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Security of Coastal Nuclear Power Stations in Relation With The State of The Sea
465
SECURITY OF COASTAL NUCLEAR POWER STATIONS IN RELATION WITH THE STATE OF THE SEA J. BERNIER
, J.
MIQUEL
Laboratoire National d'Hydraulique, Chatou (France)
ABSTRACT
The safety of a coastal power plant is concerned with two phenomena : the wind waves, and the maximum and minimum tide levels. This paper presents methods of statistical analysis for estimating the probabilities of extreme events to be taken into account by the designer. First are recalled the definitions of these phenomena, in particular the relationships existing between the maximum of N waves and the significant wave. Then the case is approached where, because of little information available, the use of either meteorological data or uncommon events recorded in a far-off past is necessary. The paper concludes with an example of statistical study of storm durations. INTRODUCTION The figure below shows a vertical cross section of a power plant bordering on the sea :
Plant
Tranqoilliration
466 The rates of flow required for the power plant cooling is pumped in the tranquillization basin, which is protected against the waves by the dike. The tranquillization basin is communicating with the sea and its level is equal to that
of the tide. A maximum residual agitation of 30 cm in the basin is consistent with the operation of the pumping station. The designer needs following complementary information
:
1. Extreme wind waves probabilities, so that the stability of the dike may be ensured against centennial events at least. 2. Maximum and minimum tide level probabilities, so that protection may be ensured against the flood (maximum level) on the one hand, against failing
of the pumps on the other (minimum level). DEFINITION OF THE WIND WAVE TAKEN INTO ACCOUNT Among the numerous statistical waves characteristics, the most frequently used for the dike design is the significant wave denoted by Hl13 (Average upper third of the greatest waves). This is the parameter that has been selected for the estimation of the wind waves risks. However, it should be indicated that many other parameters may be directly related to H
1/3
.'
1. Cartwright and Longuet-Higgins have demonstrated that in the case the wind waves follow the Gaussian model the following relation may be used
-
:
H113 = 1,6 H = 0,79 Hlllo. These results have been checked on some recordinqs (Miquel, 1975). 2. Utilizing the same assumption, Longuet-Higgins showed that the maximum of N waves is related to H1l3, and gave the expression of its mean value. Bernier, in an internal paper published at the "Laboratoire National d'Hydraulique", verified this expression. Besides, utilizing the results obtained by Cramer and Leadbetter (Cramer and Leadbetter, 1967) he could demonstrate that Hm(N) follows a law of extreme values, the mean and the standard deviation of which are :
12 log, N It appears then possible to evaluate the probabilities of H-(N)
from those
of Hi131 either directly by combining the probabilities of H113 with those of the extreme value distribution, or through simulation by reconstituting a fictitious sample in the following way
:
467 H
MAX
where
(N) = m (N) - U (N). p
[ 0,45
+ 0 , 7 8 loqe(- loge p) ]
is drawn in an uniform law on
] 0, 1 [ .
It is important to take into consideration not only the mean value but also the variability (figured by the standard deviation)
:
the neglect of this varia-
bility runs counter to safety. An exhaustive study of waves hazards should also take into account the periods. At the present, the couple (wave-period) is being studied in a frequential way (Allen, 1977) in order to assign a "probable" period to a given wave, the waves only being probabilized. Another important point, which is likely to be taken into account soon, is the storm duration
an incipient response is given farther.
:
DEFINITION OF TIDE LEVELS Definition 1 Observed maximum level -------------__-___ :
:
It is the level actually reached by
the sea. It will be denoted by HI.
Definition 2 Predicted maximum level -----------------:
:
It is the level that the sea would
reach in the absence of atmospheric perturbation
:
it is determined by the posi-
tion of the stars (astronomic tide). In France, this level is computed by the "Service Hydrographique et Oceanographique de la Marine", by summing up the ampli tudes associated with different periods, the semi-diurnal amplitude being the principal one. This level will be denoted by H
0'
Definition 3 : Tide deviation : It is the positive or negative difference ------------_-_
between H I and Ho, mainly due to meteorological conditions (pressure, wind, temperature, etc
...)
It will be denoted by
S.
