Linear Interpolation

1
Linear Interpolation

• Inverse Distance interpolation
• 10-27-00

2
Grid Data

• grid theme, is based on raster data, such as rows
  and columns of data
• can be either integer or floating point grid
  data,
• if based on integer it can have table that store
  the value of each cell
• based on floating data, it is only associated
  with Graduated Color

3
Methods

• 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
4
Computing IDW
6
Z140
Z260
4
Z440
Z350
2
2 4 6 X
Do you get 49.5 for the red square?
5
Exercise - generate a Inversion distance
weighting surface and contour

• Use the same data as last week
• turn on the point theme and have it activated
• Go to FilegtExtension and bring in Spatial
  Analyst
• Go to Surface and select Interpolate Grid

6
Output Grid Specification
You may change the cell size and compare the
results from different specifications
7
Contouring

• create a contour based on the surface from IDW

8
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

9
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

10
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
11
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

12
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

13
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

14
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
15
Semivariance
h1, h2 h3 h4
21.88 70.83 156.25 312.50
400 225 625 4
625 625 2
225 100 100 425 6
(Z1-Z1h)2
100 25 25 25 175 8
(Z2-Z2h)2
(Z3-Z3h)2
(Z4-Z4h)2
sum 2(n-h)
16
Modifications

• Tolerance - direction and distance

1m
1m
20o
5m
A
17
semivariance

• the semivariance increases as h increases
  distance increases -gt semivariance increases
• nearby points to be more similar than distant
  geographical data

18
data no longer similar to nearby values
sill
range
h
19
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
20

• 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)

21

• 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.

22
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
23

• 0.15(50)0.55(40) 0.30(25)
• 37

24
Exercise Kriging applied to Zn data

• Bring in the kriging extension from todays
  folder in GISLAB01\Class01\481_581\10-27-00
• This extension will add one item on Surface menu
  gtInterpolate Grid via kriging

25
Exercise

• Calculate contaminated Pb soil in Thiessen
  polygon exercise based on range of every 50 ppm,
  assuming bulk density of mineral soil is 2.65
  g/cm3
• compare to direct calculation of pb soil in
  ArcView