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Geostatistic Analysis

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Title: Geostatistic Analysis


1
Geostatistic Analysis
  • Lecture 6-8
  • Feb25-Mar10, 2008

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Purposes
  • Estimating or interpolating values at unsampled
    sites within the area covered by existing
    observations
  • Visiting every location is usually difficult or
    expensive.
  • Assumption spatially distributed objects are
    spatially correlated. in other words, things are
    close together tend to have similar
    characteristics (first law of geography, also
    called spatial autocorrelation).

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example
  • Few sample points to fill all cells

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Example Point elevation to surface
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Global interpolation
  • global interpolators determine a single function
    which is mapped across the whole region
  • a change in one input value affects the entire
    map
  • global algorithms tend to produce smoother
    surfaces with less abrupt changes are used when
    there is an hypothesis about the form of the
    surface, e.g. a trend

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Local interpolation
  • Local interpolators apply an algorithm repeatedly
    to a small portion of the total set of points. On
    average, values at points closer in space are
    more likely to be similar than point further
    apart (spatial autocorrelation)
  • A change in an input value only affects the
    result within the window
  • Two important steps for local interpolation
  • Define sampling neighborhood
  • Find points (samples) in the neighborhood

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If no directional influence
  • If there are no directional influences in the
    data, you want to give equal weight to sample
    points regardless of their direction from the
    prediction location. This means that you probably
    want your neighborhood to be a circle

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If directional influence
  • if there is directional influence in your data
    (such as might be caused by water draining
    downhill), then you may want to make an ellipse
    with the major axis running uphill/downhill

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Sample points
  • Once a shape is specified, you can restrict which
    sample points within the neighborhood are used.
    You do this by specifying the maximum and minimum
    numbers of points to use and by dividing the
    neighborhood into sectors. If the neighborhood is
    sectored, then the maximum and minimum
    constraints are applied to each part

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1. Explore data before interpolation
  • Before creating a surface, explore data (ED) tool
    enables you to gain a deeper understanding of the
    phenomena you are investigating so that you can
    make better decisions on issues relating to your
    data.
  • Exploring the distribution of the data, looking
    for global and local outliers, looking for global
    trends, examining spatial autocorrelation, and
    understanding the covariation among multiple
    datasets.
  • Tools includes Histogram, Voronoi Map, Normal
    QQPlot, Trend Analysis, Semivariogram/Covariance
    Cloud, General QQPlot, and Crosscovariance Cloud.
  • Only works on point and polygon layers

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1.1 Histogram
  • provides a univariate (one-variable) description
    of your data, displays the frequency distribution
    for the dataset of interest and calculates
    summary statistics.
  • measures of location
  • - mean, median, and quartiles
  • measures of spread
  • - standard deviation, variance
  • measures of shape
  • - skewness, kurtosis

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1.2 QQPlot
  • The quantile-quantile (q-q) plot is a graphical
    technique for determining if two data sets come
    from populations with a common distribution.
  • A q-q plot is a plot of the quantiles of the
    first data set against the quantiles of the
    second data set. By a quantile, we mean the
    fraction (or percent) of points below the given
    value. That is, the 0.3 quantile (or 30) is the
    point at which 30 of the data fall below and 70
    fall above that value
  • 0.3 quantile is also called 30 percentile, 0.4
    quantitle is 40 percentile
  • 25 percentile is called first quantitle
  • 50 percentile is called second quantitle (equal
    to median value)
  • 75 percentile is called third quantitle
  • 100 percentile is called fourth quantile
  • The q-q plot is used to answer the following
    questions
  • Do two data sets come from populations with a
    common distribution?
  • Do two data sets have common location and scale?
  • Do two data sets have similar distributional
    shapes?
  • Do two data sets have similar tail behavior?

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standard
  • QQPlots are graphs on which quantiles from two
    distributions are plotted relative to each other.
  • a cumulative distribution is produced by ordering
    the data and producing a graph of the ordered
    values versus cumulative distribution values

Normal QQPlot
General QQPlot
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1.3 Voronoi map
  • Voronoi maps are constructed from a series of
    polygons (thiessen polygon) formed around the
    location of a sample point.
  • The value for each polygon can be calculated
    using any of methods
  • - simple, mean, mode, cluster, entropy, median
    standard deviation, IQR (interquartile range)

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1.4 Trend analysis
  • You may be interested in mapping a trend, or you
    may wish to remove a trend from the dataset
    before using kriging. The Trend Analysis tool can
    help identify global trends in the input dataset.