Predicted tide Time
468
Generally, S is estimated by the difference between the observed HI and the calculated Ho. It will be shown farther that S can be sometimes estimated from meteorological conditions : S (P, V, etc.).
-
-
Hg and S being estimated, their values are somewhat uncertain. This uncertainty
should be allowed for in the probabilization. It can be written : H1 = Ho+ S +
E,
where (Eis the residue, the statistical characteristics of which must be given at the Same time as the estimates Ho and S . Everything said about the maxima levels can be symmetrically extended to the minima levels. WIND WAVES PROBABILITIES The sample
:
The sample of daily waves is established, namely by choosing for
each day, the surge H1,3(i)
the highest of the day. It should be made sure that
all periods of the year are equally represented in the sample, otherwise a seasonal study would be necessary. The monthly maxima method : For each month, the highest waves is selected from the sample above. The new sample {Hj}
is successively fitted to the Normal, LOT.
Normal, Extreme values distributions. The best of these fittings is chosen. Example
:
NORMAL
L06. NORMAL
EXTREME VALUES
Max. Monthly Wave
469 The “Renewal“ method
:
the shortcomings of the monthly maximum method lead us
to use a method, inspired by the study of the renewal process, which is used already for about ten years to study the rates of flow of rising rivers. Starting from the sample constituted above, the maximum wave each storm, provided that this wave
is selected in
is higher than a given threshold chosen
beforehand, and that two successive waves belong undeniably to two differing storms (independence)
:
woves
t
H’/3
1
lime
t
*
Let us take the month as a reference period. Then, two samples can be constructed :
{ Hj } is the {nk]is
set of the surges higher than the threshold,
the catalogue of the number nk of storms having exceeded the threshold in the course of the kth month.
The calculation o f the monthly probability of exceeding a value h, namely the probability of the monthly maximum H* exceeding the value h, is carried out as follows Prob Prob
Prob
:
[ H* > h]= [ H* 6 h]= +
[ H* 6
h
1
-
Prob[2
0 storm
Prob[3
1 storm
+
Prob[gr
]
=
[ H* 6 h]
Prob
>/ threshold in the course Of a month] >/ threshold and d h
storms% threshold and
+aJ
6 h]
Prob [ 3 k storms 2 threshold and
K=O
+co Prob [H*(h]=
Prob K=O
[
n=k].(Prob[HCh(H>thrt?shold])
470
Prob [H* where
-l f
>h ]= 1 -
+W
1
P(k) .Fk(h)
K=O
P(k)
is the probability of having k storms in the course of the month,
F(h)
is the probability of a storm, higher than the threshold, beino lower than or equal to h.
If h is great enough, F(h) is near to 1 and this result can be simplified to
for
{
-
> h]zl
Prob [H*
Prob [H*
+oo
1
n
=
(1 - F(h))
-
+OD
P(k) = 1 and
1
P(k). k = n
K=O
K=O
-
[ 1 + k(l - F(h))}
> h]”,
+m
1
P(k)
K=O
:
monthly average number of storms.
From the practical point of view, the nk catalogue enables P(k) or determined for the utilisation of the simplified formula two laws is used
;
n
to be
one of the following
:
Poisson’s law : P(k) = e
- A & k! k
Negative Binomial law : P(k) = k!
r(Y)
Since P(k) may considerably vary according to the month, it would be preferable, when sufficient information is available, to take as a reference period the year instead of the month. The probability F(h) is determined by the sample of the H to which are fitted the followinq laws :
i‘
471
This method has the advantage of utilizing the maximum amount of information, while warranting its homogeneity. It is possible and desirable to calculate the intervals of confidence. T I D E LEVEL PROBABILITIES
The Observed Maximum Level
:
the most simple way is, like in the case of waves
to -~ constitute the sample of daily maximum levels of the hiqh water. Then, the san
methods are used as for the wave. The result is presented in the following form
:
472
PROBABILITIES
OF HIGH WATERS IN DIEPPE
-
I
- 0,lO -
LOW WATER
-----
- 0,20 -
- 0,30 - 0,40 - q50,
I
I
I
I
1
Return
I
I l l
Period
1
1
1
I
1 1 ,
( i n years)
However, the question may arise of whether it will be safe to use only one statistical law for explaining the behaviour of a variable, which is made up of two phenomena entirely different : the astronomic tide and the tide deviation due to meteorological conditions. We decided therefore to study also these two phenomena. The Predicted Maximum Level
:
In fact, it‘s a question of a random pseudova-
riable easy to probabilize either by constitutinq directly a catalogue of predicted heights, or by using the estimates based on the semi-diurnal amplitude. The two methods can be compared in the figure below.
