Journal of Environmental Protection, 2009, 1, 50-58
Published Online November 2009 (http://www.SciRP.org/journal/jep/).
Uncertainty Analysis of Interpolation Methods in Rainfall
Spatial Distribution–A Case of Small Catchment in Lyon
Tao TAO1,*, Bernard CHOCAT2, Suiqing LIU1, Kunlun XIN1
1State key laboratory of Pollution Control and Resources Reuse, Tongji University, Shanghai, China
2U.R.G.C. Hydrology Urbane, I.N.S.A. de Lyon, 69621 Villeurbanne Cedex, France
Abstract
Quantification of spatial and temporal patterns of rainfall is an important step toward developing regional
water sewage models, the intensity and spatial distribution of rainfall can affect the magnitude and duration
of water sewage. However, this input is subject to uncertainty, mainly as a result of the interpolation method
and stochastic error due to the random nature of rainfall. In this study, we analyze some rainfall series from
30 rain gauges located in the Great Lyon area, including annual, month, day and intensity of 6mins, aiming
at improving the understanding of the major sources of variation and uncertainty in small scale rainfall in-
terpolation in different input series. The main results show the model and the parameter of Kriging should be
different for the different rainfall series, even if in the same research area. To the small region with high den-
sity of rain gauges (15km2), the Kriging method superiority is not obvious, IDW and the spline interpolation
result maybe can be better. The different methods will be suitable for the different research series, and it
must be determined by the data series distribution.
Keywords: Rainfall, Spatial Distribution, Kriging, Interpolation
1. Introduction
Precipitation is in many cases the most important input
factor in hydrological modeling [1]. The role of rainfall
is essential for urban hydrology: it is the driving phe-
nomenon of runoff mechanisms, particularly in an urban
context. Its variability constitutes a significant source of
uncertainty for hydrological modeling. Assessing rainfall
variability is an important element to developing con-
ceptual and predictive models of runoff, pollutant load-
ing, and river dynamics. Quantification of spatial and
temporal patterns of rainfall is an important step toward
developing regional water sewage models. For example,
the intensity and spatial distribution of rainfall can affect
the magnitude and duration of pollutant washoff to the
ocean [2,3], which are also an input for hydrological
models. The small size of the urban catchments and the
hydrological purposes (especially for real time applica-
tions) oblige us to consider rainfall at small scales: on the
order of 6 min in time and 5 km in space. Hence urban
hydrology requires rainfall measurements with high tem-
poral and spatial resolution.
However, this input is subject to uncertainty, as a re-
sult of measurement errors, systematic errors in the in-
terpolation method and stochastic error due to the ran-
dom nature of rainfall. In addition to the stochastic na-
ture of rainfall, the precipitation pattern may be influ-
enced by the irregular topography. The large variability
in altitude, slope and aspect may increase variability by
means of processes such as rain shading and strong
winds. Accurate estimation of the spatial distribution of
rainfall and extrapolation of point measurements over
large areas is complicated.
The best method to improve the quality of spatial
rainfall estimation is to increase the density of the moni-
toring network. However, traditionally used rain gauges
data are sparse and do not always provide adequate spa-
tial representation of rainfall. This is very costly, and in
many cases practically infeasible. And even for dense
networks, interpolation remains necessary in order to
*Corresponding author. Foundation item: The Major Science and
Technology Project-Water Pollution Control and Treatment
(NO.2008ZX7421-001)

T. TAO ET AL. 51
calculate the total rainfall over a certain area [4]. There-
fore, both the design of an adequate monitoring network
and choice of an interpolation method require insight in
the patterns of rainfall variability and the sources of un-
certainty. To provide optimal input for distributed hy-
drological modeling, the best strategy is probably a com-
bination of all available information on rainfall, includ-
ing data from hourly point observations, radar data,
denser daily measurements, physiographic factors like
elevation, and applying sophisticated interpolation or
merging methods [5].
Rainfall has been interpolated using averaged values
ranging from daily, monthly or annual aggregation levels.
