Khartoum University
Faculty of Postgraduate
Geo statistics
Instructor : Dr.Samir Mahmoud
Email: samiradm59@yahoo.com
Phone: 0909864181
1
Course administration
⚫ Lectures: Tusday 03:30 – 5:30
⚫ Classroom:
Number
⚫ Course Grading :-
Homework& Attendance 20%
Midterm
20%
Final Exam
60%
⚫ Homework : Assignments , Researches
⚫ Tests:Midterm Exam:
Final
Exam:
2
Course Objectives
⚫ To provide an overview of the Basic Components of
Geostatistics which includes:
⚫ Mesuearuring geographical Distributions
⚫ – (Semi)variogram analysis – characterization of
spatial correlation
⚫ – Kriging – Optimal interpolation; generates best
linear unbiased estimate at each location; employs
semi variogram model
References
•P. Goovaerts, 1997, Geostatisticsfor Natural Resources Evaluation,
Oxford University Press, 483 pp.
•E.H. Isaaksand R.M. Srivastava, 1989, An Introduction to Applied
Geostatistics, Oxford University Press, 561 pp.
•P.K. Kitanidis, 1997, Introduction to Geostatistics: Applications in
Hydrogeology, Cambridge University Press, 249 pages.
•R.A. Olea, 1999, Geostatisticsfor Engineers and Earth Scientists, Kluwer
Academic Publishers, 303 pp.
•Webster, R. and Oliver, M.A., 2007. Geostatisticsfor Environmental
Scientists. Statistics in Practice. John Wiley & Sons, Chichester, 330 pp.
•RossiterD.G., 2008. Spatial analysis and Geostatistics, lecture notes, ITC.
•Hengl, T. 2009, A Practical Guide to GeostatisticalMapping. The Office
for Official Publications of the European Communities Press,
Luxembourg (ISBN: 978-92-7906904-8), 270 pp.
Course Contents
1.Basic geostatistics
2.Measures of central tendency and dispersion
3.Probability
4.Measuring Geographic Distributions
5-Samplinig concept
6.Point pattern analysis
7.Spatial Analysis-Geo statistical Analyst
8.Experimental Variogram & Variogram Modeling
9.Geostatistical Estimation (Kriging& Co-Kriging)
1-Basic Geo statistics
1.1-Starting of Geo statistic
⚫ Geo statistics originated from the mining and
petroleum industries, starting with the work by Danie
Krigein the 1950's
⚫ Further was developed by Georges Matheronin the
1960's.
⚫ In both industries, geo statistics is successfully applied
to solve cases where decisions concerning expensive
operations are based on interpretations from sparse
data located in space.
1.2-Geo statistic fields
Geo statistics has since been extended to many other
¯fields in or related to the earth sciences, e.g.:
hydrogeology, hydrology, meteorology, oceanography,
geochemistry, geography, soil sciences, forestry,
landscape ecology, Geology , Mining and others
1.3-What is Geo statistics?
1.3.1-Branch of statistical sciences:
“Geo statistics can be defined as the branch of statistical
sciences that studies spatial/temporal phenomena and
locations of spatial relationships to model possible
values of variable(s) at un observed, un sampled
locations”(Caers, 2005)
1.3.2-Study of phenomena
“Geostatistics: study of phenomena that vary in space
and/or time” (Deutsch, 2002).
1.3.3-Way of describing the spatial continuity
“Geos tatistics offers a way of describing the spatial
continuity of natural phenomena and provides
adaptations of classical regression techniques to take
advantage of this continuity.”(Isaaksand Srivastava, 1989)
1.3.4-Analysis and interpretation of geoData
⚫ “Geostatistics is a subset of statistics specialized in
analysis and interpretation of geographically and
temporally referenced data”
⚫ “Geo statistics is an analytical tool for statistical
analysis of sampled field data
1.4-Statistics & Geostatistics
1.4.1-Classic statistics
Classic statistic :is generally devoted to the analysis
and interpretation of uncertainties caused by limited
sampling of a property under study.
Classic statistics: examine the statistical distribution
of a set of sampled data, geostatistics in corporates both
the statistical distribution of the sample data and the
spatial correlation among the sample data
1.4.2-Geostatistics
Is not tied to a population distribution model that
assumes, for example, all samples of a population are
normally distributed and independent from one
another.