413
FREQUENCIES OF PREDICTED LEVEL IN DIEPPE
% Frequencies of overstepping
700
8,OO
9,m
Z level
l0,OO ( in meters)
-
474 Tide differences
:
First, we constitute the catalogue of daily tide differences
obtained either by means of differences Ho - H1 on a series of observed tides, or by reconstitution from meteorological conditions ( s e e farther). Then, we proceed to the same probabilistic study as for the waves. The Sum of predicted levels and tide differences
:
We have H1 = H
0
+ S. If
Ho
and S are independent, the probability of their sum can be easily calculated by writinq Prob
:
jw +W
[ H1 >
hl] =
f
[x]=
G
[y]=Prob
where
G
[hl
-
x].
[ x < H,- < x +
Prob
f [x]. dx
dx
]
[S>y]
For our part, we found that if the coefficient of correlation between Ho and S could attain 0,3 during slight or medium storms, this coefficient is practically
zero for heavy storms by which we are particularly concerned. This result is only indicative as it corresponds to a particular case and deserves to be tested on other sites.
If the correlation is no more zero but if there exists a relationship of the kind S =
A
Ho
+ S', where
Ho and S' are independent, we can get again to the pre-
vious case by considering the independent variables
:
h1
(1 +
Ho and S ' .
The figure below enables the two methods for estimating the HI level to be compared by studying directly H 1 or by studying the sum
OF TIDE LEVELS
PROBABILITIES
IN
€3 0
DIEPPE
+
S.
-
Level (in meters )
10,90
10,ao
1420
. 1
2
3
4
5
Return
10
Period
20 30 4050 ( in y e a r s )
100
475
For the design, we take the extreme limits of these estimates to which we add confidence intervals at 7 0 % . CASE OF POOR INFORMATION
Wave data and tide data are frequently very short, rendering the statistical estimates too uncertain : additional information should then be used. Sometimes, it is fortunate to find another wave or tide series in the vicinity of the studied site. If the two series are closely related, the probability estimates of the long series can be easily transposed to the short one. If this is not the case, it is necessary then to consider other possibilities. Utilization of the meteorology
:
in the case where information, such as pres-
sure, wind, temperature in the vicinity of the site, is
available, it is possible
to establish a relationship between these data and the surqes or the tide fluctuations. As a test we tried multiple linear reqressions of the kind
n s
1/3
=
= g
where
f (P,
v, v2,
i
(P,
2
v, v ,
nO,Ap,Av, ...I
no,Ap,
P
. . .)
= temporary pressure variation
T
= temperature
V
= wind speed
A
AV,
= pressure
P
h
T,
T,
:
V
= temporary variation of the wind
Although the results are not yet exploitable for high events, they are incentive for low and medium events in so far as the obtained multiple correlation coefficients reached 0 , 8 to 0,9.
Using these relationships, we reconstructed a
fictitious sample of tide differences over a long period of time and we estimated then the probabilities resultinq from this sample. On the figure below, the obtained results can be compared with respect to the probabilities derived from observations
:
416
PROBABILITIES OF TIDE DEVIATIONS IN LE HAVRE
3000
2000
1000
500 400
300 200
100
50
40 30
-
-
Return Period (in Days)
From observations within 10 years
-
-
From meteorology
--
From observations w i t h i n 1 years
Tide Deviation ( in
20
30
40
50
70
60
80
90
100
cm
)
110
In this figure we can see that there is an acceptable compatibility between the estimates for return periods lower than 10 years. Beyond these periods, it will be necessary either to improve the statistical relationships between the meteorology and the sea states or to use mathematical prediction models. Utilization of exceptional events
:
it happens that there exist recorded data
on one or more exceptional events for which an estimate can be fixed, and which are known to be the highest within a long period of time (for instance, a century). This information is precious and may be utilized, thouqh it greatly differs from a complete catalogue of waves or tides. It allows the statistical uncertainty to be reduced and the representativity of the used sample to be proved. The detailed description of this method can be found in the references (Bernier and Miquel, 1977). It was already applied successfully to flood risk estimations
:
477
FLOOD PROBABILITIES AT HAUCONCOURT (MOSELLE 1
Return Period 1000
500 200 100
50 20
10 5 2 1
STORM DURATIONS
Recent works on random waves showed how the storm duration may affect the lifetime of dikes. Using once more the techniques applied to the study of river flow rates (Miquel and Phien BOU Pha, 1978) we can estimate, for instance, the duration probability of a storm exceeding a given surge threshold. The probabilities of the yearly sums
of storm durations can be read in the figure below
:
478
Durations
EXCEEDING A GIVEN WAVE THRESHOLD IN LE HAVRE : TOTAL ANNUAL SUMS -
Thus, in decennial year, the total
Return
duration, sum
over
Period
100 years
the year of storms exceeding the surge level of 3,5 m in Le Havre
1
about 10 days.
