Geostatistical Analyst

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Geostatistical Analyst

• an extension to ArcInfo 8.0
• aimed at performing advanced spatial data analysis

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Who Can Use the Analyst?

• The Geostatistical Analyst is a multi-discipline
  product used by many different fields including

• Environment

• Agriculture
• Exploration geology
• Meteorology
• Hydrology

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Types of Input Data

• Distributed sample points
• Continuous
• (elevation, oil spill, air pollution, a chemical
  in a lake)

• Questions answered

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What does the Geostatistical Analyst do?

• Exploratory Spatial Data Analysis (ESDA)
• Create Surface

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What it Looks Like
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What does ESDA do?

• By exploring data in different
  representations you gain a
  better understanding of the data
• Spatial and non-spatial plots
• (h-scatterplots, semivariogram, covariogram,
  crossvariogram, histograms, boxplots,
  scatterplots)

The Geostatistical Analyst ESDA Create Surface

• All plots interact with other and with the map
• (scatterplot or variogram interacting with the
  point layer in ArcMap)

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How ESDA Works

• Individual Views

The Geostatistical Analyst ESDA Create Surface

• Histogram
• QQPlot
• Semivariogram

• Tasks (grouping of views)

• Local Trends and Stationarity
• Spatial Structure and Directional Variation
• Distribution Analysis

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An ESDA Task
Task Local Outliers and Stationarity
Proportionality
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How ESDA Painting Works
The Geostatistical Analyst ESDA Create Surface
Highlight features in one window and they are
highlighted in all views, including the map
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What does Create Surface do?
The Geostatistical Analyst ESDA Create Surface

• Creates continuous surfacefrom point data
• Provides wide variety of supporting tools

• (variography, transformation, declustering,
  detrending, cross validation, validation, and
  comparison)

• Produces a variety of output surfaces

• (prediction, error of estimation, quantile map,
  and probability map)

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Create Surface Flow
Represent ArcMap
Explore ESDA
Comparison Cross Validation and Validation
side-by-side
Fit Model Select Method Set Parameters
Diagnostics Cross Validation Validation
GS Mapping
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Create Surface Flow Detail
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Data Input

• Usually because of time and cost, you can not
  measure a phenomenon at every location, thus
  samples are taken

Data Input

• Examples of samples

Method Selection

• The levels of soil and air contamination
• Meteorological fields suchas rainfall and
  temperature
• The ore grade in a mining block

Comparison
Modeling
GS Layer
Cross Validation
Validation
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Methods

• Inverse Distance Weighting
• Radial Basis Functions thin-plate spline,
  multi-quadric, inverse multi-quadric, and
  completely regularized spline
• Trend Surface Analysis global and local
  polynomial interpolation
• Kriging and Cokriging simple, ordinary,
  universal, indicator, probability, and
  disjunctive kriging and cokriging

Data Input
Method Selection
Comparison
Modeling
GS Layer
Cross Validation
Validation
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Inverse Distance Weighting

• IDW is used for a quick surface preview
• Optimal parameters can be chosen on the basis of
  minimum mean error of predictions

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Radial Basis Functions Methods

• Radial Basis Functions (RBF) are a family of
  Artificial Neural Network methods
• Used for creating surfaces from dense data
• Create smooth visually appealing contour

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Polynomial Interpolation Methods

• Polynomial interpolation is used for trend
  analysis prior to further spatial analysis such
  as kriging
• Local polynomial interpolation takes into account
  a short scale variation in addition to a
  large-scale trend

Global
Local
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Kriging/Cokriging Methods

• Linear Stationary Geostatistics
• Simple kriging
• Ordinary kriging
• Non-stationary Linear Geostatistics
• Kriging with Internal and External Trend
• Non-linear Geostatistics
• Indicator kriging
• Probability kriging
• Disjunctive kriging

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Types of Resulting Maps
Probability that Critical Level was Exceeded
Predictions
Error of Predictions
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Modeling
Data Input
Convert to Other ArcMap Layer Types
Represent
Method Selection
Fit Model
Comparison
Comparison
Modeling
GS Layers Creation
Cross Validation
Diagnostics
Validation
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Modeling IDW
Parameters

• Neighborhood search
• Number of neighbors
• Shape of neighborhood
• Angle of neighborhood

• Power

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Modeling Kriging/Cokriging
Simple and Disjunctive Kriging

• Additional Tools

• Standard Tools

• Detrending
• Declustering
• Normal score transformation
• Examination bivariate distribution
• Editing visualization and method properties
• Validation results of predictions

• Variogram andCovariance modeling
• Searching neighborhood
• Cross validation
• Visualization of the interpolation results

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Modeling Declustering

Clustered data would not have much influence on
the estimation and should be declustered to
obtain a representative cumulative distribution

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Distribution Modeling
Probability Density Distribution as a linear
superposition of Gaussian kernel functions
Cumulative Density Distribution
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Variogram and Covariance Modeling
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Searching Neighborhood

• Investigate the influence of the neighbors on
  the prediction
• Choose optimal strategy for the neighbors
  selection

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Diagnostics
Data Input
Convert to Other ArcMap Layer Types
Represent
Method Selection
Fit Model
Comparison
Comparison
Modeling
GS Layers Creation
Cross Validation
Diagnostics
Validation
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Diagnostics and Comparison

• Diagnostics provide measures for assessing how
  good a model, with its specified parameters, fits
  the data.
• It is a measure of quality within a model.
• Comparison provides measures for comparing
  between models.