Most of the earth science data (e.g., rock properties,
contaminant concentrations)
Often do not satisfy these assumptions as they can be
highly and/or possess spatial correlation
1.5-Geo statistic and Auto correlation
Auto correlation: correlation between elements of a
series and others from the same series separated from
them by a given interval. (Oxford American Dictionary)
(i.e., data values from locations that are closer together
tend to be more similar than data values from locations
that are further apart).
Example of spatially auto-correlated
parameters:
in hydrogeology: thickness, porosity, hydraulic
conductivity
1.6-Geo statistics and GIS
They both deal with Spatial Data
•There are tools allowing GIS+statistics integration
•Gis software provides tools for geo statistical analysis
Like spatial pattern analysis ,Spatial Analysis &
Geo statistical Analyst
1.7-Spatial Variable in Space
How does a variable vary in space
What controls its variation in space?
Where to locate samples to describe its spatial
variability?
How many samples are needed to represent its spatial
variability?
What is a value of a variable at some new location?
What is the uncertainty of the estimate
1.8-Nature of variables
•Many variables directly refer to processes and are
expressed in quantity per time units e.g. mm of rainfall
per year.
•In ecology: objects of interest (individual plants or
animals), often immeasurable in quantity animal species
change their location dynamically, often in
unpredictable directions and with unpredictable spatial
patterns (nonlinear trajectories); occurrence records
1.9-Examples of environmental variables
Biology :-(distribution of species and biodiversity
measures)
Soil science :- (soil properties and types)
Vegetation science:- (plant species and communities,
land cover types)
Climatology:- (climatic variables at surface and
benith/above)
Hydrology:- (water quantities and conditions)
Geology :-(rock types, rock porosity, element
concentration)
1.11-Spatial Variability
Is a result of complex processes working at the same
time and over long periods of time, rather than an effect
of a single realization of a single factor. Sum of two
components:
(a) the natural spatial variation
(b) the inherent noise.
1.10-Classification of Spatial variability
It may be classified into
• Geographical variation (2D)
• Vertical variation (3D)
• Temporal variation
• Variation at different scales (support size)
2.-Measures of central
tendency and dispersion
20
Objective
To learn how to find measures of central tendency in a set of
raw data.
Relevance
To be able to calculate the most appropriate measure of
center after analyzing the context of a study that might or
might not contain extreme values.
Central Values – Many times one number is used to describe the entire
sample or population. Such a number is called an average. There are many
ways to compute an average.
⚫ There are 4 values that are considered measures of the
center.
1. Mean
2. Median
3. Mode
4. Midrange
⚫ Mean – the arithmetic average with which you are
the most familiar.
⚫ Mean:
Ex 2
⚫ Mode values most found or
distributed
⚫ Find the mode.
⚫ A. 0, 1, 2, 3, 4 - no mode
⚫ B. 4, 4, 6, 7, 8, 9, 6, 9 - 4 ,6, and 9
Midrange
⚫ The number exactly midway
between the lowest value and
highest value of the data set. It
is found by averaging the low and
high numbers.
Example
⚫ Find the midrange of the set.
⚫ 3, 3, 5, 6, 8
Example
a) Compute the mean for the entire sample.
Measures of Variation
⚫ There are 3 values used to measure
the amount of dispersion or
variation. (The spread of the
group)
1. Range
2. Variance
3. Standard Deviation
Range
⚫ The range is the difference
between the lowest value in the
set and the highest value in the
set.
⚫ Range = High # - Low #
Example
⚫ Find the range of the data set.
⚫ 40, 30, 15, 2, 100, 37, 24, 99
⚫ Range = 100 – 2 = 98
Variance (Array)
⚫ Variance Formula
Standard Deviation
⚫ The standard deviation is the
square root of the variance.
Example – Using Formula
⚫ Find the variance.6, 3, 8, 5, 3
Find the standard deviation
Find the standard deviation
⚫ The standard deviation is the
square root of the variance.
3.Probability dispersion
36
3.1-Probability Definition
⚫ Definition: the chance or relative frequency of
occurrence of the event
⚫ •It ranges between 0 ~1
⚫ •The probabilities of all possible (mutually exclusive)
events of an experiment must sum to 1
Binomial Probability Distribution
A binomial random variable X is defined to the
number of “successes” in n independent trials
where the P(“success”) = p is constant.
