Map%20Algebra%20and%20Beyond:%20Advanced%20topics%20and%20applications%20to%20Nexrad%20Xingong%20Li%20University%20of%20Kansas%205%20November%202009

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Map Algebra and BeyondAdvanced topics and
applications to NexradXingong LiUniversity of
Kansas5 November 2009
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Major Extensions to Map Algebra

• Scott (1999) extended the original 2D MA
  operations into three dimensional raster datasets
  (volumetric MA)
• Solid earth
• Atmosphere
• Ocean
• Li and Hodgson (2004) and Wang and Pullar (2005)
  developed MA operations for vector fields (cell
  values are vectors rather than scalars)
• Aspect, surface normal
• flow and wind fields
• Mennis et al., 2005 developed cubic MA for
  spatio-temporal datasets where the third
  dimension is time
• Spatio-temporal time series
• French and Li (in press) proposed MA operations
  for the vector data model

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Map Algebra for Vector Fields

• Types of fields
• Scalar fieldseach cell stores a scalar value
• Normal, ordinal, interval, and ratio
• Vector fieldseach cell stores a vector
• 2D, 3D, Multi dimensional
• Map algebra operations on vector fields

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Mean Aspect
How to calculate the mean aspect? MeanAspect
(Aspect1 Aspect2)/2? What is the mean aspect
within landuse (or elevation) zones?
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Calculate Mean Aspect

• Whats the mean aspect of 2? and 358 ??
• (2 358) 180 ?
• Aspects are unit vectors
• How to calculate mean aspects?
• Vector algebra

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Mean Aspect by Unit Vector
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Angular Between Two Vectors
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Terrain Hillshade
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Friction and Movement Direction
The cost distance operation in ArcGIS assumes
that friction is independent of movement
direction (cost per unit distance)
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Friction and Movement Direction
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Map Algebra for the Vector Data Model

• No counterpart in the vector data model
• Have to convert vector data into raster to use
  map algebra operations
• Various problems during the conversion
• Impose an arbitrary analysis resolution

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Local Spatial Scope

• A cell in the raster data model
• A feature in the vector data model
• Two types of vector layers (focus and value
  layer)
• Each feature on the focus layer defines a local
  spatial scope of an operation
• Value layer stores the features to which the
  features on the focus layer will be spatially
  compared
• Focus and value layer can be the same

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Local Scope
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Focal Spatial Scope

• Neighborhoods for points, lines, and polygons
• Neighborhoods are not necessary polygons
• Neighborhoods can be defined based topological
  relationships among features
• Generic neighborhood could also be defined

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Neighborhoodsfor points
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Neighborhoods for lines
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Neighborhoods for polygons
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Zonal Scope

• A collection of features with the same values for
  a given field
• May become a local scope if each feature has a
  unique value in the field

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Value Feature Selection and Adjustment

• The value features and their attributes
  associated with a focus feature may partially
  overlap with the focus feature
• Four selection/adjustment options
• No adjustment on geometry and attribute
• Only on geometry
• On geometry and attribute (over value feature)
• On geometry and attribute (over spatial scope)

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Select Value Features

• Value features are selected based on the
  dimensionally extended 9-intersection model
  (DE9IM) developed by Egenhofer and Herring (1991)
  and Clementini et al. (1993)
• The within relationship (TFF)
• Geometric types which can have the within
  relationship

Local feature, neighborhood, or zone Local feature, neighborhood, or zone Local feature, neighborhood, or zone Local feature, neighborhood, or zone
Value feature Interior Boundary Exterior
Value feature Interior T F
Value feature Boundary F
Value feature Exterior
Local feature, neighborhood, or zone Local feature, neighborhood, or zone Local feature, neighborhood, or zone Local feature, neighborhood, or zone
Value feature point line polygon
Value feature point Y Y Y
Value feature line N Y Y
Value feature polygon N N Y
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Attribute Adjustment
OVER_VALUE_FEATURE
OVER_LNZ
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Operations
Operation Feature Property Output Type
Count Object Integer
Mean Attribute Double
Range Attribute Double
StdDev (Standard Deviation) Attribute Double
Maximum (Maximum Value) Attribute Double
Minimum (Minimum Value) Attribute Double
Sum Attribute Double
Product Attribute Double
Median Attribute Double
Majority Attribute same as input
Minority Attribute same as input
MaxFeature (Feature ID with maximum value) Attribute ID
MinFeature (Feature ID with minimum value) Attribute ID
MeanCentre Location Point
NNI (Nearest Neighbour Index) Location Double
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A Possible Syntax

• NewLayer FocusLayer.Operation (Scope,
  ValueLayer, Attribute, Adjustment, Normalization)

Enumeration Point Line Polygon
Local Y Y Y
Zonal (String ZoneField) Y Y Y
Radial (Double MinAngle, MaxAngle, MinRadius, MaxRadius  Double Xoffset, Yoffset) Y N N
Rectangular (Double Height, Width, RotationAngle PivoType PivotEnumeration Double Xoffset, Yoffset ) Y N N
NearestNeighbour (Integer NumOfNeighbours Double MaxDistance) Y N N
ProximalRegion Y Y Y
EuclideanBuffer (Double MinDistance, MaxDistance) Y Y Y
Connectivity (Integer Order Boolean Accumulative) N Y Y
NetworkBuffer (Double MinDistance, MaxDistance) N Y N
Generic(String NeighbourDefinitionFile) Y Y Y
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An Implementation
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Examples

• (a) NewLayer Siren.Sum (Radial (0, 0, 0, X),
  CensusBlock, POP, OVER_VALUE_FEATURE).
• (b) NewLayer SirenZone. Sum (Zonal(ID),
  CensusBlock, POP, OVER_VALUE_FEATURE).

