Chapter 5
Analysis of the Results
Even if the results have been analyzed at three different times (7600s, 9100s, 10600s), we have chosen to present here what happened at the last time, since not significant differences have been discovered passing from one solution to another.
Plotting the positions of the parcels for the three different times, it is
possible to observe that the situation does not change (see Fig.(5.1), Fig.(5.2),
Fig.(5.3)).
Analysis of the Results 75
0 0.5 1 1.5 2 2.5 3 3.5 4
x 104 0
500 1000 1500 2000 2500 3000 3500 4000 4500
x
z
t = 7600s
Figure 5.1: Plot of the parcels positions at time t=7600s. The input param- eters of this simulation are the mean ones: µ = 1φ, σ = 1.5φ, ψ = 0.7
0 0.5 1 1.5 2 2.5 3 3.5 4
x 104 0
500 1000 1500 2000 2500 3000 3500 4000 4500
x
z
t = 9100s
Figure 5.2: Parcels positions at time 9100 s.
0 0.5 1 1.5 2 2.5 3 3.5 4
x 104 0
500 1000 1500 2000 2500 3000 3500 4000 4500
x
z
t = 10600s
Figure 5.3: Parcels positions at time 10600 s.
Analysis of the Results 76
A more accurate confirmation can be found observing the size distribution of parcels in each stripe (both air and ground) at the three time instants (see Fig.(5.4)). In the considered domain, the parcels not on the ground have reached a steady regime both in the number than in the range of diameters.
The same result is valid on the ground for the size distribution only, while obviously the number of parcels deposited on the ground increases with time.
−10 0 0 10
500 1000 1500
φ
Number of parcels
Stripe 1−Air
−10 0 0 10
500 1000 1500
φ Stripe 2−Air
−10 0 0 10
500 1000 1500
φ Stripe 3−Air
−10 0 0 10
500 1000 1500
φ Stripe 4−Air
−10 0 0 10
2000 4000 6000
φ
Number of parcels
Stripe 1−Ground
−10 0 0 10
2000 4000 6000
φ Stripe 2−Ground
−10 0 0 10
2000 4000 6000
φ Stripe 3−Ground
−10 0 0 10
2000 4000 6000
φ Stripe 4−Ground
t = 7600s t = 9100s t = 10600s
Figure 5.4: Parcels size distribution plotted at three time instants. The simulation taken into account is still the one with the mean input values.
For these reasons in the next paragraphs the results of UQ and SA analysis
will be investigated only for the last instant of time (t = 10600s).
5.1 - Some Deterministic Solutions 77
5.1 Some Deterministic Solutions
Without carry out the UQ analysis, some information about what happens by changing the initial conditions can be found simply considering the sim- ulations of LPAC concerning a particular input tern. Considering the 343 quadrature points, one for each triplet of input values, it is possible to extrap- olate the deterministic solutions corresponding to the following situations:
• input value: ¯ µ, ¯ σ, ¯ ψ;
• input value: µ max , ¯ σ, ¯ ψ;
• input value: µ min , ¯ σ, ¯ ψ;
• input value: ¯ µ, σ max , ¯ ψ;
• input value: ¯ µ, σ min , ¯ ψ;
• input value: ¯ µ, ¯ σ, ψ max ;
• input value: ¯ µ, ¯ σ, ψ min ;
¯
µ, ¯ σ, ¯ ψ are the mean values of the input PDFs and they are equal to 1φ, 1.5φ and 0.7 respectively. The maximum and the minimum values of the input parameters are reported in Tab.5.1.
µ max 2φ µ min 0φ σ max 1.7φ σ min 1.3φ ψ max 0.9 ψ min 0.5
Table 5.1: Maximum and minimum values of the initial distributions.
Each input value has been analyzed and the effect of its maximum vari-
ation has been considered.
5.1 - Some Deterministic Solutions 78
5.1.1 Effect of the mean size
Fig.(5.5) shows the change in the parcels size distribution due to the variation of the mean value of the input diameters.