And quite a number of modern interpolation methods
have been proposed for rainfall. Besides geostatistical
approaches, such as ordinary Kriging, Kriging with ex-
ternal drift and co-Kriging [6,7], other techniques based
on splines [8,9] or genetic algorithms [10,11] have been
applied. These studies concluded that the Kriging
method yields a more realistic spatial behaviour of the
climatological variable of interest. The use of auxiliary
variables in order to improve the spatial interpolation of
rainfall variables has been analyzed. Some authors use
elevation as an auxiliary variable improving the spatial
interpolation of monthly rainfall data, and showing that
the interpolation of daily events is improved by the use
of elevation as a secondary variable even when these
variables show a low correlation [12].
For years various correlation and Semi-Variogram
techniques have been used to evaluate both the temporal
and spatial structure of rainfall events. Apart from corre-
lation techniques, use of a semi-variogram model for
two-dimensional (2-D) interpolation with Kriging ap-
proach has been a common practice in different engi-
neering applications, especially in the fields of mining
and hydrogeology [13]. By its definition, a semi-
variogram function has the capability of estimating the
disassociation between measurements from the different
gauge locations. In hydrologic engineering applications,
the semi-variogram development has been applied to
estimate the mean precipitation over a catchment by
Matheron (1971), Creutin and Obled (1982), Bastin,
Larent and Gevers (1984).
In this study, we analyze some rainfall series from 30
rain gauges located in the Great Lyon area (Figure 1),
including annual, monthly, daily and intensity of 6mins,
which is more useful to water drainage model. Lyon
possesses one of the densest rain gauge networks in an
urban area within Europe, having 52 gauges in an area of
460 km2. Most of the pluviometers belong to the urban
community of Lyon, with 30 tipping bucket rain gauges
working in this area. This creates a density of more than
1 pluviometer for every 15 km2. They are spread all over
the Lyon area, although with a lower density in the east-
ern part of the agglomeration. The first rain gauges were
set up in 1985, but the actual network density of the pre-
sent day was reached in 1989. The data is available every
6 minutes although this is variable according to the tip of
the bucket.
This paper focuses on three interpolation methods for
the different rainfall series. The study is aimed at im-
proving our understanding of the major sources of varia-
tion and uncertainty in small scale rainfall interpolation
in the time series with different time resolution. This is
achieved by means of a spatial characterization and data
analysis of the rainfall series. The difference in uncer-
tainty between three interpolation methods that differ
strongly in complexity, IDW, Spline and Kriging, is as-
sessed by means of cross validation and validation. This
allows us to evaluate the amount of complexity allowed
in interpolation, in view of the available data.
2. Methods
Ground-based rain gauge networks supply a reliable
source of precipitation data used in many analyses asso-
ciated with the development of rainfall models. Three
methods for the interpolation are used: IDW, Spline and
ordinary Kriging. The methods are chosen because they
represent two kinds of interpolation methods -- determi-
nistic and geostatistical methods. Each interpolated loca-
tion is given the value of the closest measurement point,
resulting in a typical polygonal pattern and discontinui-
ties at the borders of the polygons. IDW and Spline are
deterministic, while Kriging is a geostatistical method.
The Inverse Distance Weighted (IDW) and Spline
methods are referred to as deterministic interpolation
methods because they assign values to locations based on
the surrounding measured values and on specified
mathematical formulas that determine the smoothness of
the resulting surface. Inverse distance weighted (IDW)
interpolation determines cell values using a linearly
weighted combination of a set of sample points. The
weight is a function of inverse distance. The surface be-
ing interpolated should be that of a locationally depend-
ent variable. The Spline method is an interpolation
method that estimates values using a mathematical func-
tion that minimizes overall surface curvature, resulting in
a smooth surface that passes exactly through the input
points. It fits a mathematical function to a specified
number of nearest input points while passing through the
sample points. This method is best for generating gently
varying surfaces such as elevation, water table heights,
or pollution concentrations.
A second family of interpolation methods consists of
geostatistical methods, such as Kriging, that are based on
statistical models that include autocorrelation (the statis-
tical relationship among the measured points). Because
of this, not only do geostatistical techniques have the

T. TAO ET AL. 52
Figure 1. The location of rain gauges in Great Lyon.

T. TAO ET AL. 53
Figure 2. Average monthly precipitation of 30 rain gauges.
capability of producing a prediction surface, but they
also provide some measure of the certainty or accuracy
of the predictions. Geostatistical Analyst uses sample
points taken at different locations in a landscape and cre-
ates (interpolates) a continuous surface. The sample
points are measurements of some phenomenon such as
radiation leaking from a nuclear power plant, an oil spill,
or elevation heights. Geostatistical Analyst derives a
surface using the values from the measured locations to
predict values for each location in the landscape. Kriging
is an advanced, computationally intensive, geostatistical
estimation method that generates an estimated surface
from a scattered set of points with z-values [6,14,15].