0
1
2
3
4
5
6
7
8
9
Wave Threshold ( in meters) A curve of the same kind can be obtained,to describe the durations of individual storms. Indeed, such curves will be useful to designers when they will be able to take simultaneously into account both, the storm durations and their intensities. REFERENCES Allen, H., 1977. Analyse statistique des mesures de houle en differents sites du littoral franqais. Edition no 3, rapport EDF HE 46/77.01. Chatou (France) Bernier, J., Miquel, J., 1977. Exemple d'application de la theorie de la decision statistique au dimensionnement d'ouvraqe hydraulique : prise en compte de l'information het6rogGne. A.I.R.H. Baden. Cramer, Leadbetter, 1967. Stationary and related stochastic processes. Sample function properties and their applications. John Wiley. New York.
479 Miquel, J., 1975. Role et importance d'un modile statistique de la houle en vue du depouillement et du stockaqe des donnees. A.I.R.H. Sao Paulo. Miquel, J., Phien Bou Pha, B., 1977, Tempetiage : un modile d'estimation des risques d'etiage. Xime Journee de 1'Hydraulique. Toulouse.
SECURITY OF COASTAL NUCLEAR POWER STATIONS IN RELATION WITH THE STATE OF THE SEA J. BERNIER
, J.
MIQUEL
Laboratoire National d'Hydraulique, Chatou (France)
ABSTRACT
The safety of a coastal power plant is concerned with two phenomena : the wind waves, and the maximum and minimum tide levels. This paper presents methods of statistical analysis for estimating the probabilities of extreme events to be taken into account by the designer. First are recalled the definitions of these phenomena, in particular the relationships existing between the maximum of N waves and the significant wave. Then the case is approached where, because of little information available, the use of either meteorological data or uncommon events recorded in a far-off past is necessary. The paper concludes with an example of statistical study of storm durations. INTRODUCTION The figure below shows a vertical cross section of a power plant bordering on the sea :
Plant
Tranqoilliration
466 The rates of flow required for the power plant cooling is pumped in the tranquillization basin, which is protected against the waves by the dike. The tranquillization basin is communicating with the sea and its level is equal to that
of the tide. A maximum residual agitation of 30 cm in the basin is consistent with the operation of the pumping station. The designer needs following complementary information
:
1. Extreme wind waves probabilities, so that the stability of the dike may be ensured against centennial events at least. 2. Maximum and minimum tide level probabilities, so that protection may be ensured against the flood (maximum level) on the one hand, against failing
of the pumps on the other (minimum level). DEFINITION OF THE WIND WAVE TAKEN INTO ACCOUNT Among the numerous statistical waves characteristics, the most frequently used for the dike design is the significant wave denoted by Hl13 (Average upper third of the greatest waves). This is the parameter that has been selected for the estimation of the wind waves risks. However, it should be indicated that many other parameters may be directly related to H
1/3
.'
