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9 Interpolation En

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Interpolation of Hydrological Variables Josef Fürst
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Page 1: 9 Interpolation En

Interpolation of Hydrological Variables

Josef Fürst

Page 2: 9 Interpolation En

2

Learning objectives

In this section you will learn:

• Overview of most common interpolation methods

• To understand the principles of deterministic and

stochastic interpolation methods

• Ability to select the appropriate interpolation method for

a hydrological problem

• Overview of practical problems

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3

Outline

Introduction

Regionalisation and Interpolation

Principle of Interpolation

• Deterministic and statistical interpolation methods

• Global and local Interpolation

• Choice of interpolation method

Deterministic interpolation

Stochastic interpolation

• Spatial correlation

• Geostatistical interpolation

Practical problems

Page 4: 9 Interpolation En

4

Problem

A fundamental problem of hydrology is that our

models of hydrological variables assume continuity in

space (and time), while observations are done at

points.

The elementary task is to estimate a value at a given

location, using the existing observations

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5

Introduction

Hydrological data have variability in space and time

• Spatial variability is observed by a sufficient number of

stations

• Time variability is observed by recording time series

• Spatial variability can be in different range of values or

in different temporal behaviour

A continuous field v = v(x,y,z,t) is to be estimated from

discrete values vi = v(xi,yi,zi,ti)

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6

Introduction contd.

Global estimation: characteristic value for area

Point estimation: estimation at a point P = P(x,y)

We need data AND a conceptual model, how these

data are related, (i.e. a conceptual model of the

process)

If the process is well defined, only few data are

needed to construct the model

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7

Example

A groundwater table in a

confined, homogeneous,

isotropic aquifer under steady

state discharge from a well is

described by the Thiem well

formula.

Theoretically, the observation of

2 groundwater heads in different

distance from the well is

sufficient to reconstruct the

complete g.w. surface

1

212 ln

2)()(

r

r

T

Qrhrh

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8

Introduction contd.

Hydrological variables are random and uncertain

geostatistical methods

Mostly 2D consideration v = v(x,y,t)

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9

Regionalisation and Interpolation

Regionalisation: identification of the spatial

distribution of a function g, depending on local

information as well as by transfer of information from

other regions by transfer functions.

Regionalisation therefore means to describe spatial

variability (or homogeneity) of

• Model parameters

• Input variables

• Boundary conditions and coefficients

Page 10: 9 Interpolation En

10 Regionalisation and Interpolation contd.

Regionalisation includes the following tasks (and

more):

• Representation of fields of hydrological parameters and

data (contour maps)

• Smoothing spatial fields

• Identification of homogeneous zones

• Interpolation from point data

• Transfer of point information from one region to others

• Adaptation of model parameters for the transfer from

point to area

Page 11: 9 Interpolation En

11

Principles of interpolation

Given z = z(x,y) at some points we want to estimate z0

at (x0, y0)

x

y

z

(x ,y )1 1

(x ,y )2 2

(x ,y )3 3

z1

z ?0

z2

(x ,y )0 0

z3

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12

Principles of interpolation contd.

Weighted linear combination

The methods differ in the way how they establish the

weights

z can be a transformed variable, if, e.g., certain

statistical properties must be maintained

n

i

ii zwzz1

0ˆˆ

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13

Deterministic or statistical interpolation

Deterministic methods attempt to fit a surface of given

or assumed type to the given data points

• Exact

• Smoothing

Statistical (stochastic) methods treat a set of

observations as an arbitrary realisation of a 2D

stochastic process

Page 14: 9 Interpolation En

14

Example:

Precipitation data zi(t) of station I out of N stations

contain P independent events. We can interpret them

as P different scalar fields. The spatial distribution of

precipitation in a single event is a random realisation

of one 2D stochastic process.

Page 15: 9 Interpolation En

15 Deterministic or statistical interpolation contd.

Stochastic processes have a deterministic (or

structural) and a random component. The random

component can have spatial autocorrelation which is

used in interpolation.

)()()( xxfxf s

An optimal interpolation is

achieved by minimisation of

the estimation variance, which

is also used as a measure of

reliability of the interpolation.

x

x

f(x)Trend: a + bx f(x)

Page 16: 9 Interpolation En

16

Global and local interpolation

an interpolation method is working globally, if all data

points are evaluated in the interpolation.