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globe trend and anisotropy
  • globe trend is an overriding process that affects
    all measurements, can be represented by a
    mathematical formula (polynomial)
  • anisotropy is a random process that shows higher
    autocorrelation in one direction than in another
    (directional autocorrelation or directional
    influence). the reason for directional influence
    may not be known, but they can be statistically
    quantified.

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1.5 Semivariogram/covariance cloud
  • The semivariogram/covariance cloud shows the
    empirical semivariogram (half of the difference
    squared) and covariance for all pairs of
    locations within a dataset and plots them as a
    function of the distance between the two
    locations.
  • the empirical semivariogram for the (i,j)th pair
    is simply
  • 0.5(z(si)-z(sj))2, and the empirical
    covariance is the cross-product
  • where is the
    average of the data. The semivariogram/covariance
    cloud can be used to examine the local
    characteristics of spatial autocorrelation within
    a dataset and look for outliers.

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Creating Variography
Semivariogram depicts the spatial autocorrelation
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Understanding a semivariogram -range, sill, and
nugget
The distance where the model first flattens out
is known as the range The value at which the
model attains the range is called the sill The
value at which the model intercepts the y-axis is
called the nugget
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Fitting the semivariogram
  • Circular, spherical, exponential, Gaussian, and
    linear.

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Making a prediction
  • prediction based on the semivarioogram model and
    the measured values that are nearby (using search
    radius)
  • fixed search radius requires a distance and
    minimum number of points
  • variable search radius number of points needs to
    be specified. You can also specify a maximum
    distance (radius), that the search radius cannot
    exceed.

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To explore a directional influence in the
semivariogram cloud, we use the Search Direction
tools
  • the direction the pointer is facing determiners
    which pairs of data locations are plotted on the
    semivariogram.
  • lag size is the size of a lag distance, used to
    reduce the larger number of possible
    combinations. it is the size of the cells in the
    semivariogram surface.
  • number of cells is called number of lags, counted
    as the number of adjacent cells in a straight
    horizontal or vertical line from the center to
    the edge of the figure.

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1.6 Crosscovariance cloud
  • The crosscovariance cloud shows the empirical
    crosscovariance for all pairs of locations
    between two datasets and plots them as a function
    of the distance between the two locations.
  • Let z(si) denote the value at the i th location
    in dataset 1, and let y(tj) denote the value at
    the j th location in dataset 2.

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Relation between variogram and covariance
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2. Inverse Distance Weighted (IDW)
  • Each sample point has a local influence that
    diminishes with distance.
  • Weights the points closer to the processing cell
    more heavily than those farther away.
  • Operator controls how weighting is done.
  • Power.
  • High power gives more weight to closer points.
  • Radius type.
  • Considers how far away to look.
  • Barrier.
  • Search can be limited by other polygons or
    polyline.

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IDW Characteristics
  • Inverse Distance Weighting (IDW) is a quick
    deterministic interpolator that is exact. There
    are very few decisions to make regarding model
    parameters. It can be a good way to take a first
    look at an interpolated surface. However, there
    is no assessment of prediction errors, and IDW
    can produce "bulls eyes" around data locations.
    There are no assumptions required of the data.

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IDW a local interpolator
We usually uses power functions greater than 1. A
r 2 is known as the inverse distance squared
weighted interpolation.
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Adv. and disadv.
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3. Global Polynomial interpolator
  • Global Polynomial (GP) is a quick deterministic
    global interpolator that is smooth (inexact).
    There are very few decisions to make regarding
    model parameters. It is best used for surfaces
    that change slowly and gradually. However, there
    is no assessment of prediction errors and it may
    be too smooth. Locations at the edge of the data
    can have a large effect on the surface. There are
    no assumptions required of the data.

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GP interpolation fits a polynomial regression to
x,y coordinates
First order
Second order
Third order
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4. Local Polynomial interpolator
  • Global Polynomial interpolation is the only
    method in Geostatistical Analyst that does not
    use a search neighborhood. If you add the idea of
    a search neighborhood to Global Polynomial
    interpolation, you get Local Polynomial
    interpolation
  • Local Polynomial (LP) is a moderately quick
    deterministic interpolator that is smooth
    (inexact). It is more flexible than the global
    polynomial method, but there are more parameter
    decisions. There is no assessment of prediction
    errors. The method provides prediction surfaces
    that are comparable to kriging with measurement
    errors. Local polynomial methods do not allow you
    to investigate the autocorrelation of the data,
    making it less flexible and more automatic than
    kriging. There are no assumptions required of the
    data.

