9 Interpolation En
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Interpolation of Hydrological Variables
Josef Fürst
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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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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
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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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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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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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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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Introduction contd.
Hydrological variables are random and uncertain
geostatistical methods
Mostly 2D consideration v = v(x,y,t)
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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
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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
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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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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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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
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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.
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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)
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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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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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Example: Interpolation of rainfall
spatial correlation depends on time aggregation
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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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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)
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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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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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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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Spatial join (point in polygon)
assign spatial properties by spatial join
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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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TIN and linear interpolation
Surface is approximated by facets of plane triangles
Continuous surface, but discontinuous 1st derivative
##
##
#
#
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#
36.0
45.0
55.0
50.0
74.0
82.0
65.070.0
42.0
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Bi-linear interpolation
Simple and fast refinement in a 2-step interpolation
Resampling of continuous raster fields
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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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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
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30 Inverse Distance Weighting (IDW) contd.
Bull’s eye effect = 2
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31 Inverse Distance Weighting (IDW) contd.
grey:
= 0.1
red:
= 2
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32 Inverse Distance Weighting (IDW) contd.
green:
= 10
red:
= 2
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33 Inverse Distance Weighting (IDW) contd.
Interpolated values are always between Min and Max
of data
Sensitive to clustering and outliers
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Radial Basis Functions (RBF)
“rubber membranes”
supported at data points
for smooth surfaces if
many data points available
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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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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
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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
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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 )()(
)()(
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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
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40 Stochastic (geostatistical) Interpolation contd.
Kriging yields the estimated value AND the estimation
variance
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Estimated conductivity Standard deviation of estimated conductivity
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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
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42 Example: comparison of methods for interpolation of precipitation (month)
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43 Interpolation of elevation surface using different methods available in GIS: Mitas, L.,
Mitasova, H., 1999
Thiessen Polygons
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TIN
Interpolation of elevation surface using different methods available in GIS: Mitas, L.,
Mitasova, H., 1999
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IDW
Interpolation of elevation surface using different methods available in GIS: Mitas, L.,
Mitasova, H., 1999
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Kriging
Interpolation of elevation surface using different methods available in GIS: Mitas, L.,
Mitasova, H., 1999
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Topogrid (Arc/Info)
Interpolation of elevation surface using different methods available in GIS: Mitas, L.,
Mitasova, H., 1999
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RST
Interpolation of elevation surface using different methods available in GIS: Mitas, L.,
Mitasova, H., 1999
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Practical problems
Inhomogeneous density of points
• Search radius
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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
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Practical problems contd.
Special configurations of points (contour lines,
profiles, raster)
• Points along contour lines add points
0 2 4
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Practical problems contd.
• Points along profile lines
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Practical problems contd.
• Points along profile lines
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Practical problems contd.
• Points on regular grid
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Akkala et al. (2010) Interpolation techniques and associated software for environmental data. Env. Progr. & Sust. Energy (29/2) 134-141.
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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”