Kriging involves an interactive investigation of the spa-
tial behavior of the phenomenon represented by the
z-values before the best estimation method is selected for
generating the output surface.
Kriging, like most interpolation techniques, is built on
the basis that things that are close to one another are
more alike than those farther away (quantified here as
spatial autocorrelation). The semivariogram is a means to
explore this relationship. Pairs that are close in distance
should have a smaller difference than those farther away
from one another. The extent that this assumption is true
can be examined in the semivariogram. Semivariogram
measures the strength of statistical correlation as a func-
tion of distance.
The semivariogram function is defined as: Y(si, sj)
= ½ var(Z(si) - Z(sj)), where var is the variance. si and
sj are two locations, Z(si) and Z(sj) are their values.
If two locations are close to each other in terms of the
distance measure of d(si, sj), then they are expected to be
more similar, so the difference in their values Z(si) -
Z(sj), will be small. As si and sj get farther apart, they
become less similar, so the difference in their values will
become larger. Notice that the variance of the difference
increases with distance, so the semivariogram can be
thought of as a dissimilarity function.
3. Results and Discussion
The results include three parts: one is to analyze the data,
deeper understanding of data will be useful to make de-
cision of model and results analysis; two is to make deci-
sion of parameters, such as the choice of semivariogram
model, lag size, and search neighborhood, and then ac-
cording to the cross validation, and comparison of dif-
ferent models, choose the appropriate model for ordinary
Kriging interpolation; three is to use the validation to
compare the three interpolation methods in annual,
monthly, daily and intensity data, and analyze the uncer-
tainty of different methods to different rainfall series.
3.1. Data Analysis
Rainfall data collected from January 1, 1986 to Decem-
ber 31, 2005 on 30 available stations in the whole study
area were analized with interpolation technologies. Data
with different time span may also have different spatial
and temperal distribution. Furthermore, different analysis
method should be selected for different data series. In
this study, four types of data series were studied, which
were annual, monthly, daily, and maxium 6-min precipi-
tation data. Average precipitation in November were

T. TAO ET AL. 54
selected as monthly rainfall data, because precipitation in
this month is always the highest one over 20 years. Pre-
cipitation on December 2, 2003 is selected as the one-
day rainfall data, because the storm event was happened
at this period, and all stations have non-zero rainfall data.
As the distribution of data will affect the selection of
methods, we firstly analyzed the Statistical Charateristics
of rainfall data. As shown in table 1, the distribution dif-
ference of the annual rainfall is not very big, and the Cs
(Skewness Coefficient) is only 0.0557. While for the
rainfall intensity, the range of variation is bigger and Cs
is 0.194, and the biggest Cs is of the rainfall intensity.
3.2. Original Kriging Analysis
This section discusses the impact of different approaches
for the estimation of the semivariogram on the original
Kriging interpolation performance for different rainfall
data series. Because Kriging is a more complicated in-
terpolation process, one special problem for geostatistical
interpolation of whole time series is the effective and
reliable estimation of the variograms for each time step
[5], so special attention is given to the analysis of the
impact of the semivariogram estimation on the interpola-
tion performance. First we should have a deeper under-
standing of the phenomena investigated so that we can
make better decisions on issues relating to our data, and
then by cross validation and comparing the different
models, to choose an appropriate model and parameters.
1) Fit a model—to create a surface and to choose the
definition and refinement of an appropriate model.
In the process of Kriging analysis, the semivariogram
plays a central role in the analysis of geostatistical data.
A valid semivariogram model is selected and the model
parameters are estimated before Kriging analysis is per-
formed.
According the history data spatial analysis, notice that
the values of the annual rainfall change more slowly in
the north–south direction than in east–west the direction.
This is because the terrain in east–west direction changes
greatly from high to low while the north–south direction
Table 1. of the data series from 30 stations.