−10 0 0 10
500 1000 1500
φ
Number of parcels
Stripe 1−Air
−10 0 0 10
500 1000 1500
φ Stripe 2−Air
−10 0 0 10
500 1000 1500
φ Stripe 3−Air
−10 0 0 10
500 1000 1500
φ Stripe 4−Air
−10 0 0 10
2000 4000 6000 8000
φ
Number of parcels
Stripe 1−Ground
−10 0 0 10
2000 4000 6000 8000
φ Stripe 2−Ground
−10 0 0 10
2000 4000 6000 8000
φ Stripe 3−Ground
−10 0 0 10
2000 4000 6000 8000
φ Stripe 4−Ground
µ inlet = 0 µ inlet = 1 µ inlet = 2
Figure 5.5: Parcels size distribution in each region of the domain associated with the input value of σ and ψ equal to 1.5φ and 0.7 respectively, while µ varies from the minimum value of 0φ to the maximum one of 2φ. Also the mean solution associated to µ = 1φ is reported.
It is possible to observe that with the decreasing of the mean diameter of input distribution (in the Krumbein scale µ = 2φ corresponds to the smallest particles), a larger number of parcels is able to reach the right boundary of the domain, represented by Stripe 4.
Furthermore, we observe that the mean size observed in all the stripes,
except Stripe 1 in air, does not depend on the mean size of the particles
released in the atmosphere. Consequently, the mean size observed in the
air or on the ground in a particular stripe is not representative of the ini-
tial one. The statistical parameters that characterize the number of parcels
distribution are reported in tables: Tab.(5.2), Tab.(5.3), Tab.(5.4).
5.1 - Some Deterministic Solutions 79
µ=0φ
1-Air 2-Air 3-Air 4-Air
Mean [φ] 0.25 2.03 2.91 3.36
Std.Dev [φ] 1.37 0.74 0.53 0.51
Skewness 0.17 0.75 1.15 1.36
Kurtosis 0.13 0.72 0.63 4.07
Num. Par. 3445 859 297 158
1-Ground 2-Ground 3-Ground 4-Ground
Mean [φ] -0.72 1.30 2.40 2.90
Std.Dev [φ] 1.06 0.46 0.24 0.15
Skewness -0.71 0.02 -0.05 0.01
Kurtosis 0.27 -0.59 -0.81 -0.67
Num. Par. 17334 5429 754 186
Table 5.2: Statistical parameters of parcels distribution referred to the lower input value of µ, (coarse particles).
µ=1φ
1-Air 2-Air 3-Air 4-Air
Mean [φ] 1.10 2.29 3.10 3.50
Std.Dev [φ] 1.40 0.86 0.64 0.60
Skewness 0.15 0.62 0.95 1.77
Kurtosis 0.10 0.77 1.30 2.63
Num. Par. 3582 1765 795 530
1-Ground 2-Ground 3-Ground 4-Ground
Mean [φ] -0.72 1.30 2.40 2.90
Std.Dev [φ] 0.89 0.46 0.23 0.17
Skewness -0.89 -0.14 0.00 -0.03
Kurtosis 0.71 -0.63 -0.50 -0.59
Num. Par. 11146 7745 1635 504
Table 5.3: Statistical parameters of parcels distribution referred to the mean
input value of µ (1φ).
5.1 - Some Deterministic Solutions 80
µ=2φ
1-Air 2-Air 3-Air 4-Air
Mean [φ] 1.99 2.65 3.29 3.71
Std.Dev [φ] 1.44 1.00 0.80 0.75
Skewness 0.15 0.62 0.95 1.77
Kurtosis 0.13 0.72 0.63 4.07
Num. Par. 3445 859 297 158
1-Ground 2-Ground 3-Ground 4-Ground
Mean [φ] -0.72 1.30 2.40 2.90
Std.Dev [φ] 1.06 0.46 0.24 0.15
Skewness 0.14 0.71 1.11 1.20
Kurtosis 0.03 0.46 1.05 1.52
Num. Par. 3588 2768 1735 1209
Table 5.4: Statistical parameters of parcels distribution referring to the upper input value of µ (2φ).
