WFM 6202: Remote Sensing and GIS in Water Management

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WFM 6202 Remote Sensing and GIS in Water
Management
Part-B Geographic Information System (GIS)
Lecture-6 Geo-statistical Analysis

• Akm Saiful Islam

Institute of Water and Flood Management
(IWFM) Bangladesh University of Engineering and
Technology (BUET)
January, 2009
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Geo-statistical Analyst of ArcGIS

• This training will be on
• Histogram
• Normal QQ plot
• Trend Analysis
• Creating a prediction map using the
  geo-statistical wizard
• Semivariogram / covariance modeling
• Searching neighbor
• Creating a prediction standard error map
• Display Formats
• Input Data
• Groundwater well data of Dinajpur district of
  Bangladesh

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Steps in Geo-statistical Analyst

• Representation of the Data
• Explore the Data
• Fit a model (create surface)
• Perform Diagnostics

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Representation of Data

• Representing the data is a vital first step in
  assessing the validity of the data and
  identifying external factors that may ultimately
  play a role in the distribution of data.

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Explore the Data

• Distribution of the data, looking for data
  trends, looking for global and local outliers,
  examining spatial autocorrelation, understanding
  the co-variation among multiple data sets.

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Explore data

• Histogram
• Q-Q plot
• Trend Analysis
• Semivariogram
• Voronoi map
• Cross covariance

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Histogram

• Show frequency distribution as a bar graph that
  displays how often observed values fall within
  certain intervals or classes

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Normal distribution

• Skewness is zero for normal distribution

Positively skewed
Normally distributed
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Q-Q Plot

• Normal QQ Plot is created by plotting data values
  with the value of a standard normal where their
  cumulative distributions are equal

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Trend Analysis

• The Trend Analysis tool provides a
  three-dimensional perspective of the data.
• The locations of sample points are plotted on the
  x,y plane. Above each sample point, the value is
  given by the height of a stick in the z
  dimension.
• The unique feature of the Trend Analysis tool is
  that the values are then projected onto the x,z
  plane and the y,z plane as scatter plots.
• This can be thought of as sideways views through
  the three-dimensional data.
• Polynomials are then fit through the scatter
  plots on the projected planes.

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Voronoi map

• Voronoi maps are constructed from a series of
  polygons formed around the location of a sample
  point. Voronoi polygons are created so that every
  location within a polygon is closer to the sample
  point in that polygon than any other sample
  point. After the polygons are created, neighbors
  of a sample point are defined as any other sample
  point whose polygon shares a border with the
  chosen sample point.
• For example, in the following figure, the bright
  green sample point is enclosed by a polygon,
  given as red. Every location within the red
  polygon is closer to the bright green sample
  point than any other sample point (given as small
  dark blue dots). The blue polygons all share a
  border with the red polygon, so the sample points
  within the blue polygons are neighbors of the
  bright green sample point.

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Cross variance

• 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.

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Fit a Model

• A wide variety of interpolation methods available
  to create surface.
• Two main groups of interpolation techniques
• 1. Deterministic
• 2. Geo-statistical

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Interpolation techniques

• 1. Deterministic is used for creating surfaces
  from measures points based either on extent of
  similarity (Inverse Distance Weighted (IDW) or
  the degree of smoothing (radial basis functions
  and polynomials)
• 2. Geo-statistical is based on statistics and is
  used for more advanced prediction of surface
  modeling that also includes errors or uncertainty
  of prediction.

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Deterministic Methods

• Four types
• Inverse Distance Weighted (IDW)
• Global Polynomial
• Local Polynomial
• Radial Basis Functions
• Can classified into two groups
• Global uses entire data set
• Global polynomial
• Local calculates prediction from measured point
  with specified neighbors
• IDW, local polynomials, radial basis functions

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Inverse Distance Weighted (IDW)

• A window of circular shape with the radius of
  dmax is drawn at a point to be interpolated, so
  as to involve six to eight surrounding observed
  points.

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Global polynomial interpolation

• Global Polynomial interpolation fits a smooth
  surface that is defined by a mathematical
  function (a polynomial) to the input sample
  points. The Global Polynomial surface changes
  gradually and captures coarse-scale pattern in
  the data.
• Conceptually, Global Polynomial interpolation is
  like taking a piece of paper and fitting it
  between the raised points (raised to the height
  of value). This is demonstrated in the diagram
  below for a set of sample points of elevation
  taken on a gently sloping hill (the piece of
  paper is magenta).

