Table Of ContentGeoGraphic information SyStemS Zhang
Atkinson
Goodchild
Scale in Spatial
Information and
Analysis
“… an authoritative and thought-provoking account of scaling issues in
spatial information and analysis. Written by leading authorities in the field, this
book should be of considerable value to anyone working with spatial data by
providing a contemporary account of s cale, its characterisation and analysis.”
—Giles Foody, University of Nottingham, UK
“… a timely addition to the growing literature on the topic of scale and
change of support. … The authors have done a marvelous job of tackling this difficult
topic with enough breadth and depth, producing what should
become the reference in th e field for years to come.”
—Pierre Goovaerts, BioMedware, Inc.
“… A truly informative collection … will certainly promote the importance
of scale as the importance of geography has been increasingly recognized
by people while stepping into the so -called ‘Digital Earth’ big data age.”
—Hui Lin, Institute of Space and Earth Information Science,
The Chinese University of Hong Kong
“… a cutting-edge analytical survey that presents scale in its myriad contexts,
effectively linking it with cartography, spatial accuracy, spatial sampling, and
spatial statistics. As such, it will set a new standard for the next round
of state-of-the-art work con cerning geographic scale.”
—Daniel A. Griffith, University of Texas at Dallas
“… the book will serve as a long and lasting contribution to the scale aspects that
are fundamental to the many spa tial problems that are covered.”
—Alfred Stein, University of Twente, Faculty of Geo-Information
and Earth Observation (ITC), Enschede, The Netherlands
K11444
ISBN-13: 978-1-4398-2937-0
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K11444_Cover_mech.indd All Pages 3/7/14 9:52 AM
Scale in Spatial Information
and Analysis
Scale in Spatial Information
and Analysis
Jingxiong Zhang, Peter Atkinson, and Michael Goodchild
Boca Raton London New York
CRC Press is an imprint of the
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Contents
Preface.......................................................................................................................ix
Acknowledgment ......................................................................................................xi
Authors ...................................................................................................................xiii
Chapter 1 Introduction ..........................................................................................1
1.1 Issue of Scale .............................................................................1
1.2 Models of Scale .........................................................................4
1.3 Scaling Up and Down ................................................................9
1.4 Book Chapters .........................................................................12
Chapter 2 Geographic Representations ...............................................................17
2.1 Geo-Atoms ..............................................................................20
2.2 Geo-Fields ...............................................................................21
2.3 Geo-Objects .............................................................................25
2.4 Hierarchical Data Structures ..................................................28
2.4.1 Quadtrees ...................................................................28
2.4.2 Hierarchical TINs .......................................................29
2.5 Discussion ................................................................................31
2.5.1 Geo-Dipoles and Spatial Interactions ........................31
2.5.2 Temporal Dimension ..................................................32
2.5.3 Scale Dependence of Representations ........................32
Chapter 3 Geospatial Measurements ..................................................................35
3.1 Framework for Spatial Sampling .............................................36
3.1.1 Spatial Sampling Design ............................................36
3.1.2 Data Supports .............................................................43
3.1.3 Spatial Resolution in Remotely Sensed Images .........44
3.1.3.1 Rayleigh Criterion .......................................45
3.1.3.2 MTF and Its Equivalent Bandwidth ...........45
3.1.3.3 Effective Radiometric Resolution Element ...46
3.2 Optical Remote Sensing and Resolution .................................47
3.2.1 Spatial and Spectral Responses ..................................47
3.2.2 Geometric Characteristics ..........................................50
3.3 Microwave Remote Sensing and Resolution ...........................55
3.3.1 Resolution ...................................................................55
3.3.2 Radar Signals, Imaging, and Distortions ...................58
3.3.3 Passive Microwave .....................................................62
3.4 Discussion ................................................................................64
v
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Chapter 4 Geostatistical Models of Scale ...........................................................67
4.1 Geostatistical Fundamentals and Variograms .........................68
4.1.1 Random Fields ............................................................68
4.1.2 Variograms and Covariance Functions ......................70
4.1.3 Indicator Variograms and Transition
Probability- Based Models ..........................................75
4.2 Variogram Regularization and Deregularization ....................77
4.2.1 Regularization ............................................................77
4.2.2 Deconvolution of Regularized Variograms ................80
4.3 Statistics for Determining Measurement Scales .....................82
4.4 Discussion ................................................................................87
Chapter 5 Lattice Data and Scale Models ...........................................................91
5.1 Lattice Data .............................................................................93
5.2 Spatial Autocorrelation and Its Measures ...............................95
5.3 Local Models ...........................................................................99
5.4 Discussion ..............................................................................103
Chapter 6 Geostatistical Methods for Scaling ..................................................107
6.1 Kriging ..................................................................................109
6.2 Indicator Approaches.............................................................116
6.3 Upscaling by Block Kriging ..................................................118
6.4 Downscaling by Area-to-Point Kriging ................................120
6.5 Geostatistical Inverse Modeling ............................................124
6.6 Discussion ..............................................................................126
Chapter 7 Methods for Scaling Gridded Data ..................................................129
7.1 Upscaling ...............................................................................131
7.1.1 Statistical Approaches ..............................................131
7.1.2 Mechanistic Approaches ..........................................134
7.2 Downscaling ..........................................................................137
7.2.1 Deconvolution ..........................................................137
7.2.2 Super-Resolution Mapping .......................................141
7.2.3 Subpixel Mapping ....................................................144
7.3 Discussion ..............................................................................145
Chapter 8 Multiscale Data Conflation ..............................................................147
