A Comparison of Spatial Interpolation Methods for Estimation of Average Electromagnetic Field Magnitude

Progress In Electromagnetics Research M, Vol. 14, 135–145, 2010
A COMPARISON OF SPATIAL INTERPOLATIONMETH-
ODS FOR ESTIMATION OF AVERAGE ELECTROMAG-
NETIC FIELD MAGNITUDE
M. Azpurua and K. Dos Ramos
Instituto de Ingenier´ıa
Venezuela
Abstract—Several georeferenced measurements of electric field were
done in a pilot area of Caracas, Venezuela, to verify that the
magnitude of radio frequency electromagnetic fields is below the human
exposure limits, recommended by the International Commission on
Non-Ionizing Radiation Protection. The collected data were analyzed
using geographical information systems, with the objective of using
interpolation techniques to estimate the average electromagnetic field
magnitude, to obtain a continuous dataset that could be represented
over a map of the entire pilot area. This paper reviews the three
methods of interpolation used: SPLINE, Inverse Distance Weighting
(IDW) and KRIGING. A statistical assessment of the resultant
continuous surfaces indicates that there is substantial difference
between the estimating ability of the three interpolation methods and
IDW performing better overall.
1. INTRODUCTION
There is no single preferred method for data interpolation. Aspects
of the algorithm selection criteria need to be based on the actual
data, the level of accuracy required, and the time and/or computer
resources available. In the absence of criteria for selecting among the
available techniques, this paper compares three spatial interpolation
techniques — SPLINE, Inverse Distance Weighting (IDW), and
KRIGING — with the goal of determining which method creates the
best representation of reality for measured electric field intensity. The
benefits and limitations of these commonly used interpolation methods
are discussed in this paper.
Received 31 August 2010, Accepted 17 September 2010, Scheduled 26 September 2010
Corresponding author: M. Azpurua (bazpurua@fii.gob.ve).

136 Azpurua and Dos Ramos
The data used in this assessment are average electric field intensity,
measured over a frequency range of 100 kHz to 6GHz using a isotropic
E-field probe, taking 35 samples per second during a 6 minutes period
in each of the 206 measurements points. It was considered a hybrid
data acquisition pattern random — systematic of uniform distribution.
Therefore, a matrix of data points spaced approximately 100m from
each other is fixed over the 2.64 km2 pilot area.
Selecting an appropriate spatial interpolation method is funda-
mental to surface analysis since different methods of interpolation can
result in different surfaces and ultimately different results. Statistical
techniques are used to evaluate the three interpolation methods against
independently collected data [1].
2. SPATIAL INTERPOLATION
Interpolation is a method or mathematical function that estimates the
values at locations where no measured values are available. Spatial
interpolation assumes the attribute data are continuous over space.
This allows the estimation of the attribute at any location within
the data boundary. Another assumption is the attribute is spatially
dependent, indicating the values closer together are more likely to
be similar than the values farther apart. These assumptions allow
for the spatial interpolation methods to be formulated. The goal of
spatial interpolation is to create a surface that is intended to best
represent empirical reality thus the method selected must be assessed
for accuracy [2].
3. SPATIAL INTERPOLATION METHODS
The techniques assessed here include the deterministic interpolation
methods of SPLINE and Inverse Distance Weighting (IDW) and the
stochastic method of KRIGING. Each method selected requires that
the exact data values for the sample points are included in the final
output surface. These spatial interpolation methods have various
decision parameters. The selected techniques, SPLINE, IDW and
KRIGING, are not all of the interpolation methods used in spatial
analysis; but are the most common techniques that are available in
GIS software.
3.1. SPLINES
SPLINES method estimates values using a mathematical function that
minimizes the total surface curvature, resulting in a smooth surface

