geostatistics gly 560: gis for earth scientists. 2/22/2016ub geology gly560: gis introduction...

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)


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


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


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”


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)


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


Realizations


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



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

ii

ii

ii zz

zzz

xx

xx

xx

xxxx n1

After Kitanidis (Intro. To Geostatistics)



.midpoint)or avg. (e.g. point, single by the drepresente is

interval thewhere

,)()(2

1)(ˆ

:computeThen

.)(),( ts,measuremen of pairs contains and, is interval k thewhere

intervals, into distances separation break thecommonly We

1

th

k

ukii

lk

N

iii

kk

iik

uk

lk

h

hh

zzN

h

zzNhh

k

xx

xx

xx



Small-Scale Variation: Discontinuous Case

Correlation smaller than sampling scale:Z2 = cos (2 x / 0.001)



Correlation larger than sampling scale:Z2 = cos (2 x / 2)

Small-Scale Variation:Parabolic Case



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)


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


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


Variogram Models


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.


Examples of Kriging

Universal Exponential 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

geostatistics gly 560: gis for earth scientists. 2/22/2016ub geology gly560: gis introduction...

Documents