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Chapter 16 - Spatial Interpolation

- Triangulation
- Inverse-distance
- Kriging (optimal interpolation)

What is Interpolation?

- Predicting the value of attributes at unsampled

sites from measurements made at point locations

within the same area or region - Predicting the value outside the area -

extrapolation - Creating continuous surfaces from point data -

the main procedures

Types of Spatial Interpolation

- Global or Local
- Global-use every known points to estimate unknown

value. - Local use a sample of known points to estimate

unknown value. - Exact or inexact interpolation
- Exact predict a value at the point location

that is the same as its known value. - Inexact (approximate) predicts a value at the

point location that differs from its known value. - Deterministic or stochastic interpolation
- Deterministic provides no assessment of errors

with predicted values - Stochastic interpolation offers assessment of

prediction errros with estimated variances.

Classification of Spatial Interpolation Methods

Global

Local

Stochastic

Deterministic

Stochastic

Deterministic

Regression (inexact)

Kriging (exact)

Trend surface (inexact)

Thiessen (exact) Density estimation(inexact) Inver

se distance weighted (exact) Splines (exact)

Global Interpolation

- Global use all available data to provide

predictions for the whole area of interest, while

local interpolations operate within a small zone

around the point being interpolated to ensure

that estimates are made only with data from

locations in the immediate neighborhood. - Two types of global Trend surface and regression

methods

Trend Surface Analysis

- Approximate points with known values with a

polynomial equation. - See Box 16.1
- Local polynomial interpolation uses a sample of

known points, such as convert TIN to DEM

Local, deterministic methods

Define an area around the point

Find data point within neighborhood

Choose model

Evaluate point value

Thiessen Polygon (nearest neighbor)

- Any point within a polygon is closer to the

polygons known point than any other known

points. - One observation per cell, if the data lie on a

regular square grid, then Thiessen polygons are

all equal, if irregular then irregular lattice of

polygons are formed - Delauney triangulation - lines joining the data

points (same as TIN - triangular irregular

network)

Thiessen polygons

Delauney Triangulation

Example data set

- soil data from Mass near the village of Stein in

the south of the Netherlands - all point data refer to a support of 10x10 m, the

are within which bulked samples were collected

using a stratified random sampling scheme - Heavy metal concentration measured

Exercise create Thiessen polygon for zinc

concentration

- Create a new project
- Copy g\classes\4650_5650\data\3-22\Soil_poll.dbf

and import it to the project. - After importing the table into the project, you

need to create an event theme based on this table - Go to Tools gt Add XY Data and make sure the

Easting is shown in X and Northing is in

Y. (Dont worry the Unknown coordinate - Click on OK then the point theme will appear on

your project.

This is what you might see on screen

Create a polygon theme

- The next thing you need to do is provide the

Thiessen polygon a boundary so that the computing

of irregular polygons can be reasonable - Use ArcCatalog to create a new shapefile and name

it as Polygon.shp - Add this layer to your current project.
- Use Editor to create a polygon.

Creating Polygon Theme

Notes 1)Remember to stop Edits, otherwise your

polygon theme will be under editing mode all the

time2)Remember to remove the selected points

from the Soil_poll_data.txt. If you are done

so, your Thiessen polygons will be based on the

selected points only.

Extent and Cell Size

- Go to Spatial Analyst gt Options and click on

tab and use Polygon as the Analysis Mask. - If the Analysis Mask is not set, the output layer

will have rectangular shape.

Thiessen Polygon from Spatial Analyst

- Select Spatial Analyst gt Distance gt Allocation.
- In Assign to, select soil_poll Event and
- Change the default cell size to 0.1
- click OK to create cell in temporary folder.

(No Transcript)

Join Tables

- Join soil_poll Events to Alloc3 grid file by

ObjectID in Alloc3 and OID in soil_poll

Events. - Click Advanced button. Two options are

available for joining tables. - Open Attribute of Alloc3 (name may vary) and

view the joined fields.

Zinc Concentration

Symbolize the grid with two-color ramp based on

Zn concentration

Density Estimation

- Simple method divide total point value by the

cell size - Kernel estimation associate each known point

with a kernel function, a probability density

function.

Exercise

- Compute density of the sampling points from

previous dataset. - If you use the xy-event points for calculation,

you might receive error message. - Convert this layer to shape file before using

Density function from Spatial Analyst. - Select your cell size (such as 1,) and search

radius as 5.

Density function output

Inverse Distance Weighted

- the value of an attribute z at some unsampled

point is a distance-weighted average of data

point occurring within a neighborhood, which

compute

estimated value at an unsampled point n number

of control points used to estimate a grid

point kpower to which distance is

raised ddistances from each control points to an

unsampled point

Computing IDW

6

Z140

Z260

4

Z440

Z350

2

Use k 1

2 4 6 X

Do you get 49.5 for the red square?

