Applied Geostatistics
Miles Logsdon
mlog@u.washington.edu
Mimi D’Iorio
mimid@u.washington.edu
•"An Introduction to Applied Geostatistics" by
Edward H. Isaaks and R. Mohan Srivastava,
Oxford University Press, 1989.
•"Spatial Data Analysis: Theroy and Practice" by
Robert Haining, Cambridge University Press,
1993.
•"Statistics for Spatial data" by Noel a. c. Cressie,
Wiley & Sons, Inc. 1991.
Introduction to Geostatistics
D
Z(s)
• D is the spatial domain or area of
interest
• s contains the spatial coordinates
• Z is a value located at the spatial
coordinates
{Z(s): s  D}
Geostatistics: Z random; D fixed,
infinite, continuous
Lattice Models: Z random; D fixed,
finite, (ir)regular grid
Point Patterns: Z  1; D random,
finite
GeoStatistics
-A way of describing the spatial
continuity as an essential feature of
natural phenomena.
•Univariate
- The science of uncertainty which
attempts to model order in disorder.
•Bivariate
- Recognized to have emerged in the
early 1980’s as a hybrid of
mathematics, statistics, and mining
engineering.
•Spatial Description
- Now extended to spatial pattern
description
Univariate
•One Variable
•Frequency (table)
•Histogram (graph)
•Do the same thing (i.e count of
observations in intervals or classes
•Cumulative Frequency (total “below”
cutoffs)
Summary of a histogram
Measurements of location (center of
distribution
n
mean (m µ x )
median
mode

n
i 1
n
2
st . d . 
Measurements of shape (symmetry
& length
1 n
coefficient of skewness
coefficient of variation
i
  1 / n  xi   
Measurements of spread (variability)
variance
standard deviation
interquartile range
x
CS 
i 1
2
x  


n
i 1
i

2
2
IQR 
Q Q
3
3

CV 

1
Bivariate
p 
Scatterplots
Yin
Correlation

n
1
n
p
i 1


xi 
x
x


p 
X in
yi 
y

 y
Linear Regression
y  ax  b
slope
constant
a

p

y
x
b   y  a  x
Autocorrelation
Values at locations that are near to each
other are more similar than values at
locations that are farther apart.
Spatial Description
- Data Postings = symbol maps
(if only 2 classes = indicator map
- Contour Maps
- Moving Windows => “heteroscedasticity”
(values in some region are more variable than in others)
- Spatial Continuity
(h-scatterplots
* Xj,Yj
Spatial lag = h = (0,1) = same x, y+1
h=(0,0) h=(0,3) h=(0,5)
tj
hij=tj-ti
* Xi,Yi
correlation coefficient
(i.e the correlogram, relationship of p with h
*
(0,0)
ti
Lags
Variograms: How do we estimate them?
Binning Lags
Variograms: How do we estimate them?
1
1
2
2
3
4
4
3
Let’s review:
VECTOR
1
3 OR
2
2
2
p 
Yin
p 
-Data Postings => symbol maps
-Contour Maps
•Moving Windows => “heteroscedasticity”
•Spatial Continuity h-scatterplots
X in
3
15
10
5
RASTER
Geostatistics
Univariate Bivariate Spatial Description -
1
4
12
11
Values at locations that
are near to each other are
more similar than values
at locations that are
farther apart.
= Autocorrelation
Spatial Lag = h = distance
Lag bins
1
2
3
4
Definitions
Variograms: What are they?
Covariance
C (h)  cov( Z (s), Z (s  h))
Autocorrelation
 (h)  C (h) / C (0)
Variogram
2 (h)  var( Z (s)  Z (s  h))
•Correlogram = p(h) = the relationship of the correlation
coefficient of an h-scatterplot and h (the spatial lag)
•Covariance = C(h) = the relationship of the coefficient of
variation of an h-scatterplot and h
•Semivariogram = variogram =  ( h) = moment of inertia
moment of inertia =
1 n

2n i  1
x  y 
i
2
i
OR: half the average sum difference between the x and y pair
of the h-scatterplot
OR: for a h(0,0) all points fall on a line x=y
OR: as |h| points drift away from x=y
Isotropy
Variograms: What are their features?
Anisotropy
Variograms: What are their features?
Anisotropy
Variograms: What are their features?
Anisotropy
Variograms: What are their features?
Structured Process in
Geostatistics
Represent the
Represent the
Data
Data
Explore the Data
Explore the Data
Fit a Model
Fit a Model
Perform
Perform
Diagnostics
Diagnostics
Compare the
Compare the
Models
Models
Physiognomy / Pattern / structure
Composition = The presence and amount of each
element type without spatially explicit measures.
Proportion, richness, evenness, diversity
Configuration = The physical distribution in space and
spatial character of elements.
Isolation, placement, adjacency
** some metrics do both **
Types of Metics
Area Metrics
Patch Density, Size and Variability
Edge Metrics
Shape Metrics
Core Area Metrics
Nearest-Neighbor Metrics
Diversity Metrics
Contagion and Interspersion Metrics
Shape Metrics
perimeter-area relationships
Shape Index (SHAPE) -- complexity of patch compared
to standard shape
vector uses circular; raster uses square
Mean Shape Index (MSI) = perimeter-to-area ratio
Area-Weighted Mean Shape Index (AWMSI)
Landscape Shape Index (LSI)
Fractal Dimension (D), or (FRACT)
log P = 1/2D*log A; P = perimeter, A = area
P = sq.rt. A raised to D, and D = 1 (a line)
as polygons move to complexity P = A, and D -> 2
A few fractal metrics



Double log fractal dimension (DLFD)
Mean patch fractal (MPFD)
Area-weighted mean patch fractal dimension (AWMPFD)
Contagion, Interspersion and Juxtaposition
When first proposed (O’Neill 1988) proved incorrect, Li &
Reynolds (1993) alternative
Based upon the product of two (2) probabilities
Randomly chosen cell belongs to patch “i”
Conditional probability of given type “i” neighboring cells
belongs to “j”
Interspersion (the intermixing of units of different patch
types) and Juxtaposition (the mix of different types
being adjacent) index (IJI)
Changing patterns
Month
NP
LPI
LSI
MPFD
IJI
January
21.00
28.46
7.79
1.35
66.89
February
98.00
25.08
9.64
1.27
65.57
March
92.25
21.61
9.65
1.29
67.23
April
93.73
18.99
8.43
1.26
70.12
May
84.00
25.45
9.04
1.29
68.67
June
103.33
15.00
9.39
1.27
71.96
July
82.86
25.03
9.38
1.29
70.63
August
24.10
26.23
7.96
1.33
72.40
September
20.78
26.78
7.96
1.34
70.18
October
22.08
25.78
7.97
1.35
65.60
November
20.80
29.94
7.95
1.37
67.21
December
21.43
32.32
7.57
1.34
67.23
Flying