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Local Polynomial interpolation

• While Global Polynomial interpolation fits a
  polynomial to the entire surface, Local
  Polynomial interpolation fits many polynomials,
  each within specified overlapping neighborhoods.
  The search neighborhood can be defined using the
  search neighborhood dialog

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Radial Basis Functions (RBF)

• 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, multi-quadric
  function, and inverse multi-quadric function.
• RBF methods are a form of artificial neural
  networks.

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Geo-statistical Methods

• Kriging and Co-kriging
• Algorithm
• Ordinary -A variety of kriging which assumes that
  local means are not necessarily closely related
  to the population mean, and which therefore uses
  only the samples in the local neighbourhood for
  the estimate. Ordinary kriging is the most
  comrnonly used method for environmental
  situations.
• Simple - A variety of kriging which assumes that
  local means are relatively constant and equal to
  the population mean, which is well-known. The
  population mean is used as a factor in each local
  estimate, along with the samples in the local
  neighborhood. This is not usually the most
  appropriate method for environmental situations.
• Universal -
• Indicator
• Probability
• Disjunctive
• Output Surfaces
• Prediction and prediction standard error
• Quantile
• Probability and standard errors of indicators

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Kriging

• Kriging is a geostatistical method for spatial
  interpolation.
• It can assess the quality of prediction with
  estimated prediction errors.
• It uses statistical models that allow a variety
  of map outputs including predictions, prediction
  standard errors, probability, etc.

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Interpolation using Kriging
Kriging weights
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Semivariogram

• The semivariogram functions quantifies the
  assumption that things nearby tend to be more
  similar than things that are farther apart.
  Semivariogram measures the strength of
  statistical correlation as a function of
  distance.
• Semivariance
• Y(h) ½ (Z(xi) - Z(xj)2
• Covarience Sill Y(h)

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Types of semivariogram models

• Geostatistical Analyst provides the following
  functions to choose from to model the empirical
  semivariogram
• Circular
• Spherical
• Tetraspherical
• Pentaspherical
• Exponential
• Gaussian
• Rational Quadratic
• Hole Effect
• K-Bessel
• J-Bessel
• Stable

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Semi-variogram Models
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Trend

• An example of a global trend can be seen in the
  effects of the prevailing winds on a smoke stack
  at a factory (below).
• In the image, the higher concentrations of
  pollution are depicted in the warm colors (reds
  and yellows) and the lower concentrations in the
  cool colors (greens and blues).
• Notice that the values of the pollutant change
  more slowly in the eastwest direction than in
  the northsouth direction.
• This is because eastwest is aligned with the
  wind while northsouth is perpendicular to the
  wind.

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Detrending tool
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Anisotropy

• Anisotropy is a characteristic of a random
  process that shows higher autocorrelation in one
  direction than another.
• The following image shows conceptually how the
  process might look.
• Once again, the higher concentrations of
  pollution are depicted in the warm colors (reds
  and yellows) and the lower concentrations in the
  cool colors (greens and blues).
• The random process shows undulations that are
  shorter in one direction than another.

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Accounting for Anisotrophy
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Searching Neighbor

• The points highlighted in the data view give
  an indicator of the weights (absolute value in
  percent) associated with each point located in
  the moving window. The weights are used to
  estimate the value at the unknown location which
  is at the center of the cross hair.

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Data transformation
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Declustering method

• There are two ways to decluster your data by the
  cell method and by Voronoi polygons. Samples
  should be taken so they are representative of the
  entire surface. However, many times the samples
  are taken where the concentration is most severe,
  thus skewing the view of the surface.
  Declustering accounts for skewed representation
  of the samples by weighting them appropriately so
  that a more accurate surface can be created.

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Bi-variate normal distribution
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Output Surface
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Cross Validation

• Cross-validation uses all of the data to estimate
  the trend and autocorrelation models. It removes
  each data location, one at a time, and predicts
  the associated data value.

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Various Surface produced using ordinary kriging
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Model comparison

• Comparison helps you determine how good the model
  that created a geostatistical layer is relative
  to another model.

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Display Format
Contours
Filled contour
Grids
Combination of contours Filled contour and hill
shade
Hill shade
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Exercise on Geo-statistical Analyst

• Data from 21 Groundwater observation Wells as
  shape file gwowell_bwdb.shp
• Weekly data from December to May for 1994 to 2003
• Upazilla shape file upazila.shp
• Tasks
• Represent data
• Explore data
• Fit Model
• Diagnostic output
• Create output maps