZ and time) is the next major frontier. Currently, time is handled as a series of stored map layers that can be animated to view changes on the landscape. Add predictive modeling to the mix and proposed management actions (e.g., timber harvesting and subsequent vegetation growth) can be introduced to look into the future. Tomorrow’s data structures will accommodate time as a stored dimension and completely change the conventional mapping paradigm that will spawn entirely new analytical capabilities (2020s).
Mapping focus
Data/Structure focus
Analysis focus
…about every decade
The Early Years
Computer Mapping
(1970s)
(Berry)
Overviewl Session on Precision Ag?

13. Joseph K. Berry, MGUG 2014 Opening
So Where to Head from Here?
Joseph K. Berry, MGUG 2014 Opening
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Beyond Mapping Compilation Series
…nearly 1000 pages and more than 750 figures in the Series provide a comprehensive and longitudinal perspective of the underlying concepts, considerations, issues and evolutionary development of modern geotechnology (RS, GIS, GPS).
This paradigm shift, however, is not without peril.  A portion of the GIS community firmly holds that a didactic approach is needed to first educate potential users on the fundamental nature of the data and cartographic principles.  Similarly, a portion of the STEM community firmly holds that map-ematics is simply a collection of computer algorithms with negligible supporting mathematical theory and that the idea of a spatial distribution of a map variable is not a statistically valid concept. 
The “perfect geotechnology storm” is the result of he convergence of four critical enabling technologies; 1) the personal computers’ dramatic increase in computing power, 2) the maturation of GPS and RS (remote sensing) technologies, 3) a ubiquitous Internet and 4) the general availability of digital mapped data. Also, keep in mind that geotechnology is in its fourth decade—
- the 1970s saw Computer Mapping automate the drafting process through the introduction of the digital map;
- the 80s saw Spatial Database Management link digital maps to descriptive records;
- the 90s saw the maturation of Map Analysis and Modeling capabilities that moved mapped data to effective information by investigating spatial relationships; and finally,
the 00s focused on Multimedia Mapping emphasizing data delivery through Internet proliferation of data portals and advanced display mechanisms involving 3D visualization and virtual reality environments, such as in Google and Virtual Earths.
The “era of maps as data” (Where is What?) is rapidly giving way to the “age of spatial information” where mapped data and analytical tools directly support decision-making (Why, So What and What If?). The key for developing successful solutions beyond data delivery lies in domain expertise as much, if not more, than mapping know-how. This means that outreach across campus is as important (and quite possibly more important) than honing courses for training core GIS professionals. It suggests less flagship/toolbox software systems and more custom/tailored packages solving well-defined spatial problems that stimulate “thinking with maps.”
For geotechnology to truly become a fabric of society and spatial reasoning an integral part of critical-thinking and problem-solving the analytical concepts and procedures need to be enveloped and taught within the diverse set of STEM tents on campus.  For example, the rise and fall of GIS in natural resources education, in large part the result of GIS as something outside the normal tapestry of wildlife, forestry, recreation knowledge sets.  To be effective and fully brought into the fold, map analysis principles need to be exposed from the inside of applied disciplines.
An analogy is the evolution of statistics as a “pure” and “applied” discipline.  Stat courses have moved out from statistics departments to tailored courses within many (most?) majors on campus, such as NR, agriculture, public health, international studies and business, that all have their own stat courses.  Statistics departments haven’t withered and died; rather they focus on advancing the field, with the teaching of applied aspects most often left to the individual disciplines.  At some point we (GIS community) need to back-away from the didactic path requiring students to learn geographic/cartographic basics (e.g., geodes, projections, etc) before they are allowed to run with the wealth of preconditioned mapped data out there– move some of the focus away from geotechnology as a “technological tool” and toward the perspective of geotechnology as an “analytical tool.” 
 If the idea of a Lat/Lon as the “universal spatial key” takes hold, it seems to make less a comprehensive understanding of cartographic theory (with midterm test questions in a course primarily designed for GIS specialists) less necessary for developing and using a spatially-enabled database.  The analogy is holding the requirement that “all who employ regression” must be able to derive its equations through least squares theory and be able to write an essay on how it corresponds to maximum likelihood theory.
I suspect sometime beyond my lifetime, the view of “maps as data” will take firm hold within the STEM disciplines and a recognized “map-ematics” will prevail.  Heck, at their essence maps are just organized set of numbers …and we do math/stat things to these map variables to better understand the patterns and spatial relationships inherent in the data. 
Someday generating a map of the variation in a data set (deriving the spatial distribution of mapped data in geographic space) will be as second nature as computing its average (deriving the central tendency of the data in number space).
So What’s the Point? – Is GIS Technology Ahead of Science?; GIS Evolution and Future Trends; Spatial Modeling in Natural Resources; Lumpers and Splitters Propel GIS; The Softer Side of GIS
Joseph K. Berry
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