1. Cartwright and Longuet-Higgins have demonstrated that in the case the wind waves follow the Gaussian model the following relation may be used
-
:
H113 = 1,6 H = 0,79 Hlllo. These results have been checked on some recordinqs (Miquel, 1975). 2. Utilizing the same assumption, Longuet-Higgins showed that the maximum of N waves is related to H1l3, and gave the expression of its mean value. Bernier, in an internal paper published at the "Laboratoire National d'Hydraulique", verified this expression. Besides, utilizing the results obtained by Cramer and Leadbetter (Cramer and Leadbetter, 1967) he could demonstrate that Hm(N) follows a law of extreme values, the mean and the standard deviation of which are :
12 log, N It appears then possible to evaluate the probabilities of H-(N)
from those
of Hi131 either directly by combining the probabilities of H113 with those of the extreme value distribution, or through simulation by reconstituting a fictitious sample in the following way
:
467 H
MAX
where
(N) = m (N) - U (N). p
[ 0,45
+ 0 , 7 8 loqe(- loge p) ]
is drawn in an uniform law on
] 0, 1 [ .
It is important to take into consideration not only the mean value but also the variability (figured by the standard deviation)
:
the neglect of this varia-
bility runs counter to safety. An exhaustive study of waves hazards should also take into account the periods. At the present, the couple (wave-period) is being studied in a frequential way (Allen, 1977) in order to assign a "probable" period to a given wave, the waves only being probabilized. Another important point, which is likely to be taken into account soon, is the storm duration
an incipient response is given farther.
:
DEFINITION OF TIDE LEVELS Definition 1 Observed maximum level -------------__-___ :
:
It is the level actually reached by
the sea. It will be denoted by HI.
Definition 2 Predicted maximum level -----------------:
:
It is the level that the sea would
reach in the absence of atmospheric perturbation
:
it is determined by the posi-
tion of the stars (astronomic tide). In France, this level is computed by the "Service Hydrographique et Oceanographique de la Marine", by summing up the ampli tudes associated with different periods, the semi-diurnal amplitude being the principal one. This level will be denoted by H
0'
Definition 3 : Tide deviation : It is the positive or negative difference ------------_-_
between H I and Ho, mainly due to meteorological conditions (pressure, wind, temperature, etc
...)
It will be denoted by
S.
Predicted tide Time
468
Generally, S is estimated by the difference between the observed HI and the calculated Ho. It will be shown farther that S can be sometimes estimated from meteorological conditions : S (P, V, etc.).
-
-
Hg and S being estimated, their values are somewhat uncertain. This uncertainty
should be allowed for in the probabilization. It can be written : H1 = Ho+ S +
E,
where (Eis the residue, the statistical characteristics of which must be given at the Same time as the estimates Ho and S . Everything said about the maxima levels can be symmetrically extended to the minima levels. WIND WAVES PROBABILITIES The sample
:
The sample of daily waves is established, namely by choosing for
each day, the surge H1,3(i)
the highest of the day. It should be made sure that
all periods of the year are equally represented in the sample, otherwise a seasonal study would be necessary. The monthly maxima method : For each month, the highest waves is selected from the sample above. The new sample {Hj}
is successively fitted to the Normal, LOT.
Normal, Extreme values distributions. The best of these fittings is chosen. Example
:
NORMAL
L06. NORMAL
EXTREME VALUES
Max. Monthly Wave
469 The “Renewal“ method
:
the shortcomings of the monthly maximum method lead us
to use a method, inspired by the study of the renewal process, which is used already for about ten years to study the rates of flow of rising rivers. Starting from the sample constituted above, the maximum wave each storm, provided that this wave
is selected in
is higher than a given threshold chosen
beforehand, and that two successive waves belong undeniably to two differing storms (independence)
:
woves
t
H’/3
1
lime
t
*
Let us take the month as a reference period. Then, two samples can be constructed :
{ Hj } is the {nk]is
set of the surges higher than the threshold,
the catalogue of the number nk of storms having exceeded the threshold in the course of the kth month.
The calculation o f the monthly probability of exceeding a value h, namely the probability of the monthly maximum H* exceeding the value h, is carried out as follows Prob Prob
Prob
:
[ H* > h]= [ H* 6 h]= +
[ H* 6
h
1
-
Prob[2
0 storm
Prob[3
1 storm
+
Prob[gr
]
=
[ H* 6 h]
Prob
>/ threshold in the course Of a month] >/ threshold and d h
storms% threshold and
+aJ
6 h]
Prob [ 3 k storms 2 threshold and
K=O
+co Prob [H*(h]=
Prob K=O
[
n=k].(Prob[HCh(H>thrt?shold])
470
Prob [H* where
-l f
>h ]= 1 -
+W
1
P(k) .Fk(h)
K=O
P(k)
is the probability of having k storms in the course of the month,
F(h)
is the probability of a storm, higher than the threshold, beino lower than or equal to h.
