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Wind speed and direction are fundamental data for many application fields, such as power generation and hydrological modelling. Wind measurements are usually few and sparse; hence, spatial interpolation of wind data is required. However, in mountainous areas with complex orography, accurate interpolation of wind data should consider topographic effects. Due to computational constraints, fully physically based methods that solve thermodynamic and mass conservation equations in three dimensions cannot be applied for long-time simulations or very large areas, while fast empirical methods seem more suitable. The aim of this work is to compare fast empirical methods to interpolate wind speed against a physically based full atmospheric model in order to assess the impact of the introduced approximation in estimating the wind field and the potential evapotranspiration. Comparison is carried out over the area of the upper Po River basin, a predominantly alpine region located in northern Italy. Results show that empirical topographic correction can increase accuracy of interpolated wind speed in areas with complex topography, but it requires about 50% more computational time than simpler empirical methods that do not consider topography.
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RESEARCH ARTICLE
Wind speed interpolation for evapotranspiration
assessment in complex topography area
Giovanni Ravazzani
1
&Alessandro Ceppi
1
&Silvio Davolio
2
Received: 5 July 2019 /Accepted: 11 October 2019/
#Springer Nature Switzerland AG 2020
Abstract
Wind speed and direction are fundamental data for many application fields, such
as power generation and hydrological modelling. Wind measurements are usually
few and sparse; hence, spatial interpolation of wind data is required. However, in
mountainous areas with complex orography, accurate interpolation of wind data
should consider topographic effects. Due to computational constraints, fully phys-
ically based methods that solve thermodynamic and mass conservation equations
in three dimensions cannot be applied for long-time simulations or very large
areas, while fast empirical methods seem more suitable. The aim of this work is
to compare fast empirical methods to interpolate wind speed against a physically
based full atmospheric model in order to assess the impact of the introduced
approximation in estimating the wind field and the potential evapotranspiration.
Comparison is carried out over the area of the upper Po River basin, a predom-
inantly alpine region located in northern Italy. Results show that empirical
topographic correction can increase accuracy of interpolated wind speed in areas
with complex topography, but it requires about 50% more computational time
than simpler empirical methods that do not consider topography.
Keywords Wind speed interpolation .Complex topography .Evapotranspiration assessment
Introduction
Wind speed data are fundamental for evapotranspiration assessment in hydrological
studies (Ravazzani et al. 2012). For application to large river basins, characterised by
Bulletin of Atmospheric Science and Technology
https://doi.org/10.1007/s42865-019-00001-5
*Alessandro Ceppi
alessandro.ceppi@polimi.it
1
Department of Civil and Environmental Engineering, Politecnico di Milano, Piazza Leonardo da
Vinci, 32 Milan, Italy
2
National Research Council of Italy, Institute of Atmospheric Sciences and Climate, Via Piero Gobetti,
101 Bologna, Italy
a limited number of available wind observation sites, data interpolation becomes
necessary to estimate wind spatial variability (Cheng and Georgakakos 2011). How-
ever, in complex topography areas, interpolation is difficult because of spatial varia-
tion in wind velocity caused by slopes, canyons or valleys, and of the sheltering and
diverting effects of terrain (Ryan 1977;Rotachetal.2015). Several works deal with
interaction of meteorological variables with complex topography (Lussana et al. 2018;
Corbari et al. 2009). Proposed methods for the wind field can be summarised as
follows (Liston and Sturm 1998): (1) applying a physically based numerical weather
prediction model which satisfies all relevant momentum and continuity equations
(Ercolani et al. 2015); (2) applying an atmospheric model in which only mass
continuity is satisfied (Wagenbrenner et al. 2016); (3) interpolating wind speed and
direction observations in conjunction with empirical windtopography relationships
(Ryan 1977).
Numerical weather prediction models have been run successfully for specific test cases
that imply specialised model configurations and require technical expertise and access to
computing resources (Seaman et al. 2012; Ching et al. 2014;Helbigetal.2017).
Mass conserving models have been applied to small-scale high spatial resolution domain
and, even though less demanding than numerical weather models, they still require substantial
resources when applied to larger scales (Forthofer et al. 2014).
