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Map Algebra and Beyond:Advanced topics and applications
to Nexrad
Xingong LiUniversity of Kansas5 November 2009
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
Map Algebra for Vector Fields
• Types of fields– Scalar fields—each cell stores a scalar value
• Normal, ordinal, interval, and ratio
– Vector fields—each cell stores a vector• 2D, 3D, Multi dimensional
• Map algebra operations on vector fields
Mean Aspect
Aspect1 Aspect2
How to calculate the mean aspect?
MeanAspect = (Aspect1 + Aspect2)/2?
What is the mean aspect within landuse (or elevation) zones?
Calculate Mean Aspect
• What’s the mean aspect of 2 and 358 ?– (2 + 358) = 180 ?
• Aspects are unit vectors• How to calculate mean aspects?
– Vector algebra
Mean Aspect by Unit Vector
30 and330
x
y
=330 =30
0 aspect mean
0/180)arctan(
0/)tan(
0.866 2 / 0.866)(0.866y
0 2 / 0.5)-0.5(x
0.866) ,5.0())30cos(),30(sin(
)866.0 ,5.0())330cos(),330(sin(
p
yx
Angular Between Two Vectors
)B Aarccos(
B A where
|B| |A|
B A)cos(
:)( B and )( A vector obetween tw Angle ,,
baba
bbaa
yyxx
yxyx
AB
x
y
Terrain Hillshade
Friction and Movement Direction
The cost distance operation in ArcGIS assumes that friction is independent of movement direction (cost per unit distance)
Friction and Movement Direction
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
missing polygons
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
Local Scope
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
Neighborhoodsfor points
Neighborhoods for lines
Neighborhoods for polygons
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
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)
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 (“T*F**F***”)• Geometric types which can have the ‘within’ relationship
Local feature, neighborhood, or zone
Value feature
Interior Boundary Exterior
Interior T * F
Boundary * * F
Exterior * * *
Local feature, neighborhood, or zone
Value feature
point line polygon
point Y Y Y
line N Y Y
polygon N N Y
Attribute Adjustment
OVER_VALUE_FEATURE
OVER_LNZ
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
A Possible SyntaxNewLayer = 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
An Implementation
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).
Examples(b) NewLayer = Subwatersheds. Majority (Local(), RadarCells,
PRECIP, ON_GEOMETRY, Area)(c) NewLayer = Subwatersheds. Sum (Local(), RadarCells, PRECIP,
OVER_LNZ)
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
Spatiotemporal Map AlgebraCubic local functions
Cubic Focal functions
Mennis, J., Viger, R., and Tomlin, D. 2005, “Cubic map algebra functions for spatiotemporal
analysis”. Cartography and Geographic Information Science, 32(1): 17- 32.
Cubic Zonal Operations
vary only in space vary only in time vary both in space and time
Antecedent Precipitation and Water Quality• Explore the relationship between water sample quality
and antecedent rainfall (precipitation occurred before water samples were taken)
Water Samples in Space and Time• 1049 water samples were collected from 89 locations at
different times (from 1992 to 1999)
Defining Spatiotemporal Zones
1227a
1224b
Zone = flow length + antecedent time
Total Amount of Phosphorous vs. Antecedent Precipitations
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
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
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
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
Time Series Data Animation
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 t+1
A Storm Example
20 40 60 80 100 120 140 160
10
20
30
40
50
60
70
80 1
2
3
4
5
6
7
8
9
10
Projected on x-y plane
20 40 60 80 100 120 140 160
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
Projected on to the x-time plane
10 20 30 40 50 60 70 80
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
Projected on to the y-time plane
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
20 40 60 80 100 120 140 160
10
20
30
40
50
60
70
80
STORM : 2
Data Processing and Software Tools
Warm Season Storm Spatio-temporal Characteristics• Warm season: April to September• 04/01/96—09/30/2006• 519,562 storms
Precipitat ion Number of Storms
Year20062005200420032002200120001999199819971996
Pre
cip
ita
tio
n (
mm
)
26,000,000
24,000,000
22,000,000
20,000,000
18,000,000
16,000,000
14,000,000
12,000,000
10,000,000
8,000,000
6,000,000
4,000,000
2,000,000
0
Nu
mb
er o
f Sto
rms
55,000
50,000
45,000
40,000
35,000
30,000
25,000
20,000
15,000
10,000
5,000
0
Temporal Characteristics (Annual)
Precipitat ion Number of Storms
Month987654
Pre
cipi
tati
on (
mm
)
50,000,000
45,000,000
40,000,000
35,000,000
30,000,000
25,000,000
20,000,000
15,000,000
10,000,000
5,000,000
0
Num
ber of Storms
130,000
120,000
110,000
100,000
90,000
80,000
70,000
60,000
50,000
40,000
30,000
20,000
10,000
0
Temporal Characteristics (diurnal)
Precipitat ion Number of Storms
Hours2220181614121086420
Pre
cipi
tati
on (
mm
)
19,000,000
18,000,000
17,000,000
16,000,000
15,000,000
14,000,000
13,000,000
12,000,000
11,000,000
10,000,000
9,000,000
8,000,000
7,000,000
6,000,000
5,000,000
4,000,000
3,000,000
2,000,000
1,000,000
0
Num
ber of Storms
60,000
55,000
50,000
45,000
40,000
35,000
30,000
25,000
20,000
15,000
10,000
5,000
0
Spatial Characteristics
Total number of storms that occurred during the 11 year period
Spatial Characteristics
Total amount of storm precipitation in mm during the 11 year period
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
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
0 2 4 6 8 10 12 14
x 105
-6.3
-6.2
-6.1
-6
-5.9
-5.8
-5.7
x 106
Length represents movement speed. Start point is precipitation-weighted centroid.
Storm Movement
1000
2000
300030
210
60
240
90270
120
300
150
330
180
0
2000
4000
600030
210
60
240
90270
120
300
150
330
180
0
Directional distribution of storms (left) and storm precipitation (right)
Storm Movement
100
200
300
40030
210
60
240
90270
120
300
150
330
180
0
50
100
15030
210
60
240
90270
120
300
150
330
180
0
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).
Generalize Storm Life• The maximum precipitation path for each storm was used as a
generalization of the storm graph• Identified based on the Dijkstra’s shortest-path algorithm
where precipitation is the weight
10 20 30 40 50 60 70 80 90
20
30
40
50
60
70
80
Generalized Storm Track Examples• Storm centroid time• Storm average movement speed (km/hour)
Summary• Several extensions to the original MA have been
introduced– 3D– Vector fields (still a raster)– 2D+time– 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
Acknowledgments
• Dr. Donna Tucker and graduate student Keith French and Tingting Xu
• KU Big 12 Fellowship