Progress In Electromagnetics Research M, Vol. 14, 2010 137
that passes exactly through the sampled points. While there are
more entry points specified, the greater the influence of distant points
and the smoother the surface. Advantages of splining functions are
that they can generate sufficiently accurate surfaces from only a few
sampled points and they retain small features. A disadvantage is
that they may have different minimum and maximum values than
the data set and the functions are sensitive to outliers due to the
inclusion of the original data values at the sample points. This is
true for all exact interpolators, which are commonly used in GIS, but
can present more serious problems for SPLINE since it operates best
for gently varying surfaces, i.e., those having a low variance [3]. For
the mathematical formulation of the SPLINES interpolation method,
it can be consulted [3].
3.2. Inverse Distance Weighting (IDW)
Inverse Distance Weighting is based on the assumption that the
nearby values contribute more to the interpolated values than distant
observations. In other words, for this method the influence of a known
data point is inversely related to the distance from the unknown
location that is being estimated. The advantage of IDW is that it
is intuitive and efficient. This interpolation works best with evenly
distributed points. Similar to the SPLINE functions, IDW is sensitive
to outliers. Furthermore, unevenly distributed data clusters results in
introduced errors [4].
The simplest form of IDW interpolation is called, Shepard
method [4] and it uses weight function wi given by (1):
wi = h
−p
i
n∑
j=0
h−pj
(1)
where p is an arbitrary positive real number called the power parameter
(typically p = 2) and hj are the distances from the dispersion points
to the interpolation point, given by:
hi =
√
(x− xi)2 + (y − yi)2 (2)
where (x, y) are the coordinates of the interpolation point and (xi, yi)
are the coordinates of each dispersion point. The weight function varies
with a value of unity at the dispersion point to a value close to zero as
the distance to the dispersion point increase. The weight functions are
normalized as a sum of the weights of the unit. Then, the interpolated

138 Azpurua and Dos Ramos
value of the electric field E(x, y) is given by:
E(x, y) =
n∑
j=0
wjE(xj , yj) (3)
In order to improve the computational time is possible to set
bounds to the dispersion points that contribute to the calculation of
the interpolated value, to all those dispersion points within a given
search radius centered on the interpolated point. For the particular
application developed in this work, it was determined that the most
appropriate search radius was 500m, so that the computation times
were manageable.
3.3. KRIGING
Similar to IDW, KRIGING uses a weighting, which assigns more
influence to the nearest data points in the interpolation of values
for unknown locations. KRIGING, however, is not deterministic
but extends the proximity weighting approach of IDW to include
random components where exact point location is not known
by the function. KRIGING depends on spatial and statistical
relationships to calculate the surface. The two-step process of
KRIGING begins with semivariance estimations and then performs the
interpolation. Some advantages of this method are the incorporation
of variable interdependence and the available error surface output.
A disadvantage is that it requires substantially more computing and
modeling time, and KRIGING requires more input from the user [1].
KRIGING belongs to the family of linear least squares estimation
algorithms. The aim of KRIGING is to estimate the value of an
unknown real-valued function, E, at a point, (x, y), given the values of
the function at some other points, {(x1, y1), (x2, y2), . . . , (x3, y3)}. A
KRIGING estimator is said to be linear because the predicted value
E(x, y) is a linear combination that may be written as:
E(x, y) =
n∑
j=0
λjE(xj , yj) (4)
The weights λi are solutions of a system of linear equations which
is obtained by assuming that E is a sample-path of a random process
F (x, y), and that the error of prediction
ε(x, y) = F (x, y)−
n∑
j=0
λjE(xj , yj) (5)

Progress In Electromagnetics Research M, Vol. 14, 2010 139
is to be minimized in some sense. For instance, the so-called simple
kriging assumption is that the mean and the covariance of F (x, y) is
known and then, the KRIGING predictor is the one that minimizes
the variance of the prediction error [1].
4. METHODOLOGY
4.1. Study Site
The measurements were taken in a pilot zone of Caracas, Venezuela
(Figure 1) with an approximate area of 2.64 km2, which represents
0.609% of the total geographical area of the Caracas. A total of 206
measurements points, were selected over the pilot zone. This area is
characterized by a dynamic economic-business activity which is evident
given the presence of shopping centers, office buildings of the national
telephone operators and other telecommunications companies. On the
other hand, this area also boasts a large number of hospitals and
schools, which is of interest to know the impact of electromagnetic
fields.
Figure 1. Study site.

