Lecture8_interpolationf_2011

7/28/2019 Lecture8_interpolationf_2011

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Spatial Interpolation

GEOS 5350

Demers, M.N., Geographic Information Systems

Chang, Kang-tsung, Introduction to geographicinformation systems


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Spatial interpolation is theestimation the value of

properties at unsampled

sites

within the area covered by

existing observations (controlpoints).

Calculates some property of

the surface at a given point


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Classifications of interpolations

1- Global & Local interpolation2- Exact interpolation & Inexact Interpolation

3 -

Deterministic and Stochastic


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1 -

Global and Local Interpolation

Global interpolation:

uses all available control points

Adequate for terraines

that do not show abrupt

variations

Assumes good spatial autocorrelation on regionalscales

More generalized estimationsLocal interpolation:

uses a sample of control points

Adequate for terraines

that show abrupt variations

Assumes good spatial autocorrelation on local scales

More local estimations


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2 - Exact interpolation & InexactInterpolation

Exact interpolation

Predicts a value at control points that is the same asthe observed values.

The interpolation produces a surface that passes bythe control points


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Continue

Inexact InterpolationPredicts a value for the control points that differ

from the observed value


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3 - Deterministic and StochasticDeterministic interpolation

No assessment of errors with the predictedvalues

Stochastic interpolation

Offers assessment of errors with predictedvalues. These methods assume random

errors


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Examples


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First-order trend surface (polynomial)

I - Global(a) First-order trend surface


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Example

(1)

Set up 3 equations

(2)

Re-write in matrix format

(3)

Calculate

X = 377

Y = 318

X2

= 29007

Y2

= 20714

XY = 23862

YZ = 4445.8

XZ = 5044

X Y Z X2 Y2 XY XZ YZ

69 76 20.82 4761 5776 5244 1437 1582.32

59 64 10.91 3481 4096 3776 643.7 698.24

75 52 10.38 5625 2704 3900 778.5 539.76

86 73 14.6 7396 5329 6278 1256 1065.8

88 53 10.56 7744 2809 4664 929.3 559.68

377 318 67.27 29007 20714 23862 5044 4445.8


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Continue

(4) Plug in values for 5 points

(5) Solve for b coefficients:Multiply inverse of left matrix byright matrix

(6) Use the b coefficients tocalculate z

for any point

(X,Y) (69, 67)


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(b) Higher-order trend surface

First order polynomials

(inclined surface) can notrepresent the complexnatural surfaces.

A cubic or third ordermodels can betterrepresent such surfaces(e.g., hills, valleys)

Third-order trend surface

(nine coefficients)


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Example


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II - LocalIt is all about mechanisms for the selection of a suite of control points

Closest points Points within a certain radius


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(a) - Trends (polynomials) could belocal

Local polynomials as opposedto global polynomials could be

used as well for betterrepresentation of surfaces


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(b) -

Theissen

Polygons

Also called proximal

method

Attempts to weight data points

by area

Commonly used for

precipitation data


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Continue

Triangles are drawn connecting control points (e.g.,stations) using the Delaunay triangulation technique

(also used for TIN)

Lines are drawn perpendicular to sides of triangles at

their midpoints

Polygons are defined by intersections of these lines

Values for control points are assigned to enclosingpolygons


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(c)- Density EstimationSimple density functions:

Number of points/cell size

(e.g., 10,000 m2)(shown inshades of grey)

Size of circle centered atcenter of cell size

Other methods include

Kernel density estimation

Kernel density function is a

commonly used alternative


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(d)- Inverse Distance Weighted (IDW)Interpolation

Estimated value at point 0

Is the z value at control point i

Distance between point I and point 0

The larger the k, the greater the influence

of neighboring points.

S number of used points


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Continue

Zi d i d i2

1/(di2

) Zi x 1/(d i2

)

20.82 18 324 0.0031 0.06426

10.91 20.88 435.97 0.0023 0.02502

10.38 32.31 1043.9 0.0010 0.00994

14.6 36.05 1299.6 0.0008 0.01123

10.56 47.2 2227.8 0.0004 0.00474

SUM 0.0076 0.11520


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Examplezi Between

points

Distance

(di

)

20.82 0,1 18

10.91 0,2 20.88

10.38 0,3 32.31

14.6 0,4 36.05

10.56 0,5 47.20

Assuming k = 2


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All values are within themaximum & minimum values ofknown points

Small enclosed isolines

are

typical of this method

Annual precipitation surface created by

inverse distance squared


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(e) - Radial Basis Functions (RBF) Splines

A large group of interpolation methods

Exact interpolators The difference between them is how thesurface fits between the control points Each RBF also has a parameter thatcontrols the smoothness of generated

surface Differences between the outputs of thesemethods are small


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Continue

Exact Function

Good for large smoother surfaces

Doesnt work well with abruptchanges


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Continue

Creates a surface passing by control point

and has the least possible change in slopeat all points

Unlike the IDW method, predicted values arenot limited to the Max and Min dictated by

data.


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Kriging Next Lecture

Geostatistical

Methods

Ordinary Kriging Simple Kriging

Universal Kriging

Indicator Kriging Probability Kriging Disjunctive Kriging Cokriging


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Integrated Seismic Risk Map of

Egypt

Generate Egypts first seismic risk map using a GIS

approach


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Red Sea-related Seismicity


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Seismic Risks & Hazards

Seismic hazard

---

Strength and frequency

of shaking from earthquakes,

Seismic risk ---- The chance of losing humanlife and/or property because of earthquakeground shaking.


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Maps in the GISSeismic hazard map ---- Probability of occurrenceof seismic ground shaking within a certain time

frameFault hazard map -- Surface & subsurface faultzones subject to re-activation,

Amplification map -- Distribution of alluviumdeposits that amplify ground shaking,Liquefaction map

---

Areas susceptible (shallow

groundwater, seismically active) to soil (sand/silt)liquefaction,

Population density map

---

Areas subject to

increased risk of loss in human life and property.


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Seismic

Hazard Map

Probability of occurrence of seismic ground shaking of a

certain intensity (peak ground acceleration PGA) in a

specified time interval (250 years)


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Distribution

Map

Frequency Map

# of earthquakes/cell

H d F lt M


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Hazard Fault MapFaults proximal to earthquake epicenters &

could be reactivated under current stress

regimes

H d F lt M


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Hazard Fault Map

Intersection of the fault & distribution maps


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Amplification Map

Earthquake ground motions areamplified by alluvium soil deposits

S il H d Li f i M


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Soil Hazard Liquefaction Map

Intersection of earthquake coverage map withcoverage map for soils that are saturated, porous,and have shallow water table (i.e., Nile deposits)


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Population Risk Map

Highly populated areas affected by seismicity

(low to high PGA)


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Lecture8_interpolationf_2011