GEOSTAT2013_Geostatistics_1

GEOSTAT 2013 - A. Brenning Geostatistics 1 1 / 47

GEOSTAT 2013:Geostatistics (1)

Alexander Brenning

University of Waterloo, Canada

Todays class

Introduction

Intrinsic RandomFunctions

SemivariogramModeling


Introduction

Intrinsic Random Functions

Semivariogram Modeling

Introduction

. IntroductionOverview

History

Current Research

Case Study

MotivationSemivariogramCloudEmpiricalSemivariogram

Case Study:Empirical Svgm.

Directional Svgm.

Wrap-up




Overview

Introduction

. OverviewHistory

Current Research

Case Study



Directional Svgm.

Wrap-up




Geostatistics deals with measurements of one or severalnumeric variables at point locations.

Analysis of the spatial dependence structure Kriging interpolation Kriging & regression

Interpolation with an underlying trend Regression in the presence of spatial autocorrelation

Geostatistical simulation

History

Introduction

Overview

. HistoryCurrent Research

Case Study



Directional Svgm.

Wrap-up




Since 1911: Early spatial analyses of agricultural data(Mercer & Hall, 1911; Youden & Mehlich, 1937) and forestry(Langsaetter, 1926; Matern, 1960; Jowett, 1955)

Kolmogorov, 1941: Optimal spatial interpolation

Concepts equivalent to variogram and kriging Motivated by meteorological applications Could not be applied: lack of computer power

1950s: Daniel G. Krige, South African mining engineer,develops a technique that is now called kriging

Predict block ore grades based on spatially autocorrelatedpoint samples

1960s: Georges Matheron (Ecole des Mines,Fontainebleau) formulates geostatistical theory

See Webster & Oliver, 2007

Current Research

Introduction

Overview

History

. Current ResearchCase Study



Directional Svgm.

Wrap-up




Current research topics include:

Space-time geostatistics (Christakos, 2000) Machine-learning techniques in geostastistics (Kanevski et

al., 2008) Geostatistics for massive, multi-scale data

e.g. combining multiple sources of remote sensing data ordownscaling remote sensing data

Multipoint geostatistics

Case Study: Soil Contamination

Introduction

Overview

History

Current Research

. Case StudyMotivationSemivariogramCloudEmpiricalSemivariogram


Directional Svgm.

Wrap-up




Top soil heavy metalconcentrations (in ppm),and soil and landscapevariables

At 155 locations on afloodplain of the riverMeuse / Maas,Netherlands

Bulk samples from an areaof approx. 15m x 15m

Sample dataset of thegstat package)

Kriging prediction of cadmium

Meuse river (photo: pbase.com)

Motivation

Introduction

Overview

History

Current Research

Case Study

. MotivationSemivariogramCloudEmpiricalSemivariogram


Directional Svgm.

Wrap-up




Observations at locations near to each other are, on average,more similar than observations at locations remote from eachother.

This can be measured using some similarity or dissimilaritymeasure:

Correlation correlogram(in time series analysis: autocorrelation function)

Semivariance semivariogram (in geostats)

Semivariogram Cloud

Introduction

Overview

History

Current Research

Case Study

Motivation

.SemivariogramCloud

EmpiricalSemivariogram


Directional Svgm.

Wrap-up




Measurements ziat locations xi:

Distance Semivariance|xj xi| (zj zi)2/2

1 (3 7)2/2 = 83 (16 7)2/2 = 40.5. . . . . .

(N locations N(N 1) pairs)

Semivariogram cloud:Plot semivariance againstdistance.

Empirical Semivariogram

Introduction

Overview

History

Current Research

Case Study

MotivationSemivariogramCloud

.EmpiricalSemivariogram


Directional Svgm.

Wrap-up




1. Group all pairs of sampleswith distance htogether in a class Nh.

2. Average the semivariancewithin each class Nh:

(h) =1

2 |Nh|

(i,j)Nh

(zizj)2

Semivariogram cloud:

Empirical semivariogram:

Case Study: Empirical Semivariogram

Introduction

Overview

History

Current Research

Case Study


.Case Study:Empirical Svgm.

Directional Svgm.

Wrap-up




Semivariogram of log(zinc concentration)

Directional Semivariograms

Introduction

Overview

History

Current Research

Case Study



. Directional Svgm.Wrap-up




Use pairs of samples withapproxly same orientationto estimate an empiricalsemivariogram

Explore possibledependence on thedirection (anisotropy)

E.g. air pollution moresimilar in wind directionthan perpendicular towind direction

Wrap-up of motivation

Introduction

Overview

History

Current Research

Case Study



Directional Svgm.