Results Annual(mm) Month(mm) Day(mm) Intensity of 6min(mm/h)
Mean 798.966 105.565 63.479 94.062
Minimum 719.64 77.44 46.6 66
Maximum 877.98 130.87 86.2 140.5
Sd(yEr±) 44.5 9.226 10.111 18.28
Cs 0.0557 0.0874 0.159 0.194
Sd(yEr±): Standard Deviation
Figure 3. Semivariogram result of annual data.

T. TAO ET AL. 55
Table 2. Interpolation model parameters and results of different rainfall series.
Model and results Annual Month Day Intensity
Model spherical exponential spherical Gaussian
Anisotrop yes yes no yes
Root-Mean-Square(mm) 33.36 8.311 6.48 15.29
Average Standard Error(mm) 38.24 8.945 6.776 16.48
Mean Standardized -0.01779 -0.05089 -0.007049 -0.0018
Root-Mean-Square Standardized 0.898 0.9329 0.985 0.936
is more flat. So the anisotropy method is chose. Anisot-
ropy is a characteristic of a random process that shows
higher autocorrelation in one direction than another.
Annual semivariograms are inferred from annual data
considering anisotropic behavior using automatic and
manual fitting procedures. Figure 3 shows the semi-
variogram result of the annual precipitation data, and the
parameters of different rainfall series are shown in Table
2.
2) Perform diagnostics—the output surface using
cross-validation method, which will help understand how
well the model predicts the values at unmeasured loca-
tions.
Cross-validation uses the following idea—removes
one data locations and then predicts their associated data
using the data at the rest of the locations. In this way, we
compared the predicted value to the observed value and
obtained useful information about some of our previous
decisions on the Kriging model. The most rigorous way
to assess the quality of an output surface is to compare
the predicted values with those measured in the field.
Cross-validation is used to determine "how good" the
model is. The goal should be to have standardized mean
prediction errors near 0, small root-mean-squared predic-
tion errors, average standard error near root-mean-
squared prediction errors, and standardized root-mean-
squared prediction errors near 1.
Table 2 lists the different rainfall series estimation
models, parameters and results that are compared here.
Figure 4 shows the standardized error of Kriging in-
terpolation on annual rainfall data. We may conclude
from the analysis result that, for the annual rainfalls se-
ries from 30 stations, the majority of results of Kriging
interpolation is very good with ±5% limits of relative
errors. Although the maximum and minimum interpola-
tion errors are a little high, the results are reasonably
accepted, because Kriging interpolation is based on the
measured value so that the predicted value can not sur-
pass this scope.
Figure 4. Standardized error of Kriging in annual rainfall.
Figure 5. Error of Kriging interpolation in monthly rain-
fall(mm).
Figure 6. Error of Kriging interpolation in intensity rain-
fall(mm).
Figure 7. Error of Kriging interpolation in daily rain-
fall(mm).

T. TAO ET AL. 56
Figure 8. Measured value and predicted value of IDW for
annual rainfall.
Figure 9. Measured value and predicted value of Kriging
for annual rainfall.
The same spatial analysis was carried out in monthly
rainfall, daily rainfall and rainfall intensity data, as
shown in Figure 5, Figure 6, and Figure 7. The result
shows that interpolation on the monthly data series is
obviously better than the results on the intensity series
and the daily series. The relative errors of monthly series
are about within ±8% limits, and we also noticed that the
Cs of monthly data serie is obviously smaller than that of
the intensity data serie and the daily data seire. Therefore,
for the Kriging interpolation method, where there is
smaller data variation, there is better interpolation result.
3.3. Results of Three Interpolation Methods
The annual rainfall interpolation results of Kriging and
IDW method are shown in Figure 8 and Figure 9. From
the regression line of interpolation result, we could see
that both methods show good outputs, while the Kriging
result (the regression line slope is 0.38) is slightly better
than IDW (the regression line slope is 0.317). More de-
tailed analysis results are shown in Table 3-1. For
Kriging method, the relative error and std_dev of results
are 3.9 and 2.273, while for IDW the values are 2.192
and 3.959. It is believed that the difference of results
from these two methods is acceptable. Although neither
of these two methods has remarkable advantage in han-
dling with the average annual rainfall data, the spline
method will surely get worse results than the above two
methods.
Table 3-1. Validation results of three methods in annual
rainfall.