If h is great enough, F(h) is near to 1 and this result can be simplified to
for
{
-
> h]zl
Prob [H*
Prob [H*
+oo
1
n
=
(1 - F(h))
-
+OD
P(k) = 1 and
1
P(k). k = n
K=O
K=O
-
[ 1 + k(l - F(h))}
> h]”,
+m
1
P(k)
K=O
:
monthly average number of storms.
From the practical point of view, the nk catalogue enables P(k) or determined for the utilisation of the simplified formula two laws is used
;
n
to be
one of the following
:
Poisson’s law : P(k) = e
- A & k! k
Negative Binomial law : P(k) = k!
r(Y)
Since P(k) may considerably vary according to the month, it would be preferable, when sufficient information is available, to take as a reference period the year instead of the month. The probability F(h) is determined by the sample of the H to which are fitted the followinq laws :
i‘
471
This method has the advantage of utilizing the maximum amount of information, while warranting its homogeneity. It is possible and desirable to calculate the intervals of confidence. T I D E LEVEL PROBABILITIES
The Observed Maximum Level
:
the most simple way is, like in the case of waves
to -~ constitute the sample of daily maximum levels of the hiqh water. Then, the san
methods are used as for the wave. The result is presented in the following form
:
472
PROBABILITIES
OF HIGH WATERS IN DIEPPE
-
I
- 0,lO -
LOW WATER
-----
- 0,20 -
- 0,30 - 0,40 - q50,
I
I
I
I
1
Return
I
I l l
Period
1
1
1
I
1 1 ,
( i n years)
However, the question may arise of whether it will be safe to use only one statistical law for explaining the behaviour of a variable, which is made up of two phenomena entirely different : the astronomic tide and the tide deviation due to meteorological conditions. We decided therefore to study also these two phenomena. The Predicted Maximum Level
:
In fact, it‘s a question of a random pseudova-
riable easy to probabilize either by constitutinq directly a catalogue of predicted heights, or by using the estimates based on the semi-diurnal amplitude. The two methods can be compared in the figure below.
413
FREQUENCIES OF PREDICTED LEVEL IN DIEPPE
% Frequencies of overstepping
700
8,OO
9,m
Z level
l0,OO ( in meters)
-
474 Tide differences
:
First, we constitute the catalogue of daily tide differences
obtained either by means of differences Ho - H1 on a series of observed tides, or by reconstitution from meteorological conditions ( s e e farther). Then, we proceed to the same probabilistic study as for the waves. The Sum of predicted levels and tide differences
:
We have H1 = H
0
+ S. If
Ho
and S are independent, the probability of their sum can be easily calculated by writinq Prob
:
jw +W
[ H1 >
hl] =
f
[x]=
G
[y]=Prob
where
G
[hl
-
x].
[ x < H,- < x +
Prob
f [x]. dx
dx
]
[S>y]
For our part, we found that if the coefficient of correlation between Ho and S could attain 0,3 during slight or medium storms, this coefficient is practically
zero for heavy storms by which we are particularly concerned. This result is only indicative as it corresponds to a particular case and deserves to be tested on other sites.
If the correlation is no more zero but if there exists a relationship of the kind S =
A
Ho
+ S', where
Ho and S' are independent, we can get again to the pre-
vious case by considering the independent variables
:
h1
(1 +
Ho and S ' .
The figure below enables the two methods for estimating the HI level to be compared by studying directly H 1 or by studying the sum
OF TIDE LEVELS
PROBABILITIES
IN
€3 0
DIEPPE
+
S.