5.1.2 Effect of the standard deviation
Keeping constant the mean diameter and the sphericity, Fig.(5.6) reveals
the effects of the extreme variation of the standard deviation of the initial
distribution. According to the classification proposed in Tab.(3.1), the initial
distribution is poorly sorted, which means that the variability of the parcels
dimensions around the mean value is high. The statistical values of the
frequency distributions are presented in the table below. (see Tab.(5.5),
Tab.(5.6), Tab.(5.7)).
5.1 - Some Deterministic Solutions 81
−10 0 0 10
500 1000 1500
φ
Number of parcels
Stripe 1−Air
−10 0 0 10
500 1000 1500
φ Stripe 2−Air
−10 0 0 10
500 1000 1500
φ Stripe 3−Air
−10 0 0 10
500 1000 1500
φ Stripe 4−Air
−10 0 0 10
2000 4000 6000
φ
Number of parcels
Stripe 1−Ground
−10 0 0 10
2000 4000 6000
φ Stripe 2−Ground
−10 0 0 10
2000 4000 6000
φ Stripe 3−Ground
−10 0 0 10
2000 4000 6000
φ Stripe 4−Ground
σ inlet = 1.3 σ inlet = 1.5 σ inlet = 1.7
Figure 5.6: Parcels size distribution associated to µ = 1φ and ψ = 0.7 (mean input value), while σ varies from its minimum to its maximum value, which are 1.3ψ and 1.7ψ respectively.
σ=1.3φ
1-Air 2-Air 3-Air 4-Air
Mean [φ] 1.06 2.15 2.96 3.35
Std.Dev [φ] 1.23 0.77 0.54 0.50
Skewness 0.16 0.61 1.11 1.45
Kurtosis 0.12 0.58 1.21 3.08
Num. Par. 3621 1728 676 430
1-Ground 2-Ground 3-Ground 4-Ground
Mean [φ] -0.15 1.39 2.41 2.89
Std.Dev [φ] 0.78 0.45 0.23 0.17
Skewness -0.86 -0.14 0.00 0.08
Kurtosis 0.67 -0.59 -0.60 -0.57
Num. Par. 10911 8629 1608 441
Table 5.5: Output results referred to the input triplet: µ = 1φ, σ = 1.3φ and
ψ = 0.7.
5.1 - Some Deterministic Solutions 82
σ=1.5φ
1-Air 2-Air 3-Air 4-Air
Mean [φ] 1.10 2.29 3.10 3.50
Std.Dev [φ] 1.40 0.86 0.64 0.60
Skewness 0.15 0.62 0.95 1.77
Kurtosis 0.10 0.77 1.30 2.63
Num. Par. 3582 1765 795 530
1-Ground 2-Ground 3-Ground 4-Ground
Mean [φ] -0.72 1.30 2.40 2.90
Std.Dev [φ] 0.89 0.46 0.23 0.17
Skewness -0.89 -0.14 0.00 -0.03
Kurtosis 0.71 -0.63 -0.50 -0.59
Num. Par. 11146 7745 1635 504
Table 5.6: Output results referred to the input triplet: µ = 1φ, σ = 1.5φ and ψ = 0.7.
σ=1.7φ
1-Air 2-Air 3-Air 4-Air
Mean [φ] 1.13 2.42 3.24 3.63
Std.Dev [φ] 1.57 0.95 0.76 0.71
Skewness 0.19 0.83 1.14 1.38
Kurtosis 0.08 0.94 1.16 2.52
Num. Par. 3540 1796 917 623
1-Ground 2-Ground 3-Ground 4-Ground
Mean [φ] -0.47 1.41 2.43 2.91
Std.Dev [φ] 1.01 0.46 0.23 0.17
Skewness -0.91 -0.14 -0.07 -0.17
Kurtosis 0.73 -0.65 -0.48 -0.70
Num. Par. 11319 6997 1638 482
Table 5.7: Output results referred to the input triplet: µ = 1φ, σ = 1.7φ and
ψ = 0.7.