140 Azpurua and Dos Ramos
4.2. Data Collection Process
During the process of data collection, measurements were performed
according to the following considerations:
• Time considerations: Measurements were taken over a period of 30
days, from February 15th to March 15th of 2010. Measurements
were performed only on working days (from Mondays to Fridays).
Each measurement was taken between 8:00 am and 5:00 pm.
Previous investigations suggest this is the more suitable time
interval to take this kind of measurements [5].
• Geographical considerations: For each measurement point
geographical coordinates were taken using a GPS navigation unit.
• Measurements considerations: Each measurement was carried out
averaging the magnitude of the electric field intensity, measured
35 times per second, using an isotropic E-field probe during
six (6) minutes, as indicated by ICNIRP [6]. The instrument
measurement bandwidth was 100 kHz–6GHz. To minimize the
interference caused by the operators, all the cellphones and
personal radiation sources were disconnected. Finally, it is
assumed and procured that measurements are taken in the far
field zone.
4.3. Data Processing and Representation Tools
The spatial analysis and the geographical data representation carried
out in this work, was mainly supported by the usage of two informatics
tools: gvSIG 1.9 and Past 2.02. GvSIG is a Geographic Information
System (GIS), that is, a desktop application designed for capturing,
storing, handling, analyzing and deploying any kind of referenced
geographic information in order to solve complex problems, while Past
is a data analysis package that includes common statistical, plotting
and modeling functions. Both, gvSIG and Past are free software tools
distributed under the GNU/GPL license.
4.4. Assessment Methods
Different measures of fit may be used to determine how well an
interpolated map represents the observed data. With most methods,
some measure may be constructed of the closeness of the interpolated
values E(x, y) to the values Ei observed at control sites Øi. In
this work, it was calculated the mean absolute error “MAE”, the
mean squared error “MSE” and the Euclidean distance “D”, between
a set of control points (on which measurements of electric field

Progress In Electromagnetics Research M, Vol. 14, 2010 141
intensity were done, but not included in the interpolation process)
and the interpolated results. The number of control points taken was
approximately 10% of the total number of measurement points, which
are twenty (20), randomly distributed over the study site perimeter
taking into account that the control point shall not be located closer
to 20m from a previous measured point. The equations used in those
calculations were:
MAE = 1n
n∑
i=1
|E(x, y)− Ei| (6)
MSE = 1n
n∑
i=1
(E(x, y)− Ei)2 (7)
D =
√√√√
n∑
i=1
(E(x, y)− Ei)2 (8)
Additionally, a month later, 10% of the measurements were
repeated under similar conditions to the original ones, following the
same measurement protocol, to ensure the results are valid over time
without suffering significant changes. It was found that the maximum
difference in the electric field intensity measured for one point, was
about 0.5V/m. Therefore, it was fixed as a criterion, that only the
interpolated results with an error lower than 0.5V/m will be accepted
as valid.
5. RESULTS
The interpolation process was carried out using the methods: IDW,
SPLINES and KRIGING. The results of each interpolation process
were represented over the study site map. The mapping result of the
(a) (b) (c)
Figure 2. Interpolated maps of the E-field intensity average
magnitude. (a) IDW, (b) SPLINES, (c) KRIGING.

142 Azpurua and Dos Ramos
Table 1. Results of the measures of fit applied to the interpolation
methods.
MEASURES
OF FIT
MAE
(V/m)
MSE
(V/m)2 D(V/m)
MaxError
(V/m)
MinError
(V/m)
IDW 0.17 0.05 1.01 0.55 0.008
KRINGING 0.74 0.73 3.81 1.66 0.111
SPLINE 0.89 1.11 4.71 1.98 0.034
E-field average magnitude is shown in Figure 2, in which the red dots
are the dispersion points. The E-field intensity average magnitude is
represented using a color scale where the lower values are represented
using blue, and the higher values using red. The interpolated values
are valid for the points inside the perimeter of the study site, even
when the algorithms that calculate the raster layer, considers all pixels
within rectangle defined by the base map.
The measures of fit carried out, yielded the results shown in
Table 1. It shows that within the interpolation methods used, the
IDW method is the one that best estimated the measurement results
of the electric field average magnitude.
The results show that the average magnitude of the electromag-
netic fields is mainly determined by the distance between the sources
and the observation point. Furthermore, the results also show that
the obstacles represented by the typical buildings and similar have
no important influence on the magnitude of the average electric field
intensity.
Also it’s important to compute the number of control points whose
interpolated results are within the acceptance margin. For the IDW
method only two (2) of twenty (20) control points are lightly outside
the acceptance interval. Similarly, only 9 and 5 of the 20 control points
met the acceptance criteria for KRIGING and SPLINES interpolation
methods, respectively. This fact it’s shown in Figure 3 using a scatter
plot. Figure 3 also shows that KRIGING and SPLINES methods have
a very poor relationship with the actual E-field average magnitude, due
to the large dispersion observed around the perfect prediction line.
When the error distribution is represented in histograms
(Figure 4), it could be seen that the IDW interpolation method
has a superior performance than KRIGING and SPLINES methods.
Additionally, for each method, the errors are concentrated around its
mean value, and the error density diminishes as it gets apart of its
mean value.
Finally, we elaborated a raster layer of the average electric field

