interpolation kriging
TRANSCRIPT
-
7/28/2019 Interpolation Kriging
1/38
Kriging
Using Geostatistical Analyst, ESRI
Chang, Kang-tsung, 2006, Introduction toGeographic Information Systems,
-
7/28/2019 Interpolation Kriging
2/38
So Far
We dealt with deterministic models that
can not provide estimates foraccuracy/certainty in predictions.
Kriging is a stochastic model that provides
estimates for accuracy/certainty in
predictions.
-
7/28/2019 Interpolation Kriging
3/38
Kriging
Spatial variation consists of 3 components
Random spatially correlated component
A drift or structure/trend
Random error term
-
7/28/2019 Interpolation Kriging
4/38
How to measure the spatially
correlated component?
Uses Semivariogram to measure spatially
correlated component, also called spatial
autocorrelation
-
7/28/2019 Interpolation Kriging
5/38
Creating a Semivariogram?
Where:
= semivariance between point Xi and Xj
h = distance separating Xi and Xj
Z = attribute value (height, ore quality, etc)
First what is semi-variance between two point separated by distance h?
-
7/28/2019 Interpolation Kriging
6/38
Semivariance between one point
and all other points
If I calculate all possible
semivariance for point A
(red dot), I get 11 points
A
-
7/28/2019 Interpolation Kriging
7/38
Repeat for Semivariance
calculations for every possible pair
Much more semivariance
computations if we repeat
for all possible pairs
-
7/28/2019 Interpolation Kriging
8/38
If all pairs are considered, we get a
Semivariogram cloud
Semivariogram allows us to investigate spatial dependence
If there is spatial dependence points that are closer together will have smallsemi-variance and vice versa
Because all pairs are plotted on the semivariogram, the semivariogram
becomes difficult to manage/interpret
Distance
-
7/28/2019 Interpolation Kriging
9/38
Binning is the solution
A process that is used to average semivariance data by distance and
direction
First group pairs of sample points into lag classes. If lag size is 2000
meters, lag classes are: (
-
7/28/2019 Interpolation Kriging
10/38
Continue
Third: For each group of pairs with similar lag and direction (in thesame grid cell), we compute the average semivariance between
sample points separated by lag h
Where: = average semi-variance between sample points separated by lag h
N = is the number of pairs of sample points sorted by direction in the bin
-
7/28/2019 Interpolation Kriging
11/38
Semivariogram after binning
On a semivariogram, if spatial dependence exist, semivariance
is expected to increase with distance
Same lag, different directions
-
7/28/2019 Interpolation Kriging
12/38
Anisotropy
Anisotropy is a term describing the existence of directional differences in spatialdependence
From semivariogram data, directional dependence can be extracted if it exists
low
Plume
Pollutant concentration high
Wind direction
-
7/28/2019 Interpolation Kriging
13/38
Continue
Fourth: To use the semivariogram as an interpolator, the semivariogramdata must be fitted with a mathematical function/model
-
7/28/2019 Interpolation Kriging
14/38
Semivariogram
Measures the variability of data with
respect to spatial distribution Looks at variance between pairs of data
points over a range of separation scales Forms the basis of every geostatistical
study
-
7/28/2019 Interpolation Kriging
15/38
Semiveriogram conditions:
Stationarity
The entire dataset can be described with one statistical model
Under the condition of stationarity, you can see the distance ofcorrelation in your data
h
(h)
Correlated at any distance
Correlated at a max distance
-
7/28/2019 Interpolation Kriging
16/38
Semiveriogram types
Many functions to choose from; Geostatistical analyst provides 11 models
Examples include: Gaussian, Linear, Spherical, Circular, and Exponential
Popular models are the spherical and exponential
Spatial dependencelevels after a certain
distance
Spatial dependenceDecreases exponentially
-
7/28/2019 Interpolation Kriging
17/38
Nugget (C0
):
Semivariance
at distance 0: Represents microscale
variations or measurement errors
Range (a) : Distance at which semivariance
levels off: Represents the spatially correlated portion
Sill (C0
+ C1
):
Semivariance
at which leveling takes place
N.B.: Every data set will have unique model features (model parameters)
C0
C1
C0 &C1
(h)a
Semivariogram: Model Parameters
-
7/28/2019 Interpolation Kriging
18/38
Fifth: Extract model parameters (C0, C1, and a) for the dataset under
consideration from the semivariogram. This will allow us to calculate amodeled semivariance between any 2 points knowing the distance hseparating the two points.
Extract model parameters
C0
C1
C0 &C1
(h)a
-
7/28/2019 Interpolation Kriging
19/38
Continue
Sixth: Use the model parameters to extract semivariance values for anypoint by solving a set of simultaneous equations
Let us consider the following Problem:
Knowing three points: 1, 2, 3, we want to use krigging to estimate point 0
-
7/28/2019 Interpolation Kriging
20/38
How?
