Transcript Spatial Interpolation - University at Buffalo

Geostatistics
GLY 560:
GIS for Earth Scientists
Introduction
Premise:
One cannot obtain error-free estimates of
unknowns (or find a deterministic model)
Approach:
Use statistical methods to reduce and
estimate the error of estimating unknowns
(must use a probabilistic model)
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UB Geology GLY560: GIS
Estimator of Error
• We need to develop a good estimate of an
unknown. Say we have three estimates of
an unknown:
Want toestimateunknown,Tˆ0





2
2
1 ˆ
1 ˆ
1 ˆ
  T0  T1  T0  T2  T0  T3
3
3
3
2
where 0 is themean square error
2
0
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UB Geology GLY560: GIS

2
Estimator of Error
• An estimator that minimizes the mean
square error (variance) is called a “best”
estimator
• When the expected error is zero, then the
estimator is called “unbiased”.
Want toestimateunknown,Tˆ0





2
2
1 ˆ
1 ˆ
1 ˆ
  T0  T1  T0  T2  T0  T3
3
3
3
where 02 is themean square error
2
0
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UB Geology GLY560: GIS

2
Estimator of Error
• Note that the variance can be written
more generally as:
n
Tˆ0   i Ti
i 1
where n is thenumber of measurements
and 1 , 2 ,....n are coefficients or weights
• Such an estimator is called “linear”
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UB Geology GLY560: GIS
BLUE
An estimator that is
• Best: minimizes variance
• Linear: can be expressed as the sum
of factors
• Unbiased: expects a zero error
…is called a BLUE
(Best Linear Unbiased Estimator)
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UB Geology GLY560: GIS
BLUE
• We assume that the sample dataset is
a sample from a random (but
constrained) distribution
• The error is also a random variable
• Measurements, estimates, and error
can all be described by probability
distributions
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UB Geology GLY560: GIS
Realizations
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UB Geology GLY560: GIS
Experimental Variogram
• Measures the variability of data with
respect to spatial distribution
• Specifically, looks at variance
between pairs of data points over a
range of separation scales
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UB Geology GLY560: GIS
Experimental Variogram
Consider n measurements, z (x1 ), z (x 2 )...z (x n ),
where x is an arrayof coordinates of themeasurement points.
1
We plot thesquare difference: z (x i )  z (xi )
2
against the separationdistance x i  xi ,
where x i and xi are measurement pairs, and
denotesthedistancebetween the points(magnitudeof the vector)
After Kitanidis (Intro. To Geostatistics)
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UB Geology GLY560: GIS
Experimental Variogram
We commonly break theseparationdistancesintointervals,

wherethe k th intervalis hkl , hku

and containsN k pairs of measurements,z (x i ), z (xi ).
Then compute:
ˆ (hk ) 
1
2N k
Nk
 z(x )  z(x ),
i
i
i 1
wherethe interval
hkl  x i  xi  hku
is represented by thesinglepoint,hk (e.g. avg.or midpoint).
After Kitanidis (Intro. To Geostatistics)
7/21/2015
UB Geology GLY560: GIS
Small-Scale Variation:
Discontinuous Case
Correlation smaller than sampling scale:
Z2 = cos (2 p x / 0.001)
After Kitanidis (Intro. To Geostatistics)
7/21/2015
UB Geology GLY560: GIS
Small-Scale Variation:
Parabolic Case
Correlation larger than sampling
scale:
Z2 = cos (2 p x / 2)
After Kitanidis (Intro. To Geostatistics)
7/21/2015
UB Geology GLY560: GIS
Stationarity
• Stationarity implies that an entire
dataset is described by the same
probabilistic process… that is we can
analyze the dataset with one statistical
model
(Note: this definition differs from that given by Kitanidis)
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UB Geology GLY560: GIS
Stationarity and the Variogram
• Under the condition of stationarity, the
variogram will tell us over what scale the
data are correlated.
Correlated at any distance
(h)
Uncorrelated
Correlated at a max distance
h
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UB Geology GLY560: GIS
Variogram for Stationary Dataset
Semi-Variogram
function
•Range: maximum
distance at which data
are correlated
•Nugget: distance over
which data are
absolutely correlated or
unsampled
Range
Sill
Nugget
Separation Distance
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UB Geology GLY560: GIS
•Sill: maximum
variance ((h)) of data
pairs
Variogram Models
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Kriging
• Kriging is essentially the process of using
the variogram as a Best Linear Unbiased
Estimator (BLUE)
• Conceptually, one is fitting a variogram
model to the experimental variogram.
• Kriging equations may be used as
interpolation functions.
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UB Geology GLY560: GIS
Examples of Kriging
Universal
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Exponential
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Circular
Final Thoughts
• Kriging produces nice (can be exact)
interpolation
• Intelligent Kriging requires
understanding of the spatial statistics
of the dataset
• Should display experimental variogram
with Kriging or similar methods
7/21/2015
UB Geology GLY560: GIS