. Wrap-upIntrinsic RandomFunctions



Results of variogram analysis are of practical importance:

Nugget measurement error, apply smoothing insteadof interpolation?

Range search radius of kriging interpolation Range optimize spatial sampling design

This was just a rough, empirical introduction motivating thesemivariogram.

Now we will use a more formal approach to link thisempirical motivation with geostatistical models.

Intrinsic Random Functions

Introduction

.Intrinsic RandomFunctions

Regionalized variable

Probability Distrib.

Stationarity

CovarianceSecond-orderStationarity

Correlogram

Semivariogram

Intrinsic Stationarity

Semi- and Covgm.

Isotropy

Svgm. Shape

Interpretation




Introduction


.Regionalizedvariable


Stationarity


Correlogram

Semivariogram


Semi- and Covgm.

Isotropy

Svgm. Shape

Interpretation



A value z(x) R at a location x D is considered to be arealization of a random variable Z(x).

D is the domain, e.g. the (infinite) set of all point locationsin the study area. Here: D R2.

The set (z(x))xD = {z(x) : x D} of values is called aregionalized variable.

It is a realization of the random function or random field

Z = (Z(x))xD ,

the infinite family of all random variables at points x D. However, we only observe a finite set of n regionalized

values

z1 = z(x1), . . . , zn = z(xn)

from one particular realization (z(x))xD of Z.

Probability Distributions

Introduction



.ProbabilityDistrib.

Stationarity


Correlogram

Semivariogram


Semi- and Covgm.

Isotropy

Svgm. Shape

Interpretation



A random variable Z(x) is characterized by its probabilitydistribution function Fx:

P (Z(x) z) = Fx(z) Fx(z) is the probability that an outcome of Z(x) is less or

equal a given value z. In general, these distribution functions can potentially be

incredibly complex, and even more so the joint distributionfunction of several random variables Z(x1), . . . , Z(xk).

We have to make some assumptions to be able to modelthese distributions using semivariograms etc.

Stationarity

Introduction




. StationarityCovarianceSecond-orderStationarity

Correlogram

Semivariogram


Semi- and Covgm.

Isotropy

Svgm. Shape

Interpretation



In general, random variables Z(x1), Z(x2) at differentlocations may follow different probability distributions. (almost) impossible to handle!

Different types of stationarity make life easier. Strict stationarity means that any multiple-point

distribution function depends only on the location of pointsrelative to each other.

The joint distribution is the same throughout the entiredomain.

Intrinsic stationarity and second-order stationarity areweaker forms of stationarity:

only based on mean and covariance/variogram only based on pairs of points (two-point geostats)

Covariance

Introduction




Stationarity

. CovarianceSecond-orderStationarity

Correlogram

Semivariogram


Semi- and Covgm.

Isotropy

Svgm. Shape

Interpretation



Variation of the random field Z in space can be described byits mean or expected value m,

m(x) = E(Z(x)), x D,and its covariance function C,

C(x,x+h) = E [(Z(x) EZ(x)) (Z(x+h) EZ(x+h))] ,for all x,x+h D.Note that C(x,x) = Var(Z(x)) is the variance.

The covariance function C is stationary if it only dependson the distance (vector) between points, i.e. if it can bewritten as

C(x,x+h) = C(h).

Second-order Stationarity

Introduction




Stationarity

Covariance

.Second-orderStationarity

Correlogram

Semivariogram


Semi- and Covgm.

Isotropy

Svgm. Shape

Interpretation



Second-order stationarity (or weak stationarity) describes atype of independence of location:

The mean is constant:

m(x) = m for all x D And the covariance function only depends on the distance

vector h between an arbitrary pair of points x,x+h D:C(x,x+ h) = C(h) for all x,x+h D.

C(h) is called the covariogram.

Correlogram

Introduction




Stationarity


. CorrelogramSemivariogram


Semi- and Covgm.

Isotropy

Svgm. Shape

Interpretation



The correlogram of a second-order stationary randomfunction is

(h) =C(h)

C(0),

where the variance C(0) = Var(Z(x)) is constant in space. (h) may vary between 1 and +1.

Semivariogram

Introduction




Stationarity


Correlogram

. SemivariogramIntrinsic Stationarity

Semi- and Covgm.

Isotropy

Svgm. Shape

Interpretation



Instead of the covariance function or the correlogram,geostatisticians use the semivariogram

(x,x+h) =1

2Var (Z(x) Z(x+h)) ,

The variogram is simply 2.

The semivariogram is sometimes referred to as variogram(e.g. in the gstat package) this may cause confusion!

Note: (x,x) = 0.