Relative Error of Annual rainfall
Method
Low High Mean Standard
deviation
0.5128.186 3.900 2.273
1.0437.498 3.959 2.192
Kriging-ordinary
IDW Spline
0.322 10.342 5.848 3.168
Table 3-2. Validation results of three methods in monthly
rainfall.
Relative Error of Monthly rainfall
Method
Low High Mean Standard
devaition
0.001321.24 7.052 5.638
0.6327 20.017 6.698 4.528
Kriging-ordinary
IDW Spline
0.0196 21.132 6.068 5.454
Table 3-3. Validation results of three methods in the daily
rainfall.
Relative Error of Daily rainfall
Method
Low High Mean Standard
deviation
0.33720.01 7.282 5.139
0.127 22.159 6.817 6.410
Kriging-ordinary
IDW Spline
0.001 20.951 8.969 7.981
Table 3-4. Validation results of three methods in the inten-
sity (6mins).
Relative Error of Intensity (6mins)
Method
Low High Mean Standard
deviation
2.640 25.691 11.829 7.046
1.759 34.533 12.217 8.961
Kriging-ordinar
y
IDW Spline 5.902 52.392 16.699 14.213
For the average monthly rainfall data, result from
spline method is comparatively better(shown in Table
3-2), the relative error and std_dev of which are respec-

T. TAO ET AL. 57
tively 6.068 and 5.454, while the Kriging result (7.052,
5.638) is inferior to the IDW result. For daily rainfall
data, IDW method is the best one (shown in Table 3-3),
while spline is obviously worst. For intensity of
6–minute data series, none of the three methods produce
high-quality result. Kriging is comparatively good, while
the average relative error approaches 11.829 (shown in
Table 3-4).
3.4. Uncertainty in Interpolation
These results suggest that most of the uncertainty in-
volved in the interpolation is related to the different basic
data statistics. In this view, Kriging is not a better
method, as it relies strongly on the assumption of sta-
tionarity in the means and thus, a lack of external trends.
In fact, for different rainfall series, it is difficult to de-
termine which method is better. It is not changed with
the variation of measured value, its essence should be
closely related to the space and temporal distribution of
different series. For this study region, because the rain-
fall station distributed in high density with a small area,
the Kriging method cannot show an obvious advantage.
At the same time, results from kriging and IDW interpo-
lation are not much different. But for the spline method,
the interpolation results in daily rainfall and intensity are
not very good. According to statistics analysis of data,
we can consider that the spline method is not favorable
for data with big Cs .
Although some researchers proposed that in order to
reduce the uncertainty of spatial rainfall information, the
basic way is to introduce other relative variations with
high sample density [6], and to integrate them in inter-
polation methods. But this suggestion is only suitable for
the big area with low distribution density of rain gauges.
The choice of those relative variations and their integra-
tion with interpolation methods will be one of the main
directions of the future research in rainfall interpolation.
4. Conclusions
In this study, different interpolation approaches are
compared for the different precipitation series. Special
attention is given to the impact of the variogram estima-
tion approach on the interpolation performance. The
main results can be summarized as follows:
a) As a kind of geostatistical interpolation methods,
the application of the Kriging method obtained the wide-
spread promotion, but in the application process it still
had the choice of multi parameters and the model. For
the different rainfall series in the same area, its model
and the parameter choice of Kriging method may be dif-
ferent.
b) The impact of the semivariogram on interpolation
performance is also discussed in this paper. And anisot-
ropy and isotropy are all present in the data, also leading
to the little difference in prediction performance. The
best results can be obtained using an automatic fitting
procedure except isotropic and anisotropic varigrams
from all precipitation series.
c) Given the small region with high density of rain
gauges (15km2), the Kriging method superiority is not
obvious, and to some data series, IDW and the spline
interpolation result can be better. Therefore we may ob-
tain, to the different rainfall series, they will be suitable
for the different methods, and it must be determined by
the data distribution.
In this paper, we have analyzed the connection be-
tween data variance and the different interpolation
methods, but this kind of data variance that we analyzed
is only in the magnitude, not involved spatial distribution.
From the characteristic of spatial interpolation, the spa-
tial distribution can be influential to the method selected,
so it still needs further to study in this point. Although
we concluded that the different data series possibly can
be more suitable for different method, the detailed rela-
tions between the data and method, at present still has
not been able to analyze. Therefore to the small region, if
there is no relative information to be applied, selecting
the appropriate method will be the key research point.
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