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Examples

• (b) NewLayer Subwatersheds. Majority (Local(),
  RadarCells, PRECIP, ON_GEOMETRY, Area)
• (c) NewLayer Subwatersheds. Sum (Local(),
  RadarCells, PRECIP, OVER_LNZ)

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Comparison to Raster Map Algebra

• Vector MA does not impose any arbitrary
  resolutions but simply maintain the original
  resolution of the data through its operations
• Raster MA has difficulty handling the
  neighborhoods which are defined for individual
  features or are based on the topological
  relationships between features
• The vector cartographic modeling is more
  appropriate for characterizing discrete features
  and the relationships among the features

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Spatiotemporal Map Algebra
Mennis, J., Viger, R., and Tomlin, D. 2005,
Cubic map algebra functions for spatiotemporal
analysis. Cartography and Geographic Information
Science, 32(1) 17- 32.
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Cubic Zonal Operations
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Antecedent Precipitation and Water Quality

• Explore the relationship between water sample
  quality and antecedent rainfall (precipitation
  occurred before water samples were taken)

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Water Samples in Space and Time

• 1049 water samples were collected from 89
  locations at different times (from 1992 to 1999)

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Defining Spatiotemporal Zones
Zone flow length antecedent time
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Total Amount of Phosphorous vs. Antecedent
Precipitations
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From Spatio-temporal precipitation data to
precipitation events (storms)

• The Eulerian view focuses on the change of state
  in space
• While a sequence of changes in space may portray
  the movement of an entity across the space, there
  is no explicit representation of those entities.
• no structured data object representing "a storm
• no explicit representation of behaviors that
  storms can exhibit.
• The Lagrangian view offers an alternative
  perspective that focuses on movement and uses an
  object-based approach

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Study Area and Data

• The study domain is the ABRFC (Stage III and P1
  NEXRAD products, 4 km spatial resolution, hourly
  in time)
• The precipitation data span a period of 11 years
  from 10/01/1995 to 09/30/2006

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NEXRAD (Next generation Radar)

• About 150 stations covering the entire U.S.
• Provides hourly precipitation estimate by
  combining radar, satellite, and rain gauge data
• Spatial resolution is about 4 km

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NEXRAD Data

• Precipitation data are broken down into 13
  separate geographical regions
• Each region covers a NWS-designated river basin
  (River Forecast Center)
• Temporal coverage of the dataset varies in each
  river basin
• Data can be downloaded from the NOAA website or
  from individual RFCs

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Time Series Data Animation
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Storm (Event) Extraction

• A storm (event) is defined as a contiguous
  precipitation object in space and time
• a set of connected precipitation cells delineated
  from stacked hourly NEXRAD precipitation layers.
• The algorithm is based on the component labeling
  algorithm in digital image processing
• Controlled by 3 parameters
• the minimum hourly precipitation (MHP) in a cell
• the minimum time span (MTS) of a storm
• the definition of spatial and temporal
  connectivity

0 1 0
1 1 1
0 1 0
1 1 1
1 1 1
1 1 1
0 1 0
1 1 1
0 1 0
t-1
t
t1
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A Storm Example
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Storm Tracking and Representation

• A directed graph is used to represent a storm
• Nodes are precipitation-weighted centroids of
  spatially contiguous areas receiving rainfall in
  each hour
• Directed edges indicate spatial and temporal
  linkage (split or merge) among the rainfall areas
  during the life span of the event

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Data Processing and Software Tools
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Warm Season Storm Spatio-temporal Characteristics

• Warm season April to September
• 04/01/9609/30/2006
• 519,562 storms

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Temporal Characteristics (Annual)
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Temporal Characteristics (diurnal)
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Spatial Characteristics
Total number of storms that occurred during the
11 year period
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Spatial Characteristics
Total amount of storm precipitation in mm during
the 11 year period
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Spatial Characteristics

• Precipitation-weighted centroids of the events
  were calculated and used to represent the events
  as points in space and time in storm density
  analysis
• The number of events per km2 of the 11-year
  period
• The amount of precipitation per km2 of the
  11-year period

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Storm Movement

• Precipitation-weighted mean storm movement vector
  is calculated for each storm from the directed
  graph
• All the data from 10/01/1995 to 09/30/2006

Length represents movement speed. Start point is
precipitation-weighted centroid.
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Storm Movement
Directional distribution of storms (left) and
storm precipitation (right)
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Storm Movement
Directional distribution of storms with a
duration of 3 hours (18 of all the events)
(left) and directional distribution of storms in
October (4 of all the events) (right).
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Generalize Storm Life

• The maximum precipitation path for each storm was
  used as a generalization of the storm graph
• Identified based on the Dijkstras shortest-path
  algorithm where precipitation is the weight

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Generalized Storm Track Examples

• Storm centroid time
• Storm average movement speed (km/hour)

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Summary

• Several extensions to the original MA have been
  introduced
• 3D
• Vector fields (still a raster)
• 2Dtime
• Vector data model
• Storm (or event) extraction from spatio-temporal
  snapshots
• From Eulerian to Lagrangian view
• Still need a generic analysis framework for
  spatio-temporal data beyond MA

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Acknowledgments

• Dr. Donna Tucker and graduate student Keith
  French and Tingting Xu
• KU Big 12 Fellowship