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  • Local polynomial interpolation creates a surface
    from many different formulas, each of which is
    optimized for a neighborhood
  • The neighborhood shape, maximum and minimum
    number of points, and sector configuration can be
    specified. In addition, as with IDW, the sample
    points in a neighborhood can be weighted by their
    distance from the prediction location. Thus,
    local polynomial interpolation produces surfaces
    that better account for local variation.

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5. Radial Basis Functions
  • Radial Basis Functions (RBF) are moderately quick
    deterministic interpolators that are exact. They
    are much more flexible than IDW, but there are
    more parameter decisions. There is no assessment
    of prediction errors. The method provides
    prediction surfaces that are comparable to the
    exact form of kriging. Radial Basis Functions do
    not allow you to investigate the autocorrelation
    of the data, making it less flexible and more
    automatic than kriging. Radial Basis Functions
    make no assumptions about the data.
  • RBF seems like a rubber membrane that is fitted
    to each of the measured data points while
    minimizing the total curvature of the surface.
    Because the surface must pass through each
    sampled point, radial basis functions are exact
    interpolators

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Radial basis functions (Spline)
  • Radial basis functions (RBF) methods are a series
    of exact interpolation techniques, that is, the
    surface must go through each measured sample
    value.
  • There are five different basis functions
    thin-plate spline, spline with tension,
    completely regularized spline, multiquadric
    function, and inverse multiquadric spline
  • RBFs are conceptually similar to fitting a rubber
    membrane through the measured sample values while
    minimizing the total curvature of the surface.

IDW will never predict values above the maximum
measured value or below the minimum measured
value. Spline can
http//www.math.ucla.edu/baker/java/hoefer/Spline
.htm
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Radial basis functions
Wi
Bias or error
Wi
When running spline in Spatial Analyst weight
- regularized 0, 0.001, 0.01, 0.1, 0.5
the higher the weight,
the smoother the surface - tension 0, 1,
5, 10 the higher, the
coarser number of points - used in the
calculation. the more points. the
smoother the surface
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6. Kriging
  • Kriging is a moderately quick interpolator that
    can be exact or smoothed depending on the
    measurement error model. It is very flexible and
    allows you to investigate graphs of spatial
    autocorrelation. Kriging uses statistical models
    that allow a variety of map outputs including
    predictions, prediction standard errors,
    probability, etc. The flexibility of kriging can
    require a lot of decision-making. Kriging assumes
    the data come from a stationary stochastic
    process (1. constant mean throughout the region,
    or 2. variance of differences between any two
    samples is independent of position, but depends
    on separated distance), , and some methods assume
    normally-distributed data.

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Trend and error
  • In Kriging, a predicted value depends on two
    factors a trend and an additional element of
    variability. This is an intuitive idea with
    plenty of analogies in the real world. For
    instance, if you go from the ocean to the top of
    a mountain, you have an upward trend in
    elevation. However, there is likely to be
    variation on the wayyou will go both up and down
    when crossing valleys, streams, knobs and other
    features.
  • In Kriging, the trend part of a prediction is
    called the trend. The fluctuation part is called
    spatially-autocorrelated random error. "Error"
    doesn't mean a mistakeit just means a
    fluctuation from the trend. Z(s) µ(s) e (s)
  • Assumption one e (s) is zero, (positive errors
    and negative errors)
  • Assumption two the autocorrelation of the error
    is purely spatial it depends only on distance
    and not on any other property (such as position)
    of a location.

Autocorrelation between e(s1) and e(s1h) is the
same as the one between e(s2) and e(s2h)
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Ordinary Kriging
  • In many cases, however, there is no trend in the
    dataor, if there is one, it is weak enough that
    your predictions are just as good when you ignore
    it. Assuming that there is no trend in the data
    is mathematically equivalent to assuming that the
    data have a constant mean value.
  • If the mean is a simple constant, such as µ(s)
    µ (i.e., no trend) for all locations s, and if µ
    is unknown (you do not have prior knowledge of
    the mean value), then this is the model on which
    Ordinary Kriging is based.

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Universal Kriging
  • Sometimes, there is a trend where the data values
    change consistently with the spatial coordinates.
    Mathematically, this is represented as a linear
    regression equation on the spatial x- and
    y-coordinates. Trends that vary (do not have a
    constant mean), and for which the regression
    coefficients are unknown, form models for
    Universal Kriging.

