Environmental Engineering Reference
In-Depth Information
EOFs (F
1
superimposing the individual pdfs available for
the measurement period between surveys; N is
the number of individual wave measurements
between surveys; i is an index; H is the wave
height; and H
rms0
is the root-mean-squared wave
height in deep water (with H
rms
0.707H
s
, and
the '0' subscript referring to quantities measured
in deep water or 'offshore').
The superposition carried out in Equation 16.2
implies that all the individual pdfs derived from
the wave measurements (H
rms0
) have the same
weight. Figure 16.7b shows the corresponding
composite pdfs for the nearshorewave height valid
for the time period between surveys and obtained
by summing over a large number of Rayleigh pdfs
according to Equation 16.2.
F
3
) for the wave pdfs, which explain
more than 96% of the variation.
The shape functions defined by the EOF anal-
ysis can be interpreted as various 'modes' of var-
iation in analogy with Fourier analysis. The first
eigenfunction represents the 'best fit' at describing
the variation in the data. The second eigenfunc-
tion represents the 'best fit' to the deviations of
the data from the first eigenfunction, and so on.
The number of local maxima and minima in the
eigenfunctions increases with the order of the
eigenfunction. Thus the third eigenfunction is
expected to have a more oscillatory behaviour
than either the first or the second.
The first EOF describing the profiles (E
1
) re-
flects the presence of a single bar that receives
contribution from areas seaward of it. E
2
charac-
terizes the changes in the bar of the profile, and E
3
may be related to the exchange of material across
the profile during major storm events. It is impor-
tant to take into account that the behaviour in the
three EOFsmodes after 500m is reasonably stable.
Additionally, the temporal EOFs may be analysed
to determine trends of profile changes and oscil-
latory cycles. The EOFs associated with the wave
pdfs (Fig. 16.8b) mainly represent seasonal varia-
tions in the wave climate and the effect of severe
storms.
While some interpretation of the patterns iden-
tified by the EOF has been given here, it should be
borne inmind that this is a subjective process. The
EOF analysis is an entirely statistical procedure
that does not incorporate physical understanding
to constrain the fitting procedure. Attempting
an interpretation of the physical processes on the
basis of EOF results alone is not recommended.
Applying CCA to the two datasets (Figs 16.7a
and 16.7b) produced a maximum correlation of
0.49 between U
1
and V
1
(temporal amplitudes
of the first CCA modes) (Fig. 16.9a). Figures
16.9b and 16.9c representing theCCAmodes show
that material moved between the bar area and
the foreshore is associated with an increase in the
probability of higher waves in the pdf and vice
versa. Erosion will occur in the inshore section
due to higher waves and the material will be
deposited in the area of the bar, if this bar exists.
Predictions of profile changes based on
Rayleigh distribution
The data were represented by truncated forms of
their EOF expansions in order to reduce the noise
in the records, before performing the CCA analy-
sis. In general, three to five EOF modes were
sufficient to represent most of the variation in the
datasets. In this study, five EOF modes were used
but only three are plotted. Table 16.2 summarizes
the results for the first threemodes, after themean
has been subtracted from the raw data.
The first three spatial EOFs (E
1
E
3
) obtained
from the beach profiles are displayed in
Figure 16.8a. Together they explain about 70% of
the variation in the data (the time mean was
subtracted before analysis in all data sets). Corre-
spondingly, Figure 16.8b shows the first spatial
Table 16.2
Percentage of variance for the first three
empirical orthogonal function (EOF) modes
Eigenfunction
number
% Variance
pro
les
% Variance
wave pdf
1
-
-
2
34
82
3
25
12
4
11
2
Total
70
96