Local interpolation techniques use only data points in

a certain neighbourhood of the

estimated point

2-step procedure:

densification

r

x0

y0z0

x

y

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17

Choice of interpolation method

depends primarily on the nature of the variable and its

spatial variation

Examples: Rainfall, groundwater, soil physical

properties, topography

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18

Example: Interpolation of rainfall

spatial correlation depends on time aggregation

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19

Example: Groundwater data

groundwater tables have smooth surface, but trend!

Hydrogeological information is highly random, has

faults, few points with “good” data

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20

Example: soil physical properties

Highly random: infiltration rate, soil water content,

hydraulic conductivity

geostatistical methods

few points with “good” data use of additional “soft”

information: soil maps, correlation with other data

(elevation, slope)

Page 21: 9 Interpolation En

21

Example: topography

Elevation of a ground point can be measured at any

time, repeated measures, etc...

Exact interpolation

properties of a terrain surface see DEM

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22

Deterministic interpolation methods

Polynomials

Spatial join (point in polygon)

Thiessen polygons

TIN and linear interpolation

Bi-linear interpolation

Spline

Inverse Distance Weighting (IDW)

Radial basis functions

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23

Polynomials

jn

j

i

ij

n

i

s yxcyxf00

),(n

i

i

is xcxf0

)(

ycxccyxfs 210),(2

54

2

3210),( ycxycxcycxccyxfs

12

)3(nnnk

f(x)

x1 x2 xx3 x4

•General:

•Plane:

•2. Order:

•# of coefficients

•Over- and undershoots

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24

Spatial join (point in polygon)

assign spatial properties by spatial join

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25

Thiessen polygons

Thiessen polygons, Voronoi Tesselation

a point in the domain receives the value of the closest

data point

step-wise function

##

##

#

#

##

#

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26

TIN and linear interpolation

Surface is approximated by facets of plane triangles

Continuous surface, but discontinuous 1st derivative

##

##

#

#

##

#

36.0

45.0

55.0

50.0

74.0

82.0

65.070.0

42.0

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27

Bi-linear interpolation

Simple and fast refinement in a 2-step interpolation

Resampling of continuous raster fields

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28

Splines

Spline estimates values using a mathematical

function that minimizes overall surface curvature,

resulting in a smooth surface that passes exactly

through the input points.

Conceptually, it is like bending a sheet of rubber to

pass through the points while minimizing the total

curvature of the surface.

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29

Inverse Distance Weighting (IDW)

Default method in many software packages = 2

Bull’s eye effect

controlled by exponent

N

i i

N

i i

ii

h

h

yxz

yxz

1 0,

1 0,

001

),(

),(ˆ 22

0,0, ii dh

Page 30: 9 Interpolation En

30 Inverse Distance Weighting (IDW) contd.

Bull’s eye effect = 2

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Page 31: 9 Interpolation En

31 Inverse Distance Weighting (IDW) contd.

grey:

= 0.1

red:

= 2

Page 32: 9 Interpolation En

32 Inverse Distance Weighting (IDW) contd.

green:

= 10

red:

= 2

Page 33: 9 Interpolation En

33 Inverse Distance Weighting (IDW) contd.

Interpolated values are always between Min and Max

of data

Sensitive to clustering and outliers

Page 34: 9 Interpolation En

34

Radial Basis Functions (RBF)

“rubber membranes”

supported at data points

for smooth surfaces if

many data points available

Page 35: 9 Interpolation En

35

Stochastic (geostatistical) Interpolation

Analysis of the spatial correlation in the random

component of a variable

Optimum determination of weights for interpolation

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36

Stochastic (geostatistical) Interpolation contd.

Experimental

semivariogram

things nearby tend to be

more similar than things

that are farther apart

huu

ji

ji

uZuZhN

h 2* ))()(()(2

1)(

0 200 400 600 800 1000 1200 1400 1600 1800

Lag Distance

0

50

100

150

200

250

300

350

Va

rio

gra

m

24

38

50

84

80 84

86 106

126

124

153

167159

181

181

181

177

183186

180

201

222200

Page 37: 9 Interpolation En

37 Stochastic (geostatistical) Interpolation contd.

Theoretical semivariogram: fit function through

empirical s.v.