8.1 Multivariate Geostatistics ......................................................149
8.2 Image Fusion .........................................................................154
8.2.1 Conventional Approaches.........................................154
8.2.2 Multiresolution Representations Based on Wavelets ...155
8.2.3 Wavelet-Based Image Fusion ...................................161
Contents vii
8.3 Other Multiscale Methods .....................................................163
8.4 Discussion ..............................................................................166
Chapter 9 Scale in Terrain Analysis .................................................................169
9.1 Digital Elevation Data and Their Scales ...............................171
9.2 Terrain Derivatives ................................................................176
9.2.1 Slope and Aspect ......................................................176
9.2.2 Curvature ..................................................................180
9.2.3 Secondary Properties ...............................................186
9.3 Models of Scale in Topography .............................................187
9.4 Methods for Scaling Terrain Variables .................................193
9.4.1 Upscaling ..................................................................193
9.4.2 Downscaling .............................................................196
9.5 Discussion ..............................................................................198
Chapter 10 Scale in Area-Class Mapping ...........................................................201
10.1 Area-Class Mapping .............................................................204
10.1.1 Discriminant-Space Model ......................................204
10.1.2 Methods for Mapping Area Classes .........................209
10.2 Spatial Scales and Patterns in Area Classes ..........................213
10.3 Scaling Area-Class Information ............................................219
10.3.1 Upscaling ..................................................................219
10.3.2 Downscaling .............................................................220
10.4 Discussion ..............................................................................223
Chapter 11 Information Content .........................................................................227
11.1 Information Theory ...............................................................230
11.2 Information Content in Remotely Sensed Images .................238
11.2.1 Informational Characteristics of Images ..................238
11.2.2 Information Conveyed in Images with
Additive Noise ..........................................................242
11.2.3 Information in Images with Multiplicative Noise ....247
11.3 Image Resolution and Information Content ..........................250
11.3.1 Effects of Resolution on Information Content .........250
11.3.2 Information Capacity and Super Resolution ............253
11.4 Information Content in Map Data .........................................255
11.5 Discussion ..............................................................................261
Chapter 12 Uncertainty Characterization ...........................................................265
12.1 Accuracy Metrics and Assessment ........................................268
12.1.1 Accuracy in Interval/Ratio Fields ............................268
12.1.2 Accuracy in Area Classes ........................................271
viii Contents
12.1.3 Positional Accuracy ..................................................273
12.1.4 Confidence Intervals and Hypothesis Testing .........274
12.2 Geostatistical Approaches to Validation ...............................278
12.3 Analytical Approaches to Error Propagation ........................282
12.4 Geostatistical Simulation .......................................................287
12.4.1 Simulation over a Point Support ...............................287
12.4.2 Point-Support Simulation Conditional to
Areal Data ................................................................291
12.4.3 Simulation over Blocks .............................................293
12.5 Discussion ..............................................................................296
Epilogue ................................................................................................................299
References .............................................................................................................309
Preface
Scale can be a frustrating concept. It has multiple meanings even in a single scientific
domain, and there is significant variation in meaning across domains. It underlies
how we as humans perceive and act in the world, from the scale of our own bodies
to that of our immediate surroundings, to the scale of life experiences, and to the
scale of the cosmos. It can function as a noun or as a verb, with different sets of
meanings in the two cases. It is implicit in the definitions of many properties that we
work with as environmental or social scientists. Statements such as “Minnesota has
10,000 lakes” from that state’s license plate, “The coastline of Italy measures 7600
km” (CIA 2009), “This is the flattest spot in the United States,” or “This area has the
world’s highest population density” are all scale specific, and yet we frequently make
or accept statements such as these without specifying scale or even recognizing the
importance of scale in them.
This book attempts to resolve some of these issues and to alleviate some of this
frustration by taking a rigorous, scientific approach to scale and its various meanings
in relation to the geographic world. It covers methods for measuring aspects of scale
and for changing scale through upscaling and downscaling. It discusses scale in rela-
tion to the various systems that are now available for acquiring geographic data, for
modeling environmental and social processes, and for visualizing phenomena and
their distributions over the surface and near the surface of the Earth.
This is not the first book to appear on these issues, and citations of many of the
previous ones can be found at various points in this book. It will certainly not be the
last, because geographic technologies are advancing rapidly, along with their impor-
tance in numerous areas of human activity, and because our thinking and theorizing
about scale are also in a constant state of flux. New techniques for downscaling, new
methods of analysis, new models of spatial and temporal scale, and new theories of
how scale affects and is affected by social and environmental processes are all likely
to emerge in the next few years.
Almost two decades ago the Alexandria Digital Library was funded by the US
National Science Foundation as an early attempt to build an online system for search-
ing, evaluating, and retrieving geographic datasets—in effect, an Internet-based map
library (Smith et al. 1996). One of our first concerns in developing the library was to
prioritize search criteria: what would people consider most important in searching
for online data? First, clearly, was coverage in the form of a bounding box defined
by latitude and longitude; second was theme, or the topic of the data; third was time,
or the date of validity of the data. Missing from this list was scale: the degree of
detail, since all geographic data must be generalized, sampled, or abstracted in some
way, the real world being infinitely complex. Scale was considered too difficult for
the average user to conceptualize, too difficult to characterize, and not of sufficient
importance. We, the authors, hope this book will move us some distance in the direc-
tion of resolving these issues and of promoting scale to the importance it deserves.
Level of detail is surely an inescapable property of any digital representation of any
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