Notation: X ~ BIN(n,p)
Binomial
distribution General Formula :
.
Example on Probability using binomial
distribution
⚫
⚫
⚫
⚫
⚫
⚫
If the Percentage pass of student in a course is 0.8
And we have 15 students find the :Probability for all student to pass
Probability of 8 student
Probability of 6 student
Probability of no student
q=1 - 0.8=0.2
Binomial distribution: example
⚫ If I toss a coin 20 times, what’s the probability of
getting of getting 2 or fewer heads?
4.Measuring Geographic
Distributions
Definition
⚫ Measuring the distribution of a set of features allows
⚫
⚫
⚫
⚫
you to calculate a value that represents a characteristic
of the distribution, such as the center, Directional
Distribution , Linear Directional Mean , Mean Center ,
Median Center & Standard Distance
The Measuring Geographic Distributions toolset
addresses questions such as:
Where's the center?
What's the shape and orientation of the data?
How dispersed are the features?
Measuring Geographic Distributions
toolset
Tool
Description
Central Feature
Identifies the most centrally located feature in a point,
line, or polygon feature class.
Directional Distribution
Creates standard deviational ellipses to summarize the
spatial characteristics of geographic features: central
tendency, dispersion, and directional trends.
Linear Directional Mean
Identifies the mean direction, length, and geographic
center for a set of lines.
Mean Center
Identifies the geographic center (or the center of
concentration) for a set of features.
Median Center
Identifies the location that minimizes overall Euclidean
distance to the features in a dataset.
Standard Distance
Measures the degree to which features are concentrated
or dispersed around the geometric mean center.
Central Feature (Spatial Statistics)
Definition:
⚫ Identifies the most centrally located feature in a point,
line, or polygon feature class.
• Syntax:
• CentralFeature_stats (Input_Feature_Class, Output_Feature_Class,
Distance_Method, {Weight_Field}, {Self_Potential_Weight_Field},
{Case_Field})
Parameter Explanation
Parameter
Explanation
Data Type:
Input_Feature_
Class
Containing a distribution of features from which to
identify the most centrally located feature.
Feature
Layer
Output_Feature_
Class
Contain the most centrally located feature in the Input
Feature Class
Feature
Class
Distance_Method Specifies how distances are calculated from each feature
to neighboring features.
EUCLIDEAN_DISTANCE —The straight-line distance
between two points
MANHATTAN_DISTANCE —The distance between two
points measured along axes at right angles
String
Weight_Field
(Optional)
The numeric field used to weight distances in the
origin-destination distance matrix.
Field
Self_Potential_
Weight_Field
(Optional)
The field representing self-potential—the distance or
weight between a feature and itself.
Field
Case_Field
(Optional)
Field used to group features for separate central feature
computations. (integer, date, or string) type.
Field
Directional Distribution
Definition :
Creates standard deviational ellipses to summarize the
spatial characteristics of geographic features: central
tendency, dispersion, and directional trends
Syntax:
DirectionalDistribution_stats (Input_Feature_Class,
Output_Ellipse_Feature_Class, Ellipse_Size, {Weight_Field},
{Case_Field})
Parameter Explanation
Parameter
Explanation
Data
Type
Input_Featur
e_Class
A feature class containing a distribution of features for which the
standard deviational ellipse will be calculated.
Feature
Layer
Output_Ellip
se_Feature_C A polygon feature class that will contain the output ellipse feature.
lass
Ellipse_Size
The size of output ellipses in standard deviations. The default ellipse
size is 1; valid choices are 1, 2, or 3 standard deviations.
•1_STANDARD_DEVIATION
•2_STANDARD_DEVIATIONS
•3_STANDARD_DEVIATIONS
Feature
Class
String
Weight_Field The numeric field used to weight locations according to their
(Optional)
relative importance.
Field
Case_Field
(Optional)
Field used to group features for separate directional distribution
calculations. The case field can be of integer, date, or string type.
Field
Parameter
Explanation
Data
Type
Linear
Directional
Mean
Definition :
Identifies the mean direction, length, and
geographic center for a set of lines
• Syntax
• DirectionalMean_stats (Input_Feature_Class, Output_Feature_Class,
Orientation_Only, {Case_Field})
Parameter
Explanation
Parameter
Explanation
Input_Featur The feature class containing vectors for which
e_Class
the mean direction will be calculated.