Introduction




Stationarity


Correlogram

Semivariogram

.IntrinsicStationarity

Semi- and Covgm.

Isotropy

Svgm. Shape

Interpretation



Consider the drift of the process:

E (Z(x+ h) Z(x)) . A semivariogram is called stationary if it only depends on

the distance vector h between pairs of points:

(x,x+ h) = (h) for all x,x+h D. A random field with zero drift and a stationary semivariogram

is defined to be intrinsically stationary (or just intrinsic). Second-order stationarity implies intrinsic stationarity. Intrinsic stationarity does not imply second-order stationarity!

Semivarigram and Covariogram

Introduction




Stationarity


Correlogram

Semivariogram


.Semi- andCovgm.

Isotropy

Svgm. Shape

Interpretation



The covariance function and semivariogram of a random fieldare related to each other:

2(x,x+h) = C(x,x) + C(x+h,x+h) 2C(x,x+h). If the random field is second-order stationary, then

(h) = C(0) C(h). If the semivariogram of an intrinsic random field is bounded by a finite

value (), then it has a stationary covariance function

C(h) = () (h) +M, (1)

where M > 0 is a constant (i.e. independent of h). Note that random

functions with different covariograms may have the same semivariogram.

Isotropy

Introduction




Stationarity


Correlogram

Semivariogram


Semi- and Covgm.

. IsotropySvgm. Shape

Interpretation



Note that h has so far been considered to be a vector. In general, the semivariance and covariance may depend on

the direction of h.

directional semivariograms If the semivariogram depends only on the distance h = |h|,

but not on the orientation of h, then it is called isotropic.

Shape of the Semivariogram

Introduction




Stationarity


Correlogram

Semivariogram


Semi- and Covgm.

Isotropy

. Svgm. ShapeInterpretation



The svgm of an isotropic intrinsic random function is a simple graph

this is its typical shape and structure:

Note that (0) = 0! A nugget effect 2nug is present if (h) > 0 for h very close to zero.

The sill 2 = () is the semivar. level of the plateau (if it exists). The range r is the lag distance where the svgm reaches the sill.

Interpretation of the Semivariogram

Introduction




Stationarity


Correlogram

Semivariogram


Semi- and Covgm.

Isotropy

Svgm. Shape

. InterpretationSemivariogramModeling


sill: If the svgm is unbounded, then the random field is NOTsecond-order stationary.

range (or autocorrelation range): Observations areconsidered to be uncorrelated beyond this distance.What uncorrelated means, depends however on the scale.

nugget effect 2nug: represents microscale variation andmeasurement error not a continuous surface!

Overall, the characteristics of the svgm near h = 0 are ofcritical importance for characterizing the random function.

Semivariogram Modeling

Introduction


.SemivariogramModeling

Svgm. Models

Spherical Svgm.

Exponential Svgm.

Nugget Svgm.

Svgm. Cloud

Empirical Svgm.

Robust Estimator

Directional Svgm

Anisotropy

Svgm. Fitting


Semivariogram Models

Introduction



. Svgm. ModelsSpherical Svgm.

Exponential Svgm.

Nugget Svgm.

Svgm. Cloud

Empirical Svgm.

Robust Estimator

Directional Svgm

Anisotropy

Svgm. Fitting


The empirical svgm is NOTa valid svgm.: A function hasto honour certainmathematical properties inorder to be a valid svgm.

A semivariogram model isa function

(h; )

that gives a valid svgm forany parameter = (1, . . . , k).

is estimated from theempirical svgm.

Many parameters moreflexible, but harder to fit.

Svgm models can becombined by adding them.

Spherical Semivariogram

Introduction



Svgm. Models

. Spherical Svgm.Exponential Svgm.

Nugget Svgm.

Svgm. Cloud

Empirical Svgm.

Robust Estimator

Directional Svgm

Anisotropy

Svgm. Fitting


The one size fits all svgm. (?) The spherical semivariogram without nugget effect is:

sph(h;2, r) =

{2(3h2r h

3

2r3

)if h < r,

2 otherwise.

Sill 2, range r; stationary

Exponential Svgm.

Introduction



Svgm. Models

Spherical Svgm.

.ExponentialSvgm.

Nugget Svgm.

Svgm. Cloud

Empirical Svgm.

Robust Estimator

Directional Svgm

Anisotropy

Svgm. Fitting


exp(h;2, r) =

{2exp(h/r) if h > 0,0 if h = 0.

Similar to the spherical svgm. Converges to the sill as h.

Use a practical range corresponding to 95% of the fullsill to describe the model.

More convex shape than spherical svgm.

Nugget Semivariogram

Introduction



Svgm. Models

Spherical Svgm.