For long-run simulations of hydrological processes in large river basins, simple empirical
models are preferable in order to limit simulation time (Ravazzani et al. 2014). While several
studies have compared the general accuracy of wind speed interpolation methods, few research
efforts have been specifically addressed towards comparing the effectiveness of empirical
spatial interpolators of wind speed observations for assessing evapotranspiration.
Palomino and Martin (1995) compared two interpolation techniques: one based on
the inverse distance squared weighted, the other on the inverse absolute elevation
difference, at six meteorological towers in the south of Spain; they concluded that the
topographical elevation difference is an important variable that can improve the spatial
interpolation of wind fields. Luo et al. (2008) compared four deterministic and three
geostatistical methods of spatial interpolation to determine their suitability for esti-
mating daily mean wind speed surfaces, from data recorded at nearly 190 locations
across England and Wales. González-Longatt et al. (2015) used a simple geostatistical
Kriging method to interpolate horizontal wind speed and an orographic correction to
account for changes on terrain elevation, in order to assess wind electrical power
resource in Venezuela.
In this study, three spatial empirical interpolation techniquesThiessen polygon
(Thiessen 1911), inverse distance weighting (IDW) (Shepard 1968), and a modified
IDW accounting for topographic properties (IDW-TOP) adapted from MicroMet
(Liston and Elder 2006)are evaluated, and their accuracy for evapotranspiration
estimate in complex topography areas is assessed.
After providing details about the interpolation methods in The interpolation
techniques,Case study: the upper Po River basindescribes the application site
and available data. The wind field reference scenario and performance analysis are
presented in The wind fields reference scenario derived from meteorological model
and Performance analysis, respectively. Results and discussiondiscusses the
results, and concluding remarks are presented in Summary and conclusions.
Bulletin of Atmospheric Science and Technology
Materials and methods
The interpolation techniques
Thiessen polygons
The Thiessen polygon method is one of the simplest techniques, still widely used in hydro-
logical studies, also referred to as the nearest neighbour method or Voronoi tessellation. It
predicts the attributes of unsampled points based on those of the nearest sampled point.
Polygons are drawn according to the distribution of the sampled data points, with one polygon
per data point, which is then located in the centre of the polygon (Hartkamp et al. 1999). The
drawback of this method is that it produces an abrupt transition between boundaries.
Inverse distance weighting
The IDW is a deterministic estimation method whereby values at unsampled points are
determined by a combination of values at known sampled points. Weighting of nearby points
is strictly a function of distance. The assumption is that values closer to the unsampled location
are more representative of the value to be estimated. This method produces a gradual change of
interpolated surface. The weight (λ)oftheith unsampled point is computed as:
λi¼1=dp
i
n
i¼1
1=dp
i
ð1Þ
where diis the distance from the known point to the unsampled point, nis the total number of
known points used in interpolation and pis a positive real number, called the power parameter.
Inverse distance weighting with topographic correction
The IDW-TOP method is adapted from the MicroMet model by Liston and Elder (2006).
Observed values of wind speed W(m s1) and direction θare first converted into zonal u
(m s1) and meridional v(m s1) wind components using:
u¼Wsinθð2Þ
v¼Wcosθð3Þ
Then, uand vcomponents are independently interpolated to a regular grid using the IDW. The
resulting values are converted back to wind speed using:
W¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u2þv2
pð4Þ
These gridded values are then modified to account for topographic variations, multiplying by
an empirically based weighting factor (Ww):
Ww¼1þγsΩsþγcΩcð5Þ
Bulletin of Atmospheric Science and Technology
where Ωsand Ωcare the topographic slope and curvature, respectively, in the direction of the
wind, and γsand γcare positive constants which weight the relative influence of Ωsand Ωcon
modifying the wind speed.
The terrain-modified wind speed (Wt(m s1)) is calculated from:
Wt¼WwWð6Þ
Case study: the upper Po River basin
The area of interest is the upper Po River basin, in northern Italy, closed at the confluence with
Ticino River. It covers 37,200 km2(Fig. 1), of which 13,700 km2(36.8%) has an elevation
higher than 1000 m, 10,800 km2(29%) has an elevation in the range 3001000 m, and
12,700 km2(34.2%) lies below 300 m. This is predominantly an alpine region that is bounded
on three sides by mountain chains. The environmental monitoring of this area is managed by
the Regional Environmental Agencies of Piedmont, and includes 95 sites where wind speed
and direction are hourly measured, as reported in Fig. 1.
