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940 L. Liberti et al. / Renewable Energy 50 (2013) 938e949

Fig. 2. Ground tracks of satellites considered in model validation. Thick gray lines identify Jason-1 and Jason-2 tracks. Black lines partially overlying the gray ones symbolize Topex-
Poseidon tracks while thin dashed lines represent Envisat and ERS-2 tracks.

2.2. Validation against satellite data slope ¼ Pn 1 xi yi : (4)
Pin¼ 1 xixi
Satellite radar altimeters provide Hs measurements in areas far
i¼

from the coast where wave buoys are not normally located. Hs We also considered the Willmott index of agreement (d) [24]
which is defined as:
measures from satellite were used to evaluate the overall behavior

of the model over the entire Mediterranean Sea. Table 1 lists the "#
Xn Xn Ày0i
satellite delivering altimeter measurements that were in operation d ¼ 1À ðyi À xiÞ2= À x0iÁ2 : (5)

during the simulated period while Fig. 2 shows the satellite tracks i¼1 i¼1

over the Mediterranean over which measures are periodically

performed. Satellite data was downloaded from the AVISO web site where y0i ¼ yi À x, xi0 ¼ xi À x and x is the average of the observed
values. The d index is bounded in the interval [0,1]. A value of 1
[22] and processed to remove outliers according the method implies perfect match between model and observations. Values in
the table show good agreement between model and satellite Hs
described by Queffeulou and Bentamy [23]. This procedure is based with small biases not exceeding 0.15 m, best fit lines with slopes
above 0.9 and d values above 0.92. A similar analysis is presented in
on the statistical analysis of the differences between consecutive a recent paper [25] where the performances of various 0.1 reso-
lution wave models in the Mediterranean Sea were evaluated, over
data points along the tracks. Following this preliminary step, Hs a two months period, by forcing them with wind fields with various
resolutions ranging from 10 to 40 km. The results of the study, in
values were matched to model values by extracting the model terms of bias, rmse, slope and si are similar to the ones we found.
For instance, the best fit line slopes of the comparison between
output from the grid cell enclosing the satellite point at the nearest model and satellites Jason and ERS-2 data is reported to lie in range
0.7e0.99; the root mean square error obtained by comparing
available time. No temporal or spatial interpolation of model or satellite data with the WAM model forced with 10 km ALADIN
winds is reported to be around 0.55 with bias of about 0.3. Fig. 3
satellite data was performed. Table 2 reports for each satellite the

values of indices generally used to evaluate model performance. We

included in our analysis the root mean square error (rmse), the bias

between model and measures (bias), the scatter index (si) and the

slope of the best fit line passing through the origin (slope). Given

a series of n model values yi and corresponding measures xi the

indices are calculated as follows:

bias ¼ 1 Xn À xiÞ; (1)
n ðyi (2)

i¼1

rmse ¼ uutvffinffiffiffiÀffi1ffiffiffiffi1ffiffiffiffiiffiXffi¼ffinffiffi1ffiffiðffiffiyffiffiffiiffiffiÀffiffiffiffiffixffiffiiffiffiÞffi2ffiffi; 12
10 500

si ¼ rmse (3) 8 250
Xn ; 100
1 yi 6Hs - Model 50
n (m) 10
i¼1 4
Entries

Table 2 25
Statistics of satellite and model significant wave height (Hs) comparison for the
entire Mediterranean. 0
0
Satellite Samples Bias (m) Rmse (m) Slope si d 0 2 4 6 8 10 12

Topex-Poseidon 457,000 À0.128 0.331 0.912 0.279 0.957 Hs - Satellite
Jason-1 910,133 À0.028 0.362 0.979 0.304 0.951 (m)
Jason-2 242,766 0.366 1.018 0.303 0.950
Envisat 695,768 0.024 0.385 0.921 0.310 0.943 Fig. 3. Scatter plot of model vs. Jason-1 Hs for the entire Mediterranean. Value pairs are
ERS-2 363,336 À0.141 0.426 0.962 0.400 0.929 grouped in 0.25 m wide bins, corresponding areas are painted according to the number
À0.011 of entries in each bin. Dashed line is the best fit line between model and satellite data
points.