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2/22/2016UB Geology GLY560: GIS Estimator of Error We need to develop a good estimate of an unknown. Say we have three estimates of an unknown:
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Geostatistics
GLY 560: GIS for Earth Scientists
05/05/23 UB Geology GLY560: GIS
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)
05/05/23 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:
error squaremean theis where
ˆ31ˆ
31ˆ
31
ˆ unknown, estimate Want to
20
2
30
2
20
2
1020
0
TTTTTT
T
05/05/23 UB Geology GLY560: GIS
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”.
error squaremean theis where
ˆ31ˆ
31ˆ
31
ˆ unknown, estimate Want to
20
2
30
2
20
2
1020
0
TTTTTT
T
05/05/23 UB Geology GLY560: GIS
Estimator of Error
•Note that the variance can be written more generally as:
or weights tscoefficien are ,...., andtsmeasuremen ofnumber theisn where
ˆ
21
10
n
i
n
ii TT
•Such an estimator is called “linear”
05/05/23 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)
05/05/23 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
05/05/23 UB Geology GLY560: GIS
Realizations
05/05/23 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
05/05/23 UB Geology GLY560: GIS
Experimental Variogram
vector) theof (magnitude points ebetween th distance thedenotes
and pairs,t measuremen are and where
, distance separation eagainst th
)()(21 :difference square plot the We
points.t measuremen theof scoordinate ofarray an is where ),()...(),( ts,measuremenn Consider 2
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After Kitanidis (Intro. To Geostatistics)
05/05/23 UB Geology GLY560: GIS
Experimental Variogram
.midpoint)or avg. (e.g. point, single by the drepresente is
interval thewhere
,)()(2
1)(ˆ
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.)(),( ts,measuremen of pairs contains and, is interval k thewhere
intervals, into distances separation break thecommonly We
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After Kitanidis (Intro. To Geostatistics)
05/05/23 UB Geology GLY560: GIS
Small-Scale Variation: Discontinuous Case
Correlation smaller than sampling scale:Z2 = cos (2 x / 0.001)
After Kitanidis (Intro. To Geostatistics)
05/05/23 UB Geology GLY560: GIS
Correlation larger than sampling scale:Z2 = cos (2 x / 2)
Small-Scale Variation:Parabolic Case
After Kitanidis (Intro. To Geostatistics)
05/05/23 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)
05/05/23 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.
(h)
h
Correlated at any distance
Correlated at a max distance
Uncorrelated
05/05/23 UB Geology GLY560: GIS
Variogram for Stationary Dataset
Nugget
Range
Sill
Separation Distance
Sem
i-Var
iogr
am
func
tion
•Range: maximum distance at which data are correlated•Nugget: distance over which data are absolutely correlated or unsampled•Sill: maximum variance ((h)) of data pairs
05/05/23 UB Geology GLY560: GIS
Variogram Models
05/05/23 UB Geology GLY560: GIS
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.
05/05/23 UB Geology GLY560: GIS
Examples of Kriging
Universal Exponential Circular
05/05/23 UB Geology GLY560: GIS
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
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