5.1 - Some Deterministic Solutions 83
5.1.3 Effect of the sphericity
The last deterministic case concerns the variation of the sphericity (see Fig.(5.7)). For this analysis the statistical parameters are reported in Tab.(5.8), Tab.(5.9), Tab.(5.10).
−10 0 0 10
500 1000 1500
φ
Number of parcels
Stripe 1−Air
−10 0 0 10
500 1000 1500
φ Stripe 2−Air
−10 0 0 10
500 1000 1500
φ Stripe 3−Air
−10 0 0 10
500 1000 1500
φ Stripe 4−Air
−10 0 0 10
2000 4000 6000 8000
φ
Number of parcels
Stripe 1−Ground
−10 0 0 10
2000 4000 6000 8000
φ Stripe 2−Ground
−10 0 0 10
2000 4000 6000 8000
φ Stripe 3−Ground
−10 0 0 10
2000 4000 6000 8000
φ Stripe 4−Ground
ψ inlet = 0.5 ψ inlet = 0.7 ψ inlet = 0.9
Figure 5.7: Parcels size distribution in each region of the domain associated
with the input value of µ and σ equal to 1φ and 1.5φ respectively, while ψ
varies from the minimum value of 0.5 to the maximum one of 0.9.
5.1 - Some Deterministic Solutions 84
ψ=0.5
1-Air 2-Air 3-Air 4-Air
Mean [φ] 1.01 1.76 2.77 3.26
Std.Dev [φ] 1.46 1.07 0.72 0.65
Skewness 0.15 0.26 1.02 1.30
Kurtosis -0.04 0.51 1.04 2.15
Num. Par. 3685 2645 1168 718
1-Ground 2-Ground 3-Ground 4-Ground
Mean [φ] -1.02 0.661 2.05 2.59
Std.Dev [φ] 0.86 0.76 0.25 0.20
Skewness -0.75 -0.55 -0.08 -0.11
Kurtosis 0.16 -0.17 -0.50 -0.81
Num. Par. 4030 11934 2382 718
Table 5.8: Output values referred to the input triplet: µ = 1φ, σ = 1.5φ and ψ = 0.5.
ψ=0.7
1-Air 2-Air 3-Air 4-Air
Mean [φ] 1.10 2.29 3.10 3.50
Std.Dev [φ] 1.40 0.86 0.64 0.60
Skewness 0.15 0.62 0.95 1.77
Kurtosis 0.10 0.77 1.30 2.63
Num. Par. 3582 1765 795 530
1-Ground 2-Ground 3-Ground 4-Ground
Mean [φ] -0.72 1.30 2.40 2.90
Std.Dev [φ] 0.89 0.46 0.23 0.17
Skewness -0.89 -0.14 0.00 -0.03
Kurtosis 0.71 -0.63 -0.50 -0.59
Num. Par. 11146 7745 1635 504
Table 5.9: Output values referred to the input triplet: µ = 1φ, σ = 1.5φ and
ψ = 0.7.
5.1 - Some Deterministic Solutions 85
ψ=0.9
1-Air 2-Air 3-Air 4-Air
Mean [φ] 1.26 2.65 3.34 3.70
Std.Dev [φ] 1.37 0.76 0.60 0.57
Skewness 0.09 0.89 1.23 1.47
Kurtosis 0.19 1.00 1.31 3.02
Num. Par. 3236 1246 589 418
1-Ground 2-Ground 3-Ground 4-Ground
Mean [φ] 0.03 1.88 2.73 3.15
Std.Dev [φ] 0.98 0.35 0.20 0.15
Skewness -0.83 0.01 -0.08 0.05
Kurtosis 0.48 -0.64 -0.61 -0.68
Num. Par. 15556 5414 1131 364
Table 5.10: Output values referred to the input triplet: µ = 1φ, σ = 1.5φ and ψ = 0.9.