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Diagnostics Cross Validation (How good is the
model?)
Statistics and graphics quantify how good the
model fits
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Diagnostics Validation (How good are the
predictions?)
The predictions are compared to a subset of
samples that are excluded from the interpolation.
Both statistics and maps are provided.
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Comparison (How good is this model relative to
another?)
Comparison allows you to answer questions such
as, for my data set does Ordinary Kriging provide
better predictions than IDW
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Summary
Represent ArcMap
Explore ESDA
Comparison Cross Validation and Validation
side-by-side
Fit Model Select Method Set Parameters
Diagnostics Cross Validation Validation
GS Mapping
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Summary

• The Geostatistical Analyst performs Exploratory
  Spatial Data Analysis and Creates Surfaces
• Allows for analysis of distributed sample points
  of continuous data
• Does so in an easy to use, interactive, Wizard
  environment, that provides reliable defaults

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Summary

• Allows for the output of various output surfaces
  (e.g., estimation map, probability map)
• Fully integrated with ArcMap
• The output surfaces can be used as is or as input
  into larger models to make more informed decisions

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A Deterministic Process
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A Deterministic Process
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A Random Process

• The flipping of a coin
• Random variable
• Rules
• Independent process

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A Deterministic Process

• A global polynomial interpolation
• Fitting a piece of paper or rubber membrane
  through the points
• Minimize variance
• Order of polynomial

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A Deterministic Process

• Local polynomial
• Fit a polynomial to smaller windows
• Order of polynomial

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A Deterministic Process

• Inverse Distance Weighted (IDW)
• Neighborhood size
• Number of points to consider in a neighborhood
• Power

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A Random Process

• Random but dependent
• The Rules

The Rule If the second and third flips are heads
then the last coin is the same as the first else
the last is different
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A Random Process

• First coin is Heads and the second is Tails
• First is Tails and the second is Heads

The Rule If the second and third flips are heads
then the last coin is the same as the first else
the last is different
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General Formula for Geostatistics
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Assumption for Geostatistics

• Stationarity
• The mean is the same throughout the area
• That the variance between any two points is the
  same for any specified distance (lag)

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Example Ordinary KrigingThe input sample data
Values (1,5) 100 (3,4) 105 (1,3)
105 (4,5) 100 (5,1) 115
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Example Ordinary KrigingThe equation
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Example Ornary KrigingThe empirical variogram
Values (1,5) 100 (3,4) 105 (1,3)
105 (4,5) 100 (5,1) 115
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Example Ordinary KrigingBinning the data
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Example Ordinary KrigingFitting the model
Variance
150 120 90 60 30
Empirical
Weighted Least Square Regression line Passing
through 0
Fitted
LagVar 13.5304h
1 2 3
4 5 6
Distance
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Example Ordinary KrigingThe C (gamma) matrix
C w D
Distance Slope 2.236 13.5304
Lagrange multiplier
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Example Ordinary KrigingThe inverse matrix
w D C-1
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Example Ordinary KrigingVariogram dialogue
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Example Ordinary KrigingThe D vector
w D C-1
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Example Ordinary KrigingWeights and predictions
w D C-1
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Example Ordinary KrigingThe search neighborhood
dialogue



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Example Ordinary KrigingKriging weights
Values (1,5) 100 (3,4) 105 (1,3)
105 (4,5) 100 (5,1) 115
(0.4623)
(0.00434)
(0.07622)
102.50
(0.47376)
(-0.01662)
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Example Ordinary KrigingThe error estimation
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Example Ordinary KrigingKriging weights
Values (1,5) 100 (3,4) 105 (1,3)
105 (4,5) 100 (5,1) 115
(-0.05)
(-0.06)
(0.17)
(0.17)
112.1
(0.76)
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Different Kriging Models

• Ordinary stationarity and constant mean
  (unknown)
• Simple stationarity and constant mean (known)
• Kriging of residuals take account trend, krige
  residuals
• Universal with internal trend take account trend
  but from covariant (e.g., elevation)

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Different Kriging Models

• Indicator stationarity, convert into a indicator
  (0 - 1) map
• Probability similar to indicator (0 - 1) but
  cokrige on original data
• Disjunctive linear combination of functions

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Summary of Calculations

• Deterministic vs Geostatistics
• Sample points and want to predict unknown
• C w D
• C matrix is structure derived from variography
• Pairing of points, variance, binning, fit model

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Summary of Calculations

• D matrix from spatial configuration of points
• Weights derived from C-1 D (structure and
  configuration
• Prediction and error surfaces
• Different models (e.g. Simple, Ordinary,
  Universal Kriging

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