Output_Feat
ure_Class
A line feature class that will contain the features
representing the mean directions of the input
feature class.
Data Type
Feature Layer
Feature Class
•DIRECTION —The From and To nodes are
Orientation_ utilized in calculating the mean (default).
Boolean
Only
•ORIENTATION_ONLY —The From and To node
information is ignored.
Case_Field
(Optional)
Field used to group features for separate
directional mean calculations. The case field can Field
be of integer, date, or string type.
Parameter
Explanation
Data Type
Mean Center
Identifies the geographic center (or the center of
concentration) for a set of features.
Syntax
MeanCenter_stats (Input_Feature_Class, Output_Feature_Class, {Weight_Field},
{Case_Field}, {Dimension_Field})
Parameter Explanation
Parameter
Explanation
Data Type
Input_Feature_
Class
A feature class for which the mean center will
be calculated.
Feature Layer
Output_Feature
_Class
A point feature class that will contain the
features representing the mean centers of the
input feature class.
Feature Class
Weight_Field
(Optional)
The numeric field used to create a weighted
mean center.
Field
Case_Field
(Optional)
Field used to group features for separate mean
center calculations. The case field can be of
Field
integer, date, or string type.
Dimension_Fiel
d
(Optional)
A numeric field containing attribute values
from which an average value will be
calculated.
Field
Parameter
Explanation
Data Type
Median Center
Definition :
Identifies the location that minimizes overall
Euclidean distance to the features in a dataset.
Syntax
MeanCenter_stats (Input_Feature_Class, Output_Feature_Class, {Weight_Field},
{Case_Field}, {Dimension_Field})
Parameter Explanation
Parameter
Explanation
Data Type
Input_Feature_
Class
A feature class for which the mean center will be Feature
calculated.
Layer
Output_Feature
_Class
A point feature class that will contain the
features representing the mean centers of the
input feature class.
Feature
Class
Weight_Field
(Optional)
The numeric field used to create a weighted
mean center.
Field
Case_Field
(Optional)
Field used to group features for separate mean
center calculations. The case field can be of
integer, date, or string type.
Field
Dimension_Fiel
d
(Optional)
A numeric field containing attribute values from
Field
which an average value will be calculated.
Parameter
Explanation
Data Type
Standard Distance
Measures the degree to which features are
concentrated or dispersed around the geometric
mean center.
Syntax
StandardDistance_stats (Input_Feature_Class,
Output_Standard_Distance_Feature_Class, Circle_Size, {Weight_Field}, {Case_Field})
Parameter Explanation
Parameter
Explanation
Data
Type
Input_Feature_
Class
A feature class containing a distribution of features for which
the standard distance will be calculated.
Feature
Layer
Output_Standa
rd_Distance_Fe
ature_Class
A polygon feature class that will contain a circle polygon for
each input center. These circle polygons graphically portray
the standard distance at each center point.
Feature
Class
The size of output circles in standard deviations. The default
circle size is 1; valid choices are 1, 2, or 3 standard deviations.
•1_STANDARD_DEVIATION
•2_STANDARD_DEVIATIONS
•3_STANDARD_DEVIATIONS
String
Weight_Field
(Optional)
The numeric field used to weight locations according to their
relative importance.
Field
Case_Field
(Optional)
Field used to group features for separate standard distance
calculations. The case field can be of integer, date, or string
type.
Field
Parameter
Explanation
Data
Type
Circle_Size
5-Sampling Concept
57
5.1-Sampling Concept
⚫ We want to know the attributes of some population Mean,
median, range, variance, distribution. . .
⚫ •It is usually not possible to observe all individuals
•Sampling means that we only observe a portion of the
population
5.2-Reasons for Sampling
⚫ Sampling can save money.
⚫ Sampling can save time.
⚫ For given resources, sampling can applied in be
scope of the data set.
⚫ Because the research process is sometimes
destructive, the sample can save product.
⚫ If accessing the population is impossible; sampling
is the only option.
© 2002 Thomson /
South-Western
Slide 7-59
5.3-Reasons for Taking a Census(calculate
populations)
⚫ Eliminate the possibility that a random sample is not
representative of the population.
⚫ The person authorizing the study is uncomfortable
with sample information.