From the previous data it emerges that the characteristics of the initial
distribution are maintained only in Stripe 1-Air. The input parameters that
mainly influence the parcels distribution seem to be µ and ψ. With the chang-
ing in the mean diameter of the input distribution, it emerges that the total
number of parcels present on the domain varies a lot, since it pass from about
14000 units for µ = 0φ, to about 30000 parcels for µ = 2φ. It is interesting
to observe that the trend of the mean diameter of the parcels distribution
on the ground is approximately independent by the initial sorting, but i is
especially related to the sphericity. Also the standard deviation computed
in the stripes (both in air than on ground) is not particularly influenced by
the initial one. It is possible to observe that the increase of the sphericity
does not significantly affect the total number of parcels (30000 units), but it
is a leading parameter concerning the way in which the parcels are divided
on the domain.
5.2 - Polynomial Response Functions 86
5.2 Polynomial Response Functions
While the previous results gives an initial idea of the dependence of the parcels size distribution from the 3 investigated parameters at the inlet, to obtain a full picture of the variability of the results and their sensitivities we rely on the UQ analysis. Entering in the merits of the UQ analysis, first of all it is possible to plot the polynomials which actually are the output of the PCEg method. Fig.(5.8) shows the polynomial response function associated to the mean particles diameter in the Stripe 1-Air. Since the polynomial is a function of the three input variables, each plot has been made keeping con- stant one parameter and varying the other two in their range of definition. In this way it is possible to observe how the input uncertainty spreads through the output. In this particular case we find the previous results concerning the variation of the mean diameter in Stripe 1-Air, variation which accurately reproduces the input uncertainty.
A different situation is reported in Fig.(5.9) where the output parameter taken into account is the mean diameter of the parcels in Stripe 1-Ground.
Here the input variability referring to µ is not maintained and the sphericity
shows its leading role, while the initial standard deviation seems to not be
particularly influential.
5.2 - Polynomial Response Functions 87
−1 −0.5 0 0.5 1
−1
−0.5 0 0.5 1 0 0.5 1 1.5 2 2.5
σ ψ
Mean diameter Stripe 1−Air
(a) Polynomial response function plotted for µ = 1φ, while σ and ψ vary in their input range.
−1 −0.5 0 0.5 1
−1
−0.5 0 0.5 1 0 0.5 1 1.5 2 2.5
µ ψ
Mean diameter Stripe 1−Air
(b) Polynomial response function plotted for σ = 1.5φ, while µ and ψ vary in their input range.
−1 −0.5 0 0.5 1
−1
−0.5 0 0.5 1 0 0.5 1 1.5 2 2.5
µ σ
Mean diameter Stripe 1−Air
(c) Polynomial response function plotted for ψ = 0.7, while µ and σ vary in their input range.
Figure 5.8: Sketch of the polynomial response function associated to the
mean value of the parcels diameter in Stripe 1-Air. The domain reported on
x-y axis refers to the range of existence of Legendre polynomials[-1,+1].
5.2 - Polynomial Response Functions 88
−1 −0.5 0 0.5 1
−1
−0.5 0 0.5
−1.5 1
−1
−0.5 0 0.5
σ ψ
Mean diameter Stripe 1−Ground
(a) Polynomial response function plotted for µ = 1φ, while σ and ψ vary in their input range.
−1 −0.5 0 0.5 1
−1
−0.5 0 0.5
−1.5 1
−1
−0.5 0 0.5
µ ψ
Mean diameter Stripe 1−Ground
(b) Polynomial response function plotted for σ = 1.5φ, while µ and ψ vary in their input range.