The wind field reference scenario derived from the meteorological model
The wind speed field used as the reference scenario is obtained from the output of the
MOLOCH meteorological model (Malguzzi et al. 2006; Trini Castelli et al. 2019). MOLOCH
is a non-hydrostatic, fully compressible, convection-resolving model that integrates the set of
atmospheric equations employing a hybrid terrain-following vertical coordinate and a latitude
longitude rotated grid. Details on model physics and on numerical schemes can be found in
Buzzi et al. (2014) and Davolio et al. (2017). MOLOCH is run daily at CNR ISAC to provide
operational forecasts for the following 45 h. In the present study, only the first 24 h of each run
was considered and concatenated in order to create a continuous time series of hourly winds
for 2183 h, from 1 September to 30 November 2015, at 1500-m spatial resolution.
Fig. 1 Geographical location (on the left) and topography of the study area as from the digital elevation model
used by the MOLOCH meteorological model, and wind measurement sites (on the right)
Bulletin of Atmospheric Science and Technology
Performance analysis
The performance of empirical methods to interpolate wind speed observations was assessed in terms
of ability to reproduce the MOLOCH output, considered as the benchmark solution, starting from a
set of sparse points. To this purpose, for each of the first 24 h of daily MOLOCH run, the uand v
wind components of all model grid cells that include one monitoring station were extracted and
converted to wind speed and direction, as depicted in Fig. 2. This procedure allowed reconstructing a
continuous time series of virtual wind speed and direction data for the 95 stations over the upper Po
River basin. These data act as pseudo-observations in this analysis, since the use of real wind speed
and direction observations to test interpolation techniques was not feasible. In fact, the huge number
of missing data in the observations prevented having available a continuous time series of
homogeneous set of stations for a sufficient long period. In addition, the use of MOLOCH output
avoids the inclusion of possible inaccuracies of real observations in the interpolation results. When
IDW-TOP method was applied, the same elevation model implemented in the MOLOCH model
was used. Therefore, an interpolation method perfectly performing should reproduce exactly the
MOLOCH output.
In the present study, the performance of interpolation methods is evaluated through the average
normalised root mean square error (NRMSE) and the relative bias (RB) respectively defined as:
NRMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
n
n
i¼1
xibxi

r
1
n
n
i¼1b
xi
ð7Þ
RB ¼1
n
n
i¼1
xib
xi
xi
ð8Þ
Fig. 2 Diagram of the procedure to reconstruct pseudo time series of continuous wind speed and direction data
extracted from the MOLOCH maps
Bulletin of Atmospheric Science and Technology
Fig. 4 Computational time required by Thiessen, IDW and IDW-TOP to interpolate wind speed fields used in
this analysis. Percentages show the increase of computational time with respect to the Thiessen method
Fig. 3 Wind speed simulated by the MOLOCH model, and interpolated with Thiessen, IDW and IDW-TOP
Bulletin of Atmospheric Science and Technology
where nis the total number of interpolated cells in the domain, and xiand b
xiare the interpolated
and modelled values of wind speed, respectively, in the ith cell. The NRMSE has a range
between 0 and with a best score equal to 0, while the RB has a range between −∞and
with a best score equal to 0.
Results and discussion
The wind speed and direction data are interpolated using the three techniques presented in
section The interpolation techniques(Fig. 3). In terms of required computational time, IDW
and IDW-TOP results of 7.7% and 49.8% are more expensive than the Thiessen method,
respectively (Fig. 4).
For each analysed hour, NRMSE and RB are computed by comparing the interpolated map with
the original map of the MOLOCH model, and dividing the study area into three belts at different
elevations: (i) below 300 m, (ii) between 300 and 1000 m and (iii) above 1000 m, which
approximately correspond to flat, hilly and mountainous regions, respectively. The average of these
indices is shown in Table 1. Results show that in the area below 300 m, where topography is not
significant, IDW is the method that provides the best results with both indices closer to 0. In the area
above 1000 m, where topography plays a relevant role, IDW and IDW-TOP have a similar NRMSE
value, but IDW-TOP presents a lower RB in absolute value. In both cases, the Thiessen method is
affected by a significantly larger error than the others. Finally, in the hilly area between 300 and
1000 m, the best score has been obtained with the IDW method, characterised by lower values of
both NRMSE and RB skill scores, in comparison with those of IDW-TOP and Thiessen methods.