Exponential Svgm.

. Nugget Svgm.Svgm. Cloud

Empirical Svgm.

Robust Estimator

Directional Svgm

Anisotropy

Svgm. Fitting


Describes measurement error or microscale variation. Only used in combination with other svgm models.

(h;2nug) =

{0 if h = 0,2nug otherwise.

Semivariogram Cloud

Introduction



Svgm. Models

Spherical Svgm.

Exponential Svgm.

Nugget Svgm.

. Svgm. CloudEmpirical Svgm.

Robust Estimator

Directional Svgm

Anisotropy

Svgm. Fitting


For each pair of points (xi,xj), we can estimate thesemivariance individually as

(xi,xj) = (zi zj)2/2.The scatter plot of (xi,xj) against hij = |xi xj | is calledthe semivariogram cloud.

Empirical Semivariogram

Introduction



Svgm. Models

Spherical Svgm.

Exponential Svgm.

Nugget Svgm.

Svgm. Cloud

. Empirical Svgm.Robust Estimator

Directional Svgm

Anisotropy

Svgm. Fitting


AKA sample svgm or experimental svgm

Method of moments estimator: Let

Nh ={(i, j) {1, . . . , n}2 : |xi xj | h

}be the set of all pairs of points approximately distance hapart. Estimate the semivariance at distance h by

(h) =1

2|Nh|

(i,j)Nh

(zi zj)2

The empirical semivariogram is NOT a valid semivariogrammodel!

Empirical Svgm. Meuse data

Introduction



Svgm. Models

Spherical Svgm.

Exponential Svgm.

Nugget Svgm.

Svgm. Cloud

. Empirical Svgm.Robust Estimator

Directional Svgm

Anisotropy

Svgm. Fitting


Empirical semivariograms of log(zinc) in the Meuse data set:

Robust Estimator

Introduction



Svgm. Models

Spherical Svgm.

Exponential Svgm.

Nugget Svgm.

Svgm. Cloud

Empirical Svgm.

. Robust EstimatorDirectional Svgm

Anisotropy

Svgm. Fitting


Robust: stable if data departs from the model assumptions,

e.g. if random function is contaminated by anunknown process producing outliers

A robust estimator proposed by Cressie & Hawkins (1980):

(h) =1

2B(h)med

{|zi zj |1/2 : (xi,xj) Nh

},

where B(h) 0.457 corrects for bias. Large differences |zi zj | have a smaller impact on this

estimator than when using (zi zj)2 The median (med) is more robust than mean

Directional Semivariogram

Introduction



Svgm. Models

Spherical Svgm.

Exponential Svgm.

Nugget Svgm.

Svgm. Cloud

Empirical Svgm.

Robust Estimator

. Directional SvgmAnisotropy

Svgm. Fitting


The empirical svgms estimators introduced so far areomnidirectional svgms.

Now replace the set Nh of pairs approx. distance h apart bya restricted version Nh; containing only those pairs (xi,xj)where xi xj is oriented approximately in direction .

Enough data? Use directional semivariograms only if enoughdata are available.

Directional Semivariogram

Introduction



Svgm. Models

Spherical Svgm.

Exponential Svgm.

Nugget Svgm.

Svgm. Cloud

Empirical Svgm.

Robust Estimator

. Directional SvgmAnisotropy

Svgm. Fitting


Directional semivariograms of log(zinc) in the Meuse data set:

Modeling Anisotropy

Introduction



Svgm. Models

Spherical Svgm.

Exponential Svgm.

Nugget Svgm.

Svgm. Cloud

Empirical Svgm.

Robust Estimator

Directional Svgm

. AnisotropySvgm. Fitting


Geometric anisotropy:

Can be dealt with by transforming the coordinate space.

Local anisotropy:

Anisotropy may follow nonlinear sediment structures,drainage networks,. . .

Hard to explore, hard to model. . .

Fitting the Semivariogram

Introduction



Svgm. Models

Spherical Svgm.

Exponential Svgm.

Nugget Svgm.

Svgm. Cloud

Empirical Svgm.

Robust Estimator

Directional Svgm

Anisotropy

. Svgm. Fitting


Goal: Minimize the error e() of the svgm model with respect tothe empirical svgm as a function of the model parameters .

Svgm fitting by eye

Usually not too bad, sometimes not avoidable

Ordinary-least-squares (OLS) estimation:

e() =k

j=1

((hj) (hj))2,

where h1, . . . , hk are the bins of the empirical svgm.

This estimator does not account for varying numbers ofpairs |Nhj | how can we do better?


Introduction



Svgm. Models

Spherical Svgm.

Exponential Svgm.