Where:
Problem: Knowing three points: 1, 2, 3, we want to estimate
semivariance values and solve equations to determine value at
point 0
(hij) is the semivariance between known point I & J
(hi0) is the semivariance between known point I & point to be estimated
Lagrange multiplier to ensure minimum estimation of errorMore like a fudge factor
(hij) is the semivariance between known point I & J
(hi0) is the semivariance between known point I & point to be estimated
Lagrange multiplier to ensure minimum estimation of errorMore like a fudge factor
(hij) is the semivariance between known point I & J
(hi0) is the semivariance between known point I & point to be estimated
Lagrange multiplier to ensure minimum estimation of errorMore like a fudge factor
-
7/28/2019 Interpolation Kriging
21/38
Thus
Derive semivariance between known pairs of known
points (1,2,3) and each of the known point and unknown
point knowing model parameters
Derive weights solving by solving the simultaneous
equations
Plug in weights in the equation below to calculate thevalue of the unknown point (Z0)
Z0
= Z1
W1
+ Z2
W2
+ Z3
W3
-
7/28/2019 Interpolation Kriging
22/38
What if we had > 3 points
(1) Apply Matrix algebra to solve simultaneous Equations
-
7/28/2019 Interpolation Kriging
23/38
= Estimated value
= Known value at point x
= Weight associated with point x
S = number of sample points used in estimation
Where:
(2) Estimator equation becomes
What if we had > 3 points
-
7/28/2019 Interpolation Kriging
24/38
How is this different from Inverse
Distance MethodKrigging IDW
Calculation of weightsinvolve
(1) Variance between point to beestimated and Known points
(2) Variance between Known points
(1) Variance between pointto be estimated and
Known points
Measuring reliability A measure for the reliability of the
estimate
None
-
7/28/2019 Interpolation Kriging
25/38
Kriging Types in ArcGIS
Ordinary kriging
Simple kriging Universal kriging
Indicator kriging Probability kriging
Disjunctive kriging Cokriging
-
7/28/2019 Interpolation Kriging
26/38
From before
Spatial variation consists of 3 components
Random spatially correlated component
A trend component
Random non correlated component(error term)
-
7/28/2019 Interpolation Kriging
27/38
Components of Kriging
The value of z depends on: (1) trend component, (2) random autocorrelatedcomponent, and (3) random non-correlated component (for simplicity notrepresented in figures)
Z(s) = (s) + (s)Where:
Z = Value at point s
= Trend component value at point s (first order or second order polynomial)
= Random, autocorrolated component
-
7/28/2019 Interpolation Kriging
28/38
Ordinary Kriging
Assumes there is no trend
Assumes m(s) is unknown and constant
Focuses on the spatially correlated component
-
7/28/2019 Interpolation Kriging
29/38
Simple Kriging
Assumes (s) , the mean of data set is known and is constant
Assumes there is no trend component
In the majority of cases this is unrealistic assumption
-
7/28/2019 Interpolation Kriging
30/38
Indicator Kriging
(s) is constant, and unknown
Values are binary (1 or 0)
Example, a point is forest or non forest
-
7/28/2019 Interpolation Kriging
31/38
Universal Kriging
Assumes z values change because of a drift (trend) in addition toautocorrelation.
(s) is not constant
Trend component expressed as a 1st order (plane) or 2nd orderpolynomials (quadratic surface)
Kriging is performed on residual after the trend is removed
-
7/28/2019 Interpolation Kriging
32/38
Cokriging
Adds second variable which is correlated
with the primary function It assumes the correlation between the
variables can be used to improve theprediction of the primary variable
Example better kriging of precipitation can
be done if elevation is included as a
secondary variable
-
7/28/2019 Interpolation Kriging
33/38
Kriging outputs & assumptions
Prediction map
Prediction map reports the mean
Assumes a normal population at each point & thus,
-
7/28/2019 Interpolation Kriging
34/38
Kriging outputs & assumptions
Probability maps show the probability of finding a value at
a location(e.g., > threshold value of 1) Assumes a normal population at each point
5%
35%
50%
-
7/28/2019 Interpolation Kriging
35/38
Kriging Outputs
Quantile map: Quantile map showing the probability for each location forhaving an extreme value (highest 5%)
Assumes a normal population at each point
-
7/28/2019 Interpolation Kriging
36/38
Kriging outputs
Error maps refer to how good we can map
these predictions
-
7/28/2019 Interpolation Kriging
37/38
Error Assesment
-
7/28/2019 Interpolation Kriging
38/38
How can we choose the most
appropriate method for interpolation?
Cross validation:(1) Remove a known point from data set
(2) Use remaining points to estimate the value at the point removed
(3) Compare the estimated to known value
(4) Repeat for all points & calculate root mean squares
(5) The method that produces the least difference is selected (closer
to 1)