Exercise - generate a Inversion distance

weighting surface and contour

- Spatial Analyst gt Interpolate to Raster gt Inverse

Distance Weighted - Make sure you have set the Output cell size to 0.1

Contouring

- create a contour based on the surface from IDW

IDW and Contouring

Problem - solution

- Unsampled point may have a higher data value than

all other controlled points but not attainable

due to the nature of weighted average an average

of values cannot be lesser or greater than any

input values - solution - Fit a trend surface to a set of control points

surrounding an unsampled point - Insert X and Y coordinates for the unsampled

point into the trend surface equation to estimate

a value at that point

Splines

- draughtsmen used flexible rulers to trace the

curves by eye. The flexible rulers were called

splines - mathematical equivalents - localized - piece-wise polynomial function p(x) is

Spline - math functions

- piece-wise polynomial function p(x) is
- p(x)pi(x) xiltxltxi1
- pj(xi)pj(xi) j0,1,,,,
- i1,2,,,,,,k-1

i1

x1

xk1

x0

xk

break points

Spline

- r is used to denote the constraints on the spline

(the functions pi(x) are polynomials of degree m

or less - r 0 - no constraints on function

Exercise create surface from spline

- have point data theme activated
- select Surface gt Interpolate Grid
- Define the output area and other parameters
- Select Spline in Method field, Zn for Z

Value Field and regularized as type

Kriging

- Comes from Daniel Krige, who developed the method

for geological mining applications - Rather than considering distances to control

points independently of one another, kriging

considers the spatial autocorrelation in the data

semivariance

20

Z1 Z2 Z3 Z4 Z5

10

20 30 35 40 50

10 20 30 40 50

Zi values of the attribute at control

points hmultiple of the distance between control

points nnumber of sample points

Semivariance

h1, h2 h3 h4

21.88 91.67 156.25 312.50

(Z1-Z1h)2

100 25 25 25 175 8

225 100 100 425 6

400 225 625 4

625 625 2

(Z2-Z2h)2

(Z3-Z3h)2

(Z4-Z4h)2

sum 2(n-h)

Modifications (in real world)

- Tolerance - direction and distance needed to be

considered

10m

1m

20o

5m

A

semivariance

- the semivariance increases as h increases

distance increases -gt semivariance increases - nearby points to be more similar than distant

geographical data

data no longer similar to nearby values

sill

range

h

kriging computations

- we use 3 points to estimate a grid point
- again, we use weighted average

w1Z1 w2Z2w3Z3

estimated value at a grid point

Z1,Z2 and Z3 data values at the control

points w1,w2, and w3 weighs associated with

each control point

- In kriging the weighs (wi) are chosen to minimize

the difference between the estimated value at a

grid point and the true (or actual) value at that

grid point. - The solution is achieved by solving for the wi

in the following simultaneous equations - w1?(h11) w2?(h12) w3?(h13) ?(h1g)
- w1?(h12) w2?(h22) w3?(h23) ?(h2g)
- w1?(h13) w2?(h32) w3?(h33) ?(h3g)

- w1?(h11) w2?(h12) w3?(h13) ?(h1g)
- w1?(h12) w2?(h22) w3?(h23) ?(h2g)
- w1?(h13) w2?(h32) w3?(h33) ?(h3g)
- Where ?(hij)semivariance associated with

distance bet/w control points i and j. - ?(hig) the semivariance associated with the

distance bet/w ith control point and a grid

point. - Difference to IDW which only consider distance

bet/w the grid point and control points, kriging

take into account the variance between control

points too.

Example

distance

1 2 3 g

Z1(1,4)50

0 3.16 2.24 2.24

1 2 3 g

Z3(3,3)25

0 2.24 1.00

0 1.41

Z(2,2)?

0

Z2(2,1)40

w10.00w231.6w322.422.4 w131.6w20.00w322.410.

0 w122.4w222.4w30.0014.1

?

?10h

h

- 0.15(50)0.55(40) 0.30(25)
- 37

Homework 6 due next Thursday midnight.

- Task 1 Chapter 16 tasks
- Task 2
- Calculate volume of contaminated Pb soil in

Thiessen polygon exercise based on range of every

50 ppm, assuming soil density of 1.65 g/cm3 and

only the top 1-foot soil is considered. - Use IDW to compute the volume of the contaminated

Pb - Use Kriging (if its working) to compute

concentration of Pb - Compare these three methods and see the

differences (use same output cell size for all

three methods) - In Doc file, describe your selection of cell

size, search radius and results from different

choice of cell sizes (if you have time to create

layers with different cell sizes.