Spatial Data Analysis

Unit 5:

Spatial Data Analysis

H.P. Nachtnebel IWHW-BOKU

Environmental Risk Analysis Data Analysis

Goals and Structure

• Goal: development of tools for spatio-temporal data analysis

• Introduction

• Methodology (spatial analysis)

• Application

• Uncertainty

• Summary and conclusions

Introduction

• Environmental data exhibit often a spatial and a temporal correlation

e.g. the spreading of a pollution plume in a groundwater system

the movement of a thunderstorm over a basin

the leaching of pesticides through the soil to the groundwater

Environmental Risk Analysis Data Analysis

Introduction

• Spatial variability can be detected by a monitoring network

• Temporal variability can be detected by frequent sampling at a location

Examples: CO2-concentration

Temporal scale

Examples: CO2-concentation

Environmental Risk Analysis Data Analysis

Spatial scale

Examples: fallout

Examples: fallout

Environmental Risk Analysis Data Analysis

Examples: natural pollution

Environmental Risk Analysis Data Analysis

Spatial data analysis

• Several concepts are applicable to both spatial and temporal data sets

• Some definitions:

- interpolation: estimation of a value within a domain covered by observations

- extrapolation: estimation of a value outside the domain

- regionalisation: identifying properties within a domain (could be also achieved by transferring information from another region to the domain of interest)

Spatial data analysis

• Is scale dependent

• Resolution depends on monitoring system

• Analysis can be based on statistical and/or physical models

Environmental Risk Analysis Data Analysis

Methods and concepts

(1) Thiessen or Voronoi diagrams

monitoring points transects

Within one unit a specific but constant value

Environmental Risk Analysis Data Analysis

Thiessen or Voronoi diagram

Methods and concepts (2) Linear interpolation

Monitoring points (x_i,y_i) and observations Z_i(x_i,y_i)

x y

Z(x,y)

?

cy bx

a y

x

Z( , )   

1 1

1 1

1(x , y ) a bx cy

Z   

2 2

2 2

2(x , y ) a bx cy

Z   

3 3

3 3

3(x , y ) a bx cy

Z   

Environmental Risk Analysis Data Analysis

Methods and concepts

(2) Linear interpolation: Isolines

• Linear changes of Z within each element

• Discontinuity at the boundaries

Methods and concepts (3) nonlinear interpolation

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7 6

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10

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7

10

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9 8

Isolines are generated by linear interpolation at

several grid points and then the connecting line

is smoothed

Environmental Risk Analysis Data Analysis

Methods and concepts

(3) nonlinear interpolation: Inverse distance

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w  c

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Inverse Distance Weighting (IDW)

• Bull’s eye effect  = 2

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Environmental Risk Analysis Data Analysis

Inverse Distance Weighting (IDW)

grey:

 = 0.1

red:

 = 2

Inverse Distance Weighting (IDW)

green:

 = 10

red:

 = 2

Environmental Risk Analysis Data Analysis

Methods and concepts Kriging

• Was developed by D. Krige (1951) in mining application in South Africa

• Methodological improvements by Matheron

(1960) and subsequently applied by Delhomme (1978), Journel (1978)…..

• It provides an estimate and the estimation variance (uncertainty)

• Several extensions from Ordinary Kriging:

indicator kriging, external drift kr., universal kr., co-kriging, fuzzy kriging,…

Methods and concepts Ordinary Kriging

• Basic assumptions: stationarity in space

mean and co-variance are constant within a given domain

what we observe in nature is the realisation of a random field

complete information is included in the co-variance or variogram

Kriging is a BLUE estimator (Best Linear Unbiased Estimator)

Environmental Risk Analysis Data Analysis

Methods and concepts Ordinary Kriging

Z(x,y)

x y

Select a point (location i=1) Define a distance h +-h

Estimate the variance of all the points within that distance Z(x,y) measurement value at location x,y

h h

Methods and concepts Ordinary Kriging

Z(x,y)

x y

Select a point (location i=1) Define a distance h +-h

Estimate the variance of all the points within that distance Continue with another point (location i=2)

Repeat it for all points

Then you have an estimate of the variance (h) Enlarge the distance h and start again

h h

Environmental Risk Analysis Data Analysis

Estimation with Kriging

• Establish an empirical variogram

• Fit a theoretical variogram

) (

* )]

( )

( ) [

( 2

1 ₂

h x

Z x

h Z

N _{x x h} ^{i j}

j i











Methods and concepts Ordinary Kriging

Variance (h)

Empirical Variogram

Theoretical Variogram

h (Distanz)

Environmental Risk Analysis Data Analysis

Methods and concepts Ordinary Kriging

Varianz

d₁ d₂ d₃

Empirisches Variogramm

Theoretisches Variogramm

h (Distanz)

• Variogram has 3 parameters (max)

Estimation and its variance

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Environmental Risk Analysis Data Analysis

Simple example

• 2 observation: Z₁(x₁=1) =2, Z₂(x₂=-2)=4

• A linear variogram (h)=IhI

• Estimate Z(x=0) and the estimation variance

0₁ + 3₂ +  = 1 3₁ + 0₂ +  = 2

₁ + ₂ = 1

₁ = 0,6667, ₂ = 0,3333  = 0 Z*(x=0) = 2,6667 ² = 1.3333

) (

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x^{i j i}

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(

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i n

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i Z x

x

Z   

 

Types of variograms

Theoretical Variograms

Environmental Risk Analysis Data Analysis

Applications

3410500 3411000 3411500 3412000 3412500 3413000 3413500 3414000 5470000

5470500 5471000 5471500 5472000 5472500 5473000 5473500 5474000

55 60 65 70 75 80 85 90 95 100 105

3410500 3411000 3411500 3412000 3412500 3413000 3413500 3414000 5470000

5470500 5471000 5471500 5472000 5472500 5473000 5473500 5474000

11 12 13 14 15 16 17 18

Estimated conductivity Standard deviation of estimated conductivity

Observation and interpolation

• Observed values have a spatial variance

• Interpolated values are smoother (smaller variance)

• To simulate a random field conditional simulation is required (same value at

observation points) and same variance as in reality

Environmental Risk Analysis Data Analysis

Summary and conclusions

• Several interpolation/extrapolation techniques were presented

• Linear and nonlinear

• Deterministic and stochastic

• Kriging is a BLUE estimator

it provides an estimate and the estimation variance

• Can be used to develop a monitoring system

• Can be used to simulate spatial structures