The same mark has been obtained considering the whole basin. This result confirms that IDW is a
high-quality method in terms of computational cost and performances.
The meteorological forcing fields are then used to calculate the potential reference evapo-
transpiration ET0, using the FAO-56 Penman-Monteith equation (Allen et al. 1998). The
calculation is performed by both using the MOLOCH model meteorological fields, and replacing
Table 1 Performance of interpolation methods (Thiessen, IDW and IDW-TOP) applied to wind speed on the
whole basin, and in the three areas characterised by different elevations: below 300 m, between 300 and 1000 m,
above 1000 m
Whole basin < 300 m 3001000 m > 1000 m
NRMSE RB NRMSE RB NRMSE RB NRMSE RB
Thiessen 0.873 0.122 0.447 0.027 0.748 0.076 1.151 0.290
IDW 0.669 0.060 0.390 0.026 0.595 0.069 0.853 0.161
IDW-TOP 0.674 0.093 0.414 0.122 0.603 0.104 0.842 0.052
Table 2 Relative bias in assessing cumulated reference evapotranspiration using the three interpolation methods
(Thiessen, IDW and IDW-TOP) on the whole basin and in the three areas with different elevations
Whole basin < 300 m 3001000 m >1000 m
RB RB RB RB
Thiessen 0.015 0.008 0.01 0.079
IDW 0.016 0.004 0.002 0.061
IDW-TOP 0.057 0.054 0.074 0.02
Bulletin of Atmospheric Science and Technology
the wind speed fields with those interpolated with the three methods. Table 2shows RB of
cumulative evapotranspiration estimate, while Fig. 5shows the evolution of the cumulated ET0
for the area below 300 m and for the area above 1000 m. Note that the IDW-TOP reduces the error in
the complex topography area, but in the other areas it generates a larger underestimation than the
other two methods. In fact, in flat area (< 300 m) and over hilly terrains (3001000 m), the best
scores are achieved with the IDW method, and even over the entire basin IDW performs better than
IDW-TOP in terms of absolute values: 0.016 vs 0.057. This is another hint: the IDW-TOP
methodology does not justify the huge computational cost.
Summary and conclusions
In this study, three empirical methods to interpolate wind speed are evaluated over the upper
Po River basin in northern Italy, in order to assess the impact on estimating potential
evapotranspiration. Thiessen polygons, IDW and IDW with topographic correction (IDW-
TOP) were used to interpolate sparse wind speed and direction data (pseudo-observations)
extracted at real station locations from meteorological fields obtained by numerical meteoro-
logical model (MOLOCH) simulations. The wind field over the entire area obtained through
the three interpolation methods was compared against the model wind field to evaluate
interpolation performance.
Results show that IDW-TOP is able to increase the accuracy of wind speed interpolation
only in the area where topography plays a relevant role (higher elevations). In other areas (as
plain or hill), simple IDW is better than the other two methods. When interpolated fields are
used to compute potential evapotranspiration, IDW-TOP shows a good match with evapo-
transpiration computed with the original MOLOCH wind data, while it introduces underesti-
mations in other areas. In terms of computational efficiency, IDW and IDW-TOP require 7.7%
and 49.8% more than the Thiessen method, respectively, for interpolating wind speed.
However, the Thiessen method results are far less accurate than those of the other two.
Fig. 5 Cumulated reference evapotranspiration (ET0) computed with meteorological fields simulated by the
MOLOCH model (black line) and the wind speed interpolated with Thiessen (green line), IDW (red line) and
IDW with topographic correction (IDW-TOP in blue) in the area with elevation lower than 300 m (left), and in
the area with elevation higher than 1000 m (right)
Bulletin of Atmospheric Science and Technology
Acknowledgements We thank Prof. Glen E. Liston for kindly providing the original Fortran code of the
MicroMet model from which the program used in this analysis was derived.
Funding information This research was funded by Italian Ministry of University and Research within the
project Reconciling precipitation with runoff: the role of understated measurement biases in the modelling of
hydrological processes(http://www.precipitation-biases.it/).
Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of interest.
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Bulletin of Atmospheric Science and Technology
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