−1 −0.5 0 0.5 1
−1
−0.5 0 0.5
−1.5 1
−1
−0.5 0 0.5
µ σ
Mean diameter Stripe 1−Ground
(c) Polynomial response function plotted for ψ = 0.7, while µ and σ vary in their input range.
Figure 5.9: Mean parcels diameter associated to Stripe 1-Ground. The
domain reported on x-y axis refers to the range of existence of Legendre
5.3 - Statistical Analysis of the Response Functions 89
5.3 Statistical Analysis of the Response Func- tions
5.3.1 Cumulative and Probability Distribution Functions
The polynomial response functions can be also used to evaluate the variation of the output values (see Tab.(3.4)) for a wide number of initial conditions, without running the simulations again, but simply using the polynomials as
"emulators" of the Lagrangian code. In this way the computational time is extremely reduced. We have set DAKOTA in order to obtain 10000 sam- ples of the output quantities for a likewise number of random input triplets.
These samples are then used to compute the statistical parameters of each re- sponse value (i.e. mean value, standard deviation, skewness, kurtosis) and to evaluate the Cumulative Distribution Functions (CDFs) and the Probability Distribution Functions (PDFs) of the response values.
In the input file of DAKOTA it is possible to specify the probability levels at which the CDFs are evaluated: [0.01; 0.05; 0.25; 0.50; 0.75; 0.95; 0.99].
Fig.(5.10) shows the CDFs and the PDFs of the mean parcels diameter
computed in every stripes both in the air and on the ground. The red curve of
each plot represents the input distribution, which has been set as an uniform
distribution over the range of [0φ; 2φ].
5.3 - Statistical Analysis of the Response Functions 90
−2 0 0 2 4
0.2 0.4 0.6 0.8 1
µ Stripe 1−Air
CDF
−2 0 0 2 4
0.2 0.4 0.6 0.8 1
µ Stripe 2−Air
CDF
−2 0 0 2 4
0.2 0.4 0.6 0.8 1
µ Stripe 3−Air
CDF
−2 0 0 2 4
0.2 0.4 0.6 0.8 1
µ Stripe 4−Air
CDF
−2 0 0 2 4
0.2 0.4 0.6 0.8 1
µ Stripe 1−Ground
CDF
−2 0 0 2 4
0.2 0.4 0.6 0.8 1
µ Stripe 2−Ground
CDF
−2 0 0 2 4
0.2 0.4 0.6 0.8 1
µ Stripe 3−Ground
CDF
−2 0 0 2 4
0.2 0.4 0.6 0.8 1
µ Stripe 4−Ground
CDF
(a)
0 2 4
0 0.5 1 1.5 2
Stripe 1−Air
µ
0 2 4
0 0.5 1 1.5 2
Stripe 2−Air
µ
0 2 4
0 0.5 1 1.5 2
Stripe 3−Air
µ
0 2 4
0 0.5 1 1.5 2
Stripe 4−Air
µ
−2 0 0 2 4
0.5 1 1.5 2 2.5
Stripe 1−Ground
µ
−2 0 0 2 4
0.5 1 1.5 2 2.5
Stripe 2−Ground
µ
−2 0 0 2 4
0.5 1 1.5 2 2.5
Stripe 3−Ground
µ
−2 0 0 2 4
0.5 1 1.5 2 2.5
Stripe 4−Ground
µ
(b)
Figure 5.10: CDFs (a) and PDFs (b) related to the mean parcels diameter.
5.3 - Statistical Analysis of the Response Functions 91
From the figure it is possible to observe that the input variability of µ is reproduced only in Stripe 1-Air, while, going away from the inlet, the mean diameter not only becomes smaller but also the original variability is lost (see (Tab.5.11)). This is evident since the CDFs tend to became steeper and they are localized around a particular value of µ, similarly the PDFs histograms of the mean diameter are shifted to higher values of µ (which means finer particles according to the Krumbein scale), and their dispersion decreases going away from the inlet. A quite good representation of the mean diam- eter input distribution can be obtained sampling in Stripe 2-Ground, which means from 20Km to 30Km from the inlet.