© 2002 Thomson /
South-Western
Slide 7-60
5.4-Population Frame
⚫ A list, map, directory, or other source used to
represent the population
⚫ Over registration -- the frame contains all
members of the target population and some
additional elements
Example: using the chamber of commerce
membership directory as the frame for a target
population of member businesses owned by
women.
© 2002 Thomson /
South-Western
Slide 7-61
Under registration
⚫ -- the frame does not contain all members of the
target population.
Example: using the chamber of commerce
membership directory as the frame for a target
population of all businesses.
5.5-Random & Nonrandom Sampling
⚫ Random sampling
• Every unit of the population has the same probability of
being included in the sample.
• A chance mechanism is used in the selection process.
• Eliminates bias in the selection process
• Also known as probability sampling
© 2002 Thomson /
South-Western
Slide 7-63
Nonrandom Sampling
• Every unit of the population does not have the same
probability of being included in the sample.
• Open the selection bias
• Not appropriate data collection methods for most
statistical methods
• Also known as nonprobability sampling
5.6-Random Sampling Techniques
⚫ Simple Random Sample
⚫ Stratified (layer)Random Sample
⚫ Proportionate
⚫ not Proportionate
⚫ Systematic (have methodology)Random Sample
⚫ Cluster (or Area) Sampling
© 2002 Thomson /
South-Western
Slide 7-65
5.7-Simple Random Sample
⚫ Number each frame unit from 1 to N.
⚫ Use a random number table or a random number
generator to select n distinct numbers between 1 and
N, inclusively.
⚫ Easier to perform for small populations
⚫ dificults for large populations
© 2002 Thomson /
South-Western
Slide 7-66
5.8-Stratified (classified)Random Sample
⚫ Population is divided into non overlapping sub
⚫
⚫
⚫
⚫
populations called strata
A random sample is selected from each stratum
Potential for reducing sampling error
Proportionate -- the percentage of thee sample
taken from each stratum is proportionate to the
percentage that each stratum is within the
population
Disproportionate -- proportions of the strata within
the sample are different than the proportions of the
strata within the population
© 2002 Thomson /
South-Western
Slide 7-67
5.9-Cluster( grouped in area) Sampling
⚫ Population is divided into non overlapping clusters or
areas
⚫ Each cluster is a miniature(small pieces), or
microcosm, of the population.
⚫ A subset of the clusters is selected randomly for the
sample.
⚫ If the number of elements in the subset of clusters is
larger than the desired value of n, these clusters may be
subdivided to form a new set of clusters and subjected
to a random selection process.
© 2002 Thomson /
South-Western
Slide 7-68
5.10-Cluster Sampling Advantages and disadvantages
●
●
Advantages
• More convenient (suitable)for geographically dispersed
populations
• Reduced travel costs to contact sample elements
• Simplified administration of the survey
• Unavailability of sampling frame prohibits using other
random sampling methods
Disadvantages
• Statistically less efficient when the cluster elements are
similar.
• Costs and problems of statistical analysis are greater
than for simple random sampling
© 2002 Thomson /
South-Western
Slide 7-69
5.10-Sampling Property :To know the attributes of some population
Mean, Median, Range, Variance, Distribution . . .
It is usually not possible to observe all individuals ,
Sampling means that we only observe a portion of the
population
5.10.2-Sampling allow inferences
The law of large numbers:
The larger the sample compared to the population, the
more the sample parameters approach the population
parameters
2. The concept of probability sampling:
Each individual has some defined chance of being
sampled
3. The concept of representativeness:
selected individuals are “typical” of the population.
5.10.3-Spatial Sampling
Sampling in space:
When the location of the sampled individual is
recorded and used in the analysis.
Extra inferences possible from spatial sampling:
• Prediction at un sampled locations
• Inference of spatial dependence: local or regional trends
• Point-patterns: Dispersion / clustering
• Directional statistics: alignment
9.Geostatistical
Interpolating , Kriging&
IDW(inverse distance
weighted)
Interpolation concept
⚫ Interpolation predicts values for cells in a raster from a
limited number of sample data points. It can be used
to predict unknown values for any geographic point
data, such as elevation, rainfall, chemical
concentrations, and noise levels.
Interpolating an elevation surface
A typical use for point interpolation is to create an
elevation surface from a set of sample measurements.