0 200 400 600 800 1000 1200 1400 1600 1800

Lag Distance

0

50

100

150

200

250

300

350

Va

rio

gra

m

Page 38: 9 Interpolation En

38 Stochastic (geostatistical) Interpolation contd.

Ordinary Kriging

n

i

iijiji

n

i

n

j

n

j

j

ijij

n

j

i

n

ii

xxxxx

nixxxx

xVxV

111

2

1

1

1

)(2)()(

1

equations) of (system ,...,1 )()(

)()(

Page 39: 9 Interpolation En

39 Stochastic (geostatistical) Interpolation contd.

Kriging goes through a two-step process:

1. variograms and covariance functions are created to

estimate the statistical dependence (called spatial

autocorrelation) values, which depends on the model

of autocorrelation (fitting a model),

2. prediction of unknown values

Page 40: 9 Interpolation En

40 Stochastic (geostatistical) Interpolation contd.

Kriging yields the estimated value AND the estimation

variance

3410500 3411000 3411500 3412000 3412500 3413000 3413500 34140005470000

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 34140005470000

5470500

5471000

5471500

5472000

5472500

5473000

5473500

5474000

11

12

13

14

15

16

17

18

Estimated conductivity Standard deviation of estimated conductivity

Page 41: 9 Interpolation En

41 Stochastic (geostatistical) Interpolation contd.

problems of kriging

• Assumption of stationarity is not justified in many

hydrological variables

• Spatial trends

enhancements of kriging

• Universal Kriging (spatial trends)

• Indicator Kriging (inhomogeneities)

• Probabilistic Kriging (data with errors)

• Co-kriging (using correlation to other variables)

• External drift kriging

Page 42: 9 Interpolation En

42 Example: comparison of methods for interpolation of precipitation (month)

Page 43: 9 Interpolation En

43 Interpolation of elevation surface using different methods available in GIS: Mitas, L.,

Mitasova, H., 1999

Thiessen Polygons

Page 44: 9 Interpolation En

44

TIN

Interpolation of elevation surface using different methods available in GIS: Mitas, L.,

Mitasova, H., 1999

Page 45: 9 Interpolation En

45

IDW

Interpolation of elevation surface using different methods available in GIS: Mitas, L.,

Mitasova, H., 1999

Page 46: 9 Interpolation En

46

Kriging

Interpolation of elevation surface using different methods available in GIS: Mitas, L.,

Mitasova, H., 1999

Page 47: 9 Interpolation En

47

Topogrid (Arc/Info)

Interpolation of elevation surface using different methods available in GIS: Mitas, L.,

Mitasova, H., 1999

Page 48: 9 Interpolation En

48

RST

Interpolation of elevation surface using different methods available in GIS: Mitas, L.,

Mitasova, H., 1999

Page 49: 9 Interpolation En

49

Practical problems

Inhomogeneous density of points

• Search radius

Page 50: 9 Interpolation En

50

Practical problems contd.

• Over- and undershoots: 2 close points define a steep

gradient which has long range influence if distance to

next points is large

Page 51: 9 Interpolation En

51

Practical problems contd.

Special configurations of points (contour lines,

profiles, raster)

• Points along contour lines add points

0 2 4

Page 52: 9 Interpolation En

52

Practical problems contd.

• Points along profile lines

Page 53: 9 Interpolation En

53

Practical problems contd.

• Points along profile lines

Page 54: 9 Interpolation En

54

Practical problems contd.

• Points on regular grid

Page 55: 9 Interpolation En

Akkala et al. (2010) Interpolation techniques and associated software for environmental data. Env. Progr. & Sust. Energy (29/2) 134-141.

55

Page 56: 9 Interpolation En

56

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57

Summary and conclusions

Interpolation is a matter of weighting the data points

The nature of the variable determines the method of

interpolation

Deterministic methods

Stochastic (geostatistical) methods

• Analysis of spatial correlation

• Optimum interpolation (BLUE)

• Reliability of interpolation (variance)

GIS interpolation often simplistic, “smooth maps”


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