1-Air 2-Air 3-Air 4-Air
µ mean [φ] 1.12 2.27 3.08 3.51
µ std. dev [φ] 0.53 0.35 0.22 0.19
1-Ground 2-Ground 3-Ground 4-Ground
µ mean [φ] -0.39 1.33 2.41 2.90
µ std. dev. [φ] 0.38 0.40 0.20 0.16
Table 5.11: Mean value and standard deviation referred to the PDFs associ- ated to the mean parcels diameter in every cell.
In order to reconstruct the input variability of the ash deposit and to
study the ash dispersion, it is fundamental to investigate also the behavior
of the sorting of the parcels, in other words the dispersion of the parcels
dimensions around the mean value must be considered. In Fig.(5.11) are
reported the CDFs and the PDFs of the standard deviation of the parcels
frequency distributions in the domain. At the release location for σ it has
been assumed a uniform PDF over the range [1.3φ; 1.7φ] (red curves on the
figures). Concerning the air situation, the initial poorly sorted distribution
is conserved only in Stripe 1-Air, in the other cells the parcels distributions
tend to become moderately sorted (the standard deviation decreases), even if
it continues to have a good range of variation (see Fig.(5.11b)). Overall it is
possible to say that, going away from the inlet, the parcels are finer and more
uniform in their size distribution, even if a small dispersion around the mean
5.3 - Statistical Analysis of the Response Functions 92
value is kept. On the ground the situation is quite different from the air, since
not only the input variability is immediately lost (the deposit is moderately
sorted in Stripe 1-Ground and becomes well sorted in Stripe 4-Ground), but
also, differently from the air, the slope of the CDFs curves for the standard
deviation changes and becomes steeper. This means that, while it is possible
to reconstruct the input mean diameter distribution by sampling in Stripe
2-Ground, the input standard deviation can not be detected in a particular
stripe. In order to retrieve the input standard deviation of the ash cloud
particles, it is fundamental to take into account the overall ash deposit and,
even in this case, the lack of a right information about the air situation can
compromises the measurement. The statistical parameters that describes the
trend of the standard deviation in the domain are reported in Tab.(5.12).
5.3 - Statistical Analysis of the Response Functions 93
0 1 2
0 0.2 0.4 0.6 0.8 1
σ Stripe 1−Air
CDF
0 1 2
0 0.2 0.4 0.6 0.8 1
σ Stripe 2−Air
CDF
0 1 2
0 0.2 0.4 0.6 0.8 1
σ Stripe 3−Air
CDF
0 1 2
0 0.2 0.4 0.6 0.8 1
σ Stripe 4−Air
CDF
0 1 2
0 0.2 0.4 0.6 0.8 1
σ Stripe 1−Ground
CDF
0 1 2
0 0.2 0.4 0.6 0.8 1
σ Stripe 2−Ground
CDF
0 1 2
0 0.2 0.4 0.6 0.8 1
σ Stripe 3−Ground
CDF
0 1 2
0 0.2 0.4 0.6 0.8 1
σ Stripe 4−Ground
CDF
(a)
0 1 2
0 1 2 3 4 5
Stripe 1−Air
σ
0 1 2
0 1 2 3 4 5
Stripe 2−Air
σ
0 1 2
0 1 2 3 4 5
Stripe 3−Air
σ
0 1 2
0 1 2 3 4 5
Stripe 4−Air
σ
0 1 2
0 10 20 30
Stripe 1−Ground
σ
0 1 2
0 10 20 30
Stripe 2−Ground
σ
0 1 2
0 10 20 30
Stripe 3−Ground
σ
0 0.5 1 1.5 2
0 5 10 15 20 25 30