Nugget Svgm.

Svgm. Cloud

Empirical Svgm.

Robust Estimator

Directional Svgm

Anisotropy

. Svgm. Fitting


Iteratively reweighted least-squares (WLS) estimation(Cressie, 1985): In iteration i,

ei() =k

j=1

w(i)j (

(hj) (hj ; i))2,

where w(0)j = |Nhj |, and w(i)j = |Nhj |((hj; i))2 for i > 0.

More pairs more weight Smaller semivariance more weight Good fit near the origin.

Default method in the R package gstat


Introduction



Svgm. Models

Spherical Svgm.

Exponential Svgm.

Nugget Svgm.

Svgm. Cloud

Empirical Svgm.

Robust Estimator

Directional Svgm

Anisotropy

. Svgm. Fitting


Restricted (or Residual) Maximum Likelihood (REML)estimation: Kitanidis (1985)

Based on more sophisticated mathematical arguments Depends on the assumption of a Gaussian (i.e. normal)

distribution More often used in conjunction with spatial linear models

fitted by (restricted) maximum likelihood estimation May be slow.

Svgm. Fitting Meuse data

Introduction



Svgm. Models

Spherical Svgm.

Exponential Svgm.

Nugget Svgm.

Svgm. Cloud

Empirical Svgm.

Robust Estimator

Directional Svgm

Anisotropy

. Svgm. Fitting


Fitted spherical semivariogram of log(zinc) in the Meuse data set:

Class Wrap-up

Introduction



Svgm. Models

Spherical Svgm.

Exponential Svgm.

Nugget Svgm.

Svgm. Cloud

Empirical Svgm.

Robust Estimator

Directional Svgm

Anisotropy

. Svgm. Fitting


Overview of applications of geostatistics Introduction to the mathematical concepts of intrinsic and

stationary random functions

Prerequisite for kriging!

Exploring, interpreting and modeling semivariograms

Ready for variogram analysis and kriging!

Next class will be more fun with less math!

Some Basic Probability Theory

Introduction



.

Some BasicProbabilityTheory

Expected value

Variance

Covariance


Expected value

Introduction



Some BasicProbability Theory

. Expected valueVariance

Covariance


The expected value (mean) of a suitable random variableX is the average of its outcomes weighted by its probabilitydistribution.

If the probability density function f of X exists,

E(X) =

R

tf(t)dt.

Note that E(aX + b) = aE(X) + b,E(X + Y ) = E(X) + E(Y ), i.e. the expectation is linear.

Variance

Introduction




Expected value

. VarianceCovariance


The variance of a suitable random variable X is the meansquared variation of X around its mean value:

Var(X) = E(X EX) = E(X2) (EX)2. The variance is not linear: Var(aX + b) = a2Var(X),

E(X + Y ) = E(X) + E(Y ), i.e. the expectation is linear. For uncorrelated X, Y , we have

Var(X + Y ) = VarX +VarY . Standard deviation: (X) =

VarX

Covariance

Introduction




Expected value

Variance

. Covariance


The covariance of a pair of suitable random variables X, Yis defined as:

Cov(X,Y ) = E(X EX)(Y EY ). It measures how X and Y vary together.

Positive: the higher, the higher Negative: the higher, the lower Zero: uncorrelated

X,Y independent X,Y uncorrelated X,Y uncorrelated ; X,Y independent Correlation: (X,Y ) = Cov(X,Y )/

VarX VarY

IntroductionOverviewHistoryCurrent ResearchCase Study: Soil ContaminationMotivationSemivariogram CloudEmpirical SemivariogramCase Study: Empirical SemivariogramDirectional SemivariogramsWrap-up of motivation

Intrinsic Random FunctionsRegionalized variableProbability DistributionsStationarityCovarianceSecond-order StationarityCorrelogramSemivariogramIntrinsic StationaritySemivarigram and CovariogramIsotropyShape of the SemivariogramInterpretation of the Semivariogram

Semivariogram ModelingSemivariogram ModelsSpherical SemivariogramExponential Svgm.Nugget SemivariogramSemivariogram CloudEmpirical SemivariogramEmpirical Svgm. Meuse dataRobust EstimatorDirectional SemivariogramDirectional SemivariogramModeling AnisotropyFitting the SemivariogramFitting the SemivariogramFitting the SemivariogramSvgm. Fitting Meuse data

Some Basic Probability TheoryExpected valueVarianceCovariance

GEOSTAT2013_Geostatistics_1

Documents

brenning geostatistics

directional svgm

empirical svgm

geostatistics deals

current research topics

multiscale data

soil heavy metalconcentrations

autocorrelatedpoint