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Simple Kriging and indicator Kriging
  • A known constant mean for the entire dataset,
    then you have the model for Simple Kriging.
  • look at the left side of the equation, Z(s)
    µ(s) e(s). You can do some useful math
    operations on Z(s). For example, suppose you want
    to predict the probability that Z(s) is above or
    below some threshold value, such as 0.12 ppm for
    ozone concentration. You can transform Z(s) to an
    indicator variable, where it gets the value 0 if
    Z(s) is below the threshold and 1 if it is above
    it. This is called Indicator Kriging.

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Ordinary Kriging
Simple Kriging
Universal Kriging
Indicator Kriging
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7. Cokriging
  • Cokriging is a moderately quick interpolator that
    can be exact or smoothed depending on the
    measurement error model. Cokriging uses multiple
    datasets and is very flexible, allowing you to
    investigate graphs of cross-correlation and
    autocorrelation. Cokriging uses statistical
    models that allow a variety of map outputs
    including predictions, prediction standard
    errors, probability, etc. The flexibility of
    cokriging requires the most decision-making.
    Cokriging assumes the data come from a stationary
    stochastic process, and some methods assume
    normally-distributed data.

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8. definitions of output maps (for Kriging
methods)
  • Prediction maps (interpolated maps) estimate
    values at locations where measurements have not
    been taken.
  • Standard error maps (the square root of the
    variance of a prediction) show the distribution
    of prediction error for a surface. Error tends to
    be highest in places where there is little or no
    sample data.
  • Quantile maps show the values with 100 percent
    probability that the true values are less than
    the quantile map values.
  • Probability maps show the odds that the true
    value at a location is greater than a threshold
    value. The probability of exceeding a threshold
    is determined from predicted values, the error
    distribution, and the specified threshold value

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Kriging family and output maps
  • Transformation and trend removal can help justify
    assumptions of normality and stationarity

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creating a prediction map
  • while applying a transformation
  • while using detrending
  • while considering error for nugget

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9. Performing diagnostics
  • before produce a final surface, you should know
    how well the model predicts the values at unknown
    locations. cross-validation and validation help
    you make an informed decision as to which model
    provides the best predictions.
  • cross-validation uses all of the data to estimate
    the autocorrelation model. Then it removes each
    data location, one at a time, and predicts the
    associated data value. compare the predicted
    value with measured value
  • validation first remove part of the data (test
    dataset) using Create Subset tool, and then uses
    the rest of the data (training dataset) to
    develop the trend and autocorrection models to be
    used for prediction
  • in both methods, graphs and summary statistics
    used for diagnostics are the same predicted,
    prediction error (predicted-measured),
    standardized error (error/estimated kriging
    standard error), normal QQPlot (standardized
    error and standard normal distribution)

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create subset for validation
demo
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basic rules for good predicts
  • Mean error should be close to 0,
  • RMS (root-mean-square) error, average standard
    error, and mean standardized error should be as
    small as possible,
  • Root mean square stardardized error should be
    close to 1
  • uncertainty of prediction standard errors
    average estimated standard error versus RMS
    prediction error.
  • - equal, good
  • - larger than RMS, overestimate
  • - less than RMS, underestimate
  • or RMS standardized error
  • - 1, good,
  • - lt1, overestimate
  • - gt1, underestimate

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10. Compare model results
  • model surfaces can be compared using
    cross-validation statistics
  • previous basic rules for good predict still use
    in here

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11. Displaying geostatistical layers
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12. Comparison of different mathods
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here stochastic is geostatistic
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Summary
  • Kriging performs statistical analysis of the
    error in its predictions. This allows it to
    create four kinds of surfaces prediction,
    standard error, quantile, and probability.
  • Prediction maps estimate values at locations
    where measurements have not been taken. (All
    interpolators make prediction maps.)
  • Standard error maps show the distribution of
    prediction error for a surface. Error tends to be
    highest in places where there is little or no
    sample data.
  • Quantile maps show the values that the true
    values are unlikely to exceed.
  • Probability maps show the odds that the true
    value at a location is greater than a threshold
    value.
  • The various interpolation methods (Inverse
    Distance Weighting, Global Polynomial, Local
    Polynomial, Radial Basis Functions, and Kriging)
    offer trade-offs in speed, flexibility, and their
    advantages and disadvantages.
  • Fast interpolators produce output surfaces
    quickly, but are not as good at capturing subtle
    surface variations.
  • Exact interpolators predict values equal to the
    observed value at all sampled locations. Smooth
    interpolators do not.
  • Flexible interpolators allow users to fine-tune
    the output, while inflexible interpolators allow
    users to avoid making lots of choices

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Main references
  • ESRI book using ArcGIS Geostatistical Analyst
  • ESRI visual campus at campus.esri.com
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