In the following graphic, each symbol in the point layer
represents a location where the elevation has been
measured. By interpolating, the values for each cell
between these input points will be predicted.
Kriging origin
• The origin of the word kriging is from D.G. Krige, a
South African mining engineer who in the 1950’s
developed empirical methods for predicting grades at
un sampled locations using the known grades of
sampled at nearby sites
How Kriging works
⚫ Kriging is an advanced geostatistical procedure that
generates an estimated surface from a scattered set of
points with z-values.
⚫ Unlike other interpolation methods in the
Interpolation toolset, to use the Kriging tool
effectively involves an interactive investigation of the
spatial behavior of the phenomenon represented by
the z-values before you select the best estimation
method for generating the output surface.
The IDW (inverse distance weighted)
⚫ The IDW (inverse distance weighted) and Spline
interpolation tools are referred to as deterministic
interpolation methods because they are directly based
on the surrounding measured values or on specified
mathematical formulas that determine the
smoothness of the resulting surface.
Kriging Assumption
⚫ Kriging assumes that the distance or direction between
sample points reflects a spatial correlation that can be used
to explain variation in the surface.
⚫ The Kriging tool fits a mathematical function to a specified
number of points, or all points within a specified radius, to
determine the output value for each location.
⚫ Kriging is a multistep process; it includes exploratory
statistical analysis of the data, variogram modeling,
creating the surface, and (optionally) exploring a variance
surface.
⚫ Kriging is most appropriate when you know there is a
spatially correlated distance or directional bias in the data.
It is often used in soil science and geology.
The kriging formula
⚫ Kriging is similar to IDW in that it weights the
surrounding measured values to derive a prediction for
an unmeasured location. The general formula for both
interpolators is formed as a weighted sum of the data:
Creating a prediction surface map
with kriging(PRCATCAL)
⚫ To make a prediction with the kriging interpolation
⚫
⚫
⚫
⚫
⚫
method, two tasks are necessary:
Uncover the dependency rules.
Make the predictions.
To realize these two tasks, kriging goes through a two-step
process:
It creates the variograms and covariance functions to
estimate the statistical dependence (called spatial
autocorrelation) values that depend on the model of
autocorrelation (fitting a model).
It predicts the unknown values (making a prediction).
Kriging methods(PRACTICAL)
⚫ There are two kriging methods: ordinary and universal.
⚫ Ordinary kriging is the most general and widely used of the
⚫
⚫
⚫
⚫
⚫
kriging methods and is the default.
It assumes the constant mean is unknown.
This is a reasonable assumption unless there is a scientific
reason to reject it.
Universal kriging assumes that there is an overriding trend in
the data—for example, a prevailing wind—and it can be
modeled by a deterministic function, a polynomial.
This polynomial is subtracted from the original measured
points, and the autocorrelation is modeled from the random
errors.
.
Variography model
Fitting a model, or spatial modeling, is also known
as structural analysis, or variography.
In spatial modeling of the structure of the measured
points, you begin with a graph of the empirical
semivariogram, computed with the following equation
for all pairs of locations separated by distance h:
The semivariogram formula
Semivariogram(distanceh) = 0.5 *
average((valuei – valuej)2)
⚫ The formula involves calculating the difference squared
between the values of the paired locations.
⚫ The image below shows the pairing of one point (the red
point) with all other measured locations. This process
continues for each measured point
Calculating the
difference
squared between
the paired
locations
⚫ compute the average semivariance for all pairs of points
that are greater than 40 meters apart but less than 50
meters.
⚫ The empirical semivariogram is a graph of the averaged
semivariogram values on the y-axis and the distance (or
lag) on the x-axis (see diagram below).
Empirical semivariogram graph
example(PRACTICAL)
Spatial autocorrelation quantifies a basic principle
of geography: things that are closer are more alike
than things farther apart.
Fitting a model to the empirical
semivariogram
⚫ To fit a model to the empirical semivariogram, select a
function that serves as your model—for example, a
spherical type that rises and levels off for larger
distances beyond a certain range (see the spherical
model example below).
⚫ There are deviations of the points on the empirical
semivariogram from the model; some points are above
the model curve, and some points are below.
⚫ However, if you add the distance each point is above
the line and add the distance each point is below the
line, the two values should be similar.