-
Level (in meters )
10,90
10,ao
1420
. 1
2
3
4
5
Return
10
Period
20 30 4050 ( in y e a r s )
100
475
For the design, we take the extreme limits of these estimates to which we add confidence intervals at 7 0 % . CASE OF POOR INFORMATION
Wave data and tide data are frequently very short, rendering the statistical estimates too uncertain : additional information should then be used. Sometimes, it is fortunate to find another wave or tide series in the vicinity of the studied site. If the two series are closely related, the probability estimates of the long series can be easily transposed to the short one. If this is not the case, it is necessary then to consider other possibilities. Utilization of the meteorology
:
in the case where information, such as pres-
sure, wind, temperature in the vicinity of the site, is
available, it is possible
to establish a relationship between these data and the surqes or the tide fluctuations. As a test we tried multiple linear reqressions of the kind
n s
1/3
=
= g
where
f (P,
v, v2,
i
(P,
2
v, v ,
nO,Ap,Av, ...I
no,Ap,
P
. . .)
= temporary pressure variation
T
= temperature
V
= wind speed
A
AV,
= pressure
P
h
T,
T,
:
V
= temporary variation of the wind
Although the results are not yet exploitable for high events, they are incentive for low and medium events in so far as the obtained multiple correlation coefficients reached 0 , 8 to 0,9.
Using these relationships, we reconstructed a
fictitious sample of tide differences over a long period of time and we estimated then the probabilities resultinq from this sample. On the figure below, the obtained results can be compared with respect to the probabilities derived from observations
:
416
PROBABILITIES OF TIDE DEVIATIONS IN LE HAVRE
3000
2000
1000
500 400
300 200
100
50
40 30
-
-
Return Period (in Days)
From observations within 10 years
-
-
From meteorology
--
From observations w i t h i n 1 years
Tide Deviation ( in
20
30
40
50
70
60
80
90
100
cm
)
110
In this figure we can see that there is an acceptable compatibility between the estimates for return periods lower than 10 years. Beyond these periods, it will be necessary either to improve the statistical relationships between the meteorology and the sea states or to use mathematical prediction models. Utilization of exceptional events
:
it happens that there exist recorded data
on one or more exceptional events for which an estimate can be fixed, and which are known to be the highest within a long period of time (for instance, a century). This information is precious and may be utilized, thouqh it greatly differs from a complete catalogue of waves or tides. It allows the statistical uncertainty to be reduced and the representativity of the used sample to be proved. The detailed description of this method can be found in the references (Bernier and Miquel, 1977). It was already applied successfully to flood risk estimations
:
477
FLOOD PROBABILITIES AT HAUCONCOURT (MOSELLE 1
Return Period 1000
500 200 100
50 20
10 5 2 1
STORM DURATIONS
Recent works on random waves showed how the storm duration may affect the lifetime of dikes. Using once more the techniques applied to the study of river flow rates (Miquel and Phien BOU Pha, 1978) we can estimate, for instance, the duration probability of a storm exceeding a given surge threshold. The probabilities of the yearly sums
of storm durations can be read in the figure below
:
478
Durations
EXCEEDING A GIVEN WAVE THRESHOLD IN LE HAVRE : TOTAL ANNUAL SUMS -
Thus, in decennial year, the total
Return
duration, sum
over
Period
100 years
the year of storms exceeding the surge level of 3,5 m in Le Havre
1
about 10 days.
0
1
2
3
4
5
6
7
8
9
Wave Threshold ( in meters) A curve of the same kind can be obtained,to describe the durations of individual storms. Indeed, such curves will be useful to designers when they will be able to take simultaneously into account both, the storm durations and their intensities. REFERENCES Allen, H., 1977. Analyse statistique des mesures de houle en differents sites du littoral franqais. Edition no 3, rapport EDF HE 46/77.01. Chatou (France) Bernier, J., Miquel, J., 1977. Exemple d'application de la theorie de la decision statistique au dimensionnement d'ouvraqe hydraulique : prise en compte de l'information het6rogGne. A.I.R.H. Baden. Cramer, Leadbetter, 1967. Stationary and related stochastic processes. Sample function properties and their applications. John Wiley. New York.
479 Miquel, J., 1975. Role et importance d'un modile statistique de la houle en vue du depouillement et du stockaqe des donnees. A.I.R.H. Sao Paulo. Miquel, J., Phien Bou Pha, B., 1977, Tempetiage : un modile d'estimation des risques d'etiage. Xime Journee de 1'Hydraulique. Toulouse.