Progress In Electromagnetics Research M, Vol. 14, 2010 143
strength using the IDW method and this layer is superimposed on a
base map of the pilot area in order to obtain the representation shown
in Figure 5.

Perfect prediction
IDW Method
KRIGING Method
SPLINE Method
Upper acceptance limit
Lower acceptance limit
Comparison of Spatial Interpolation Methods IDW, KIRGING and SPLINE
for Estimation of Average Electric Fiels Magnitude
Es
tim
ate
d E
lec
tric
Fie
ld
Av
era
ge
Ma
gni
tud
e (V
/m)
Measured Electric Field Average Magnitude (V/m)
3
2.5
2
1.5
1
0.5
00 0.5 1 1.5 2 2.5 3
r i i
I t
I t
I
r t li it
li i
Figure 3. Scatter plot for the comparison of the results obtained
using the interpolation methods IDW, KRIGING and SPLINES for
the calculation of the average electric field intensity magnitude.
Figure 4. Error distribution in the calculation of the E-field intensity
average magnitude using the interpolation methods IDW, KRIGING
and SPLINES.

144 Azpurua and Dos Ramos
|E| (V/m)
0-0.25
0.26-0.5
0.51-0.75
0.76-1
1.01-1.25
1.26-1.5
1.51-1.75
1.76-2
2.01-2.25
2.26-2.5
2.51-2.75
2.76-3
3.01-3.25
3.26-3.5
9.75-10
Figure 5. Geographical representation of the average electric field
intensity magnitude.
6. CONCLUSION
This study has shown that IDW interpolation method is most likely
to produce the best estimation of a continuous surface of the average
magnitude of electric field intensity. The IDW method exactness was
superior to the one shown by the SPLINES and KRIGING techniques.
This could be justified not only because the magnitude of the radiated
electromagnetic fields in free space decreases with the inverse of
distance (free space loss) but also because in a urban environment,
the ground irregularity and clutter, increase the effect of absorption,
shadowing, scattering, divergence, and defocusing of the diffracted
waves, contributing to the attenuation of the electromagnetic fields.
That is why attenuation and, in consequence, the electromagnetic field
average magnitude, at a defined height, can be described on a statistical
basis considering all those effects combined together. Therefore, for the
far field region, the expected field strength varies as 1/Dn, where D
in the distance from source, and n is a dimensionless number, greater
than 1, that typically varies from 1, 3 for open country areas to about
2, 8 for heavily built-up urban areas [7].
In our test case, an n = 2, which correspond with the power
parameter of the IDW interpolation method, was determined as
the more suitable, and it is coherent with the expected n = 2.2
proposed in [8] for the perdition of propagation of interference from

Progress In Electromagnetics Research M, Vol. 14, 2010 145
industrial radio frequency equipment. Hence, the average magnitude
of the electromagnetic fields is mainly determined by the distance
between the sources and the observation point; and the statistically
determined power parameter that resumes the effects of absorption,
shadowing, scattering, divergence, and defocusing of the diffracted
waves, showing that for our particular case, the environmental
electromagnetic field distribution produce little impact over the human
exposure to electromagnetic fields. Furthermore, the results also show
that specific obstacles represented by the typical buildings and similar
have no important influence on the magnitude of the average electric
field intensity.
Nevertheless, it should be noted that IDW interpolation method
should be tuned adjusting the power parameter and the search radius
to improve accuracy, in each specific case. It also should be considered
the incorporation of an altimetric model of the radiation sources and
measurement points, as a way to improve the interpolation results.
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