Interpolation of Hydrological Variables
Josef Fürst
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
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
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
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)
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
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
8
Introduction contd.
Hydrological variables are random and uncertain
geostatistical methods
Mostly 2D consideration v = v(x,y,t)
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
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
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
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
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
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.
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)
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
17
Choice of interpolation method
depends primarily on the nature of the variable and its
spatial variation
Examples: Rainfall, groundwater, soil physical
properties, topography
18
Example: Interpolation of rainfall
spatial correlation depends on time aggregation
19
Example: Groundwater data
groundwater tables have smooth surface, but trend!
Hydrogeological information is highly random, has
faults, few points with “good” data
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)
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
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
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
24
Spatial join (point in polygon)
assign spatial properties by spatial join
25
Thiessen polygons
Thiessen polygons, Voronoi Tesselation
a point in the domain receives the value of the closest
data point
step-wise function
##
##
#
#
##
#
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
27
Bi-linear interpolation
Simple and fast refinement in a 2-step interpolation
Resampling of continuous raster fields
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.
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
30 Inverse Distance Weighting (IDW) contd.
Bull’s eye effect = 2
#
#
#
# #
##
#
##
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
##
#
#
##
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
# #
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
# #
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
# #
##
#
##
#
#
#
#
#
#
#
#
##
#
#
#
#
##
#
#
##
#
#
#
##
#
#
#
##
#
#
##
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
# #
#
#
#
##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
# ##
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
##
#
#
#
#
#
#
#
#
#
31 Inverse Distance Weighting (IDW) contd.
grey:
= 0.1
red:
= 2
32 Inverse Distance Weighting (IDW) contd.
green:
= 10
red:
= 2
33 Inverse Distance Weighting (IDW) contd.
Interpolated values are always between Min and Max
of data
Sensitive to clustering and outliers
34
Radial Basis Functions (RBF)
“rubber membranes”
supported at data points
for smooth surfaces if
many data points available
35
Stochastic (geostatistical) Interpolation
Analysis of the spatial correlation in the random
component of a variable
Optimum determination of weights for interpolation
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
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
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 )()(
)()(
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
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
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
42 Example: comparison of methods for interpolation of precipitation (month)
43 Interpolation of elevation surface using different methods available in GIS: Mitas, L.,
Mitasova, H., 1999
Thiessen Polygons
44
TIN
Interpolation of elevation surface using different methods available in GIS: Mitas, L.,
Mitasova, H., 1999
45
IDW
Interpolation of elevation surface using different methods available in GIS: Mitas, L.,
Mitasova, H., 1999
46
Kriging
Interpolation of elevation surface using different methods available in GIS: Mitas, L.,
Mitasova, H., 1999
47
Topogrid (Arc/Info)
Interpolation of elevation surface using different methods available in GIS: Mitas, L.,
Mitasova, H., 1999
48
RST
Interpolation of elevation surface using different methods available in GIS: Mitas, L.,
Mitasova, H., 1999
49
Practical problems
Inhomogeneous density of points
• Search radius
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
51
Practical problems contd.
Special configurations of points (contour lines,
profiles, raster)
• Points along contour lines add points
0 2 4
52
Practical problems contd.
• Points along profile lines
53
Practical problems contd.
• Points along profile lines
54
Practical problems contd.
• Points on regular grid
Akkala et al. (2010) Interpolation techniques and associated software for environmental data. Env. Progr. & Sust. Energy (29/2) 134-141.
55
56
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”